Braid groups are almost co-Hopfian

2004/03/08 by Robert W. Bell, Dan Margalit, Bell, Robert W. +1
Mathematics · #20F36 #57M07 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #math.GR #math.GT #msc:20F36 #msc:57M07

paper · pdf · doi:10.48550/arxiv.math/0403145

27 pages, 7 figures, improved exposition, minor corrections

openalex publication_date 2004/03/08 · arxiv created 2005/06/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Automorphisms of surface braid groups
• Braids and Permutations

Related

• Geometric representations of the braid groups
• Braid Groups are Linear
• Braid groups and discrete diffeomorphisms of the punctured disk
• Abstract commensurators of braid groups
• Geometric representations of the braid group on a nonorientable surface