Math Seminars
Rostislav Grigorchuk
Texas A&M University
Spectra of Fractal Groups and Related Topics
The spectrum of a graph is the spectrum of the discrete Laplace operator
associated to this graph. The spectrum of a finitely generated group is
the spectrum of the Cayley graph of the group. More generally, one can
consider the spectrum of a Schreier graph associated to a group and a
subgroup.
The spectral theory of graphs and groups is extremely interesting
subject related to many other fascinating topics (Ramanujan graphs and
expenders, Ihara zeta function, Poisson and Martin boundary, amenability
and random walks, reduced C*-algebras and idempotents).
Many questions in operator theory, operator K-theory, theory of
representations and in abstract harmonic analysis can be reduced to a
questions about spectral properties of some associated graph or a group.
For instance, the famous criterion of H.Kesten states that a finitely
generated group is amenable if and only if its spectral radius is 1.
In my talk I will give an overview of results about spectra of Cayley
Graphs and spectra of Shreier graphs associated to Fractal groups. These
groups will be generated by finite automata and will be of branch type.
We will describe a method of computation of a spectra based on use of
self-similarity properties of a group and of related objects and on the
idea of introducing extra parameters in the spectral problem. The last
idea leeds unexpectedly to the invariance of the multidimensional
spectrum with respect to some rational mapping $f$ of the same dimension.
Thus the spectrum becomes an $f$- invariant set. The rational mappings
which arise in this way are very interesting and some of them remind of
the Henon map. They posses a property that we call integrability and
which we use for computation of von Neimann-Kesten-Serre spectral
measure and of Ihara zeta function which can be defined for infinite
graphs as well. Some interesting examples of computation of Ihara zeta
function of infinite graphs and groups will be considered at the end of
the talk.