KTH/SU Mathematics Colloquium

07-09-26

Tsachik Gelander, The Hebrew University

The uniform Tits alternative and some applications

In his celebrated 1972 paper J. Tits proved a fundamental dichotomy for linear groups, known today as the Tits alternative. Jointly with E. Breuillard we established several results improving those of Tits. In the talk I will concentrate mainly on the uniform version which state that given a finitely generated non-virtually solvable linear group G, there is a constant m=m(G) s.t. for any generating set S of G, one can find generators of a free group F_2 in the ball of radius m w.r.t S. I will also explain some of the most important applications:
  1. Growth: Eskin-Mozes-Oh theorem about uniform exponential growth, as well as some improvements, e.g. uniformity of Cheeger constants.
  2. Dynamic: (A) Non-amenable linear groups are uniformly non-amenable (B) Connes-Sullivan conjecture on amenable actions (originally proved by Zimmer).
  3. Riemannian foliations: Carrier conjecture on growth of leaves. I will also explain the topological version of Tits alternative whose connected (Archimedean) case follows from the uniform version (but proved earlier), and some applications to finite group theory.