Braid groups are almost co-Hopfian

Details Braid groups are almost co-Hopfian Braid groups are almost co-Hopfian

2004-03-08

arxiv.org/abs/math/0403145v2

Mathematics

Geometric Topology

27 pages, 7 figures, improved exposition, minor corrections

Scientific paper

Let B_n be the braid group on n > 3 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 B_n+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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