# A locally hyperbolic 3-manifold that is not homotopy equivalent to any hyperbolic 3-manifold

## TL;DR

The paper constructs a specific locally hyperbolic 3-manifold with a particular fundamental group property, demonstrating it cannot be homotopy equivalent to any complete hyperbolic 3-manifold, thus providing a counterexample in geometric topology.

## Contribution

It introduces a novel example of a locally hyperbolic 3-manifold that defies homotopy equivalence to any complete hyperbolic 3-manifold, highlighting new complexities in 3-manifold topology.

## Key findings

- Constructed a locally hyperbolic 3-manifold with no divisible subgroups in its fundamental group.
- Proved this manifold is not homotopy equivalent to any complete hyperbolic 3-manifold.
- Provides a counterexample impacting the understanding of hyperbolic 3-manifold classification.

## Abstract

We construct a locally hyperbolic 3-manifold $M$ such that $\pi_ 1(M)$ has no divisible subgroups. We then show that $M$ is not homotopy equivalent to any complete hyperbolic manifold.