Equivariant dissipation in non-archimedean groups

Abstract

We prove that, if a topological group $G$ has an open subgroup of infinite index, then every net of tight Borel probability measures on $G$ UEB-converging to invariance dissipates in $G$ in the sense of Gromov. In particular, this solves a 2006 problem by Pestov: for every left-invariant (or right-invariant) metric $d$ on the infinite symmetric group $\mathrm{Sym}(\mathbb{N})$, compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups $\left(\mathrm{Sym}(n),d\!\!\upharpoonright_{\mathrm{Sym}(n)},\mu_{\mathrm{Sym}(n)}\right)_{n \in \mathbb{N}}$ equipped with the restricted metrics and their normalized counting measures dissipates, thus fails to admit a subsequence being Cauchy with respect to Gromov's observable distance.