Embedding groups into acyclic groups
Martin Palmer, Xiaolei Wu
TL;DR
The paper develops a family of acyclic groups by embedding Thompson-like groups into acyclic hosts via topological groupoids. It establishes that labelled Thompson groups V(G) and twisted Brin–Thompson groups S V_G are acyclic by realising them as topological full groups of étale groupoids and applying Li’s bridge between groupoid and full-group homology, aided by Künneth and spectral sequence arguments. Consequently, it derives strong embedding results: every group of type F_n embeds quasi-isometrically into an acyclic group of the same type with no proper finite-index subgroups, and every finitely generated group embeds into a 2-generated simple acyclic group. The work also produces first examples of acyclic groups of type F_n but not F_{n+1}, including simple ones, and identifies a rich supply of universally boundedly acyclic groups, with implications for finiteness properties and group embeddings in geometric group theory.
Abstract
We show that labelled Thompson groups and twisted Brin--Thompson groups are all acyclic. This allows us to prove several new embedding results for groups. First, every group of type $F_n$ embeds quasi-isometrically as a subgroup of an acyclic group of type $F_n$ that has no proper finite-index subgroups. This improves results of Baumslag--Dyer--Heller ($n=1$) and Baumslag--Dyer--Miller ($n=2$) from the early 80s, as well as a more recent result of Bridson ($n=2$). Second, we show that every finitely generated group embeds quasi-isometrically as a subgroup of a $2$-generated, simple, acyclic group. Our results also allow us to produce, for each $n\geqslant 2$, the first known example of an acyclic group that is of type $F_n$ but not $F_{n+1}$. These examples can moreover be taken to be simple. Furthermore, our examples provide a rich source of universally boundedly acyclic groups.