# Embedding mapping class groups of orientable surfaces with one boundary component

Abstract : We denote by $S_{g,b,p}$ an orientable surface of genus $g$ with $b$ boundary components and $p$ punctures. We construct homomorphisms from the mapping class groups of $S_{g,1,p}$ to the mapping class groups of $S_{g',1,(b-1)}$, where $b\geq 1$. These homomorphisms are constructed from branched or unbranched covers of $S_{g,1,0}$ with some properties. Our main result is that these homomorphisms are injective. For unbranched covers, this construction was introduced by McCarthy and Ivanov~\cite{IM}. They proved that the homomorphisms are injective. A particular cases of our embeddings is a theorem of Birman and Hilden that embeds the braid group on $p$ strands into the mapping class group of $S_{(p-2)/2,2,0}$ if $p$ is even, or into the mapping class group of $S_{(p-1)/2,1,0}$ if $p$ is odd. We give a short proof of another result of Birman and Hilden \cite{BH} for surfaces with one boundary component.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-00490852
Contributor : Lluis Bacardit <>
Submitted on : Wednesday, July 18, 2012 - 9:58:33 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Friday, October 19, 2012 - 3:07:22 AM

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punts_finits.pdf
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• HAL Id : hal-00490852, version 3

### Citation

Lluis Bacardit. Embedding mapping class groups of orientable surfaces with one boundary component. 2012. ⟨hal-00490852v3⟩

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