Type: Article

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

Hernández-Hernández, Jesús;

Abstract:

Let S be an orientable, connected surface of finite topological type, with genus g?2 , empty boundary and complexity at least 2; we prove that any graph endomorphism of the curve graph of S is actually an automorphism. Also, as a complement of the author’s previous results, 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 C(S) . The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leininger’s finite rigid sets. We prove as a consequence that Aramayona and Leininger’s rigid set also exhausts C(S) via rigid expansions. The combinatorial rigidity results follow as an immediate consequence, based on the author’s previous results.