Using methods of Marina Ratner, this paper proves that if ergodic zero-entropy translations on finite-volume homogeneous spaces of two connected semisimple Lie groups are measure-theoretically isomorphic, then they are isomorphic for algebraic reasons. The paper includes applications to the study of certain natural actions of semisimple Lie groups.
This paper extends my generalization [1] of M. Ratner's rigidity theorem to homogeneous spaces of groups that need not be semisimple. This involves study of the "semisimple splittings" of Lie groups, developed by L. Auslander and others.
This paper extends my work [1,2] on the rigidity of group actions on homogeneous spaces to the rigidity of foliations of double-coset spaces. For example, this generalizes work of L. Flaminio to yield a theorem on the rigidity of horospherical foliations on various frame bundles over finite-volume locally-symmetric spaces of non-positive sectional curvature.
The cosets of any connected Lie subgroup of a Lie group G give rise to a foliation of any homogeneous space of G. Extending work of D. Benardete, the paper shows in certain interesting cases that if the foliations by two unimodular subgroups are topologically equivalent, then they are equivalent for algebraic reasons.
This note gives simple necessary and sufficient conditions for two homogeneous real polynomials of two variables to be equivalent under a continuous (or differentiable, or real-analytic) change of variables.
Lattices and parabolic subgroups are the obvious examples of cocompact subgroups of a connected, semisimple Lie group with finite center. Using an argument of C. C. Moore, this paper shows that every cocompact subgroup is, roughly speaking, a combination of these. I subsequently learned that the main part of this result had already been proved by M. Goto and H.-C. Wang.
Roughly speaking, this paper shows that if U is a nilpotent, unipotent subgroup of a Lie group G, and H is a closed subgroup of G, then every measurable quotient of the U-action on H\G is a double-coset space L\G/K. The proof relies on M. Ratner's classification of the ergodic U-invariant probability measures on H\G.
If H is a subgroup of finite index in SL(n,Z) with n > 2, this paper shows that every continuous action of H on the circle or the real line factors through an action of a finite quotient of H. The same is true if H is any arithmetic subgroup of any simple algebraic group G over Q (the rational numbers) with Q-rank(G) > 1.
This paper shows that every quotient of a nonsingular action with minimal self-joinings has a simple form. Previously, this was known for actions of the infinite cyclic group Z, and for measure-preserving actions of any group.
This paper shows that the geometric structure of any tessellation of any simply connected solvmanifold is closely related to the geometry of some compact homogeneous space. In particular, geometrically-defined flows on solvtessellations are finitely covered by natural flows on solvmanifolds.
This paper shows that if D is a "Zariski dense" lattice in a simply connected, solvable Lie group G, then every finite-dimensional representation of D virtually extends to a representation of G. By combining this with work of Margulis on lattices in semisimple groups, a similar result is obtained for lattices in many groups that are neither solvable nor semisimple.
Let A and B be n by n matrices of determinant 1 over a field K, with n > 2 or |K|> 3. We show that if A is not a scalar matrix, then B is a product of matrices similar to A. Analogously, we conjecture that if a and b are elements of a semisimple algebraic group G over a field of characteristic zero, and if there is no normal subgroup of G containing a but not b, then b is a product of conjugates of a. The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. This implies that the only conjugation-invariant subsemigroups of a semisimple Lie group with finite center are the normal subgroups.
Let G be a connected, solvable linear algebraic group over a number field K, let S be a finite set of places of K that contains all the infinite places, let O be the ring of S-integers of K, and let A be the ring of S-adeles of K. We define a certain closed subgroup H of G(A) that contains G(O), and prove that G(O) is a superrigid lattice in H, by which we mean that finite-dimensional representations of G(O) more-or-less extend to representations of H. Furthermore, we note that a superrigidity theorem for many non-solvable S-arithmetic groups can be proved by combining our main theorem with the Margulis Superrigidity Theorem.
I [10] previously showed that every Zariski-dense lattice in any simply connected, solvable Lie group is superrigid. This paper shows that the restriction to lattices is unnecessary: the same result holds for any Zariski-dense, discrete subgroup.
Suppose L is a semisimple Levi subgroup of a connected Lie group G, and c is a GL(n,R)-valued Borel cocycle for the action of G on a Borel space X. Assume L has finite center, and that the real rank of every simple factor of L is at least two. We show that if L is ergodic on X, and the restriction of c to L is cohomologous to a homomorphism (modulo a compact group), then, after passing to a finite cover of X, the cocycle c itself is cohomologous to a homomorphism (modulo a compact group).
We consider the following three prehomogeneous vector spaces (G,V):In each case, we find an explicit irrationality condition on the vector x that implies the G(Z)-orbit of x is dense, and interpret the result number theoretically. This answers a question of Yukie, for the three prehomogeneous vector spaces that we consider.
- the 3rd exterior power of the 6-dimensional representation of GL(6,Q);
- the 3rd exterior power of the 7-dimensional representation of GL(7,Q), augmented by the scalar action of GL(1,Q) on this resulting vector space; and
- the exterior square of the 2n-dimensional representation of GL(2n,Q).
We also prove a similar result for many cases where G1 and
G2 are neither solvable nor semisimple. This paper, like my
previous paper [4]
on the subject, is based on ideas of D. Benardete.
It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, R.Kulkarni and T.Kobayashi constructed examples that are not obvious when G = SO(2,2n) or SU(2,2n). H.Oh and D.Witte constructed additional examples in both of these cases, and obtained a complete classification when G = SO(2,2n). We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G = SU(2,2n). This includes the construction of another family of examples.
The main results are obtained from methods of Y.Benoist and
T.Kobayashi: we fix a Cartan decomposition G = KAK, and study the
intersection of KHK with A. Our exposition generally assumes only the
standard theory of connected Lie groups, although basic properties of
real algebraic groups are sometimes also employed; the specialized
techniques that we use are developed from a fairly elementary level.
Unipotent flows are well-behaved dynamical systems. In particular, Marina Ratner has shown that the closure of every orbit for such a flow is of a nice algebraic (or geometric) form. After presenting some consequences of this important theorem, these lectures explain the main ideas of the proof. Some algebraic technicalities will be pushed to the background.Chapter 1 is the main part of the book. It is intended for a fairly general audience, and provides an elementary introduction to the subject, by presenting examples that illustrate the theorem, some of its applications, and the main ideas involved in the proof.
Chapter 2 gives an elementary introduction to the theory of entropy, and proves an estimate used in the proof of Ratner's Theorem. It is of independent interest.
Chapters 3 and 4 are utilitarian. They present some basic facts of ergodic theory and the theory of algebraic groups that are needed in the proof. The reader (or lecturer) may wish to skip over them, and refer back as necessary.
Chapter 5 presents a fairly complete (but not entirely rigorous) proof of the measure-theoretic version of Ratner's Theorem. (We follow the approach of G.A.Margulis and G.Tomanov.) Unlike the other chapters, it is rather technical.
The first four chapters can be read independently, and should be largely accessible to second-year graduate students. All four are needed for Chapter 5. A reader who is familiar with ergodic theory and algebraic groups, but not unipotent flows, may skip Chaps. 2, 3, and 4 entirely, and read only Sects. 1.4-1.7 of Chap. 1 before beginning Chap. 5.
Let G be the real points of a semisimple algebraic Q-group, let Γ be an arithmetic subgroup of G and let T be the real points of an R-split torus in G. We prove that if there is a divergent T-orbit in G/Γ, and Q-rank(G) > 1, then the dimension of T is not larger than Q-rank(G). This provides a partial answer to a question of G.Tomanov and B.Weiss.
We present unpublished work of D.Carter, G.Keller, and E.Paige on bounded generation in special linear groups. Let n be a positive integer, and let A = O be the ring of integers of an algebraic number field K (or, more generally, let A be a localization OS-1.) If n = 2, assume that A has infinitely many units.
We show there is a finite-index subgroup H of SL(n,A), such that every matrix in H is a product of a bounded number of elementary matrices. We also show that if T is any element of SL(n,A), and T is not a scalar matrix, then there is a finite-index, normal subgroup N of SL(n,A), such that every element of N is a product of a bounded number of conjugates of T.
For n > 2, these results remain valid when SL(n,A) is replaced by any of its subgroups of finite index.
Let Γ be an irreducible lattice in a connected, semisimple Lie group G with finite center. Assume that the real rank of G is at least two, that G/Γ is not compact, and that G has more than one noncompact simple factor. We show that Γ has no orientation-preserving actions on the real line. (In algebraic terms, this means that Γ is not right orderable.) Under the additional assumption that no simple factor of G is isogenous to SL(2,R), applying a theorem of E.Ghys yields the conclusion that any orientation-preserving action of Γ on the circle must factor through a finite, abelian quotient of Γ.
The proof relies on the fact, proved by D.Carter, G.Keller, and E.Paige, that SL(2,A) is boundedly generated by unipotents whenever A is a ring of integers with infinitely many units. The assumption that G has more than one noncompact simple factor can be eliminated if every noncocompact lattice of higher real rank is virtually boundedly generated by unipotents.
Let Γ be a finitely generated, amenable group. We prove that if Γ has a nontrivial, orientation-preserving action on the real line, then Γ has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Γ has a faithful action on the circle, then some finite-index subgroup of Γ has the property that all of its nontrivial finitely generated subgroups have infinite, cyclic quotients. It also means that every left-orderable, amenable group is locally indicable. This answers a question of Peter Linnell.
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be interpreted as a statement about the existence of totally geodesic subspaces of finite-volume, noncompact, locally symmetric spaces of higher rank.) Another formulation of the result states that if G is any isotropic, almost simple algebraic group over Q (the rational numbers), such that the real rank of G is greater than 1, then G contains an isotropic, almost simple Q-subgroup H, such that H is quasisplit, and the real rank of H is greater than 1.
This is an expository paper. It is well known that a linear transformation can be defined to have any desired action on a basis. From this fact, one can show that every group homomorphism from Zk to Rd extends to a homomorphism from Rk to Rd, and we will see other examples of discrete subgroups H of connected groups G, such that the homomorphisms defined on H can ("almost") be extended to homomorphisms defined on all of G. This is related to a very classical topic in geometry, the study of linkages.
The study of group actions on manifolds is the meeting ground of a variety of mathematical areas. In particular, interesting geometric insights can be obtained by applying measure theoretic techniques. These notes provide an introduction to some of the important methods, major developments, and open problems in the subject. They are slightly expanded from lectures of Robert J. Zimmer at a CBMS Conference at the University of Minnesota, Minneapolis, in June, 1998. The main text presents a perspective on the field as it was at that time, and comments after the notes of each lecture provide suggestions for further reading, including references to recent developments, but the content of these notes is by no means exhaustive.
This expository paper describes the various methods that have yielded partial results on the conjecture that if n > 2, then no lattice in SL(n,R) has a faithful action on the circle (by homeomorphisms). Topics include amenability, Kazhdan's property (T), bounded cohomology, bounded generation, and the Reeb-Thurston Stability Theorem.