• Corpus ID: 235727379

On spectral properties of the Schreier graphs of the Thompson group $F$

@inproceedings{Dudko2021OnSP,
  title={On spectral properties of the Schreier graphs of the Thompson group \$F\$},
  author={Artem Dudko and Rostislav I. Grigorchuk},
  year={2021}
}

• A. Dudko, R. Grigorchuk
• Published 2 July 2021
• Mathematics

The Thompson’s group F is one of the most famous and most important groups related to many areas of mathematics (see the survey [3]). The question about amenability of this group remains open for more than 50 years despite many attempts to solve it. By a remarkable Kesten’s criterion of amenability a group G is amenable if and only if 1 belongs to the spectrum of the Markov operator associated to a symmetric random walk on G. This criterion also holds for graphs of uniformly bounded degree (see [4]). Amenability of the Thompson’s group F (which is equivalent to amenability of the Cayley graph of F ) would imply that every Schreier graph of F is amenable. This motivates studying spectral properties of such graphs. This article is devoted to studying spectral properties of the family of Schreier graphs associated to the action of F on the interval [0, 1] (see [3]). A Schreier graph of a group G is determined by a triple (G,H, S), where H is a subgroup of G and S is a system of generators for G. These graphs naturally arise in association with an action of G on a set X . Given x ∈ X one can introduce the Schreier graph Γx = Γ(G,Gx, S), where Gx is the stabilizer of x. An interesting and important case is when a subgroup H < G is maximal and of infinite index or when H is weakly maximal (i.e. H has infinite index in G and is maximal among the groups with this property). Study of spectral, combinatorial and dynamical properties of Schreier graphs associated with weakly maximal subgroups of branch groups of intermediate growth constructed by the second author got a lot of attention and led to many unexpected results and connections. Among them are