UNIPOTENT FLOWS ON HOMOGENEOUS SPACES OF SL2(C) A THESIS submitted to the Faculty of Science of the University of Bombay for the degree of Master of Science BY NIMISH A. SHAH TATA INSTITUTE OF FUNDAMENTAL RESEARCH BOMBAY 400 005 1992 Preface In 1987, G.A. Margulis settled a 60 years old conjecture due to A. Oppen- heim about values of quadratic forms at integeral points, using ergodic theory on homogeneous spaces. This was achieved by proving a particular case of a conjecture of Raghunathan about orbit closures for unipotent actions on homogeneous spaces. In 1988, the National Board of Higher Mathematics or- ganized an instructional conference in ergodic theory, which provided me a good background to appreciate the significance of Raghunathan’s conjecture. After the conference S.G. Dani suggested me to read the work of Margulis and give lectures on it. He said that it would be a good idea to verify Raghu- nathan’s conjecture for the group SL2(C), using the new techniques developed by him and Margulis, and write my Master’s thesis on it. While the group structure of SL2(C) is easy to understand, the dynamics of unipotent flows on its homogeneous spaces illustrate many of the interest- ing ergodic theoretic phenomena. The study of this case provided me a good insight into Raghunathan’s conjecture in the general case. Some of the impor- tant results used in the proof are based on the work of Garland and Raghu- nathan about fundamental domains of noncocompact lattices in semisimple groups of real rank 1. In turn, their work depends on the theorem of Kazhdan and Margulis about existence of unipotent elements in noncompact lattices in semisimple groups. The main endeavor in writing this thesis is to present the whole theory for the group SL2(C), using only the language of linear algebra and topological groups. I wish to thank Professor S.G. Dani for introducing me the subject of er- godic theory on homogeneous spaces. Each discussion with him gave me new insights into the problems I was studying. I also thank him for suggesting several corrections in the thesis. Many of the papers about flows on homo- geneous spaces use the machinery of algebraic groups and discrete subgroups. I was always helped by Professor Gopal Prasad in understanding the results used in these papers and learning the proofs. I express my thanks to him. I would like to thank Professor M.S. Raghunathan for always encouraging me, and telling me about many interesting results during several conversations in- cluding many at tea tables. His book on discrete subgroups of Lie groups was my most helpful source. Nimish A. Shah i Contents Introduction 1 1 Topological groups and homogeneous spaces 8 1.1 Definitions and immediate consequences . 8 1.2 Invariant measures on topological groups . 13 1.3 Invariant measures on homogeneous spaces . 17 1.4 Closed subgroups of Rn and Rn/Zn ............... 22 1.5 Covolumes of discrete subgroups in euclidean spaces . 26 2 The group SL2(C) and its homogeneous spaces 30 2.1 Exponential map, Lie algebras, and Unipotent subgroups . 30 2.2 Subgroups of SL2(C)....................... 37 2.3 The invariant measure on SL2(C)/SL2(Z[i]) . 44 2.4 Ergodic properties of transformations on homogeneous spaces 47 3 L-homogeneous spaces of SL2(C) 55 3.1 A result of Kazhdan and Margulis . 55 3.2 Non-divergence of unipotent trajectories . 62 3.3 Volumes of compact orbits of horospherical subgroups . 65 3.4 A lower bound for the relative measures of a large compact set on unipotent trajectories . 70 3.5 Description of noncompact L-homogeneous spaces of SL2(C). 76 4 Proof of Raghunathan’s conjecture for SL2(C) 81 4.1 Minimal invariant sets . 81 4.2 Topological limits and inclusion of orbits-I . 84 4.3 Specialization to the case of SL2(C)............... 87 4.4 Topological limits and inclusion of orbits-II . 91 4.5 Closures of orbits of N1 ...................... 94 4.6 Closures of orbits of subgroups containing a nontrivial unipotent element .............................. 101 Bibliography 110 ii Introduction A subgroup action on a homogeneous space is a classical object of study in ergodic theory. In this thesis we shall study in detail the homogeneous spaces of G = SL2(C) admitting finite G-invariant measures, and verify Raghunathan’s conjecture about closures of orbits of unipotent subgroups of G acting on these spaces. A subgroup Γof a locally compact group G is called a lattice in G, if Γis discrete and the homogeneous space G/Γadmits a finite G-invariant measure. A linear transformation T of a finite dimensional vector space V is called n unipotent, if (T 1V ) = 0 for some n N. Let G be a Lie− group and let denote∈ its Lie algebra. An element u G is called an Ad-unipotent elementG of G, if the linear automorphism Ad u of∈ is unipotent. A subgroup U G is called a unipotent subgroup of G, if all ofG elements of U are unipotent⊂ elements of G. Raghunathan’s conjecture : Let G be a Lie group, Γ be a lattice in G, and U be a unipotent subgroup of G. Then for any x G/Γ, there exists a closed subgroup L of G containing U such that ∈ Ux = Lx. Moreover, the orbit Lx admits a finite L-invariant measure. For an abelian group G, the conjecture follows from elementary arguments. For a nilpotent group G, the conjecture follows from a result of Green [G] about unitary representations of nilpotent groups. For a solvable group G, a result due to Mostow [R, Sect.3.3] says that if N is the maximal connected normal unipotent subgroup of G, then every N-orbit in G/Γis compact. In this case every connected unipotent subgroup of G is contained in N, and hence the verification of the conjecture reduces to the nilpotent case. In the case of semisimple groups the conjecture is of particular significance. For G = SL2(R), it was proved by Hedlund [He] that if U is a nontrivial one- parameter unipotent subgroup of G then any orbit of U in G/Γis either dense or periodic; the periodic orbits exist if and only if G/Γis noncompact. In fact, for a cocompact lattice Γ, Furstenberg [F] proved that any U-invariant probability measure on G/Γis G-invariant, and hence it is unique. 1 Let G be a Lie group. For g G, define ∈ n n U(g)= u G : g ug− as n . { ∈ →∞ → ∞} Then U(g) is called the horospherical subgroup of G associated to g. Note that U(g) is a closed connected unipotent subgroup of G. Now let G be a semisimple group (with trivial center and no compact factors) and Γbe a cocompact lattice in G. It was proved by Veech [V] and Bowen [B] that every orbit of a nontrivial horospherical subgroup of G is dense in G/Γ. In the noncompact case it was proved by Margulis [M1] that if G = SLn(R), Γ=SLn(Z), and u(t) t R is a one-parameter unipotent subgroup of G, then ∈ for any x G/Γ,{ there} exists a compact set C G/Γsuch that the set (C) of return times,∈ defined as ⊂ R (C)= n N : u(n)x C , R { ∈ ∈ } is unbounded. In [D1, D3, D4] this result was strengthened by Dani, who proved, in particular, that for a suitable compact set C, the set (C) of return times has positive density in the set of positive integers. UsingR the strenghened verson, in [D6] he verified Raghunathan’s conjecture when Γis any lattice in a Lie group G and U is a horospherical subgroup of G. This generalizes Veech’s result for noncocompact lattices. A major source of interest in Raghunathan’s conjecture is the fact, observed by Raghunathan, that a very special case of the conjecture implies the validity of the following conjectue of A. Oppenheim. Oppenheim’s conjecture : Let Q be a nondegenerate, indefinite quadratic form on Rn, n 3. Suppose that Q is not multiple of a rational form. Then Q(Zn) is dense≥ in R. The conjecture can be reduced to the case of n =3.LetQ be the quadratic form 2x x x2, and define 1 3 − 2 H = SO(Q)= g SL (R):Q(gx)=Q(x) for all x R3 . { ∈ 3 ∈ } In [M2], Margulis proved that every relatively compact orbit of H in the space SL3(R)/SL3(Z) is compact. From this result it is not difficult to deduce Op- penheim’s conjecture using Mahler’s criterion and the standard techniques of algebraic groups. For an elementry proof of Oppenheim conjecture see [DM3, M5]. Note that H0 is not a unipotent subgroup, but it is generated by unipotent elements of G. The following conjecture was formulated by Margulis [M3]. 2 Generalized Raghunathan conjecture: Let G be a Lie group and Γ be a lattice in G. Let U be a subgroup of G generated by unipotent elements of G contained in U. Then for any x G, there exists a closed subgroup L of G containing U such that Ux = Lx,∈ and the orbit Lx admits a finite L-invariant measure. In his proof Margulis has developed new techniques to study orbit closures of unipotent subgroups. Generalizing these techinques and the results about returning of a unipotent trajectory to a compact set with positive density, in [DM1, DM2] Dani and Margulis proved that (i) any H-orbit in SL3(R)/Γis either dense or closed, and (ii) Raghunathan’s conjecture holds for the orbits of unipotent subgroups of H acting on SL3(R)/Γ, where Γis a lattice in SL3(R).