Tatiana Smirnova-Nagnibeda, Université de Genève

Abstract:
Given an action of a group $G$ on a set $X$, and a generating set $S $ of $G$, one can define the Schreier graph $\Gamma(G,S,X)$ with the vertex set $X$ and the edge set consisting of pairs of vertices $(x,y)$ such that there exists $s\in S\cup S^{-1}$ with $s\cdot x=y$. We shall be interested in self-similar groups, defined by their actions by automorphisms on a regular rooted tree. Such an action preserves the levels of the tree and induces also an action of $G$ on the boundary of the tree. First examples of corresponding families of finite and infinite Schreier graphs were considered by R.Grigorchuk et al as a source of interesting new examples of spectral computations. I shall report on recent progress in our understanding of this class of graphs, with both new examples and general results.