Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2407.07703

Embedding groups into boundedly acyclic groups

Fan Wu·Xiaolei Wu·Mengfei Zhao·Zixiang Zhou
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that the \sϕ-labeled Thompson groups and the twisted Brin--Thompson groups are boundedly acyclic. This allows us to prove several new embedding results for groups. First, every group of type $F_n$ embeds quasi-isometrically into a boundedly acyclic group of type $F_n$ that has no proper finite index subgroups. This improves a result of Bridson and a theorem of Fournier-Facio--Löh--Moraschini. Second, every group of type $F_n$ embeds quasi-isometrically into a $5$-uniformly perfect group of type $F_n$. Third, using Belk--Zaremsky's construction of twisted Brin--Thompson groups, we show that every finitely generated group embeds quasi-isometrically into a finitely generated boundedly acyclic simple group. We also partially answer some questions of Brothier and Tanushevski regarding the finiteness property of $ϕ$-labeled Thompson group $V_ϕ(G)$ and $F_ϕ(G)$.

Explore related subjects

math.GRmath.KT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fan Wu, Xiaolei Wu, Mengfei Zhao, Zixiang Zhou. 2025-04-13. Embedding groups into boundedly acyclic groups. https://arxiv.org/abs/2407.07703

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Margulis-Soifer theorem for one-relator groups

We establish the Margulis-Soifer dichotomy for one-relator groups: every one-relator group is either virtually solvable or has a maximal subgroup of infinite index. We also present examples of one-relator groups with and without free maximal subgroups of infinite index, as well as examples that possess both free and non-free infinite index maximal subgroups. Triviality of the Frattini subgroup is also shown for all non-solvable one-relator groups. We close the paper with a short list of questions.

math.GR↗
arXiv2026

Finite quotients of spherical Artin groups

We show the smallest non-abelian quotients of spherical and affine Artin groups are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. We deduce irreducible spherical Artin groups are determined by their finite quotient groups.

math.GR↗
arXiv2026

Cosets with constant characteristic polynomial

Let H be a linear group. We show that if there is an invertible matrix x such that all the elements of xH share the same characteristic polynomial then H is virtually solvable. There are plenty of applications that will be presented in future paper. Here, we discuss some applications to the generalized Weigold conjecture and present an alternative straightforward proof of the Formanek--Procesi nonlinearity theorem for Aut(F_n), n>2, over every field. When n>5 our non-linearity proof gives a stronger result than the original Formanek--Procesi theorem.

math.GR↗