Big Torelli groups: Generation and commensuration
| Autor: | Javier Aramayona, Tyrone Ghaswala, Autumn Kent, Alan McLeay, Jing Tao, Rebecca Winarski |
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| Přispěvatelé: | Ministerio de Economía y Competitividad (España) |
| Rok vydání: | 2019 |
| Předmět: |
010102 general mathematics
Geometric Topology (math.GT) Group Theory (math.GR) 01 natural sciences Mathematics::Geometric Topology 20F65 57M50 Mathematics - Geometric Topology Mathematics::Group Theory 0103 physical sciences FOS: Mathematics Discrete Mathematics and Combinatorics 010307 mathematical physics Geometry and Topology 0101 mathematics Mathematics - Group Theory |
| Zdroj: | Digital.CSIC. Repositorio Institucional del CSIC instname |
| Popis: | For any surface $\Sigma$ of infinite topological type, we study the Torelli subgroup ${\mathcal I}(\Sigma)$ of the mapping class group ${\rm MCG}(\Sigma)$, whose elements are those mapping classes that act trivially on the homology of $\Sigma$. Our first result asserts that ${\mathcal I}(\Sigma)$ is topologically generated by the subgroup of ${\rm MCG}(\Sigma)$ consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman, Powell, and Putman we deduce that ${\mathcal I}(\Sigma)$ is topologically generated by separating twists and bounding pair maps. Next, we prove the abstract commensurator group of ${\mathcal I}(\Sigma)$ coincides with ${\rm MCG}(\Sigma)$. This extends the results for finite-type surfaces of Farb-Ivanov, Brendle-Margalit and KIda to the setting of infinite-type surfaces. Comment: Made changes suggested by the referee. To appear in Groups, Geometry, and Dynamics |
| Databáze: | OpenAIRE |
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