Diameters in box spaces

Vortrag von Prof. Dr. Alain Valette

Datum: 06.12.17  Zeit: 15.45 - 16.45  Raum: ETH HG G 43

If $G$ is a finitely generated residually finite group, and $(N_i)_{i>0}$ is a decreasing sequence of finite index normal subgroups with trivial intersection, we study the diameter of $G/N_i$ ​as a function of the order $|G/N_i|$. For $0<\alpha\geq 1$, we say (after Breuillard and Tointon) that the box space $\square_(N_i)G$ has property $D_\alpha$, if the diameter of $G/N_i$ grows at least like $|G/N_i|^\alpha$. We show that a box space has property $D_1$ if and only if G is virtually cyclic; and that a group G admits some box space with property $D_\alpha$ (for some $\alpha>0$) if and only if G virtually maps onto $\mathbb{Z}$. For the lamplighter group and for a lattice in SOL, we provide explicit examples of box spaces with and without property $D_\alpha$.