On virtual embeddings of braid groups into mapping class groups of surfaces

TL;DR

This paper establishes precise conditions under which finite index subgroups of braid groups can be embedded into the mapping class groups of surfaces, advancing understanding of their algebraic and geometric relationships.

Contribution

It provides a necessary and sufficient criterion for embedding braid groups into surface mapping class groups, clarifying their structural connections.

Findings

Derived a complete criterion for embeddings

Characterized the algebraic conditions for embeddings

Enhanced understanding of braid and surface group relationships

Abstract

In this article, we give a necessary and sufficient condition for embedding a finite index subgroup of Artin's braid group into the mapping class group of a connected orientable surface.