Connectivity properties of group actions on non-positively curved spaces I: Controlled connectivity and openness results

TL;DR

This paper investigates the topological properties of group actions on non-positively curved spaces, introducing a new invariant to analyze openness of certain classes of actions, with implications for understanding their structure.

Contribution

It introduces a novel 'controlled topology' invariant for group actions on CAT(0) spaces, establishing openness results for cocompact actions with specific stabilizer properties.

Findings

Cocompact actions form an open subset in the space of all actions.

Actions with discrete orbits and stabilizers of type F_n form an open subset.

The new invariant simplifies analysis when orbits are discrete.

Abstract

Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Isometric actions of G on M are (by definition) points in the space R := Hom(G, Isom(M)) with the compact open topology. Sample theorems: 1. The cocompact actions form an open subset of R. 2. The cocompact actions with discrete orbits whose point-stabilizers have type F_n form an open subset of the subspace of R consisting of all actions with discrete orbits. (F_1 means finitely generated, F_2 means finitely presented etc.) The key idea is to introduce a new "controlled topology" invariant of such actions - dependent on n - which is unfamiliar when the orbits are not discrete but which becomes familiar (cf 2.) when the orbits are discrete. (This is the first of two papers.)