Graph morphisms and exhaustion of curve graphs of low-genus surfaces

TL;DR

This paper extends previous results to low-genus surfaces, proving that certain graph morphisms of curve graphs are induced by surface homeomorphisms, using finite subgraphs and rigid expansions.

Contribution

It demonstrates that all graph endomorphisms of curve graphs of low-genus surfaces are automorphisms and that graph morphisms are induced by homeomorphisms, extending prior rigidity results.

Findings

Any graph endomorphism of the curve graph is an automorphism.

Graph morphisms between curve graphs are induced by surface homeomorphisms.

Rigid sets exhaust the curve graph via rigid expansions.

Abstract

This work is the extension of the results by the author in [7] and [6] for low-genus surfaces. Let $S$ be an orientable, connected surface of finite topological type, with genus $g \leq 2$, empty boundary, and complexity at least $2$; as a complement of the results of [6], we prove that any graph endomorphism of the curve graph of $S$ is actually an automorphism. Also, as a complement of the results in [6] we prove that under mild conditions on the complexity of the underlying surfaces any graph morphism between curve graphs is induced by a homeomorphism of the surfaces. To prove these results, we construct a finite subgraph whose union of iterated rigid expansions is the curve graph $\mathcal{C}(S)$. The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid sets in [2]. Similarly to [7], a consequence of our proof is that Aramayona and Leininger's rigid set also exhausts the curve graph via rigid expansions, and the combinatorial rigidity results follow as an immediate consequence, based on the results in [6].