Abstract twisted Brin--Thompson groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912999335264256 |
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| author | Fournier-Facio, Francesco Wu, Xiaolei Zaremsky, Matthew C. B. |
| author_facet | Fournier-Facio, Francesco Wu, Xiaolei Zaremsky, Matthew C. B. |
| contents | Given a group $G$ acting faithfully on a set $S$, one gets a simple group denoted $SV_G$, 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 $SV_G$. While the resulting group is not simple, since $SV_G$ surjects onto $SV_{G/\ker(G \curvearrowright S)}$, we prove that every proper normal subgroup of $SV_G$ lies in the kernel of this surjection, so $SV_G$ 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 of additional results about abstract twisted Brin--Thompson groups, some of which are new even in the faithful case: they are all uniformly perfect, have property NL and property FW$_\infty$, are boundedly acyclic and $\ell^2$-invisible, and are $C^*$-simple as soon as they have trivial amenable radical. Along the way we formulate a new general criterion for $\ell^2$-invisibility that is interesting in its own right. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24687 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Abstract twisted Brin--Thompson groups Fournier-Facio, Francesco Wu, Xiaolei Zaremsky, Matthew C. B. Group Theory Given a group $G$ acting faithfully on a set $S$, one gets a simple group denoted $SV_G$, 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 $SV_G$. While the resulting group is not simple, since $SV_G$ surjects onto $SV_{G/\ker(G \curvearrowright S)}$, we prove that every proper normal subgroup of $SV_G$ lies in the kernel of this surjection, so $SV_G$ 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 of additional results about abstract twisted Brin--Thompson groups, some of which are new even in the faithful case: they are all uniformly perfect, have property NL and property FW$_\infty$, are boundedly acyclic and $\ell^2$-invisible, and are $C^*$-simple as soon as they have trivial amenable radical. Along the way we formulate a new general criterion for $\ell^2$-invisibility that is interesting in its own right. |
| title | Abstract twisted Brin--Thompson groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2603.24687 |