Equivariant dissipation in non-archimedean groups

TL;DR

This paper proves that certain topological groups with open subgroups of infinite index exhibit dissipation of probability measures, solving a longstanding problem related to symmetric groups and Gromov's observable distance.

Contribution

It establishes a general dissipation result for nets of measures on groups with open subgroups of infinite index, addressing Pestov's 2006 problem.

Findings

Dissipation occurs in groups with open subgroups of infinite index.

Finite symmetric groups with compatible metrics dissipate, lacking Gromov's Cauchy subsequences.

The result applies to nets of measures converging to invariance in topological 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.