Splittings of One-Ended Groups with One-Ended Halfspaces

TL;DR

This paper explores the structure of one-ended groups through the lens of halfspaces derived from group splittings, revealing conditions under which these groups can be decomposed into simpler components and their topological properties.

Contribution

It introduces the concept of halfspaces in group splittings, proves that splittings can be refined to have one-ended halfspaces, and links these structures to topological invariants of the group.

Findings

Splittings can be refined so all halfspaces are one-ended.

One-ended groups often have JSJ splittings with one-ended halfspaces.

Certain splittings imply the group has nontrivial second cohomology and is not simply connected at infinity.

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 $n\geq3$.