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Embedding groups into boundedly acyclic groups

Fan Wu, Xiaolei Wu, Mengfei Zhao, Zixiang Zhou

TL;DR

This work shows that φ-labeled Thompson groups $V_φ(G)$ and twisted Brin–Thompson groups are boundedly acyclic, enabling a suite of embedding results: any group of type $F_n$ embeds quasi-isometrically into a boundedly acyclic group of type $F_n$ with no proper finite index subgroups, into a $5$-uniformly perfect $F_n$ group, and into finitely generated boundedly acyclic simple groups. The authors develop a comprehensive framework built on wreath recursions, $G$-matrices, paired forest diagrams, and Stein–Farley complexes to transfer finiteness properties, compute bounded cohomology, and prove BA for these groups. They further show that topological full groups with extremely proximal actions are boundedly acyclic, yielding quasi-isometric embeddings of any finitely generated group into finitely generated boundedly acyclic simple groups, and they establish ℓ^2-invisibility for these host groups. Collectively, these results advance the understanding of bounded cohomology in Thompson-type groups and provide new, robust embedding mechanisms with strong finiteness and cohomological consequences, including the construction of boundedly acyclic groups with unsolvable word problems. The work also connects to Thompson’s Splinter groups and Röver–Nekrashevych constructions, highlighting a rich interplay between dynamics on the Cantor set, geometric group theory, and cohomological vanishing phenomena.

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 $F_φ(G)$.

Embedding groups into boundedly acyclic groups

TL;DR

This work shows that φ-labeled Thompson groups and twisted Brin–Thompson groups are boundedly acyclic, enabling a suite of embedding results: any group of type embeds quasi-isometrically into a boundedly acyclic group of type with no proper finite index subgroups, into a -uniformly perfect group, and into finitely generated boundedly acyclic simple groups. The authors develop a comprehensive framework built on wreath recursions, -matrices, paired forest diagrams, and Stein–Farley complexes to transfer finiteness properties, compute bounded cohomology, and prove BA for these groups. They further show that topological full groups with extremely proximal actions are boundedly acyclic, yielding quasi-isometric embeddings of any finitely generated group into finitely generated boundedly acyclic simple groups, and they establish ℓ^2-invisibility for these host groups. Collectively, these results advance the understanding of bounded cohomology in Thompson-type groups and provide new, robust embedding mechanisms with strong finiteness and cohomological consequences, including the construction of boundedly acyclic groups with unsolvable word problems. The work also connects to Thompson’s Splinter groups and Röver–Nekrashevych constructions, highlighting a rich interplay between dynamics on the Cantor set, geometric group theory, and cohomological vanishing phenomena.

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 embeds quasi-isometrically into a boundedly acyclic group of type 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 embeds quasi-isometrically into a -uniformly perfect group of type . 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 -labeled Thompson group and .
Paper Structure (18 sections, 60 theorems, 50 equations, 4 figures)

This paper contains 18 sections, 60 theorems, 50 equations, 4 figures.

Table of Contents

  1. $\phi$-labeled Thompson groups
  2. Wreath recursions
  3. $G$-matrix and the $\phi$-labeled Thompson group.
  4. Tree diagrams and G-matrices
  5. Reduction to injective wreath recursion
  6. The labeled Thompson group $V(G)$
  7. The quasi-retract from $V(G)$ to $G$
  8. Relation with Thompson's Splinter group
  9. Normal subgroups of $V(G)$ and its uniform perfection
  10. Finiteness properties of labeled Thompson groups
  11. Labeled Thompson groups and paired forest diagrams
  12. Brown's criterion and discrete Morse theory
  13. The $G$-labeled Stein--Farley complex
  14. Connectivity of the descending links
  15. Tools to compute the bounded cohomology of groups
  16. ...and 3 more sections

Key Result

Theorem 1

There exists a functor $\mathbf{groups} \to \mathbf{groups}$ associating to each group $G$ the labeled Thompson group $V(G)$ with the following properties:

Figures (4)

  • Figure 1: An example of the correspondence between partition set and trees under an expansion.
  • Figure 2: An example of correspondence between $G$-matrices and tree diagrams.
  • Figure 3: Two equivalent G-labeled paired forest diagrams where $\phi(g_2)=((g_{20},g_{21}),(1,2))$.
  • Figure 4: An example of the multiplication of two elements in $V(G)$.

Theorems & Definitions (127)

  • Theorem 1
  • Remark 2
  • Corollary 3
  • Remark 4
  • Corollary 5
  • Corollary 6
  • Remark 7
  • Corollary 8
  • Conjecture 9
  • Remark 10
  • ...and 117 more