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Braid groups are almost co-Hopfian

Robert W. Bell·Dan Margalit
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Abstract

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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Robert W. Bell, Dan Margalit. 2005-06-18. Braid groups are almost co-Hopfian. https://arxiv.org/abs/math/0403145

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