Multiplication of Weak Equivalence Classes May Be Discontinuous

2018/03/25 by Bernshteyn, Anton
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.09307

Abstract

For a countably infinite group Γ, let W_Γ denote the space of all weak equivalence classes of measure-preserving actions of Γ 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 (induced by taking products of actions) that makes W_Γ an Abelian semigroup. Burton, Kechris, and Tamuz showed that if Γ is amenable, then W_Γ is a topological semigroup, i.e., the product map W_Γ× W_Γ→ W_Γ\colon (\mathfraka, \mathfrakb) ↦ \mathfraka × \mathfrakb is continuous. In contrast to that, we prove that if Γ is a Zariski dense subgroup of SL_d(ℤ) for some d \geqslant 2 (for instance, if Γ is a non-Abelian free group), then multiplication on W_Γ is discontinuous, even when restricted to the subspace FW_Γ of all free weak equivalence classes.

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