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Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a non-Archimedean extension of Mostow's theorem by showing the amenable linear locally compact groups have property (M). However property (M) does not hold for all solvable locally compact groups: indeed, we exhibit an example of a metabelian locally compact group with a non-uniform lattice. We show that compactly generated metabelian groups, and more generally nilpotent-by-nilpotent groups, do have property (M). Finally, we highlight a connection of property (M) with the subtle relation between the analytic notions of strong ergodicity and the spectral gap., Comment: 38 pages

The authors correct an error in Proposition 3.24 of their work Divergence in lattices in semisimple Lie groups and graphs of groups, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2451–2505.

We discuss a relationship between the dynamical properties of a maximal diagonalizable group A on certain arithmetic quotients and arithmetic properties of the lattice. In particular, we consider the semigroup of all integer quaternions under multiplication. For this semigroup we use measure rigidity theorems to prove that the set of elements that are not divisible by a given reduced quaternion is very small: We show that any quaternion that has a sufficiently divisible norm is also divisible by the given quaternion. Restricting to the quaternions that have norm equal to products of powers of primes from a given list (containing at least two) we show that the set of exceptions has subexponential growth.

Let ν \nu be a probability measure on S L d ( Z ) \mathrm {SL}_d(\mathbb {Z}) satisfying the moment condition E ν ( ‖ g ‖ ϵ ) > ∞ \mathbb {E}_\nu (\|g\|^\epsilon )>\infty for some ϵ \epsilon . We show that if the group generated by the support of ν \nu is large enough, in particular if this group is Zariski dense in S L d \mathrm {SL}_d , for any irrational x ∈ T d x \in \mathbb {T}^d the probability measures ν ∗ n ∗ δ x \nu ^{* n} * \delta _x tend to the uniform measure on T d \mathbb {T}^d . If in addition x x is Diophantine generic, we show this convergence is exponentially fast.

Let Γ be a non-elementary subgroup of SL 2 ( Z ) . If μ is a probability measure on T 2 which is Γ-invariant, then μ is a convex combination of the Haar measure and an atomic probability measure supported by rational points. The same conclusion holds under the weaker assumption that μ is ν-stationary, i.e. μ = ν ∗ μ , where ν is a finitely supported, probability measure on Γ whose support suppν generates Γ. The approach works more generally for Γ SL d ( Z ) . To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Let Γ be a finitely generated group. Given a finite set of generators S of Γ, the word length lS(γ) for an element γ ∈ Γ is defined to be the smallest positive integer for which there exist s1, · · · , sn ∈ S ∪ S −1 such that γ = s1 · · · sn. For each n ∈ N, denote by BS(n) the set of elements in Γ whose word length with respect to S is at most n. It follows from the subadditive property of lS(·) that limn→∞ |BS(n)| 1/n exists, which we denote by ωS(Γ). A finitely generated group Γ is said to be of exponential growth if ωS(Γ) > 1, of polynomial growth if for some c > 0 and d ∈ N, |BS(n)| ≤ c · n d for all n ≥ 1 and of intermediate growth otherwise, for some finite generating set S of Γ. Observe that the growth type of Γ does not depend on the choice of generating set S. If a finitely generated group Γ is linear, it is known that Γ is either of polynomial growth in which case Γ is virtually nilpotent, or of exponential growth otherwise ([Tit72], [Mil68], [Wol68]).

The Oppenheim conjecture, proved by Margulis [Mar1] (see also [Mar2]), asserts that for a nondegenerate indefinite irrational quadratic form Q in n ≥ 3 variables, the set Q(Zn) is dense. In [EMM] (where we used the lower bounds established in [DM]) a quantitative version of the conjecture was established. Namely: Let ρ be a continuous positive function on the sphere {v ∈ Rn | ‖v‖ = 1}, and let Ω = {v ∈ Rn | ‖v‖ < ρ(v/‖v‖)}. We denote by TΩ the dilate of Ω by T . For an indefinite quadratic form Q in n variables, let NQ,Ω(a, b, T ) denote the cardinality of the set

A finitely generated group Γ is called representation rigid (briefly, rigid) if for every n, Γ has only finitely many classes of simple ℂ representations in dimension n. Examples include higher rank S-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.

Several aperiodic hyperbolic tiling systems consisting of a single convex tile are constructed.

Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any $n \geq 1$, there exists a constant $\epsilon(n)>0$ such that for any $0 \leq k, Comment: 18 pages

We construct lattices in Aut Tn x Aut Tm which are finitely presented, torsion free, simple groups.

We confirm a conjecture of Jens Marklof regarding the equidistribution of certain sparse collections of points on expanding horospheres. These collections are obtained by intersecting the expanded horosphere with a certain manifold of complementary dimension and turns out to be of arithmetic nature. This equidistribution result is then used along the lines suggested by Marklof to give an analogue of a result of W. Schmidt regarding the distribution of shapes of lattices orthogonal to integer vectors., Revised version. To appear in Compositio

A CAT(−1)-space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane ([B], [BriHa], [Gr]). The CAT(−1)-property is one among many possible generalizations to singular spaces of the notion of negative curvature. Important examples of CAT(−1)-spaces include Riemannian manifolds of sectional curvature k ≤ −1 and their convex subsets ([B-G-S]), metric trees and piecewise hyperbolic cell complexes ([Mou],[Da],[Hag],[Be 1],[Be 2],[B-Br]). In this paper we establish certain superrigidity results for isometric actions of a group Λ on a CAT(−1)-space in the following two settings: A. The group Λ is a subgroup of a locally compact group G with Γ < Λ < ComGΓ, where Γ < G is a sufficiently large discrete subgroup and ComGΓ = {g ∈ G : g−1Γg and Γ share a subgroup of finite index} is the commensurator of Γ in G. B. The group Λ is an irreducible lattice in G := ∏n α=1Gα(kα), where each Gα is a semisimple algebraic group defined over a local field kα. The issues addressed in this paper are motivated on one hand by earlier work of G.A. Margulis ([Ma]) dealing with the linear representation theory of Λ, where in case A, G is a semisimple group and Γ < G a lattice, and on the other hand by the results of Lubotzky, Mozes and Zimmer ([L-M-Z]) concerning isometric actions of Λ on trees, where Γ < Λ < ComGΓ, G is the group of automorphisms of a regular tree and Γ < G is a lattice. Our approach to establishing superrigidity results is based on ergodic theoretic methods developed by Margulis ([Ma],[Zi 3],[A’C-B]). In this context, the following notion of boundary of a locally compact group Γ will be useful: let B be a standard Borel space on which Γ acts by Borel automorphisms preserving a σ-finite measure class μ.

We study cocompact lattices with dense projections in a product $G_1 \times G_2$ of locally compact groups and show, under the assumption that each $G_i$ is a closed subgroup of the automorphism group $Aut(T_i)$ of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only finitely many discrete subgroups of $G_1 \times G_2$.

It is shown that a probability measure on a homogeneous space Γ\G which is invariant under a subgroup H < G which is epimorphic in a subgroup L < G is invariant under L. When L = G we obtain a subgroup H such that for any lattice Γ < G its action on Γ\G is uniquely ergodic.

Using affine Bruhat-Tits buildings, we associate certain subshifts of finite type to systems arising from the action of a Cartan subgroup of ap-adic semisimple Chevalley group on compact quotient spaces Γ\G. These are used to study the resulting dynamical systems.

The Oppenheim conjecture, proved by Margulis in 1986, states that the set of values at integral points of an indefinite quadratic form in three or more variables is dense, provided the form is not proportional to a rational form. In this paper we study the distribution of values of such a form. We show that if the signature of the form is not (2, 1) or (2, 2), then the values are uniformly distributed on the real line, provided the form is not proportional to a rational form. In the cases where the signature is (2, 1) or (2, 2) we show that no such universal formula exists, and give asymptotic upper bounds which are in general best possible. Let Q be an indefinite nondegenerate quadratic form in n variables. Let LQ = Q(Z) denote the set of values of Q at integral points. The Oppenheim conjecture, proved by Margulis (cf. [Mar]) states that if n ≥ 3, and Q is not proportional to a form with rational coefficients, then LQ is dense. The Oppenheim conjecture enjoyed attention and many studies since it was conjectured in 1929 mostly using analytic number theory methods. In this paper we study some finer questions related to the distribution the values of Q at integral points.

Let G be a Lie group and Γ be a discrete subgroup. We show that if {μn} is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G.

ForG=PGL2(ℚ p )×PGL2 ℚ we study the closures of orbits under the maximal split Cartan subgroup ofG in homogeneous spacesΓ\G. We show that if a closure of an orbit contains a closed orbit then the orbit is either dense or closed. We show the relation of this to divisibility properties of integral quaternions and other lattices.

Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $\le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. By Olshanskii-Osin-Sapir, that property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur that states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. It also gives a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy. We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has "many" periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditch's properties that are weaker than local compactness. This gives a new proof of Behrstock's result that every pseudo-Anosov element in a mapping class group is Morse. On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the $\mathbb{Q}$-rank is 1 and when the lattice is $SL_n(\mathcal{O}_S)$ where $n\ge 3$, $S$ is a finite set of valuations of a number field $K$ including all infinite valuations, and $\mathcal{O}_S$ is the corresponding ring of $S$-integers., v1: 34 pages; v2: implemented referee's comments/ The paper is accepted in Tr.AMS v3: more small changes are made, especially in Section 3, v6: added an erratum correcting Proposition 3.24 and Theorems 4.4, 4.9; v7: fixed the proof of Theorem 7.7

We show that the group of almost automorphisms of a d-regular tree does not admit lattices. As far as we know this is the first such example among (compactly generated) simple locally compact groups., Comment: 17 pages. Revised according to referee's report

We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method we give examples of free groups and of Kazhdan groups which are generated by the different states of one finite (bi-reversible) automaton. We also reproduce the theorem of Macedonska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree., 21 pages, 4 figures

Geometric and Functional Analysis, 12 (6), ISSN:1016-443X, ISSN:1420-8970

An analogue of the Howe-Moore theorem for vanishing of matrix coefficients of representations of certain semisimple Lie groups is proved for certain automorphism groups of a regular or bi-regular trees. It is also shown that for measure preserving actions of some such groups ergodicity implies mixing of all orders.

We announce the folowing result: Any finitely generated non virtually solvable linear group over a field of characteristic zero has uniform exponential growth.

We show that for Lie groups whose adjoint representation reflects their topology, mixing implies mixing of all orders. In particular we prove that mixing actions of a semisimple Lie group are mixing of all orders, answering a conjecture of B. Marcus.

The object of this work is to study the properties of dynamical systems defined by tilings. A connection to symbolic dynamical systems defined by one- and two-dimensional substitution systems is shown. This is used in particular to show the existence of a tiling system such that its corresponding dynamical system is minimal and topological weakly mixing. We remark that for one-dimensional tilings the dynamical system always contains periodic points.

Let G and $H\subset G$ be connected reductive real algebraic groups defined over $\Bbb Q$ , and admitting no nontrivial $\Bbb Q$ -characters. Let $\Gamma\subset G(\Bbb Q)$ be an arithmetic lattice in G, and $\pi:G\rightarrow\Gamma\backslash G$ be the natural quotient map. Let $\mu_H$ denote the H-invariant probability measure on the closed orbit $\pi(H)$ . Suppose that $\pi(Z(H))$ is compact, where Z(H) denotes the centralizer of H in G. We prove that the set $\{\mu_H\cdot g :\in G\}$ of translated measures is relatively compact in the space of all Borel probability measures on $\Gamma\backslash G$ , where $\mu_H\cdot g(E)=\mu_H(Eg^{-1})$ for all Borel sets $E\subset\Gamma\backslash G\$ .