Surfaces of infinite-type are non-Hopfian

Abstract

We show that finite-type surfaces are characterized by a topological analog of the Hopf property. Namely, an oriented surface $\Sigma$ is of finite-type if and only if every proper map $f:\Sigma \to \Sigma$ of degree one is homotopic to a homeomorphism.

8 pages, 4 figures. v2: incorporates the referee's comments, accepted in the Comptes Rendus Mathématique