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 Ab\'{e}rt 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 $\mathcal{W}_\Gamma \times \mathcal{W}_\Gamma \to \mathcal{W}_\Gamma \colon (\mathfrak{a}, \mathfrak{b}) \mapsto \mathfrak{a} \times \mathfrak{b}$ is continuous. In contrast to that, we prove that if $\Gamma$ is a Zariski dense subgroup of $\mathrm{SL}_d(\mathbb{Z})$ for some $d \geqslant 2$ (for instance, if $\Gamma$ is a non-Abelian free group), then multiplication on $\mathcal{W}_\Gamma$ is discontinuous, even when restricted to the subspace $\mathcal{FW}_\Gamma$ of all free weak equivalence classes.