Multiplication of Weak Equivalence Classes May Be Discontinuous

Abstract

For a countably infinite group $\Gamma$, let ${\mathcal{W}}_{\Gamma }$ denote the space of all weak equivalence classes of measure-preserving actions of $\Gamma$ on atomless standard probability spaces, equipped with the compact metrizable topology introduced by Abért and Elek. There is a natural multiplication operation on ${\mathcal{W}}_{\Gamma }$ (induced by taking products of actions) that makes ${\mathcal{W}}_{\Gamma }$ an Abelian semigroup. Burton, Kechris, and Tamuz showed that if $\Gamma$ is amenable, then ${\mathcal{W}}_{\Gamma }$ is a topological semigroup, i.e., the product map ${\mathcal{W}}_{\Gamma }\times {\mathcal{W}}_{\Gamma }\to {\mathcal{W}}_{\Gamma }:(\mathfrak{a},\mathfrak{b})\mapsto \mathfrak{a}\times \mathfrak{b}$ is continuous. In contrast to that, we prove that if $\Gamma$ is a Zariski dense subgroup of ${\mathrm{SL}}_d(\mathbb{Z})$ for some $d⩾2$ (for instance, if $\Gamma$ is a non-Abelian free group), then multiplication on ${\mathcal{W}}_{\Gamma }$ is discontinuous, even when restricted to the subspace ${\mathcal{F}\mathcal{W}}_{\Gamma }$ of all free weak equivalence classes.

14 pages; v2: minor changes following a referee report