Stability of hyperbolic groups acting on their boundaries
Group Theory
2023-08-21 v2 Geometric Topology
Abstract
A hyperbolic group acts by homeomorphisms on its Gromov boundary. We use a dynamical coding of boundary points to show that such actions are topologically stable in the dynamical sense: any nearby action is semi-conjugate to (and an extension of) the standard boundary action. This result was previously known in the special case that the boundary is a topological sphere. Our proof here is independent and gives additional information about the semiconjugacy in that case. Our techniques also give a new proof of global stability when the boundary is a circle.
Cite
@article{arxiv.2206.14914,
title = {Stability of hyperbolic groups acting on their boundaries},
author = {Kathrynn Mann and Jason Fox Manning and Theodore Weisman},
journal= {arXiv preprint arXiv:2206.14914},
year = {2023}
}
Comments
v2: 22 pages, 3 figures. Added result about cell-like fibers in the sphere case, and a new proof of a theorem of Matsumoto