Spectra of some group defined by a finite automaton
by
Rostislav I. Grigorchuk
Steklov Mathematical Institute
Coauthors: Andrzej Zuk (CNRS, Ecole Normale Supérieure de Lyon)

We are interested in the spectrum of the random walk operator for some group defined by a finite automaton and acting on a rooted tree, and for its natural finite quotients.

Let H be an infinite dimensional Hilbert space given with an isomorphism H = H \oplusH. We investigate the group G of unitary operators acting on H, which is generated by two operators a and b which act on H as follows:

a = æ ç è 0 a b 0 ö ÷ ø ,     b = æ ç è a 0 0 b ö ÷ ø .

We show that G is isomorphic with the group

æ è Õ C  C2 ö ø \rtimesC,

where C is the infinite cyclic group, C₂ is the group of order 2 and C acts by a shift.

We compute the spectrum of the operator

a + a-1 + b + b-1

acting on l²(G) and on l²(G_n) where G_n is a family of finite quotients of G associated to a natural action of G on a rooted tree.

Date received: June 28, 1999