Rigidity results for certain 3-dimensional singular spaces and their fundamental groups

TL;DR

This paper introduces hyperbolic P-manifolds, a class of 3D singular spaces, and proves rigidity results including a version of Mostow rigidity and quasi-isometric rigidity for their universal covers.

Contribution

It defines hyperbolic P-manifolds and establishes topological boundary characterizations leading to new rigidity theorems in 3D singular spaces.

Findings

Topological characterization of boundary subsets in universal covers

A version of Mostow rigidity for hyperbolic P-manifolds

Quasi-isometric rigidity results for these spaces

Abstract

In this paper, we introduce a particularly nice family of locally CAT(-1) spaces, which we call hyperbolic P-manifolds. For $X^3$ a simple, thick hyperbolic P-manifold of dimension 3, we show that certain subsets of the boundary at infinity of the universal cover of $X^3$ are characterized topologically. Straightforward consequences include a version of Mostow rigidity, as well as quasi-isometric rigidity for these spaces.