Braid groups and the co-Hopfian property

Let Bn be the braid group on n 4 strands. We prove that Bn modulo its center is co-Hopfian. We then show that any injective endomorphism of Bn is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from Bn to Bn+1 is geometric for n 7. Additionally, we obtain analogous results for mapping class groups of punctured spheres. The methods use Thurstonʹs theory of surface homeomorphisms and build upon work of Ivanov–McCarthy.