Cyclic orders and actions of Leary--Minasyan groups on coarse $\mathrm{PD}(n)$ spaces

TL;DR

This paper demonstrates that for dimensions n ≥ 3, certain torsion-free CAT(0) groups with boundaries embedded in spheres are not virtually fundamental groups of compact aspherical manifolds, using cyclic order analysis.

Contribution

It introduces a new obstruction based on cyclic orders on boundaries, showing these Leary--Minasyan groups cannot act as fundamental groups of certain manifolds.

Findings

Existence of CAT(0) groups with boundary embeddings in spheres

Obstruction from cyclic order analysis on boundary trees

These groups are not virtually fundamental groups of compact aspherical manifolds

Abstract

We show that for $n \geq 3$, there are torsion-free CAT(0) groups with visual boundaries that embed into $S^n$ but which are not virtually the fundamental group of a compact aspherical $n+1$-manifold. The groups are the CAT(0) and not bi-automatic groups constructed previously by Leary and Minasyan. The obstruction comes from analyzing certain cyclic orders on the boundary of the Bass-Serre tree, in the same manner as Kapovich-Kleiner ruled out actions of Baumslag-Solitar groups on coarse $\mathrm{PD}(3)$ spaces.