Montreal Geometric & Combinatorial Group Theory Seminar

Speaker: Ben Steinberg (Carleton)

Title:
“On a class of automata groups generalizing lamplighter groups”

Date: 3:30PM, Monday, November 10, 2003

Place: Room 1120, Burnside Hall, McGill University

Abstract:

Grigorchuk and Zuk showed that the lamplighter
group Z_2 wr Z can be

generated by a 2-state automaton. Using this automaton, they calculated

the spectral measures of random walks on the
lamplighter group's Cayley

graph. These calculations were used to show the
strong Atiyah conjecture

on L_2-betti number is false. They also showed the semigroup
generated

by their automaton is free.

One interesting property of this automaton is
that each input letter acts

as a reset on the states. The reset automata form the bottom
level in the

Krohn-Rhodes hierarchy of counter-free
automata. We study automata groups

generated by reset automata. It turns out such groups are always
similar

in nature to lamplighter groups: they are
locally finite-by-cyclic; under

mild hypotheses the semigroups generated by
these automata are free. We

also show that if G is any finite Abelian group,
then G wr Z can be

generated by a |G| state reset automaton
generalizing the example of Z_2

by Grigorchuk and Zuk. These automata share the fundamental properties

used by Grigorchuk and Zuk for their spectral
calculations and therefore

one can hope to calculate the spectra of such wreath
product groups.

Analogous automata are constructed for finite
non-Abelian groups, although

the automata group is not exactly a wreath
product.

This is joint work with Pedro Silva.