## Colloquium October 29

**Date: **Friday October 29
**Time: **16:00 to 17:00

**Place: **455 McBryde (Commons Room)

**Speaker: **Rostislav Grigorchuk of Texas A&M University

**Title: **Algebraic and spectral properties of groups generated by finite automata

### Abstract

Automata groups were defined 40 years ago but started to
play important role only two decades later after discovery that
they can give a solution to famous problems such as Burnside
problem on torsion groups or Milnor problem on group of intermediate
growth.
The development of the theory of automata groups during last two
decades showed that the number of difficult problems to which
methods and ideas of groups generated by finite automata can be
applied is much larger and that they can be used in many areas
of mathematics such as algebra, analysis, geometry, probability,
dynamics, computer science, mathematical logic and other.
Automata groups were used to solve several difficult problems in
analysis around the notion of amenability introduced by von
Neumann
in 1929 (as a result of study of algebraical roots of
Banach-Tarskii paradox), in Riemannian geometry (around Atiyah
Problem on *L*^{2}-invariants), in the
theory of profinite groups (Zelmanov Conjecture on
groups of finite width), in geometric group theory
(Gromov problem on groups of uniformly exponential growth).
There are indications that they can be useful to attack the
Kaplanskii Conjecture on idempotents and the
Kaplanskii Conjecture on the Jacobson radical, Mazur-Fontain
Conjecture in Number theory and other problems of mathematics.
We will give a survey of results and methods.

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