The ergodic commutant conjecture for sofic groups

Let GG be a sofic group, let S\mathcal{S} be the universal sofic group, and let L\mathcal{L} be the Loeb measure space. For an embedding π:GS\pi:G\to\mathcal{S}, write π(G)S\pi(G)'\cap\mathcal{S} for the subgroup of elements in S\mathcal{S} commuting with π(G)\pi(G).

Ergodic commutant conjecture. Every sofic group GG admits an embedding π:GS\pi:G\to\mathcal{S} such that the action

π(G)SL\pi(G)'\cap\mathcal{S}\curvearrowright\mathcal{L}

is ergodic.

A positive answer would remove the subamenability assumption in the theorem discussed immediately before this conjecture and would provide a route to constructing non-sofic groups via amalgamated products without amenable amalgams. The supplied text does not state that the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Ben Hayes and Srivatsav Kunnawalkam Elayavalli, “On sofic approximations of non amenable groups”, arXiv:2306.04713 (2024).

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