On the quasi-isometric rigidity of chambers and walls in cusp-decomposable manifolds

TL;DR

This paper investigates the geometric rigidity of cusp-decomposable manifolds, demonstrating that their fundamental group isomorphisms preserve their decomposed structure and that walls and pieces embed quasi-isometrically.

Contribution

It establishes quasi-isometric embeddings of walls and pieces and proves that fundamental group isomorphisms preserve the manifold's decomposition.

Findings

Walls and pieces embed quasi-isometrically in the universal cover

Isomorphisms of fundamental groups preserve the decomposition

Properties of the electric space are used to analyze rigidity

Abstract

A cusp-decomposable manifold is a manifold constructed from a finite number of complete, negatively curved, finite volume manifolds and identifying the boundaries of truncated cusps by diffeomorphisms. Using properties of the electric space of the universal cover of cusp-decomposable manifolds, we show that the inclusion of walls and pieces induces quasi-isometric embeddings. We also show that isomorphisms between fundamental groups of higher graph manifolds preserve the decomposition into pieces.