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Mapping Class Groups: Cofinite Universal Space.pdf

0811.3871v1.pdf
Preview of Mapping Class Groups: Cofinite Universal Space
🔗 Source: arxiv.org
📊 Size: 166 KB
📄 Pages: 12 pages
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Summary

The mapping class group Γg,n for surfaces of negative Euler characteristic has a cofinite universal space EΓg,n for proper actions. The approach is to construct a truncated Teichm¨uller space Tg,n(ε) by introducing a lower bound for the length of shortest closed geodesics and showing that Tg,n(ε) is a Γg,n equivariant deformation retract of the Teichm¨uller space Tg,n. The existence of such a cofinite universal space is important in the study of the cohomology of the group Γg,n. The result also implies that there are only finitely many conjugacy classes of finite subgroups of Γg,n. Furthermore, the rational Novikov conjecture in K-theory holds for Γg,n. The truncated space Tg,n(ε) is contractible and the set of fixed points (Tg,n(ε))H is contractible for a finite subgroup H ⊂Γg,n, making Tg,n(ε) a cofinite universal space for proper actions of Γg,n. The proof is based on the construction of a real analytic manifold with corners Tg,n(ε) and the existence of an equivariant deformation retraction from Tg,n to Tg,n(ε). This result was stated in [Har, §3, iv)] but the proof is incomplete, and the present purpose is to complete the considerations of [Har]. The assumption that Γ is torsion-free was used in an essential way in [Iv1] [Iv2].

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Ji and Wolpert prove that mapping class groups for surfaces of negative Euler characteristic have cofinite universal spaces for proper actions. They construct a truncated Teichmüller space and show it's a deformation retract of the full space. This leads to finite cohomology and finite conjugacy classes of finite subgroups.

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  • File Format: PDF
  • File Size: 166 KB
  • Pages: 12
  • Language: EN
  • Total Downloads: 41
  • Last Updated: 3 weeks ago

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