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:
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Growth: Eskin-Mozes-Oh theorem about uniform exponential growth, as
well as some improvements, e.g. uniformity of Cheeger constants.
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Dynamic: (A) Non-amenable linear groups are uniformly non-amenable
(B) Connes-Sullivan conjecture on amenable actions (originally proved
by Zimmer).
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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.