CAT(0) and CAT(-1) fillings of hyperbolic manifolds

TL;DR

This paper introduces new examples of hyperbolic and relatively hyperbolic groups of various dimensions by applying CAT(0)/CAT(-1) filling techniques to hyperbolic manifolds with cusps, analyzing their boundaries at infinity.

Contribution

It provides novel constructions of hyperbolic groups using CAT(0)/CAT(-1) fillings on hyperbolic manifolds, expanding the known examples in geometric group theory.

Findings

Constructed hyperbolic groups of all dimensions ≥ 4

Analyzed boundaries at infinity using Morse-theoretic techniques

Demonstrated interesting properties of the resulting groups

Abstract

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension $d$ for all $d\geq 4$. These examples result from applying CAT$(0)$/CAT$(-1)$ filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory.