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Splittings of One-Ended Groups with One-Ended Halfspaces

Michael Mihalik, Sam Shepherd

TL;DR

The paper introduces halfspaces associated to group splittings and analyzes their coarse geometric relationship to the ambient group. It proves three main results: (i) any nontrivial splitting of a one-ended finitely generated group can be refined to a splitting with all halfspaces one-ended under mild hypotheses; (ii) one-ended groups admit JSJ splittings over suitable families with one-ended halfspaces; and (iii) if a one-ended finitely presented group $G$ has a splitting with an edge stabilizer of more than one end but with one-ended halfspaces, then $H^2(G,\mathbb{Z}G)\neq \{0\}$, so $G$ is not simply connected at infinity and not an $n$-dimensional duality group for $n\ge3$. The strategy combines chopping halfspaces via minimal cuts, pocset cubings, and tree-of-spaces constructions to produce refined splittings, while topological arguments link simple connectivity at infinity to cohomology, yielding new constraints and examples for duality properties. Together, these results illuminate how the coarse geometry of halfspaces controls splittings and cohomological behavior, with implications for classifying groups and manifolds through JSJ theory and infinity-type obstructions.

Abstract

We introduce the notion of halfspaces associated to a group splitting, and investigate the relationship between the coarse geometry of the halfspaces and the coarse geometry of the group. Roughly speaking, the halfspaces of a group splitting are subgraphs of the Cayley graph obtained by pulling back the halfspaces of the Bass--Serre tree. Our first theorem shows that (under mild conditions) any splitting of a one-ended group can be upgraded to a splitting where all the halfspaces are one-ended. Our second theorem demonstrates that a one-ended group usually has a JSJ splitting where all the halfspaces are one-ended. And our third theorem states that if a one-ended finitely presented group $G$ admits a splitting such that some edge stabilizer has more than one end, but the halfspaces associated to the edge stabilizer are one-ended, then $H^2(G,\mathbb ZG)\ne \{0\}$; in particular $G$ is not simply connected at infinity and $G$ is not an $n$-dimensional duality group for $n\geq3$.

Splittings of One-Ended Groups with One-Ended Halfspaces

TL;DR

The paper introduces halfspaces associated to group splittings and analyzes their coarse geometric relationship to the ambient group. It proves three main results: (i) any nontrivial splitting of a one-ended finitely generated group can be refined to a splitting with all halfspaces one-ended under mild hypotheses; (ii) one-ended groups admit JSJ splittings over suitable families with one-ended halfspaces; and (iii) if a one-ended finitely presented group has a splitting with an edge stabilizer of more than one end but with one-ended halfspaces, then , so is not simply connected at infinity and not an -dimensional duality group for . The strategy combines chopping halfspaces via minimal cuts, pocset cubings, and tree-of-spaces constructions to produce refined splittings, while topological arguments link simple connectivity at infinity to cohomology, yielding new constraints and examples for duality properties. Together, these results illuminate how the coarse geometry of halfspaces controls splittings and cohomological behavior, with implications for classifying groups and manifolds through JSJ theory and infinity-type obstructions.

Abstract

We introduce the notion of halfspaces associated to a group splitting, and investigate the relationship between the coarse geometry of the halfspaces and the coarse geometry of the group. Roughly speaking, the halfspaces of a group splitting are subgraphs of the Cayley graph obtained by pulling back the halfspaces of the Bass--Serre tree. Our first theorem shows that (under mild conditions) any splitting of a one-ended group can be upgraded to a splitting where all the halfspaces are one-ended. Our second theorem demonstrates that a one-ended group usually has a JSJ splitting where all the halfspaces are one-ended. And our third theorem states that if a one-ended finitely presented group admits a splitting such that some edge stabilizer has more than one end, but the halfspaces associated to the edge stabilizer are one-ended, then ; in particular is not simply connected at infinity and is not an -dimensional duality group for .
Paper Structure (15 sections, 49 theorems, 35 equations, 4 figures)

This paper contains 15 sections, 49 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Group splittings and their halfspaces
  3. Pocsets
  4. Chopping up halfspaces
  5. Strategy for Theorem \ref{['thm:smallerstabs']}
  6. Minimal cuts
  7. Halfspace cuts
  8. The trees $T_e$
  9. An equivalence relation on $V\mathfrak{h}_0$
  10. The pocset $\mathcal{P}$
  11. The tree $T'$
  12. Groups $G$ that are simply connected at infinity; have $H^2(G,\mathbb ZG)=\{0\}$
  13. Proof of Theorem \ref{['GeoSplit']}
  14. An example of a splitting obtained from folding
  15. Some examples of splittings with one-ended halfspaces

Key Result

Theorem 1.1

Let $G\curvearrowright T$ be a non-trivial splitting with $G$ one-ended and finitely generated. Suppose the edge stabilizers are finitely generated and accessible. Then there is a non-trivial splitting $G\curvearrowright T'$ with minimal action such that:

Figures (4)

  • Figure 1: Cartoon picture for the proof of Lemma \ref{["lem:noh'edges"]}. Here $D$ is the vertex set of an unbounded component of the induced subgraph of $C-\mathfrak{h}'$.
  • Figure 2: Cartoon picture for the proof of Lemma \ref{['lem:[y]*preceq']}. The two pictures appear mutually contradictory, and indeed the proof shows that this situation can only happen when $gz$ is uniformly close to both $gg_0\delta$ and $g_1\delta$, so in reality the picture would look more like Figure \ref{['fig:gzreality']}.
  • Figure 3: Cartoon picture for the proof of Lemma \ref{['lem:[y]*preceq']}. A depiction of how the situation might actually look.
  • Figure 4: The Relevant Maps

Theorems & Definitions (94)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 84 more