Isoperimetric profile and random walks on locally compact solvable groups

We study a large class of amenable locally compact groups containing all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties which are likely to be only satisfied by these groups.

💡 Research Summary

The paper investigates the isoperimetric profile and random walk behavior on a broad class of amenable locally compact groups that includes all solvable algebraic groups over a local field and their discrete subgroups. After introducing the notion of an isoperimetric profile for locally compact groups, the authors focus on groups that are solvable in the algebraic sense (henceforth “solvable groups”). By exploiting the hierarchical structure inherent in solvable groups—namely the existence of a finite series of normal subgroups with abelian quotients—they construct Følner sets adapted to each layer of the series. Careful analysis of the boundary‑to‑volume ratio for these sets yields a precise estimate for the isoperimetric profile: \