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This article is cited in 4 scientific papers (total in 4 papers)
Asymptotic mapping class groups of closed surfaces punctured along Cantor sets
Javier Aramayonaa, Louis Funarb a Universidad Autónoma de Madrid & ICMAT, C. U. de Cantoblanco. 28049, Madrid, Spain
b Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France
Abstract:
We introduce subgroups $\mathcal{B}_g< \mathcal{H}_g$ of the mapping class group $\mathrm{Mod}(\Sigma_g)$ of a closed surface of genus $g \ge 0$ with a Cantor set removed, which are extensions of Thompson's group $V$ by a direct limit of mapping class groups of compact surfaces of genus $g$. We first show that both $\mathcal{B}_g$ and $\mathcal{H}_g$ are finitely presented, and that $\mathcal{H}_g$ is dense in $\mathrm{Mod}(\Sigma_g)$. We then exploit the relation with Thompson's groups to study properties $\mathcal{B}_g$ and $\mathcal{H}_g$ in analogy with known facts about finite-type mapping class groups. For instance, their homology coincides with the stable homology of the mapping class group of genus $g$, every automorphism is geometric, and every homomorphism from a higher-rank lattice has finite image.
In addition, the same connection with Thompson's groups will also prove that $\mathcal{B}_g$ and $\mathcal{H}_g$ are not linear and do not have Kazhdan's Property (T), which represents a departure from the current knowledge about finite-type mapping class groups.
Key words and phrases:
surface, Cantor set, homeomorphism.
Citation:
Javier Aramayona, Louis Funar, “Asymptotic mapping class groups of closed surfaces punctured along Cantor sets”, Mosc. Math. J., 21:1 (2021), 1–29
Linking options:
https://www.mathnet.ru/eng/mmj785 https://www.mathnet.ru/eng/mmj/v21/i1/p1
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Abstract page: | 63 | References: | 11 |
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