Big mapping class groups and the co-Hopfian property

TL;DR

This paper investigates the structure of big mapping class groups of infinite-type surfaces, providing new examples of non-co-Hopfian groups and characterizing injective homomorphisms under certain conditions.

Contribution

It constructs the first examples of injective endomorphisms of mapping class groups that are not surjective and characterizes injective homomorphisms induced by homeomorphisms under topological conditions.

Findings

Constructed uncountably many non-co-Hopfian mapping class groups.

Proved that certain injective homomorphisms are induced by homeomorphisms.

Showed that superinjective maps between curve graphs do not impose topological restrictions.

Abstract

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are the first examples of injective endomorphisms of mapping class groups (of surfaces with empty boundary) that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by homeomorphism. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.