Big Mapping Class Groups and the Co-Hopfian Property

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