Montreal Geometric & Combinatorial Group Theory Seminar
Speaker: Ben Steinberg (Carleton)
“On a class of automata groups generalizing lamplighter groups”
Date: 3:30PM, Monday, November 10, 2003
Place: Room 1120, Burnside Hall, McGill University
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.