Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups.

Crisp, John; Wiest, Bert

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Let $M$ be a surface, let $N$ be a subsurface, and let $n \leq m$ be two positive integers. The inclusion of $N$ in $M$ gives rise to a homomorphism from the braid group $B_{n} N$ with $n$ strings on $N$ to the braid group $B_{m} M$ with $m$ strings on $M$ . We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of $π_{1} N$ in $π_{1} M$ . Then we calculate the commensurator, the normalizer and the centralizer of $B_{n} N$ in $B_{m} M$ for large surface...