Speed of random walks, isoperimetry and compression of finitely generated groups

Boundaries and random walks on finitely generated infinite groups

We prove that almost every path of a random walk on a finitely generated non-amenable group converges in the compactification of the group introduced by W.J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of 1-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a...

“ Isoperimetry on Solvable Groups and Random Walks ”

We exhibit a large class of locally compact amenable groups having the same isoperimetric profile and the same probability of return for random walks. This class of groups contains closed subgroups of triangular matrices over local fields and their quotients, but some of them are not linear. Abstract : In this talk, we introduce a variant of the isoperimetric profile on groups that we compute o...

Embeddings of discrete groups and the speed of random walks

Let G be a group generated by a finite set S and equipped with the associated left-invariant word metric dG. For a Banach space X let αX(G) (respectively α # X(G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f : G → X and c > 0 such that for all x, y ∈ G we have ‖ f (x) − f (y)‖ ≥ c · dG(x, y). In particular, the Hilbert compre...

Finitely Generated Abelian Groups

Finitely Generated Semiautomatic Groups

The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups and in particular shows that every finitely generated group of nilpotency class 3 is semiautomatic.