# Isomorphisms between curve graphs of infinite-type surfaces are geometric

## TL;DR

This paper proves that any simplicial isomorphism between the curve graphs of infinite-type surfaces is geometric, induced by a homeomorphism, implying a strong link between the combinatorial and topological structures.

## Contribution

It establishes that curve graph isomorphisms for infinite-type surfaces are always geometric, extending known results from finite-type surfaces to the infinite case.

## Key findings

- S and S' are homeomorphic if their curve graphs are isomorphic.
- Any simplicial isomorphism of curve graphs is induced by a homeomorphism.
- The result generalizes finite-type surface rigidity to infinite-type surfaces.

## Abstract

Let $\phi:\mathcal{C}(S)\to\mathcal{C}(S')$ be a simplicial isomorphism between the curve graphs of two infinite-type surfaces. In this paper we show that in this situation $S$ and $S'$ are homeomorphic and $\phi$ is induced by a homeomorphism $h:S\to S'$.