Moscow Mathematical Journal
Volume 21, Issue 1, January–March 2021 pp. 1–29.
Asymptotic Mapping Class Groups of Closed Surfaces Punctured along Cantor Sets
Authors:
Javier Aramayona (1) and Louis Funar (2)
Author institution:(1) Universidad Autónoma de Madrid & ICMAT, C. U. de Cantoblanco. 28049, Madrid, Spain
(2) Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France
Summary:
We introduce subgroups Bg<Hg of the mapping class group Mod(Σg) of a closed surface of genus g≥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 Bg and Hg are finitely presented, and that Hg is dense in Mod(Σg). We then exploit the relation with Thompson's groups to study properties Bg and Hg 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 Bg and Hg 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.
2010 Math. Subj. Class. 57M50, 20F65.
Keywords: Surface, Cantor set, homeomorphism.
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