# Quasi-Isometric Rigidity of Cusp-Decomposable Manifolds

## TL;DR

This paper investigates the large-scale geometric properties of cusp-decomposable manifolds and establishes their fundamental groups' quasi-isometric rigidity, enhancing understanding of their coarse geometric structure.

## Contribution

It provides a detailed description of the universal cover's geometry and proves quasi-isometric rigidity for the fundamental groups of cusp-decomposable manifolds.

## Key findings

- Universal covers have specific coarse geometric properties.
- Quasi-isometries between universal covers are characterized.
- Fundamental groups are quasi-isometrically rigid.

## Abstract

In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguy\^{e}n Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometric rigidity for fundamental groups of cusp-decomposable manifolds.