# Limit spaces of vertex and edge replacement systems

## TL;DR

This paper introduces VERSs, a framework for graph expansions, and studies their limit spaces, providing conditions for hyperbolicity and illustrating with examples from group theory, fractals, and graph systems.

## Contribution

It defines VERSs and their limit spaces, offering new insights into their hyperbolic properties and connections to various mathematical structures.

## Key findings

- Sufficient conditions for hyperbolicity of the history graph
- Identification of the limit space as the Gromov boundary in hyperbolic cases
- Examples linking VERSs to self-similar groups, fractals, and graph systems

## Abstract

We introduce and study VERSs (vertex and edge replacement systems) as a technology of graph expansions. We consider its history graph, an augmented tree that records each graph expansion, and we provide sufficient conditions under which it is hyperbolic. When hyperbolic, its Gromov boundary is what we call the limit space of the VERS. We provide three examples from different areas of mathematics: Schreier graphs and limit spaces of finitely generated contracting self-similar groups, injective post-critically finite iterated function systems and limit spaces of edge replacement systems.