Multiplication of weak equivalence classes may be discontinuous

Abstract

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