Braid groups are almost co-Hopfian
Abstract
Let Bn be the braid group on n > 3 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. Additionally, we obtain analogous results for mapping class groups of punctured spheres. The methods use Thurston's theory of surface homeomorphisms and build upon work of Ivanov and McCarthy.
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