Questions in Geometric Group Theory — Q 4.4

Definitions and mathematical statement

Let $G$ be a finitely presented one-ended group. One-ended means that deleting any finite vertex set from a Cayley graph leaves exactly one infinite connected component. Suppose $G$ acts by homeomorphisms on a compact connected metric space $X$ containing at least three points. Put \[ X^{(3)}=\{(x_1,x_2,x_3)\in X^3:x_i\ne x_j\text{ for }i\ne j\}. \] The action is a convergence action if it acts properly on this triple space: for compact $K\subset X^{(3)}$, only finitely many $g\in G$ satisfy $gK\cap K\ne\varnothing$. A point $p\in X$ is a cut point when $X\setminus\{p\}$ is disconnected. Local connectedness means every point has a basis of connected open neighborhoods.

Determine the consequences for $G$ and its action when $X$ has a cut point. In particular, determine under what hypotheses this forces a nontrivial splitting of $G$, and whether $X$ must be locally connected in the setting of the source question.

The topology concerns the given convergence space, not an arbitrarily chosen Cayley-graph compactification. The motivating hyperbolic-boundary theorem assumes that $X$ is the Gromov boundary of a hyperbolic group; that conclusion is not silently applied to a general convergence action. If a minimality or geometric-finiteness assumption is necessary, identifying it is part of resolving the stated structural problem.

Short English statement

What do cut points and local connectedness of a convergence-action space tell us about a one-ended finitely presented group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100404 (AMR-010-0404), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 4.4 (PDF page 12). Original source indexed by AMR-010-0404. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.