# Finite index subgroups of mapping class groups

Abstract : Let $g\geq3$ and $n\geq0$, and let ${\mathcal{M}}_{g,n}$ be the mapping class group of a surface of genus $g$ with $n$ boundary components. We prove that ${\mathcal{M}}_{g,n}$ contains a unique subgroup of index $2^{g-1}(2^{g}-1)$ up to conjugation, a unique subgroup of index $2^{g-1}(2^{g}+1)$ up to conjugation, and the other proper subgroups of ${\mathcal{M}}_{g,n}$ are of index greater than $2^{g-1}(2^{g}+1)$. In particular, the minimum index for a proper subgroup of ${\mathcal{M}}_{g,n}$ is $2^{g-1}(2^{g}-1)$.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-00592181
Contributor : Luis Paris <>
Submitted on : Wednesday, May 11, 2011 - 3:17:38 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Friday, August 12, 2011 - 2:51:48 AM

### Files

MinIndSubV6.pdf
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### Identifiers

• HAL Id : hal-00592181, version 1
• ARXIV : 1105.2468

### Citation

Luis Paris, Jon A Berrick, Volker Gebhardt. Finite index subgroups of mapping class groups. 2011. ⟨hal-00592181⟩

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