Braid groups are almost co-Hopfian
Geometric Topology
2007-05-23 v2 Group Theory
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.
Cite
@article{arxiv.math/0403145,
title = {Braid groups are almost co-Hopfian},
author = {Robert W. Bell and Dan Margalit},
journal= {arXiv preprint arXiv:math/0403145},
year = {2007}
}
Comments
27 pages, 7 figures, improved exposition, minor corrections