Braid groups and the co-Hopfian property

Let B-n be the braid group on n >= 4 strands. We prove that B-n modulo its center is co-Hopfian. We then show that any injective endomorphism of B-n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B-n to Bn+1 is geometric for n >= 7. 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-McCarthy. (c) 2005 Elsevier Inc. All rights reserved.