# Strongly Contracting Geodesics in a Tree of Spaces

## TL;DR

This paper investigates the properties of strongly contracting geodesics in a tree of spaces, showing they are quasiconvex and establishing conditions under which the entire space is hyperbolic.

## Contribution

It proves that strongly contracting geodesics in vertex spaces are quasiconvex in the combined space and that uniform hyperbolicity of vertex spaces implies the whole space is hyperbolic.

## Key findings

- Strongly contracting geodesics are quasiconvex in the combined space.
- If vertex spaces are uniformly hyperbolic, then the entire space is hyperbolic.
- Vertex spaces are quasiconvex in the combined hyperbolic space.

## Abstract

Let X be a tree of proper geodesic spaces with edge spaces strongly contracting and uniformly separated from each other by a number depending on the contraction function of edge spaces. Then we prove that the strongly contracting geodesics in vertex spaces are quasiconvex in X. We further prove that in X if all the vertex spaces are uniformly hyperbolic metric spaces then X is a hyperbolic metric space and vertex spaces are quasiconvex in X.