Flows on Homogeneous Spaces. (AM-53), Volume 53 9781400882021

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Annals of Mathematics Studies Number 53

ANNALS OF MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse I.

Algebraic Theory of Numbers, by

H erm ann W e y l

3.

Consistency of the Continuum Hypothesis, by

11.

Introduction to Nonlinear Mechanics, by N.

16. Transcendental Numbers, by 19.

Fourier Transforms, by S.

K u r t G o d el

and N.

Kr ylo ff

B o g o l iu b o f f

C a r l L u d w ig S i e g e l

and

B och n er

K. C h a n d rasekh aran

20. Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. 21. Functional Operators, Vol. I, by

L efsch etz

J o h n von N e u m a n n

24. Contributions to the Theory of Games, Vol. I, edited by H. W. 25. Contributions to Fourier Analysis, edited by A. A. P. C a l d e r o n , and S. B o c h n e r

Zygm un d,

Kuhn

W.

and A.

W . T ucker

M.

T ran su e,

M o rse,

28.

Contributions tothe Theory of Games, Vol. II, edited by H. W.

and A.

W . T ucker

29.

Contributions tothe Theory of Nonlinear Oscillations, Vol. II, edited by S.

L efsc h etz

30.

Contributions tothe Theory of Riemann Surfaces, edited by L.

33.

Contributions tothe Theory of Partial Differential Equations, edited by L. n e r , and F. J o h n

34. Automata Studies, edited by C. E.

Shan n o n

and

Kuhn

A h lfo r s

et al. B ers,

S.

B och­

J. M c C arth y

36. Contributions to the Theory of Nonlinear Oscillations, Vol. Ill, edited by S.

L efsch etz

38. Linear Inequalities and Related Systems, edited by H. W.

T ucker

and A. W.

K uhn

39.

Contributions to the Theory of Games, Vol. Ill, edited by M. and P. W o l f e

40.

Contributions to the Theory of Games, Vol. IV, edited by R.

D resh er,

A. W.

T ucker

D uncan L uce

and A. W.

Contributions to the Theory of Nonlinear Oscillations, Vol. IV, edited by S.

L efsch etz

T ucker 41.

42. Lectures on Fourier Integrals, by S.

B o ch n er.

In preparation

43. Ramification Theoretic Methods in Algebraic Geometry, by S. 44. Stationary Processes and Prediction Theory, by H.

A bhyankar

F ursten berg

45. Contributions to the Theory of Nonlinear Oscillations, Vol. V,

C e s a r i, L a Sa l l e ,

and

L efsc h etz

46. Seminar on Transformation Groups, by 47.

Theory of Formal Systems, by R.

48. Lectures on Modular Forms, by R. C. 49.

A. B orel

et al.

Sm ullyan G u n n in g

Composition Methods in Homotopy Groups of Spheres, by H.

50. Cohomology Operations, lectures by N. E.

Steen r o d ,

T oda

written and revised by D. B. A.

E p s t e in

51. Lectures on Morse Theory, by J. W.

M il n o r

52. Advances in Game Theory, edited by M. 53. Flows on Homogeneous Spaces, by L. 54. Elementary Differential Topology, by

D resh er ,

L.

Sh a p l e y ,

A uslan d er,

L.

Green ,

J.

R.

F.

and A. W. Hahn,

M un kres

55. Degrees of Recursive Unsolvability, by G. E.

Sa c k s.

In preparation

T ucker

et al.

FLOWS ON HOMOGENEOUS SPACES BY

L. AUSLANDER, L. GREEN, and F. HAHN with the assistance pf L. MARKUS and W. MASSEY and an Appendix by L. GREENBERG

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

1963

Copyright © 1963, by P r i n c e t o n U n i v e r s i t y All Rights Reserved L.C. Card 62-19959

P r e ss

Printed in the United States of America by W e s t v ie w P r e s s , Boulder, Colorado P r i n c e t o n U n i v e r s i t y P r e s s O n D e m a n d E d it i o n , 198 5

INTRODUCTION In classical dynamics the action of a one-parameter group — time — on the phase space is one of the principal objects of study. Ergodic theory concerns itself with the case when the phase space is a measure space. Topological dynamics deals with topological phase spaces. Requiring that the phase space have both structures, and that they be compatible, brings us closer to the original dynamical problem. The questions asked about such flows include the following: Topological transitivity: is there a dense orbit? Minimality: is every orbit dense? Ergodicity: are there any proper invariant subsets of positive measure? Even asking that the phase space be a manifold and the measure locally Lebesgue still leads to very general, and generally unsolved, pro­ blems. In this book we consider the case when the manifold is a homogeneous space of a Lie group. Furthermore, the one-parameter group is assumed to be a subgroup of the original group, acting in a natural fashion. This en­ ables one to get a firmer grip on the behavior of the flow, since it is now part of the action of a much larger group on the phase space. As in clas­ sical and quantum dynamics, the presence of these extra "symmetries'' helps the analysis a great deal. In particular, the theory of group representa­ tions can be used with success. During the academic year 1960-1961, The National Science Foundation sponsored a conference on Analysis in the Large at Yale University. This conference brought together for an extended period of time mathematicians from various parts of the country with different backgrounds. All but the last of the papers in this book have come directly from the interaction of the people at this conference. Markus and Hahn started things off with a search for examples of minimal flows on manifolds. Auslander suggested that flows on nilmanifolds induced by one parameter groups might be minimal. The results of these investigations are contained in Chapter TV and served as a starting point for almost all of the further material in this book. As an example of the direct advantage to the authors of the nature of the conference, it was Gottschalk who suggested the study of the distal nature of flows on nilmanifolds, which plays such a central role in the material in Chapter IV. The methods used in Chapter TV, although capable of giving new in­ formation seemed to be incapable of generalization. Green arrived at this time for the second semester and, hearing of the results in Chapter IV, suggested that a study of the ergodic nature of flows on nilmanifolds might be possible by means of group representation theory. This program and its v

generalization to certain solvmanifolds is carried out in Chapters V and VII. In Chapter VI, it is shown how results on flows on nilmanifolds can be used to prove facts about flows on certain types of solvmanifolds. Chapter III contains a detailed account of the general state of affairs for one-parameter groups acting on compact three-dimensional homogeneous spaces obtained by identifying any connected, simply connected non-compaot Lie group under a discrete subgroup. In order to make the material as self contained as possible, Chapters I and II were added as a listing and dis­ cussion of some of the results already in the literature with which the reader may not be familiar. Chapter VIII is devoted to an application of the previous results concerning flows on nilmanifolds to diophantine approximations. In an Appendix, Greenberg shows how the classical work of Hedlund on minimal flows can be extended and he relates his results to the work of Mahler [i] on diophantine approximations. The authors would also like to take this opportunity to thank Professors W. H. Gottschalk, G. A. Hedlund, S. Kakutani and J. C. Oxtoby for the many stimulating conversations which contributed so much to the material in this book. We would also like to acknowledge the financial support given by The National Science Foundation’s Grants 15 5 6 5 , 11287 and the Office of Ordnance Research Contract DA-i 9 -0 2 0 -0RD-525 1*.

vi

CONTENTS INTRODUCTION..........................................................

V

CHAPTER I.

An Outline of Results on Solvmanifolds .............. L. Auslander

1

CHAPTER II.

Ergodic Theory and Group Representations ............

7

L . Green CHAPTER III.

Flows on some Three Dimensional Homogeneous Spaces

. ,

15

L. Auslander, L. Green, F. Hahn APPENDIX TO III.

On the Fundamental Group of Certain Fibre Spaces. ,

37

¥. Massey CHAPTER IV.

Minimal Flows on Nilmanifolds...................... .

^3

L. Auslander, F. Hahn, L. Markus CHAPTER V.

Nilflows, Measure Theory ...........................

,

59

L. Green CHAPTER VI.

Flows on Certain Solvmanifolds not of Type (E)

.

.. .

67

L. Auslander, F. Hahn CHAPTER VET.

Flows in Type

E

Solvmanifolds......................

73

L. Green APPENDIX TO VII.

A Theorem on Discrete Subgroups of Solvable Groups of Type

(E) ...............................

79

L. Auslander CHAPTER VIII.

An Application of Nilflows to Diophantine Approximations .......................................

81

L. Auslander, F. Hahn APPENDIX:

Discrete Groups with Dense Orbits.................. ..

85

L. Greenberg BIBLIOGRAPHY........................................................ .

vii

105

FLOWS ON HOMOGENEOUS SPACES

CHAPTER I. AN OUTLINE OF RESULTS ON SOLVMANIFOLDS by Louis Auslander Introduction The purpose of this chapter is to give an outline of the basic theorems in the theory of solvmanifolds. We will assume in presenting this material that the reader already has a basic knowledge of Lie groups and Lie algebras and the inter-relations between these two mathematical objects. Specifically, we will assume a familiarity with the material in either Chevalley [ i] or Cohn [1]. With this as our starting point, we will out­ line the basic results in the theory of solvmanifolds. Since all this material is already in the literature, we will present no proofs, but will content ourselves with giving references. We will adopt the following conventions in this paper. A Lie group need not be connected and will be denoted by G, S, N, etc. We will call a connected Lie group an analytic group and generally use the same notation as for a Lie group. We will* use to denote the identity com­ ponent of the Lie group G. The Lie algebra of a Lie group will, of course, refer to the Lie algebra of the identity component of the Lie group and, if G denotes a Lie group, we will use L(G) to denote the Lie algebra of G. We will use exp (L(G)) to denote the exponential mapping of L(G) to G. Occasionally we will use the symbol L to denote a Lie algebra without specifying any Lie group. For W 1 and W 2 subalgebras of L, [W1,W2] will denote the subalgebra generated by the bracketof an element in W 1 with an element of W p. Analogously, if G is any group and H, and are subgroups, we will use [H.,HJ to denote the group generated by -l-i all elements h lh2h1 h? for h 1 € H 1 and h2 e H2 . Definition:

Let

L

be a Lie algebra and let

L 1 =L, L

L2 = [L1,L1], ...,Lk = [ L ^ ,Lk_1].

is calledsolvable if L^ = 0 for some Let L be a Lie algebra and let L1 » L,

L

is called nilpotent if

L2 = [ L. L1

L

!/■

= o

k. L k = t L , L k~1 ] .

for some k. 1

2

FLOWS ON HOMOGENEOUS SPACES The basic examples of these types of Lie algebras are the follow­ Lie algebras of matrices of the form

ing:

G-.0 are all nilpotent. Lie algebras of matrices of the form

are all solvable. Definition: Let

G

be a group and let

G, = G, Then

G

G2 = [G,,G,1 ,...,Gk = [Gk_,,Gk_,].

is called solvable if Gk = e Let G be a group and let G 1 . G, G2 = [g ’,g ’

If Gk = e for some k, nilpotent it is solvable.

then

G

for some

k.

Gk = [G1,Gk-1]. is called nilpotent. Clearly if

G

is

Remark: Thehomomorphic image of a solvable (nilpotent) group is solvable (nilpotent). A subgroup of a solvable (nilpotent) group is solvable (nilpotent) . Example: Any group of matrices of the form

is nilpotent. Any group of matrices of the form

o *. a. is solvable. i. Basic Facts. I.

II.

Let Let a) b) c)

L(G) L(G) be the Lie algebra of the analytic group G . Then L(G) is solvable if and only if G is solvable, L(G) is nilpotent if and only if G is nilpotent. If L is a solvable Lie algebra, then [L,L] is a nilpotent ideal in L. Further, there exists a unique maximal nilpotent ideal M in L. Let G be an analytic, simply connected, solvable group. Then G is homeomorphic to euclidean space. (Chevalley [2].)

AN OUTLINE OF RESULTS ON SOLVMANIFOLDS III. a)

b) c)

IV.

V.

3

Let G be an analytic, simply connected, solvable group. Then CG,G] is an analytic, simply connected, normal subgroup of G and [G,G] is nilpotent. (Ibid.) If G is an analytic, simply connected, solvable group, then G has a unique maximal analytic normal nilpotent subgroup M. If L(G) is the Lie algebra of G, then M = L(M) , where M is the maximal nilpotent ideal of L(G) and M is the maximal ana­ lytic normal nilpotent subgroup of G. Every representation of a solvable Lie algebra over the complex numbers is equivalent to a representation of the form

(Jacobson [i 3 .) Every Lie algebra has a faithful matrix representation. Every nilpotent Lie algebra has a faithful matrix representation of the form

C'-J

(Jacobson [1].)

Remark: Not every representation of a nilpotent Lie algebra is equivalent to one of the above form. 2.

The Exponential Mapping in Analytic Solvable Groups.

Henceforth analytic solvable or nilpotent groups will always be assumed to be simply connected, unless explicitly stated to the contrary. I.

If G

is a nilpotent analytic group, then the mapping exp (L(G)) -- > G

is a homeomorphism of II.

L(G)

onto

G.

(Malcev [1].)

If G is a solvable analytic group the exponential mapping need not be onto or a homeomorphism as the following example shows. Let (r,s,t) be three real parameters and let be the three-dimensional analytic group cos 2 irt -sin 2irt 0 0

L(S^)

is given by the triple

sin 2 -nt cos 2-rrt 0

0 0 1

0

0

(a,b,c)

representing the element

FLOWS ON HOMOGENEOUS SPACES It is straight forward to verify that exp (i,b,c) = (0,0,1) if (r,s,t) aretaken as a coordinate systemin S^. Similarly the points (r,s,i) for r2 + s2 / 0, are not in the image of L(S5) under the exponential mapping. S5 is characteristic of the type of pathology which can occur in the exponential mapping of solvable analytic groups in the following sense: III. Let S be a solvable analytic group. The exponential mapping is a homeomorphism of L(S) onto S if and only if there is no homomor­ phism of S onto S^. Dixmier [2]. IV. Let G be an n dimensional solvable analytic group. Then there exist n one-parameter subgroups g^(t), i = i,...,n such that: 1. For g € G, g = g.| (1 1) ...gj^(tR) and this representation is unique where the right hand side denotes group multiplication. ?. If G± = Cg1(t^). . . tn) 11±,...,tn any real no.), then G± is a closed subgroup of G and G^ is normal in G^_1. (Iwasawa [1].) 3.

Measures and Solvmanifolds.

Definition: Let G be a solvable analytic group and let H be a closed subgroup. Then G/H is called a solvmanifold. I. Haar measure on G induces a measure on G/H. (Mostow [2].) II. If the total measure of G/H is finite in any of the above measures, then G/H is compact. (Ibid.) III. If G/H is compact, then G has a left and right invariant measure. (Ibid.) Henceforth we will restrict ourselves to compact solvmanifolds. Definition: Let G be a nilpotent analytic group and let subgroup. Then G/H is called a nilmanifold.

H

be a closed

L . Malcev’s Results on Nilmanifolds. I.

Let N be a nilpotent analytic group and let H be a closed subgroup such that N/H is compact. Then if H^ denotes the identity com­ ponent of H, Hq is a normal subgroup of N and if N* * N/I^ and r = H/Hq , we have: a) N/H is homeomorphic to N*/r. * b) N is a simply connected, analytic nilpotent group. c) r is a discrete subgroup of N*. d) r is the fundamental group of N/H. Henceforth we will restrict the discussion of nilmanifolds to the case of N/r, where N is an analytic nilpotent group and r is a subgroup such that N/r is compact. II. r n [N,N] = r1 is a discrete subgroup of [N,N] and tN#N^/r1 is compact. III. Existence of Malcev coordinates. Let N be an n dimensional nilpotent analytic group and let r be a discrete subgroup such that

AN OUTLINE OF RESULTS ON SOLVMANIFOLDS N/r is compact. such that

Then there exist

n

one-parameter groups

5

x^(t)

N = (x.j (t^) .. .xn (tn)) satisfying the conclusions of

3

IV,

and such that

r = (x, (m1) ...xn(mn )} IV.

V.

where the m^are integers, i = i,...,n. A necessary and sufficient condition for a group r to be the funda­ mental group of a compact nilmanifold is that a) r contain no elements offinite order. b) r is finitely generated. c) r is nilpotent. Let N 1 and N2 be analytic nilpotent groups and let r1 and r2 be discrete subgroups of N 1 and N2, respectively, such that /r^ and N2/r2 are compact. Let r, — a v r2

~

>

»,

I

p

v > —2

where i denotes the injection mappings in both groups and a is an isomorphism of r1 onto r2. Then there exists a unique isomorphism p of N.j onto N2 such that the diagram is commutative. Corollary i: If r is a discrete subgroup of the analytic group N such that N/r is compact, then every automorphism of r is uniquely extendable to an automorphism of N. Corollary 2: isomorphic to VI.

If M 1 and M2 are compact nilmanifolds and tt1(M2), then M 1 is homeomorphic to M2.

is

A necessary and sufficient condition that an analytic nilpotent Lie group N have a discrete subgroup r such that N/r is compact is that L(N) have rational constants of structure relative to some basis.

Remark: Since there exist nilpotent Lie groups such that the constants of structure of L(N) are not rational relative to any basis, not every nilpotent Lie group N has a discrete subgroup r such that N/r is compact. 5.

Solvmanifolds.

In solvmanifolds the immediate generalizations of ^ .1 and U .V are false. Since these served as the basic tool theorems in the study of nilmanifolds, the study of solvmanifolds has been much more involved than the study of nilmanifolds. The first major results were the following of Mostow [1]. I. Let S be a solvable analytic group and C^ a closed subgroup of GS

FLOWS ON HOMOGENEOUS SPACES

6

such that S/C is compact. Further, let N be the maximal analytic normal nilpotent subgroup of S. Assume that £ contains no proper normal analytic subgroup of S. Then CN is closed and the connected component of theidentity in is a normal subgroup in N. II. Let M 1 and M2be two compact solvmanifolds and let tt1(M1) and tt1(M2) denote the fundamental groups of M 1 and M2 respectively. Then iftt1(M-, ) is isomorphic to tt1(M2) , M 1 ishomeomorphic to M2. III. Let M 1 be a compact solvmanifold. Then M 1is the bundlespace of a fiber bundle over a torus with a nilmanifold as fiber. Corollary: If M group of M, then

is a compact solvmanifold and ir1(M) satisfies theexact sequence

is the fundamental

tt1(M)

i— >

a

—

> n, w

—

> zs —

> i

where A Is the fundamental group of a compact nilmanifold and Zs Is the integers taken s times. The next major study of solvmanifolds was that of H. C. Wang Cl1 . His results may be summarized as follows: IV.

Let

r

satisfy the exact sequence i -- > A --- > r -- > Zs -- > i

where A is thefundamental group of a compact nilmanifold and 7 s is the integers taken s times. Then there exists an analytic solv­ able Lie group S containing r as a discrete subgroup. Of course, S/P need not be compact. V. Given r satisfying the hypothesis of IV above, there exists a Lie group £> (not necessarily connected, but simply connected) such that S 3 r as a discrete subgroup with S/r compact. Further, if Sq denotes the identity component of £5, we have ,S = F • S^, where F is a finite abelian group and the dot denotes the semi-direct product. The author (Auslander [l)) , using the above, was able to prove VI. Every fundamental group of a solvmanifold is the fundamental group of a compact solvmanifold. Let ^ be an analytic solvable group and let r be a discrete subgroup of S such that S/r is compact. Then the author has char­ acterized the groups r that satisfy this condition. Since we have no need for this result in the rest of the papers, and it is reason­ ably involved, we will not give it here. 6. I.

Solvable Lie Groups Acting on Compact Nilmanifolds.

Let S be an analytic solvable group and let r be a discrete sub­ group of S such that S /v is compact. Further, let r be a discrete subgroup of the analytic nilpotent group N such that N/r is compact. Then there exists a torus group T (compact abelian group) of auto­ morphisms of N such that S C T •N, where the dot denotes the semidirect product. Further, if H is the maximal normal nilpotent analy­ tic subgroup of S, then H C N and T acts trivially on N/H. (Auslander [2].)

CHAPTER II. ERGODIC THEORY AND GROUP REPRESENTATIONS by L. Green §1. The first explicit use of infinite-dimensional group repre­ sentations in ergodic theory was by Gelfand and Fomin [1], who noticed that the geodesic flow on surfaces of constant negative curvature is an instance of the action of a one-parameter subgroup on a homogeneous (actually double­ coset) space. Their method depended on knowing all the irreducible repre­ sentations of the group, and examining in detail their restrictions to the one-parameter subgroup of the flow. Mautner [1] pointed out that for this sort of application it is not necessary to know every representation, but only the representations of sufficiently many subgroups. We shall use both of these methods. Recently several Russian mathematicians have exploited Kolmogorov’s concept of entropy, not only to prove ergodicity and mixing for certain flows but also to obtain detailed information about the spectra. Of the cases we investigate, only the geodesic flow (Chapter III, Theorem 5.3a) can be treated this way. The horocycle flow has zero entropy, and this is probably true of all the other flows we deal with. (For a description of results on entropy and for further references, see Rohlin [i].) This chapter is intended to bring together known theorems and defi­ nitions in order to make the later exposition as self-contained as possible. The only results which have not appeared in print before are Theorem C and its corollary. In §2 we review definitions from ergodic theory. §3 introduces enough group representation terminology to enable us to describe the appli­ cation to ergodic theory. Some definitions and theorems dealing with in­ duced representations are stated in §4 . Finally, in §5 , we obtain all Ir­ reducible unitary representations of two specific groups used in the sequel. §2. Consider a measure space (S,E,n), where E Is a a-field of subsets of S, and n(S) < «. An automorphism cp of (S, Z, \i) is an in­ vertible map of S onto itself which takes E onto itself such that u(q>(E)) = n(E), for every E € E. A measure-preserving flow on (S,E,n) is a homomorphism t -► cpt of the reals into the group of automorphisms. The flow is called continuous if, for any elements A and B in Z, 7

FLOWS ON HOMOGENEOUS SPACES

8

ki(cpt(A) n B) is a continuous function of t. The flow is ergodic if cpt(E) » E for all t implies ^i(E)u(S-E) = 0. (Here it is assumed, of course, that E € £. In what follows, this condition will be understood.) It turns out that cpt is ergodic if and only if, for every A and B the Cesaro means T ijr ) u(

0}

and

e_ = (Xg; s
o } . A homomorphism of S1 onto the additive reals is given "by p((a,b)) = log a. According to the description given in (5) of the previous section, we can then write down explicitly what these induced representations are: (6)

(U (a,b)f^ x) = exp

ieXb)f(x +

Theorem B . (Gelfand-Naimark) tation of S1 is equivalent to either or is a one-dimensional representation

1 °8

a) >

f * L2( R)

Every irreducible unitary represen­ U+ or U~ given by formula (6), U^(a,b) = exp (i3 log a), £ real.

Corollary. (Mautner's Lemma) . Let U: g be a unitary repre­ sentation of S1in a Hilbert space f). Let 2ft be the closed linear mani­ fold of all e § such that Uht = y for all h € H. Then the spectrum of U (exp t X) restricted to 2ft is Lebesgue; in particular, every eigenvector of U (exp t X) is in 2ft. Proof. Apply Mautner's decomposition theorem (§3) to the represen­ tation U. if = / © J)adn(a). Let Aq be the set of a such that t)a is one-dimensional, A 1 its complement. f)D and will denote the corresponding direct integrals over these sets. Since H is the commutator of

si> *0 —

For

0 ^

€ 1^, h =

(U^ta)(x) so

\|r

Thus

2ft =

(i,b) € H,

and a c A 1,

= exp (+ iexb)*a(x) ,

f)Q . Now for a e A ], (U“ (exp t X)*“)x

= *“(x + t) ,

which is the familiar translation operator in L2(R). This one-parameter group has Lebesgue spectrum, so by the result quoted at the end of § 3 , U (exp t X) hasLebesgue spectrum on ^ = 2ft .

n

FLOWS ON HOMOGENEOUS SPACES

We now derive analogous results for the group S2 whose Lie alge­ bra @ 2 has a basis (X,Y,Z) and relations [X,Z] = oY + Z, [X,Y] = Y - crZ, [Y,Z] = 0, where a is a non-zero real number. S? may be represented by pairs ((t,w)} with t real and w complex. Here exp (tX) = (t,o), exp (aY + bZ) = (o,a + ib), and the group multiplica­ tion is (t,w)(t',w') = (t + t', e(1“iaH

w 1 + w) .

S2 is the semi-direct product of the one-parameter group {(t,o)} and the two-dimensional normal abelian subgroup H = {(o,w)). The center is trivial, and, being the only abelian normal subgroup, H must be the group to use in finding the completely irreducible representations. For each complex number z we obtain a character Xz of H by the formula x ((o,w)) = exp tiW (zw)]. Let g = (t',w'). Then we calcu­ late that x§ = x , where z 1 = z. Thus each orbit, for z 4 o, Zj z is a spiral in the z-plane. Moreover, the isotropy group xz is H for z 4 0, while for z = 0 it is, of course, all of S2. We are again in the situation of the previous example, where proper extensions of the char­ acters are not needed (except for xq). However, there is now a continuum of distinct orbits. A convenient cross-section of them is the set of x^, where r is a real number between 1 and e27r/^a. The homomorphism p is of course p((t,w)) = t, so the representation induced by x^, which we denote by Ur , has the form (7 )

(U^t>w)f)(x) = exp [irS (e(1'io)x v)]f(x + t),

f c L 2(R)

Theorem C .Every irreducible unitary representation of S2 is equivalent to a Urgiven by formula (7), with r in thehalf-open inter­ val whose end points are i and e 2lT^a, or is a one-dimensional repre­ sentation U^t w) = exp (iet), p real. Corollary D .Let U be a unitary representation of S2 in a Hilbert space. Let 9)l be the largest manifold on which U restricted to H is the identity. Then U(exp tX) restricted to 9ft1 has Lebesgue spectrum; every eigenvector of U(exp tX) is in 9ft. The proof of Corollary D is almost word for word the same as the proof of Mautner's Lemma. We omit the details. In conclusion, we remark that for either @ 1 or @ 2 the Lie subalgebra corresponding to H may be defined without reference to bases as [©, t © , . . . , t @ , @ ] . . . ) j

.

K For we see that zoN = w"1 € D. Thus zoN e Z C n(S). Hence theorem is completed by applying Lemma k . k .

p

divides

N

and the

COROLLARY h . l . Let D be a discrete subgroup of G such that G/D is compact. Then there isa discrete subgroup D* of G* such that tj*D = D* and the kernel K of tj* is non-trivial and in the center of G. Further, if D* is any subgroup of D* of finite index in D* such that M = C*\G*/D* is a manifold, then the generator of K divides the EulerPoincare characteristic of M. § 5 . The Flows. Before we begin to prove the theorems listed in the table in section 1 we recall briefly the description of the geodesic and horocycle flows on a surface of constant negative curvature. We let M be the two dimensional Riemannian manifold whose space is the set of complex numbers z with positive imaginary part and whose differential metric is given by ds2 = — —

where

z = x + iy .

y

We let U be the bundle of unit tangents vectors of M. If v is a unit tangent vector at zthen there is a unique geodesic through z with v as its tangent. The geodesic flow on U is given as follows:each unit tangent v moves along the geodesic it determines in the direction of v with unit velocity. A horocycle in M is either a circle tangent to the real axis or a straight line parallel to the real axis. If v is a unit tangent at z which does not point vertically away from the real axis then there is a unique horocycle such that v is an inward normal to the horocycle. Ifv points vertically away from the real axis then the horocycle determinedby v is the line .parallel to the real axis and through the base point of v. To each unit tangent v at z we may associate a unique unit tangent v ’ at z so that v',v have the same orientation as the positivereal and positive complex axes respectively. The horocycle flow on U is described as follows: the unit tangent v at z moves with unit velocity along the horocycle it determines in the direction of v '.

FLOWS ON SOME THREE DIMENSIONAL HOMOGENEOUS SPACES

27

The group £ of linear fractional transformations which maps the upper half plane onto itself acts simply transitively on U and is thus homeomorphic to U. We let vQ be the unit tangent at i whose direction is the same as the imaginary axis and always consider the homeomorphism determined by a -*■ a(i) where a e £. Each element of £ may be written a ( z ) = §|— where a,b,c,d are real and ad - be = i . If we let G(2) be the group of two by two real matrices of determinant one there is a natural continuous mapping of G(2) -*• £ given by -*■ - g -■*- = a(z) . The kernel G(2).

H

We thus If we geodesic flow geodesic flow

of this mapping is the two subgroup

Z2 =

(j>), ("o-?)}

see that U is homeomorphic to G( 2 )/Z2. trace the movement of the initial element vQ under the and then use the transitivity of £ on U, we see that the on U is isomorphic to the flow on G(2)/Z2 induced by the

one-parameter subgroup cp(t) = ( ® e-t) ‘ the horocycle flow is isomorphic to the flow in one-parameter subgroup cp(t) = (q

same manne^ we see that G(2)/Z2 induced by the

A compact surface of constant negative curvature is covered by the hyperbolic plane M and is the orbit space M/£', where £' is a proper­ ly discontinuous subgroup of £. Thus the geodesic and horocycle flows on compact surfaces of constant negative curvature may always be realized on G(2)/Z0D, where D is a discrete subgroup of G(2) for which G( 2)/D is t compact and the flow is induced by e-t) or (o • respectively. We will use Corollary U.3 and the following theorem to take advan­ tage of known facts about the geodesic and horocycle flows. THEOREM 5.1. Let (X,T) and (Y,T) be transformation groups, with X and Y compact connected n-dimensional manifolds. Let 0 :X -► Y be a local homeomorphism onto Y such that = (0 (x))t. Under these conditions (X,T) is minimal if and only if (Y,T) is minimal. PROOF. We recall first that minimal means that each orbit is dense. This of course is equivalent to saying there are no closed proper invariant subsets. We observe that compactness of X implies that 6 is a closed mapping and thus for each A C X we have 6(A) = 6(A). If (X,T) is minimal and y e Y then there is an x e X such that 6 (x ) =y. Consequently yT= (0 (x))T = 0 (xT) and thus yT = 0 (xT) = 6 (X) = Y, so (Y,T) is minimal. If (X,T) is not minimal then there is a point in X whose orbit is not X. The orbit closure of this point must contain a minimal set M (Gottschalk-Hedlund [i]) which is not X. We observe that M contains no interior points. For, if M did have an interior point, then each point of M would be interior and M would be open as well as closed. The con­ nectivity of X would then say M = X which is not the case. Since 6 is a local homeomorphism 0 (M) has no interior and is closed. Thus,

FLOWS ON HOMOGENEOUS SPACES

28 0 (M) 4

Y

and is invariant so Y is not minimal. The previous theorem will allow us to lift the minimality property of the horocycle flow. It seems more difficult to lift ergodicity and mix­ ing. However, Mautner's proof [i] that the geodesic flow is mixing applies word for word to the generalized geodesic flows. The generalized horocycle flows are then treated by reversing the steps E. Hopf [2] used to prove mixing of the geodesic flow. LEMMA 5.2. (Mautner) Let G be any connected Lie group which has the Lie algebra 2.Ua, and let U be a continuous unitary representa­ tion of G in a Hilbert space S>. Let be the largest subspace on which U is the identity. Then U (exp tX) has absolutely continuous spectrum on . PROOF. Let G 1 and G2 be the subgroups of G corresponding to the subalgebras generated by(X,Y) and (X,Z), respectively. Each of these is a homomorphic image of the solvable, simply-connected group S of affine motions of the real line. Hence U restricted to G^ is a unitary representation of S, i = 1,2, and the corollary to Theorem B of Chapter II applies. Let 2)^ , 9K2 be the closed linear manifolds of £ consisting of all vectors Invariant under U (exp tY), U (exp tZ), respectively. By the corollary, U (exp tX) has Lebesgue spectrum on 9JL , i = 1,2. 1 1 1 1 Let cp € 9W 1 + 2tt2, and write cp = 1 + cp2, cp1 eSJ^. If a is a Borel sub­ set of the line with Lebesgue measure zero, ||E(a)cp^|| = 0 for i =_ 1,2, where E(*) is the spectral resolution for U (exp tX) . Then ||E(A)q>|| < ||E(a )i nri HI?

(

J,

( y ) D v z g n

= 0 and U0(exp Zt)y = e y for all t € T, *1\'t' — then U^exp Zt)x = e x for all t € T, where x = ^(y) . This yields the existence of the one to one correspondence between eigenvectors. The eigenspace of &2, U2(1 belonging to the eigenvalue zero is one-dimensional andthus the flow(S1/D,cp(t)) is ergodic. There are eigenvectors of ^2, U2(cp(t)), with non zero eigenvalues. Hence such eigenvectors also exist for ,U1(cp(t)) . This shows that the spectrum of the flow (S1/D,p ab t 1 bt 0.

36

FLOWS ON HOMOGENEOUS SPACES

THEOREM 7.2. Let D C N be a discrete subgroup such that N/D is compact. If cp(t) is a one-parameter subgroup of N, then the flow (N/D,cp(t)) is always distal. The flow (N/D,cp(t)) is both minimal and ergodic if a and b are rationally independent. The flow is neither minimal nor ergodic if a and b are rationally dependent. If A is a connected, simply-connected, abelian, three-dimensional Lie group, then A is isomorphic to R 3. Arguments similar to those used before show that if D is a discrete subgroup such that R 3/D is compact, then, without loss of generality, we may assume that D is the integral lattice of R 3. We thus only need the classical Kronecker Theorem on diophantine approximations (Koksma [i]) to obtain: THEOREM 7 .3 . Let cp(t) = (at,bt,ct) be a one-parameter subgroup of R J. The flow (R3/D,cp(t)) is minimal and ergodic if a,b,c are rationally independent and is neither minimal nor ergodic if a,b,c are rationally dependent.

CHAPTER III. Appendix ON THE FUNDAMENTAL GROUP OF CERTAIN FIBRE SPACES, by W. Massey §i. Introduction. Let G be a connected topological group which is an Eilenberg-MacLane Space of type K(tt,i ) with tt abelian and let B be a connected CW-complex which is an Eilenberg-MacLane Space of type K(7T,l) • Given any principal G-bundle p : E -- > B over B, we can ask, what is the fundamental group of E? The only non-trivial part of the homotopy sequence of this bundle is the following: 0 -- >

tt1(G)

>

tt1(E)

-- >

tt1(B)

>

0

This shows that 7r1(E) is a group extension over (B) = TT with kernel "[T, (G) = tr. Corresponding to this extension there is a certain 2-dimen2 i sional cohomology class k e H (TT,1 which determines the equivalence class of the extension completely. On the other hand, the characteristic class of this fibre bundle (i.e., the first obstruction to a cross section) is a 2-dimensional cohomology class c e H^BjTr^G)) = H2(TT,1 ;tt) . It is natural to conjecture that these two cohomology classes are the same. In this note, we will essentially prove this conjecture, modulo automorphisms of K2(TT,1 ;*) , ? o r the case where TT acts trivially on tt. As an application of this result, one can determine the funda­ mental group of the bundle of unit tangent vectors to a compact, orientable 2-manifold of genus > o. §2. Statement and Proof of the Theorems. Let G be an Eilenberg-MacLane Space with tt a countable abelian group. Fol­ lowing Milnor [i], we will take G to be an abelian topological group. We will consider equivalence classes of principal G-bundles over a fixed base space B which is assumed to be a CW-complex. There are two ways of introducing an addition in this set of equivalence classes of bundles: (a) The classifying space BQ is an Eilenberg space where tt is the fundamental group of G. By the theorem of Milnor [l], we may choose K(tt,2) to be an abelian topological group. Hence the homotopy classes of maps B > BQ may be given a group structure. 37

K(tt,2),

38

FLOWS ON HOMOGENEOUS SPACES

(b) Given two principal G-bundles p1 : E1-- > B and p2 : E2 -- > B, we may form the principal G x G-bundle p1 x p2 : E1 x E 2 -- > B x B; let p ': E' > B denote the G x G-bundle over B induced by the diagonal map d :B --- > B x B. Consider the homomorphism X : G x G -- > G defined by x(g1#g2) = g1 • g2 (recall that G is abelian !) . Then by definition the X-extension+ of p 1: EJ > B is the sum of the two given bundles. LEMMA. These two methods of forming the sum of principal bundles are equivalent. PROOF. Let f1 : B sifying maps for p1 : E1 -- > B the classifying map for p1 x p2

G-

>Bq and f2 : B >Bq be the clas­ and p2: E2 > B respectively. Then : E1 x E2---> B x B is the map

f1 x f2 : B x B---- > Bq x Bq since Bq x q = Bq x Bq , etc. Theclassifying map for the bundle p»: E' > Bis (f1 x f2)o d = (f1,f2) :B > Bq x Bq . The-homo­ morphism--- x : G x G --- > G induces the so-called"characteristic” map P(x) : Bq y q -> Bq and the classifying map of the X-extensionof p 1: E 1

>B

is p(X)o (f1,f2)

:B

> Bq

.

However, it is seen without difficulty that p(x) : Bq x Bq ---> Bq is homotopic to multiplicationdefined in Bq by the group structure (which is unique up to a homotopy). Thus p(x) o (f1,f2) is the homotopy class of the sum of f1 and f2 acdording to definition (a), q.e.d. .If = (E^,B,G,pi) , i = 1,2, are principal G-bundles over B, we will denote their sum by+r\2 . Recall that any cohomology class c e H^(Bq ) defines a "char­ acteristic class" c (t ) € H^(B) for any principal G-bundle = (E,B,G,p) according to the rule c(n)

= f*(c)

^ If p1 : E 1 -- > B and p2 : E2 > B are principal bundles with groups G1 and G2 respectively, and x : G 1 > G 2 is a continuous homomorphism, then the second bundle is a X-extension of the first if and only if there exists a map 0 : E1 -- > E2 such that

(a) p20 = p1 (b) For any x € E1 and g e G 1, 0 (x • g) = (0x) • (xg) It iseasily provedthat given p1 : E 1 --> B with group G 1 andthe cc tinuoushomomorphism X : G1 >G 2, there exists thex-extension p?: E2 -- > B which is unique up to equivalence; see Borel and Hirzebruch, li §6 .5 .

39

ON THE FUNDAMENTAL GROUP OF CERTAIN FIBRE SPACES >Bq is the classifying map for r\. (a) of the sum of G-bundles that

where f : B from definition (i)

c (ti1

+

ti2)

It follows readily

= 0(11,) + c(n2)

Next, we will consider principal G-bundles over a connected base space B such that tt2(B) = o. Then t t ^ (E) is a group extension of tt1(G) by (B) (this follows from the homotopy exact sequence of the given bundle) . Since G is connected, 7r1(B) acts trivially on 7T1(G) .

space

THEOREM 1 . with tt2(B)

B

If t] = o,

> TT, (G)

0

and 6 are principal G-bundles over the base then the group extension > 7T1(Ell+0)

>tt^ (B) -- > 0

is the Baer sum of the extensions 0 -- >

tt,(G)

--->

tt1

(E )

> ir, (B)

>0

0 -- >

tt, (G)

--->

tt1

(Eg)

>

>0

and tt1

(B)

PROOF. To prove this, one uses definition (b) of the sum of two G-bundles, checking that each step of the construction of the sum of the bundlescorresponds to a step in the construction of the Baer sum of two group extensions. The first step gives rise to the following exact sequence 0 --- > tr1(G x G) -- > t t , (E x Eg) > i r, (B x B) --- > 0 which may be re-written 0

— > 7T1(G)

X TT1(G)

— > 7T} (E^) X 7^ (Eg) — > 7^ (B)

X

TT, (B) — >

0

The second step gives the following diagram: TT (E )x TT^Eg) -- > TT, (B) X TT- (B)--- > 0 o _ >

V

G)

x V

G ) ^

^

>

/ I

TT, (E') --------- > ir1 (b )

Here the vertical arrows denote ''diagonal" maps. get the diagram o -- > tr1(G) x

0 ------>

T , ( G)

tt, (G)

-- >

> 0

From the third step we

tt1(E1)

-------------- > ^ ( E ^ g )

We leave it to the reader to check that these diagrams actually correspond to the various steps in the construction of the Baer sum of two group extensi ons. q .e .d . Thus we see that we have a natural homomorphism from the group

FLOWS ON HOMOGENEOUS SPACES of all principal G-bundles over B into the Baer group of extensions of tt1 (G) by 7r1 (B) (with tt1 (B)operating trivially on tt^G)). THEOREM 2 . If B is an Eilenberg-MacLane space this natural homomorphism is an isomorphism onto.

K(TT, 1)

then

2 PROOF. Let c e H (Bq,tt) denote the so-called fundamental class (here 7r= tt1 (G) = tt2 (Bq)) . Then it is well known that the operation of assigning to any principal G-bundle v\ over B the cohomology class 2 c (tj) € H (B,tt) is an isomorphism of the group of all G-bundles over B 2 onto H (B,7r) . It is also well-known that c(t^) is the obstruction to a cross section of n- Using this remark, we can easily show as follows that the natural homomorphism is an isomorphism. Let r\ = (E,B,G,p) be a principal G-bundle such that the extension 0

> TTj (G) -- > 7T1 (E) -£*_> 7T1 (B)

>0

corresponds to the zero element of the Baer group, i.e., the extension splits. Let cp : tt1 (B) -- > tt1 (E) be a splitting homomorphism, i.e., p# « cp

=

identity.

Since B = K(TT, 1) and G = K(tt,1), it follows that Therefore there exists a continuous map f :

B

E =

(E) ,1) .

> E

such that f* = cp : 7^ (B) > ir1 (E) . Since B = K(TT,1) , it follows readily that p ° f is homotopic to the identity map B -- > B. By use of the covering homotopy theorem, any homotopy of p » f withthe identity can be covered by a homotopy of f to a cross section of i\. Therefore the characteristic class c(n) vanishes, and r\ is a trivial bundle. Next, we will prove that the natural homomorphism is onto. Let 0 ------ >

tt 1 ( G)

-S ->

G -2 _ >

ir,(B)

> 0

be any group extension of tt^G) by tt^B) with tt1 (B) operating trivial­ ly on ir^G). Let E be a space of type K(G,i). As before, there exists a map f : E -- > B such that f* = cp : tt1 (E) -- > •nr1 (B) . By use of standard techniques, we may assumethat f is a fibre map in the sense of Serre (see Serre, [1], §6 .1 ). It is readily seen that thefibreis a space of type K(tt1 (G),1 ). We may then replace this fibre space by a homotopically equivalent one p': E' >B with fibre Ghaving all the desired properties. q.e.d. 2 Combining this isomorphism with the isomorphism of H (B,tO with the group of G-bundles over B, we obtain a natural isomorphism between the Baer group and H 2 (TT, 1 ;tt) . Presumably this isomorphism is the same as the usual one such as is described in Cartan and Eilenberg, [1], Chapter XIV, although I do not see how to prove this.

ON THE FUNDAMENTAL GROUP OF CERTAIN FIBRE SPACES

§ 3 . An Application. Let B be a compact, connected 2-manifold which is not homeomorphic to the 2-sphere or real projective plane and let G be a circle group. Then B is a space of type K(7r.j(B),1) and G is a space of type K(Z,1), so Theorem 2 is applicable; the group of all principal G-bundles over B is in 1-1 correspondence with the Baer group of extensions over ir. (B) by Z, with tt. (B) operating trivially on Z; 2 2 in particular, it is isomorphic to H (tt^ (B) ,1 ,Z) = H (B,Z) . The corres­ pondence is obtained by letting the extension 0 -- > ir1(G) -- > 7T1(E) -- > tt1(B) -- > 0 correspond to the characteristic class c e H2(B,Z) for any principal Gbundle p : E -- > B. In this case, it is well known that HP(B,Z) = Z or Zo, depending on whether B is orientable or not. Since the group 2 H (B,Z) admits at most one non-trivial automorphism, we see that the isomorphi sm Baer Group H2(B,Z)

=

H2(tt1(B) ,1 ,Z)

which we have just described and the usual isomorphism which is defined between these two groups (e.g. Eilenberg-Cartan, [1], Chapter XIV) are the same or the negatives of each other. This fact may be used to give a pure­ ly algebraic description of the fundamental group ir^ (E) . An interesting special case occurs when B is an orientable surface of Euler characteristic x < o and. p : E -- > B denotes the bundle of unit tangent vectors. In this case we have the following theorem describing the structure of tt1(E) : THEOREM

3.

Under the hypotheses just mentioned the group exten-

sion 0 -- > Z -- > ir1(E) -H— ;>

tt1(B)

-- > 0

is that which is determined by one of the elements + x • u e H2(tt1(B) ,Z) , where u denotes a generator of the infinite cyclic group H2(tt1(B) ,Z) .

CHAPTER IV. MINIMAL FLOWS ON NILMANIFOLDS by L. Auslander F. Hahn L. Markus § 1 . A Classical Example. In this example let G be the real n-dimensional vector space Rn , n > 1, and let H be the discrete sub­ group of all points having integral coordinates. Then M = G/H is the ntorus. Choose a Euclidean metric in G and this projects to'a locally flat Riemann metric on M. Consider a one-parameter subgroup cp:

T -- > G: t---- > cp(t)

which is specified by an initial tangent vector and let M:

(x,t)

v

at the origin of

G,

> (t) and cp(xt) , for a real x / o, corresponding to the initial vectors v and Xv, define topologically isomorphic transformation groups on M. However, there are only a countable number (Fomin [i]) of unit vectors at the origin of G which induce a trans­ formation group on M which is topologically isomorphic to that induced by v. Thus there exist a continuum of topologically different transformation groups on the torus M which are induced by one-parameter subgroups of G = Rn . b) The flow cp* is distal, that is: for each pair of distinct points x and y of M the distance d(cp£(x) , e > 0 for some

e > o and for all t e T. c) The flow cp* is pointwise almost periodic (NemytskiiStepanov [1]), that is: for each point c and foreachneighborhood N of x there exists a relatively dense real sequence such that cpt (x) e N for k = 0, + 1, + 2, ... . 1+3

1+1+

point

FLOWS ON HOMOGENEOUS SPACES d) The flow cp* x e Mand for each

is equicontinuous on M, that is: for each e > o there exists a 6 > 0 such that d(y,x) < 6

implies

d( g = x l(t1) • x2(t2)...xn (tR) is a diffeomorphism of Rn onto G. 2) each subset • xi+1(ti+1)...xR (tR)} for i = 1,...,n is a closed normal subgroup G^ of G. 3 ) the factor groups G^/Gi+1,i = 1,...,n (andGR+1 = e) are each R 1. Then (x1(t),...,xn (t)} define a system of canonical coordinates of the second kind on G. Now consider canonical coordinates of the second kind in G.

MINIMAL FLOWS ON NILMANIFOLDS

1+5

If u^Ct), i = l,...,n are the canonical coordinates of a one-parameter subgroup on G, then each u^(t) is a polynomial in t. Again write canonical coordinates of the second kind for points x = (^ ,— ,£n) and y = ••,r \n ) With product z = xy = (5 1,...,Cn). Then

for

i = 1,. .., n

^i = *i + ni + ^i^11'* ‘*,si-i ,T11'*" ,T1i-i ^ where are real polynomials.

REMARKS. It is easy to see that Q1 = 0 and also Q ^ ( ,...,ii_1,0,0,...,0) = 0 for i = i,...,n. For instance, let x,(?,)••-xn (cn) - X,(I,)...Xi(5l)xi+,(Ii+1)...Xn ( tn)Xl(ni) ••.xn(,n) or x, (£,).. •xl(?l)gi+1 where

g^+1

and

g|+1

belong to

®i+1 Sill

-

=

x, (l,) ...x^(

+ ,'1)B1+i

.

G^+1. Now write the unique expansion xi+1^Ti+1^ '’'xn (Tn^

•

Thus by uniqueness, *i and hence

D

=

*i + ^i

Q ^ ( ^ ,...,ii_1,0,...,0) = o

for each

i = 1,...,n.

DEFINITION. Let D be a discrete group. A canonical basis for is a finite collection of elements fd1,...,dn) of D such that: 1)

each d e D can be represented uniquely by d = d^l d ^ 2..•dI^ n for integers m1 ,mg, ...,mn -

2) each collection {d ^ i ...dnmn} forms an invariantsubgroup of D, for i = 1,.. .,n. 5)

each quotient group

Di/Di+1

is

Z,

for

i = 1 ,...,n

and

Dn+1 = e ‘ In his penetrating work on nilmanifolds A. Malcev [i] proves the following basic results. LEMMA. Let D be a uniform discrete subgroup of a connected, simply-connected, nilpotent Lie group G. Let the lower central series of G

be G

=

G° ) G 1 D ... )Gf

=

e

.

Then Di= G1 n D is a uniform discrete subgroup of G1 and the image of D inG/G1, i = is a uniform discrete subgroup ofG/G1 . Let d-j^.^dg in D project to a canonical basis for D/Dg and let ds+1,...,dn be any basis for the abelian group D £. Then (d, d ,d 1t...,d„} is a canonical basis for D. Thus D always has i7 y s y s+1 7 ’ n ° a canonical basis. LEMMA. Let D be a uniform discrete subgroup of a connected, simply-connected, nilpotent Lie group G. Let (d.j,...,d ) be a canonical

FLOWS ON HOMOGENEOUS SPACES

basis for D. Then there exists a system of canonical coordinates of the second kind for G, (x1(t),...,x^(t)} such that xi(l) = d^, for i = 1,...,n. THEOREM. Let M = G/D and M= G/D where D and D are uni­ form discrete subgroups of the connected, simply-connected, nilpotent Lie groups G and G,respectively. Let ® : D -- > D be an isomorphism of D onto D. Then there exists a unique diffeomorphismisomorphism of G onto G, which restricts to ® on D. COROLLARY. diffeomorphic.

Nilmanifolds with isomorphic fundamental groups are

§U. An Example. The unique nilmanifold, up to diffeomorphism, of dimension is the circle S1 . The unique nilmanifold of dimension 2 is the torus S1x S1. In dimension 3 there are two nilpotent real Lie algebras: the commutative Lie algebra, which yields only the torus S1x S1x S1 as a nil­ manifold, and the Lie algebra 9fc(3,R) of class 2 generated by the basis {e.j,e2,e5) satisfying =

^3* [6^ ,6^]

=o,

[e 2 f 6 ^]

= o

•

Representing

we use the exponential map and compute the corresponding connected, simplyconnected, nilpotent Lie group to be T^, the set of all real matrices of the form

The vectors (e1,e9,e^) define a canonical system of coordinates of the first kind, and the one-parameter subgroups X-,(t) =

/ 1 0

t 1

0 \ /1 o , x2(t) = ( 0

0 0 \ 1t ,

\ 0

0

1 /

0 1 /

\ 0

/ 1 x5(t) = ( 0 \ 0

0 1

t\ 0 )

0

1/

yield the corresponding system of canonical coordinates of the second kind. The lower central series of T^ is T5 = where the commutator subgroup G1 one-parameter subgroup x^(t).

G° 1 G1 3 e is exactly thecenter and consists of the

MINIMAL FLOWS ON NILMANIFOLDS Let D be a uniform discrete subgroup of basis d 1 , d 2 , d 3 ’ Choose this basis so that d. and canonical basis for the image of D in G/G and commutative group G 7n D. Thus

T,

having a canonical project to a is a basis for the

d, =1

and

x , y 2 * x 2y , , Then =

d 1d 2d 1 d 2 for some integer relations

k t o. i. Choose

[d^dgi

Cd,,d"1]

d^

d*

so that k > 1.

•

d id2«

■

,k 5

Now the commutation

■ d3

j-k "

3

td^.dg]

■ 3 and the fact that is in the center of D, determine the group D up to isomorphism; since there are no relations in D other than those generated by the above commutation relations. But for each k = 1 ,2 ,3 ,... the corresponding discrete group D(k) is different since the center of D(k) modulo the commutator sub­ group of D(k) is exactly Z,k ‘ . Thus each uniform discrete subgroup of T, is isomorphic' to exactly one D(k), which is generated by

Using the commutation relations obtained above, it is easy to see that each element of D(k) can be written uniquely as

57

6b2 6s c3

ab+ b 1

for integers

a,b,c.

Then a straight forward computation shows that D(k) is a uniform discrete subgroup of T,. Therefore each 3-dimensional nilmanifold M is diffeomorphic with just one T^/D(k), which has a nonabelian fundamental group D(k), or else is the torus S1x S^x S1. Consider the uniform discrete subgroups D(k) of T^ and form the direct product of D(k) with n > o integer groups D(k,n) = D(k) x Z x...x Z- Then D(k,n) is a uniform discrete subgroup of T x x Rn and M(k,n) = T^ x Rn /D(k,n) is a nilmanifold of dimension 3 + n having

FLOWS ON HOMOGENEOUS SPACES

U8

D(k,n) as fundamental group. The center modulo the commutator subgroup of D(k,n) has the torsion coefficient of k. Hence D(k,n) is not isomorphic with D(k',n) for k ^ k ’. Hence the infinitely many nilmanifolds M(k,n), for a fixed n > 0 and k = 1,2,3,..., are of different homotopy types. §5 . Winding Class as a Topological Invariant. Let M = G/D be a nilmanifold where G is a connected, simply-connected, nilpotent Lie group and D is a uniform discrete subgroup of G. Let T = R1 be the additive group of real numbers and let cp : T -- > G : be a one-parameter subgroup in cp*:

M x T -- > M :

G.

t --- > cp(t) € G

Then

(gD,t) -- > cp^(gD)

=

( G/G1, there is defined an isomorphism of D/D1 onto a uniform discrete subgroup D* of G/G1. Also (G/G^/D* is a torus group t 3 of dimension s >_ 2. ,

DEFINITION. Let M = G/D be a nilmanifold of dimension n > 2, as above. Let the one-parameter subgroup cp on G induce the flow cp* on M. Consider the torus t S = (G/G1)/D*, as above. Define the flow cpT on ts by cpT:

t

3

x T--- > tS:

(KD*,t)

> («p»(t)h)D*

,

h = gG1 e G/G1 and cp'(t)h = (cp(t)g)G1. It will be shown in section 7 that each induced flow cp* on a nilmanifold M = G/D is pointwise almost periodic, and thus recurrent as t -- > +oo and as--t -- > -«at each point Q e M. Using this knowledge, we now define the winding class of cp*. The winding class extends the clas­ sical concept of the winding number (or rotation number) for a flow on a torus surface, cf. Poincar£ [1], Schwartzman [1]. where

MINIMAL FLOWS ON NILMANIFOLDS

1*9

REMARKS. Let M = G/D be a nilmanifold of dimension n > 2 , as above. At each point Q € M we construct the fundamental group tt1 (M,Q) which is isomorphic to D. Furthermore, this isomorphism is fixed, up to an inner automorphism of tt1 (M,Q). The subgroup D 1 = D n G 1 is characteristic in D and so we call D^Q) the corresponding subgroup of ir1 (M,Q) . Now tt (M,Q) /D1 (Q) is isomorphic to D/D 1 , and in a-canonical way, since form

D/D.,

is commutative.

tt^ (M,Q)

R1

Now

tt,(M,Q)

/D, (Q)

is a Z-module and we

/D1 (Q) which we further consider as an

R 1 -module.

Since

R 1 ® tt1 (M,Q) /D1 (Q) is a real vector space of dimension s-1 define the space Pq of all one-dimensional subspaces of

s>

2,

we

R 1 ® 7r1 (M,Q) /D1 (Q) . Note that Pq~1 is diffeomorphic with the real pros—1 jective space P of dimension s-1 > 1 , once a basis is chosen in D/D1 . DEFINITION. Let the one-parameter group cp(t) on G induce a flow cp* on a nilmanifold M = G/D of dimension n > 2 , as above. At a point Q e M choose a ballcoordinate neighborhood N of Q and consider the orbit 08 at which cp£ (Q) e N. loop

definedby the

curve cp^(Q)

joining

cp£ (Q)

Then

hence an element a point

ak (Q) If

to Q.

4 ^.

on

o < t £ t^ and

determines an element of

Intt1 (M,Q)/D1 (Q) s D/D., . If^ inP^ ” 1 ,

the

line space of

is never zero for large

0,

any arc in N tt^M^)

and

it determines

R 1® D/D 1 . k,

and if

lim k

=

a(Q) exists in ' Pq-1 , then a(Q) is independent of the choice of neigh­ borhood N and the recurrence times and a(Q) is called the winding class of cp* at Q as t -- > +°°. If 1 ^ = 0 for all large k, then we say that the winding class of cp* at Q as t ---> + °° is zero. Similar definitions hold for t -- > - 00. REMARKS. If a basis is chosen such that D/D 1 = Z S, and if a(Q) exists as t — > +°°, then of(Q) determines a point in the real proo 1 jective space P and hence we can assign s real homogeneous coordinates to a(Q). Two different bases in D/D 1 determine homogeneous coordinates for ar(Q) which are transforms of one another by an integral matrix with determinant + 1 . For a flow with zero winding class we assignthe homo­ geneous coordinates (0,0,...,o), for every basis of D/D., . For a classical flow cpT on a torus t 3 , s > 2 , the winding class a(Q) exists at each point Q e ts as t — > + -». Also a(Q) is zero if and only if the flow is the zero flow. A choice of basis for defines a basis for tt^t3^ ) at every point Q 1 e t 3 .

50

FLOWS ON HOMOGENEOUS SPACES

It is easily seen that, if a(Q) o, the homogeneous coordinates of a^) are the same for all Q1 c t 3 as t — > + « and t — > - °° (and they are all the same for the two time senses) , once the basis for tt1(ts) is designated. For s = 2, the winding class a(Q) = a in homogeneous co­ ordinates, is the PoincarS Cl] winding number of the flow. Two classical flows q>^ and + ® and as t — > - ». Given a basis for D/D1 , a corresponding basis is determined in each ir} (M,Q) /D1(Q) and also in ?r1(t s ) , where t s = (G/G^/D*, as above. If a(Q) = o for t — > + «, then a(Q1) = 0 as t — > + 00 and as t — > - oo for every point Q1 € M. If a(Q) ^ o as t — > + », then the homogeneous coordinates of a(Q.j) are the same for t— > + ooand for t — > - co, and ot(Q1) = a(Q) for all Q1e M, when referred to the corres ponding bases. Moreover, the homogeneous coordinates of a(Q) are those of the torus flow cpT on t 3 , referred to the corresponding basis. PROOF. Given a basis in D/D1 “ Z s, we determine a basis in each tt1(M,Q)/D1(Q) for Q e M and in tt^t3), as discussed above. Con­ sidering G as the universal covering space of M, we note that each right multiplication of G by an element of D determines an element of D fD^ and thus s homogeneous coordinates. The distinguished basis for^ ( t S) is defined as follows. Each element d e D defines a right translation of G, which belongs to tr1(M), and also the coset dG1 defines a translation of G/G1 which belongs to tt^ t 3). Tw o elements of D, which are in the same coset of D/D-j, yield the same element of tt^ t 3). In this way tt^ t 3) is iso­ morphic with Z s. Consider a closed loop I in M, based at Q e M. Let and §2 in G lie above Q that is, the cosets and say that £ lifts to 1 joining ^ to §2 in G. Now = Cgd, 2 with di2 € D and so ^ G 1 = 52d12G1 or ^ G 1 = (§2G1) * (d ^ G 1). Thus ^1 911(1 ^ 2 project to the same point QT in t 3 = (G/G^/D*. Thus the curve T pro­ jects to a closed loop £T in t 3 , based at QT. Moreover the loop £T in t 3 has precisely the same homogeneous coordinates as has £; and £T is null homotopic in t3 just in case £ determines an element of D^Q). Return to the consideration of the one-parameter group cp(t) in G andthe induced flows cp* in M and

-

■

Therefore

(x*)p+1

=

■

( V 3#

^°(P + 1)S’

° p 3 ^ X S^

( C( P + 1 ) 3 -

(XP P+1>

• (°P3 ' Xps} =

^° (p + l)s ’

XS

’ XpS

•

x*p+i)s- Consequently we also have c(p+l)s (x(p+1)s)

We now show that

c _(x* )

=

=

x (p+l)s

’

x* . Since

©*(-s)

=

(©*(s))_1

we have (e,e)

We therefore have

x*

=

(oa, x p ( o . s,x*s)

-

0 such that if 0 < |t| < s then N2(t) is constant. Thus if |t| and |r| < s we have ct(x*) = x*. Since any numbers r and s may be written r = mp s = nq where m and n are integers and |p| and |q| < s we have ct(x*) = x* . We remark that what we have just shown is that x£ is a oneparameter subgroup of N. §i+. The Major Theorems. Let C • N be the semi-direct product just as in the previous sections. Let v : C • N --- > C be the projection and let

C • N be a one-parameter subgroup. We say that cp has maximal rank provided that v ° cp(T) does not lie in a torus whose dimen­ sion is lower than the dimension of C. We say that cp is in general posi­ tion provided v cp(T) is dense in C. We remark that general position implies maximal rank but not conversely. For the remainder of this section we let S be a solvable Lie group with a discrete subgroup D for which S/D is compact. We further assume that there is an isomorphism D — > N, where N is a nilpotent Lie group and N/image (D) is compact. cp(t)

THEOREM U.1. If the flow is in general position.

(S/D,cp(t))

has a dense orbit then

PROOF. Let S C-- C * N and let v : C • N -- > C be the projec­ tion. This projection defines a continuous map v* of C • N/D -- > C. The restriction of v * to 5/D defines a homomorphism of (S/D,cp(t)) on­ to (C,v o N be the projection map. This induces a map 7T*: C • N/D*----- > N/D* and the restriction of tr* to S/D* is a homeomorphism onto N/D . By Corollary 3.2 c^ acts trivially onx* for each real t and r. Since cp is in general position we obtain that c acts trivially on x* for each c € C and real r. We observe that 7T*( ((c,x)D*) t) = 7T*( (ct,x*) (c,x)D) = 7T*((ctc,x*x)D) = x*xD = (xD*)t . Thus

7T*

is equivariant and the theorem is complete. We add that the questions of minimality and ergodicitv of nil­ flows are completely worked out in Chapters IV and V. THEOREM U.3.

If

S

is not nilpotent then

S

is not of type

E.

PROOF. Let S C C • N as before. It follows from the proof of Theorem 1, Auslander [2] that N = H © N' where H is the maximum normal analytic nilpotent subgroup of S. There is a homomorphism x -- > cx of N' onto C such that the elements of S are precisely those elements of C • N of the form (cx ,y + x) where y e H and x e N' are arbitrary. By Lemma 2.1 we may also write N = N 1 © N2 where N? is the fixed point set of C and N1 C H. Furthermore N1 decomposes into two dimensional invariant subgroups on each of which C acts as a rotation group. We choose y e H such that it has no non-zerocomponents in the above decomposition of N1 . Choose x c N'such that x / 0 and cx = e * We will show that there is no one-parameter subgroup of S through (e,y + x). Suppose there is a subgroup cp : T -- > S such that G/[G,G]

i

G/D ------- > G/DR ------- > G/D[G,G) Set 0 = $ * o $.(Despite the above notation, we shall use right cosets to make the group representations simpler.) G acts on G/D on the right by the formula g : D g 1 — > Dg'g. There exists a natural measure on G/D invariant under this action of G. Thus, the formula (U f) (Dg1) = f(Dg'g) defines a unitary representation of G in L2(G/D), where the latter space is constructed with respect to this measure. (This is the representation of G induced by the identity representation of D.) Every one-parameter subgroup (gt) of G is of the form {exp tX) for some X e @. Each such group defines a flow on the solvmanifold G/D by restricting the action of G to the subgroup, •(g^) is a one-parameter (possibly trivial) subgroup of G/R, and hence induces a nilflow on the nilmanifold G/DR. Without danger of confusion, we shall use the same symbols for the one-parameter subgroups and for the flows they generate.

78

FLOWS ON HOMOGENEOUS SPACES

THEOREM U.l. If X is a regular element of ®, the flow {exp tX) on G/D is ergodic if and only if {®(exp tX)) is ergodic on G/DR. In the ergodic case, every eigenfunction of U(exp tX) is the lift of an eigenfunction of the flow 0(exp tX) on the torus G/D[G,Gl. PROOF. Suppose h(Dg) is an eigenfunction for U(exp tX). By Theorem 3.1, h(Dgr) = h(Dg) for all r e R. Let H(g) be'the func­ tion of G, constant on right cosets of D, whose projection to G/D is h. Then, by the normality of R, H(drg)

=

H(dgr’) •= H(dg)

=

H(g)

for all d e D, r c R. Thus, h = F • Y, where F is an eigenfunction for the flow ®(exp tX) with the same eigenvalue. In particular, h is constant under the flow if and only if F is constant under the correspond­ ing nilflow. This proves the first part of the theorem. The proof of the theorem is completed by applying the nilflow theorem (Chapter V) to 4>(exp tX) . COROLLARY: If D is a closed uniform subgroup of the solvable, non-abelian, connected, simply-connected type (E) group G, there exist ergodic flows on G/D generated by one-parameter subgroups of G. These flows are neither equicontinuous nor mixing. PROOF. The homomorphism of © onto ©/[©,©] is continuous and open in the ordinary topology of these real vector spaces. The regular elements of © form an open set whose image is consequently open, hence contains elements of any dense set. But the vectors of ©/[©,©] giving rise to ergodic (irrational) flows on the torus G/D[G,G] are dense, so at least one is the image of a regular element of ©. Any such regular preimage then induces an ergodic flow on G/D. By the theorem, the eigenfunc­ tions cannot be dense in L2(G/D), s o the spectrum is not pure point and the flow is not equicontinuous. To prove that it is not mixing, it is suf­ ficient to show that there is at least one eigenfunction other than the con­ stants, for then the spectrum cannot be purely continuous. But the existence of such a function would follow from the fact that the torus G/D[G,G] is non-trivial. If G is nilpotent, this torus is at least two-dimensional. If G is not nilpotent, D[G,G] C DN, where N is the maximal normal nilpotent subgroup of G. Since DN / G, the torus is non-trivial and there exists some point spectrum for the flow. In three dimensions there is only one type (E) solvable group which admits a discrete uniform subgroup — this is the group in Chapter III. The regular elements of its Lie algebra are precisely the elements which induce non-trivial rotations on the associated torus; since the latter is a circle, every such one-parameter group of rotations is ergodic. In this case we were able to show that periodic orbits exist, so the flow on the solvmanifold is not minimal. It seems reasonable to conjecture: if a flow on a type (E) solvmanifold S is minimal, then S is a nilmanifold and the flow is equivalent to a nilflow.

CHAPTER VII Appendix L. Auslander Let type (E) with of G, fS,Hl shs~1h~1, s € to prove that

such that

G be a connected, simply-connected solvable Lie group of a discrete uniform subgroup D. If S and H are subsets will denote the group generated by elements of the form S, h € H. Set G°° = lim [G,..., [G, [G,G] ].. .]. We want k -► oo G°° n D is uniform in G°°.

LEMMA. Let 0* be the set of (exp tX) € D for some t / 0.

X in 0, the Lie algebra of Then 0* is dense in 0.

G,

PROOF. Let : G -- > G/D be the natural projection. Every one-parameter subgroup of G is of the form exp tX for some X € 0. If 4>(exp tjX) = (exp tgX) for t1 ^ tg, then exp(t.,-t2)X e D and X e 0*. Thus is one-to-one on every one-parameter subgroup of the form exp tY, y i 0*. Suppose Y is contained in an open subset (and hence an open cone) K of 0 - 0 * . Because G is simply-connected and of type (E), the exponential map is a homeomorphism. Since is open, its restriction to exp K is a homeomorphism. Thus n are rationally independent. If e > 0 is given and if i = i,2,...,n are arbitrary real numbers, then there is a relatively dense set P of the integers such that |X1pi (c) - 0i | < e, for

mod 1

,

i = 1,2,...,n and c e P. To prove this theorem we study a particular type of nilmanifold and a discrete flow on it. The discrete flow comes from restricting a oneparameter flow to integral values of the parameter. A generic group of our type will be a matrix group N with elements of the form

where each

N^

is of the form

81

FLOWS ON HOMOGENEOUS SPACES x

x 2/2 !

1

xn_1 /(n-1) .'

x

' o

xn_2/(n-2) !

1

x

y^

X,

yn_-| ±

\

y2i : yli

1

/

A typical element of N will be written (*,yjj_) • Let D be the discrete subgroup of N whose elements are of the form (n!a,b^) , where = 1 >•••»n > are arbitrary integers. It followsfrom Malcev [1] that D\N is compact. In order to study any flows on D\N we must be familiar with the matrixmultiplication of N. It has the following form

where v = x + z uli

for

=

yli + wli

u2i

= y 2i + w2i + wlix

uji

= yji + wji + wj-l,ix + wj-2ix2/2!

+ •" + wllx‘3‘1/(j-l) I

i, j = 1,2,...,n. By a straightforward computation we see that ( t; ) is minimal. We then examine the orbit of the coset D for integral values of t. These observations are interpreted in terms of the coordinates of the group N. This interpretation forms the proof ofthe theorem. For the sake of clarity our line of reasoning is divided into two lemmas. In both these lemmas we make use of the notation of the previous disdussion. c = -n!a.

LEMMA 1. In N let d = (n!a,bj^) If (v,u^) = d • ( t;X1) C V(a),whenever !11 ( t;X^)) is a minimal set and thus each orbit is almost periodic (Gottschalk-Hedlund [1]). Thus there is a relatively dense subset S of the reals having the property that s € S implies there is a d € D for which dcpCsjX^) e V(s). Since d = (n!a,bj^) we see that |n!a - si < s . Consequently there is a real number t such that t + s = n!a and |tl< 5. This implies that dcp(s + t;X^) e V(5)q > ( t ) C V(a) . Let P be the set of all integers obtained from S in the preceding fashion. Since S is relatively dense so is P and P satisfies the conclusion of the theorem. PROOF of THEOREM 1 . a = min (e/(n •n l a^

for each

In

•^ I lajiI) aji^'^ 1) . 11 Ji i, and let fijl

=

Lemmas 1 8111(1 let

2

let

nj be the number of non-zero ni_ " be tlie num^er non-zerc

ei/n1(-1)J+ 1 a3i j!

= 0

and

if

ajl jt 0

lf aji “ 0 1

According to Lemma 2 there is a relatively dense set of integers P having the property c € P implies there is a d c D such that dcp(c;Xi) e V(a) .

81*

PLOWS ON HOMOGENEOUS SPACES Thus

|v 1

-

|n !a

i f

d

+ c|


o

fo r

I f

s >

= l , 2, . . . , n .

i

L et

be

a

set

o f any

n

re a l

o f

in ­

j =l n u m b ers. te g e rs

P

su ch

o

is

g iv e n

th e n

th e re

I V ^ C c ) |< e , fo r

c

e P

and

To

We o b serve

th e r e s t r ic tio n seeth is

o d ic o r b it

we u se

(T h e o rem o f th e

fa s h io n .

a

r e la tiv e ly

d en se

se t

Lem m a 1

s tr a ig h t-fo r w a r d .

t h a t Lem m a 2

tru e

i f

o f

1

and

( D \ N ,c p ( t ; x ^ ) )

is

a lw a y s

o f r a t io n a l in d e p e n d e n c e

th e

fa c t

U, C h a p te r co set

1

m od.

= i , 2, . . . , n .

i

PROOF. and

is

th a t

D

th a t

IV ).

and

r e m a in s

th e

W e n ow m e r e ly

is

r e s tr ic t

p r o o f o f Lem m a 2 g o e s

u n a lte r e d

and

th e

th e

are

a ll

x^

is

p o in tw is e

ou r a tte n tio n th ro u g h

p r o o f o f T h eo rem

in

th e

2 is

zero rem o ved . p e r i­ to

th e

sam e now

CHAPTER IX. D IS C R E T E G R O U P S W IT H D E N S E O R B IT S by Leon G reen b erg § 1. m e d ia te ly

be

In tr o d u c tio n . L e t

s p e c ia liz e d

D

Q = th e q u a te r n io n s )

and

e le m e n ts

in

g e n e r a l lin e a r

s in g u la r

m a tr ic e s

D-

The

o f m a tr ic e s

w ith

is

th e

a

fie ld ,

in

be

a

d iv is io n

R = th e r e a l f ie ld ,

to le t

D n ;

D n

th e

d e te r m in a n t s im p le c tic

d e n o te gro u p

s p e c ia l

1

th e

(in

(w h ic h w i l l

r in g

o f

G L ( n ,D )

c o m p le x

n x n is

th e

gro u p

g ro u p

S L ( n ,D )

sen se

o f J .

D ie u d o n n e

th e

is

th e

gro u p

im ­

fie ld

m a tr ic e s

lin e a r

S p ( 2n , D )

gro u p

r in g

C = th e

or

w ith

o f non­

* is

th e

g ro u p

[ 1] ) ;

i f

o f m a tr ic e s

D

w h ic h

n le a v e s

in v a r ia n t

th e

b ilin e a r

fo rm

( x ,y )

J

=

( x j_ y j_ + n

- x i + n ^ i^

*

i= i We s h a ll p ro ve 1.

THEOREM v e c to r

space

S u pp ose

L et

r

i f

v

over

th a t

be

G

0 S L ( n ,D ) ,

n >

0 S p ( n ,D ) ,

n even ,

d is c r e te v

^ o ,

to

rH

L et

M

eous

c o v e r in g

th a t

T h is

is

g e o d e s ic tu re fo r

is th e

an

le t

(r e a l)

V

th e be

fo llo w in g

th e o re m .

a n n -d im e n s io n a l

L ie su b g ro u p

o f

G L ( n ,D ) .

rv

is

d en se

is

a

m ap

e G |g (v )

d en se

in

case

G

w ith K

o f a 1

S.

in

th e o re m a

in be

The p resen t

p ro ved

fo r

p r o p e r tie s rv

a

d en se

G -stru c tu re co m p a ct,

and

th e

r

G ).

T hen H

o f as

A .

th e o re m

( i .e .,

M

S

c o in c id e s

c la s s

is

e q u iv a ­

fo llo w s :

has

a

o f

hom ogen­ c o v e r in g

i s im m e r s e d every

cu rve

c o n s ta n t n e g a tiv e w ith

o f

in

d en se su b se t.

[ 1] t h a t

w ith

V as

gro u p

an everyw h ere

H e d lu n d

s u rfa c e

T hen

V.

in

r e in te r p r e te d

is M)

r\G .

c e r ta in

in

a

o f G.

com p act

be

fa c to r sp ace

V .

= v ] ,

can

su b gro u p

H C G — > G /K — >

a n a lo g u e

c u rv a tu re

a c tu a lly

w h ere

d is c r e te

com p act

tr a n s itiv ity

th e o re m

co m p a ct m a n ifo ld G /K ,

D = R o r C. w ith

w ill

The

or

G

H = {g G .

1,

o f

s u ffic ie n t i f

in

space

tr a n s fo r m a tio n s th e

o r b it

th e o re m

have

d en se any

su b gro u p

th e

above

N o te le n t

(u n d e r

a

G

The

M

C o r Q,

be

b)

w h ic h

be

le tG

o f)

e ith e r

a

G

g e n e r a l v e r s io n

D = R,

L et and

m o re

a)

€ V ,

gro u p s

D

(a

H e d lu n d 's

o f cu rva­

th e o re m

= S L ( 2, R ) .

W e m ig h t a l s o

rem ark

th a t

K.

th e o re m .

85

M a h le r

[ 1] h a s p r o v e d

th e

fo llo w in g

86

FLOWS ON HOMOGENEOUS SPACES Let r

Jk) _ +

r . { J k ) , (k), , _ T n r ' I7 17 (z) ' ^ 7 21

(k)

T ia (k) x T^T ' 7ij real’

(k) (k) _ (k) (k) _ 7n 22 711 2 1 - 7

Z + 7 22

be a Fuchsian group with compact fundamental region. Let f(x,y) be a positive definite form. Then the values f (7 ^', 7 ^)) are everywhere r?A ne j»a

In

n n p it.lv p

n n m b A r fl.

Mahler also conjectured that if

f

is an indefinite form, the

values f(r|^, 7 2 ^ are dense dn ^h-e real numbers; if r is a Kleinian polyhedral group and f a (positive definite) Hermitian form, then the values f(7 ,i , ri^) are dense in the (positive) real numbers. Theorem 1 ( k) = (r^j ( k) )) not only verifies these conjectures but shows that if r = (7 satisfies the hypotheses of Theorem 1, and f(x1 ,...,xn) is any form, then the values f(7 ^ , 7 21P >•••*7 nlf^ are dense in the range of f. For if v is the vector (1 ,0 ,...,0 ), then 7 ^ (v) = (7 ^ , 7 ^ >••• * Theorem 1 says that the n-tuples (7 ^ , ?21 >' '*'7ni ' are dense in Therefore, if f is any continuous function on V, the values f (7 ^ ,•♦•,7 ^ ) are dense in the range of f.

v*

§ 2 . Strongly Transitive Groups. In this section D is R = the real field, C = the complex field or Q = the quaternions, and V is an n-dimensional right vector space over D. If A e D n , an eigenvalue x of A will mean (as usual) an element x € D (x e C if D = R) fqr which there is a vector v ^ 0 (in the complexification of V if D = R) such that Av = vx. In the case D = Q, A can have a large set of eigenvalues, for if x is an eigenvalue, then d“1xd is also. However, it may be verified (by considering V as a 2 n-dimensional space over C) that A has a set of n eigenvalues x.j,...,x (each listed with possible multiplicities) such that any eigen­ value of A is conjugate to some x^. We shall call(x.j,...,xn) a repre­ sentative set of eigenvalues for A. A will be called semi-simple if it has a representative set of eigenvalues (x1 ,. ..,xr) for which there is a basis v-j ,... ,vn of V with Ava = vaxa = 1 >'**>n) * This will be true, in particular, if no pair xa,Xg are conjugates. The basis (vp...,vn) will be called an eigenbasis for A. If v £ 0,the ray through v is the set p(v) = {w|w = vr, r realand > 0). V has a natural topology (as a vector space over R) and this induces a quotient topology on the space of rays (p(v) is an equivalence class under the relation: w ~ v if w = vr, r real and > 0). DEFINITION. A subgroup G C GL(n,D)is called strongly transi­ tive if G satisfies the following conditions.

DISCRETE GROUPS WITH DENSE ORBITS

87

ST l. G is transitive: If v and w are non-zero vectors, then there exists g e G such that g(v) = w. ST 2. G is slmost doubly transitive on rays: a)

b)

ST 3.

G of a) b)

If p 1 , p 2 are two rays and^,^2 are two °Pen sets of rays, then there is an element g € G such that gPl € n1 and g"1p 2 € If Pi >p 2 are tvo rays and n is an open set of rays, there exists an element g € G such that gp1 and gp2 € n. contains an element g which has a representative set eigenvalues (x1,..., x ) such that: The norms Ix1I,...,IXR | are distinct. Ix1I > 1 > |x2 | .

Note that SL(n,D) and Sp(2n,D) (n > 1 in the first case, D = R or C in the second case) are strongly transitive groups. Moreover, strong transitivity is inherited by larger groups in GL(n,D) . The theorem we wish to prove is the following. THEOREM 2. Let G be a locally compact, strongly transitive group, and let r be a discrete subgroup of G with compact factor space r\G. Then for any non-zero vector v e V, the orbit rv is dense in V. The proof will be broken up into several lemmas. We shall sup­ pose that a fixed Euclidean norm is chosen in V with respect to some basis n n _i_ v1,...,vn : If v = subset of

^ a=Tl

v cfia’

then

M

=(

£ or=1

•

If x

ls anM

V, |v - X|

=

inf Iv - x| x e X

LEMMA 1. Let A e D R and let (x1,...,xr) be a representative set ofeigenvalues for A. Suppose that |x1|^ |xa | (a = 2,...,n) and v1 isaneigenvector correspondingto the eigenvalue x1. For e > o there is a neighborhood n of A such thatif B e d , then B has a representative set of eigenvalues (n.j,...,nn) and an eigenvector w 1 cor­ responding to the eigenvalue n1 with 1) 2)

|na - *-a \ < e (« = 1,•••,n) |w1 - v1| < s .

PROOF. This is known for the cases D = R or C. For D = Q, it can be proved by considering V as a 2n-dimensional vector space over C. If A = (aap) € D n > we consider Aas a linear transformation on V by choosing some fixed basis t ^ . . . , ^ of V and letting n Ate =

I

Qf-1

A Vae-

If

-

Z

“.I

tad«>

then

At

■

1
o there is an open set ft containing S such that if B = (bap) € ft then, B has a representative set of eigenvalues (n-l,•••,nn) and an eigenvector w 1 corresponding to ^ with 1)

ln1

2)

lna l< e

- b 1, I < e

3)

l w1 -

,

(a = 2,...,n) ,

v1| < e

b) Any open set n which contains Scontains an open set of the form Z(S,&) = {B| ||B- S|| < &} . c) If ft is anopen set containingS, then there is an open set flgcontaining S and an open set n1 containing 1 so that C ft. containing

Let

ft(|)

PROOF, a) By Lemma 1, for each £ there is an open set ft(|) A(|)so that B e ft(|) has a representative set of eigenvalues and an eigenvector w 1 corresponding to u1 with *0 5)

I^1 “ 11 < \ lna l < e ,

6)

|w1

a = 2,...,n

,

- v1I < e

also fulfill the condition 7)

Then

,

U ft(|)

ft =

Ib^ i - £ I
»

(S

as in Lemma 2) .

Choose a number 5 > 0. By Lemma 2a there is an open set Y containing S so that if B 1 = Cbap) € y, then B 1 has an eigenvalue n ’ and a corresponding eigenvector w. with

k)

|w, - v, | < 6

.

By Lemma 2c) there is an open set ftg containing S and an open set containing 1 so that CY . Since ||A^ - S|| ---- > 0 as r -- > Lemma 2b tells us that there is an integer rQ so that A^ e dg for

n1

r > rQ . Now suppose that B € r^A1* (r > rQ) . B = = (eap) £ ft,. Let B 1 = — ^ B. Then B, = cuA^ e C C Y. Ix^ | » Therefore B 1 has an eigenvalue u1 and a corresponding eigenvector w 1 so that

3)

and

has an eigenvalue

U)

are satisfied (where =

|x1|r uj

b.« »

u,i

e.. ----- ).

Then

with corresponding eigenvector

B

w1 ,

and

90

FLOWS ON HOMOGENEOUS SPACES

K

-

< K

- I V r bnl + I lxi!r b11 - X1*I

= I |x,|r < If we choose

-

r bn

IX,

| i x j 1* + |en - 1 |

small enough so that

| + Ien x^ ix /

- x*|

.

le^ - 1 |
rQ and B € Arn1 then value and corresponding eigenvector w 1 such that 5)

in,

-

6)

|w1 - v, | < 8

B

- x^| < has an eigen­

< si*.,r .

By Lemma 1, ^ can be shosen small enough so that this is also true for r X = - 1J2J. 'JrQ . Now consider the

(n-1) x (n-1) matrix

By theinductivehypothesis, there Dn_i 30 tbab( ? o r eigenvalues

A2 = diag(\2,X.^,...,xn) .

is a neighborhood

of

1 in

a11 integers r > o) if B £ ^ n'^Ap, then B has and c o r r e s P o n d i n S eigenvectors w^,.,.,w^ with

T) |na - xS| < &|xa |r

(a = 2,...,n)

,

8)

(a = 2, ...,n)

.

’ vcJ < &

Let n2 be a compact neighborhood of implies that

1

in D R

such that

(eap) € fl2

Let X = (x € GL(n,D) | |x(v1) - v1 | < 6, x(vff) = vQ , a = 2, ... ,n) . We assert that there is a neighborhood n of 1 in GL(n,D) suchthat 9) 10)

x _1fi x C n2

for all

x € X,

n C a1

The map (x,y) -- > x_1y x (x,y € GL(n,D)) is continuous. Therefore for each x € X there is a neighborhood 0(x) of x and a neighborhood fi (1) -1 x of 1 so that m nm € n2 for m e n(x), n e nx (1) . Since X iscom­ pact, X is covered by a finite number of these neighborhoods: Cl ( x 1),..., Q (x ) . Let = fl n f| ft (1). Then ft satisfies 9 and 1 < a < r xa

DISCRETE GROUPS WITH DENSE ORBITS Now let as € n and B = ojA . We know that with corresponding eigenvector w 1 so that

There is

5)

In, -

6)

|w1 - v1 | < 5

x e X

Thus

so that

x

x(v^) = w 1 .

x -1Bx

x_1(wl) = vr

x

Since

/

=

x 1Ar x (v )

=

x-1Arx

x3

v

,

(for

31

ni x"103x

=

has an eigenvalue

.

is of the form:

X

B

< 6|X,r

corresponding eigenvector oc > 1 .

91

(ea(3) € n2,

therefore

1

a > 1) ,

has

eigenvalue n . T v ^ and

with x(v^) = va ,

FLOWS ON HOMOGENEOUS SPACES

92

But since x Bx has eigenvalue we have b 11 = , ba1 = o for

with corresponding a > 1, and ■ p

/

/

eigenvector

v 1,

r» in n

^1

G12X2 **'

0

£22X2 ‘•* e2nxn

x ]Bx

^

(n-i)

Since

•••

the matrix

“n2 e2

'22

2 2

•‘'

has eigenvalues satisfying

with corresponding eigenvectors

7)

Let

v;

n = £

2n\r n

enn\ n

“n2

8)

e


positive real numbers, such that for any measurable set n C r\G, ii(ng) = ii(n) x(g) . This is true in particular, if r is dis­ crete. We shall call a relatively invariant measure a Haar measure on r\G. If r\G has finite Haar measure, then the measure is invariant. For putting n = r\G, we have fl = fig and n(n) = n(ftg) = n(n) x(g) . Therefore x(g)

=1 . The following lemma is due to A.

Selberg [i].

LEMMA 5 (Selberg). Let G be a locally compact groupand let r be a discrete subgroup such that r\G has finite Haarmeasure. Then for anyelement g € G and any neighborhood n of 1 there is a positive in­ teger n and ® 1,®2 € n such that cD1gnoj”1 e r. PROOF: Let tr : G ---> r\G bethe projection, whichmaps each element into its r-coset. Since r\G has finite measure and the sets tr(n), ir(n)g,..., 7r(n)gn have the same positive measure, it follows that there are distinct integers r,s (say r < s) so that ir(n)gr n 7r(n) gs ^ o.

96

FLOWS ON HOMOGENEOUS SPACES

This means that there are r2a>2gs. Then

7^ ^

o>1,a>2 € ft and

«

€ r

30 that

“1

The above lemma is the only place where we shall use the discrete­ ness of r. Note that the lemma is true under the weaker assumption that r is closed and unimodular. However A. Borel [1] has shown that if G is non-compact, simple, r is closed, unimodular and r\G has finite Haar measure, then r is either the entire group G or discrete. Our feeling is that this is probably true for strongly transitive groups. For this reason we shall suppose that r is discrete. In the following lemmas G is strongly transitive and r is a discrete subgroup with compact factor space r\G. In particular r\G has finite measure, so that Lemma 5 is valid. LEMMA 6.

If

PROOF. so that By ST 3 groups) there is a set of eigenvalues

Let

y € r

is

suppose that

p

is

any ray,

Tp is

dense in the space of rays.

ft be an open set of rays.

We must show that there

yp c ft .

(condition 3 in the definition of strongly transitive semi-simple element gQ € G which has a representative (^ , •••,*-n) with distinct absolute values. We shall

|X j =

max |\ |. Let 1 < a < n a

(v.,...,v„) 1 n

eigenbasis and let x = v1 + v2 + ... + v . Let p(x) x

and let

ft'(e)

n'(e)

be a corresponding be the

be the following ray-neighborhood of

» { p(v) | t

. ^ vada, a=1

ray through

p(x): a =

|da - 1| < e,

.

By ST 2a there is e ft and g”1p € ft’(s). The _-j g e G so that gp(v-) * element g1 = ggQg has a representative set of eigenvalues •,*n) with a corresponding eigenbasis v ’ = g(v1),...,v^ = g(vR). g"1p = p(v) n where

v

p = p(v’),

= ^ vada a=i where Suppose

Then

v1

lfa

lda * 1 I < e >

a = 1,...,n.

This means that

n = ^ vida* a=l

that we change thebasis ■i a=1

i a =l

1)

v'

and

" da l
o

61 | A '" 1 In case D = R, guarantees that 7)

f1

as

m

k

T5

and

ymp e n.

> «( oc > 1) ,

' "f

k

T 5 r’ ’ < ‘

for large

m,

we have

'

n1 are real and (for small enough e) 6) For D = C or Q, n1 can be represented as

n1 = a + bh,

where a,b € R and h is pure imaginary with |h| = 1 . 0 so that a = \\x | cos 0, b = |n1 | sin 0 and 8) n1

=

I ^ K c o s 6 + h sin 0)

9)

=

|m.™| (cosm0 + h sin m0) .

^

There is an angle

If 0 is a rational multiple of ir, then v.™ = |n.,|m for some m. For large enough r, r ^ p e n. If 0 is an irrational multiple of tr, then there is a sequence m. > °° such that cos m .0 + h sin m. 0 > 1 . For 1 m. 1 1 large enough (and small e), y 1 p e ft . The following lemma is due to Hedlund [1]. LEMMA 7 (Hedlund). Let n be an open set in V. for any (non-empty) open subset E C (1, rZ is dense in V. tains a point v such that rv is dense in V. V. in

Suppose that Then n con­

PROOF. Let n . b e a countable base for the open sets in Since rn is dense in V, there is a point v1 e n which ha3 a r-imag n1 .By continuity, there is a closed ball B1 = (v| |v - v1 I < r.j) C n,

FLOWS ON HOMOGENEOUS SPACES

98 such open v2 € such this such

that all points in B1 have r-images in n1 . Since B1 contains an subset of fl, rB1 is dense in V. Therefore there is a point B 1 which has a r-image in n2. Then there is a closed ball B2 C B 1 that all points in B2 have r-images in n1 and n2. Continuing in way, we can find a sequence of closed balls B 1 0 B 2 0 ... 0 B 0 .. ., that all points in Bn have r-images in n1 ,n2,... ,nn>

00

n B_ n=l n

contains at least one point

dense in

e

n,

and it is clear that

rv

is

V. LEMMA 8.

in

v

There exists a vector

v € V

such that

rv

is dense

V.

PROOF. A subset S in V will be called convex if cS implies that the segnent[s.j,s2] = {sis = s-jO-x) + s2x,xreal, 0 < \ < 1} C S. We shall show that if Y is a convex open setin V - Co}, then rY is dense in V. The lemma will then follow from Lemma 7. Let n be an open set in V. We must show that there exists 7 € r and v € y such that y ( v ) e n. Let Yp and hp be the following open sets of rays: *p

and let 7q

70

pQ € ftp.

=

(p (v )I v € Y}

,

ftp =

(p(v)I v e ft}

,

By Lemma 6

there is

r0 € r so that

is continuous, there is an open set of rays Z C

ft

ro0o £ Tp- sinoe such that y

c v

By ST 3 there is an element g € G set of eigenvalues (x1,...,\n) such that 1)

Ix1I,...,|xn l are distinct,

2)

|x,I >

1

which has a representative

> | x2 l .

Let ( v ^ - . ^ v ^ be a corresponding eigenbasis for g. By ST b) there is an element h e G so that hp(v-), hp(v0) € Z . The element _i p k = hgh € G has the same representative set of eigenvalues ( , . . . , xn) with a corresponding eigenbasis (h(v1),h(v2),...,h(vn)). By Leninas X and 5 there is a semi-simple element y 1 € r which has a representative set of eigenvalues (ii1,... ,nn) and a corresponding eigenbasis (w1,... ,wn) so that 3)

|ua - ^1 < e|^a |r

( 1 > |n2|, and condition U) implies that p(w1),p(w2) e Z p .

DISCRETE GROUPS WITH DENSE ORBITS Since there is segment in

Y,

since

Y

p (w 1^

and

70 p ( w 2 ) €

and a point

and

riqa

[q1 X^,q2 X^] .

p2 €

Since

(a =

1

< x


n y

is convex.

Since 0

7 0 p(w.j)

7^*^)

[p1 ,pg]= {pip = p.j(i-x) + p 2 x,x real,

qi = 701(P1^ €

>

£p C Y ,

yQ

a point p 1 €

99

> ».

where r a

is real and

Thus

=

r

as

^

00 *

Since

can be represented in the form

x1 = a + ub, where a and b are real, and u is pure imaginary of norm 1 (if D = R, then b = 0 ). Since |x1 | 2 = a 2 + b 2, there is 9 such that a = |x1 |cos 9 and b = |x.j| sin 9 . Then x1 = |x.j|(cos 9 +u sin 9 ) , and x^ = |x |r (cos r0 + u sin r0). There is always a sequence rk -- > » such that cos rv0 + u sin r.0 -- > 1 . The segment r r r 71 k tq^qg] = tq.jX.jk, q ^ x ^ ] approaches the ray p(w1). Therefore, for large p e

y^

r^,

tp ^ p g ]

C

[q1 ,q?]

Y

su ch

LEM M A th a t

th e

d en se

in

ie n s e

in

ie n s e

in

9. L et

c o r r e s p o n d in g

V .

1
w th e to

The

p(w) ,

e D

it

jx |

th e re d

e le m e n t 7 c ^

1.

r ,

T hen

is w € V

€ D ,

d

so

^ 0,

d ^ is

and

> tru e ru

to

su ch

rv is

th a t

rw

th e n r(w d )

r

th a t

^ j ^ v

d . is

d

is is

d en se

in

^

[7jv ]

has a

J

W e now

th a t a

u

su ch th a t I f

= ^w j ^ *

-- >

r. a

^

so su b ­

|wj|

d e n o te s

Thus

fo r

|w

I


o f

= |w |.

[x ]

su bsequ en ce

= w d.

now

1
w .

=

^ 0.

fo llo w s ru .

ray

th a t

sequ en ce

I t o f

th e

Jv l

such that

th e re

v e c to r on

fo llo w s

d€ D ,

i

o f v e c to rs

a

[ 7 ^ .7

th e

J

{7 ^} C r

each

seq u en ce

th e n

e le m e n t

sam e u ,

so

|w ||d.|, an

F or

th e v e c to r

span nedb y x ,

v k

V .

Wj >

to

In

Thus there is a point

fi.

e ig e n v e c to r

(7i v )x r .

co n verges

J j

v erg es

e

there is a sequence

6

be

|w.d.|

V ,

seen

L et

con verges Ln

an

p r e v io u s

^ 7 j 7 w h ic h

J

th e re

and

th e

7i 7r (v) = r . | 7 j_ v | | x [ 1
W n Let Since

7 jV

7 jp(v)

> p(w),

are on the same ray,

vj

- I

V i j ’

(of = 1,...,n).

^ x af a y a=1

=

y

then

8111(1 let

vj €

it follows that = v jr j>

faj

-

This means that

where

fijrj

v^ --- > w. r^

and

--- “----- >

'V

with lv jl = lw l * Since

is real and

fij

'V

> fa as

and

y ^v

If

i -- > ”

as j

v^

> 0 .

>

Also,

101

DISCRETE GROUPS WITH DENSE ORBITS y ^v -- > o,

since

|fQj |

>

as

0

j -- > «

fi i Since------ -- > f 1 / o, f 1 ^ ^ o

(a *

1

,... ,n) .

for large

j.

Since also

f1 j > o, we have o < |f 1 j | < 1 for large j . We may suppose that this is true for all j . Then for each j there is a positive integer S j s o s. s1 J that 1 < If-jjl l^,! < l^,!- The sequence {f ^ ^ J) has a subsequence Cf1 -j^m-i

which converges to some element

f e D, f ^ 3k ^k {ly y r^v}.

,

We now consider the sequence ence 7

Sk

V

rvv

Z

0

.

^ ..3k X«f« k V

a=1

a £ 1,

For

if

ski

ak"a |fakl

Since

_ |fJ

|flk l

k 7

---

sk

>

oo,

as

if,k i sk ,

sk | 1k 1

_

lf lkM1 l I --- >

If, I

sk fak^a

it follows that

rkv -- > x ^

if =

k— >

----- > 0

and "j---f^k --- > ° aS I*,!8* as

k

—

> 00. Thus

00.

By Lemma 9 , r(x,f) is dense in V. Since a sequence of r-images of v converges to x,f, it now follows that rv is dense in LEMMA 11. The transitivity of G and the compactness of imply that for any v e V there is a sequence (7 ^ C r so that lim y^v = 0. L■

V.

r\G

PROOF. Let v ^ 0 . Since G is transitive, Gv = V - {0 }. There is a compact subset K C G so that G = rK. The set Kv is a com­ pact subset of V which has the property that for any w e V - (0 ), there is y € r with 7 w € Kv. Since Kv is compact, there is a number M so that if w € Kv, then |w| < M. Now let {X^} be a sequence in D so that |Xj | > » as 3 -- > 00. For each j there is y^ e r so that 7j(vXj) € Kv- Then I7 j (v) IIxj I = 17 j (v) I

> 0

§3. manifold M is of unit tangent translating the

311(1

< and

7 j(v)

>

0

as

l7j(v)l < ~ — |V)I

• Therefore

j — >

Geodesic Flows. The usual geodesic flow on a Riemannian a 1 -parameter group cpt of transformations of the space U vectors on M. If u € U, cp^u is obtained by parallel vector u distance t along the oriented geodesic which Is

102

FLOWS ON HOMOGENEOUS SPACES

tangent to u. The properties of this flow on locally symmetric spaces have been studied (cf. G. Hedlund [2 ], E. Hopf [2 ], F. Mautner [1 ]). We shall consider here an extension of the flow to the space F of orthonormal n-frames on the manifold M. A frame (v1,...,vn) is paral­ lel translated distance t along the geodesic determined by v1. We shall consider the particular case of a compact 3-dimensional manifold M with constant negativecurvature. Such a manifold has as uni­ versal covering space the 3-dimensional hyperbolic space H^. The isometry group of H^ is G = SL(2,C) modulo its center. If r C G is the group of covering transformations corresponding to M, the space of frames F may be identified with r\G, and the geodesic flow in F is given by rg > rghj., where

DEFINITION: A flow q>t on a space F regionally transitive if for any two open subsets number tQ, so that cpt (ft) n Z

/

0

for

is called permanently ft, Z in F, there is a

t > tQ .

THEOREM 3. Let M be a 3-dimensional compact manifold with constant negative curvature, and let F be the space of 3-frames on M. The geodesic flow in F is permanently regionally transitive. PROOF. ponding to M. r factor space r\G. sets of G, there

1)

Let r be the group of covering transformations corres­ is a discrete subgroup of G = SL(2,C) with compact We must show that if ft and Z are any two open sub­ is a number tQ so that

Ptth-t n Z

^

0

for

t > tQ, where

h^

=

Note that it is sufficient to show this for the case that ft is a neighbor­ hood of the identity. Any open set contains a set gft where ft is a neighborhood of the identity and rgftht = g(g"1rg)ftht . If r has compact factor space, so has g~1rg. If we have shown 1) for any discrete r with compact factor space and any neighborhood ft of 1 , then it follows that (g_1rg)ftht intersects g~1Z for t > tQ,therefore 2)

rgftht n Z

^

0

for

t > tQ .

Thus it is enough to prove 1) for neighborhoods ft of theidentity. let P be the subgroup of G consisting of all matrices of the form '1

o'

DISCRETE GROUPS WITH DENSE ORBITS

103

P is the subgroup of G which leaves the vector (0,1) fixed. Every isotropy group (of a vector inthe 2-dimensional complex vector space V) is of the form gPg”1 for some g € G. The theorem of the previous section states that rv is dense in V, for v ^ 0. This implies that rgPg-1 is dense in G, for any g e G. o\ _1 / 1 o °\ j € P, then h^ p ht \a 1/ V e"2ta 1 This shows that for any p € P and any neighborhood n of 1, number tQ = tQ(p,n) so that If

3)

p

=

/ !1

ht p h^1

€ n

Since r\G is compact, there is a compact set K C G for any g e G there is y e r with rg e K. For any k e K, dense in G, therefore there is y^ e r and e P so that 7 kkplck~1

n(k)

of

e £ k_1 > k

or

there is a

rk1(T>k

€

so that rkPk-1 is

continuity, there is a neighborhood

such that rk^(k)p k n E

^

0.

K is covered by a finite number of these neighborhoods: o(k1),...,n(k^). We shall denote y ^ by y^ and p^^ by p^ (i = l,...,n). Then if k € K, 5)

r1 kpi € E

By

3)

6)

ht p± h^1

for some

there is =

tQ

i.

so that € n,

for

t

For each t, there is y e r so that y t h^ = t there is some i so that 7i7-t h^ p^ = ri ri 7t % t Thus

rn h^

ht

= 7i 7t ht ht1 ®i,t ht

n E

^

0

=

> tQ, i = 1,...,n. kt e pi €

K. By 5), foreach Z. Then for t > tQ ,

ri 7tht pi e Z

•

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