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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.
MSC: 57M50,
20F65
Language: English
DOI:
10.17323/1609-4514-2021-21-1-1-29
Fulltext:
References