Abstract twisted Brin--Thompson groups
Francesco Fournier-Facio, Xiaolei Wu, Matthew C. B. Zaremsky

TL;DR
This paper generalizes twisted Brin--Thompson groups by removing faithfulness, establishing their relative simplicity, and exploring their algebraic and analytical properties, with implications for the Boone--Higman conjecture.
Contribution
It introduces an abstract version of twisted Brin--Thompson groups without faithfulness, proving their relative simplicity and embedding properties, and explores their algebraic and analytical characteristics.
Findings
Every proper normal subgroup of $SV_G$ lies in the kernel of a surjection to $SV_{G/ ext{ker}}$.
All such groups are uniformly perfect, boundedly acyclic, and $C^*$-simple with trivial amenable radical.
The paper formulates a new criterion for $ ext{l}^2$-invisibility.
Abstract
Given a group acting faithfully on a set , one gets a simple group denoted , called a twisted Brin--Thompson group. In this paper we drop the faithfulness assumption, and get an abstract version of a twisted Brin--Thompson group . While the resulting group is not simple, since surjects onto , we prove that every proper normal subgroup of lies in the kernel of this surjection, so is ``relatively simple''. The advantage is that now we can prove that every finitely presented simple group embeds in a finitely presented abstract twisted Brin--Thompson group intersecting this kernel trivially. In particular, if the Boone--Higman conjecture is true, then so is a related conjectural characterization of groups with solvable word problem, arising purely in the world of twisted Brin--Thompson groups. We also prove a variety…
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