Lattices In Simple Lie Groups Research Articles

Characterizing a vertex‐transitive graph by a large ball

It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgakopoulos. Answering a question of Cornulier, we also construct a continuum of non pairwise isometric large-scale simply connected locally finite vertex-transitive graphs. This question was motivated by the fact that large-scale simply connected Cayley graphs are precisely Cayley graphs of finitely presented groups and therefore have countably many isometric classes.

Counting Lattice Points in norm balls on higher rank simple Lie groups

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. We focus particularly on the case of the special linear groups where we give a detailed proof of error estimates which constitute the first improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991, and are nearly twice as good in some cases.

Counting lattices in simple Lie groups: The positive characteristic case

In this article we prove a conjecture of A. Lubotzky: let G = G_0(K), where K is a local field of characteristic p>5, G_0 is a simply connected, absolutely almost simple K-group of K-rank at least 2. We give the rate of growth of r_x(G) :={H< G| H a lattice in G; vol(G/H)<x}/~, where H_1\simH_2 if and only if there is an abstract automorphism t of G such that H_2 =t(H_1). We also study the rate of subgroup growth s_x(H) of any lattice H in G. As a result we show that these two functions have the same rate of growth which proves Lubotzky's conjecture. Along the way, we also study the rate of growth of the number of equivalence classes of maximal lattices in G with covolume at most x.

Weak amenability of hyperbolic groups

We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit of Haagerup and prove that for the word length distance d of a hyperbolic group, the Schur multipliers associated with the kernel r d have uniformly bounded norms for 0 &lt; r &lt; 1. We then combine this with a Bożejko–Picardello type inequality to obtain weak amenability.

Groups that do not act by automorphisms of codimension-one foliations

Let r be a finitely generated group having the property that any action of any finite-index subgroup of r by homeomorphisms of the circle must have a finite orbit. (By a theorem of E. Ghys, lattices in simple Lie groups of real rank at least 2 have this property.) Suppose that such a r acts on a compact manifold M by automorphisms of a codimension-one C 2 foliation, F. We show that if F has a compact leaf, then some finite-index subgroup of r fixes a compact leaf of F. Furthermore, we give sufficient conditions for some finite-index subgroup of r to fix each leaf of F.