| Résume | Le séminaire d'algèbres d'opérateurs aura lieu à l'IHP le temps du trimestre "algèbres d'opérateurs : approximation, ridigité et dynamique"
Résumé : Consider a sequence $(X_k)_{k>0}$ of finite, connected, d-regular graphs with size going to infinity. It is known that, if $(X_k)_{k>0}$ is an expanding sequence, then the diameter of $X_k$ is exactly logarithmic in the number of vertices of $X_k$. On the opposite, for $0<\alpha \leq 1$, say that $(X_k)_{k>0}$ has property $D_\alpha$ if the diameter of $X_k$ grows at least like $|X_k|^\alpha$: this is a strong form of non-expansion. For a finitely generated, residually finite group $G$, let $(N_k)_{k>0}$ be a filtration of $G$. The box space associated with $(N_k)_{k>0}$ is the sequence of finite Cayley graphs $(Cay(G/N_k,S))_{k>0}$ where $S$ is a fixed finite generating set of $G$. It turns out that property $D_\alpha$ for a box space is related to interesting algebraic properties of $G$, and we will survey results by Breuillard-Tointon, Khukhro-Valette, and Guo-Tointon. Now, assuming that $G$ can be realized as a subgroup of $GL_n(A)$, where $A$ is a unital, commutative ring, we have filtrations given by congruence subgroups, giving rise to arithmetic box spaces. Sometimes the growth of the diameter is more constrained on arithmetic box spaces then on general box spaces, i.e. non-arithmeticity can be detected purely metrically. We will illustrate this with lamplighter groups $L_F:=F\wr\Z$, where $F$ is a finite abelian group. This is joint work with Etienne Grezet. |
