From braid groups to mapping class groups

Abstract

In this paper, we classify homomorphisms from the braid group of $n$ strands to the mapping class group of a genus $g$ surface. In particular, we show that when $g<n-2$, all representations are either cyclic or standard. Our result is sharp in the sense that when $g=n-2$, a generalization of the hyperelliptic representation appears, which is not cyclic or standard. This gives a classification of surface bundles over the configuration space of the complex plane. As a corollary, we partially recover the result of Aramayona-Souto, which classifies homomorphisms between mapping class groups, with a slight improvement.

25 pages, 3 figures; typos corrected, references added, introduction rewritten for clarity, argument in section 6 simplified (results unchanged)

References in corpus (5)

• Braids and Permutations
• Homomorphisms of commutator subgroups of braid groups
• Homomorphisms between braid groups
• Small Quotients of Braid Groups
• Upper Bounds for Totally Symmetric Sets

Cited by in corpus (1)

• Lower central series, surface braid groups, surjections and permutations