Isoperimetric profile of subgroups and probability of return of random walks on elementary solvable groups

We introduce a notion of large-scale foliation for metric measure spaces and we prove that if X is large-scale foliated by Y , and if Y satisfies a Sobolev inequality at large-scale, then so does X. In particular the Lp-isoperimetric of Y grows faster than the one of X. A special case is when Y = H is a closed subgroup of a locally compact group X = G. The class of elementary solvable groups is the smallest class, stable under closed (compactly generated) subgroups, quotients, finite products, that contains the group of triangular matrices T (d, k), for every d ∈ N and every local field k. If G is a unimodular elementary solvable group with exponential growth, we prove that the Lp-isoperimetric profile of G satisfies jG,p(t) ≈ log t, for every 1 ≤ p ≤ ∞. As a consequence, the probability of return of symmetric random walks on such groups decreases like e−n 1/3 . We obtain a stronger result when the group is a quotient of a solvable algrebraic group over a p-adic field, namely, such a group has linear isoperimetric profiles inside balls. Among other consequences, we obtain that these groups have trivial reduced cohomology with values in the left regular representation on Lp(G), for 1 < p < ∞.