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 ${\mathcal {W}}_{\Gamma } \times {\mathcal {W}}_{\Gamma } \to {\mathcal {W}}_{\Gamma } \colon (\mathfrak {a}, \mathfrak {b}) \mapsto \mathfrak {a} \times \mathfrak {b}$ 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.