An Infinite Genus Mapping Class Group and Stable Cohomology

Communications in Mathematical Physics (Impact Factor: 2.09). 06/2005; 287(3). DOI: 10.1007/s00220-009-0728-1
Source: arXiv


We exhibit a finitely generated group $\M$ whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface $\su$ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus $g$ with $n$ boundary components, for any $g\geq 0$ and $n>0$. We construct a representation of $\M$ into the restricted symplectic group ${\rm Sp_{res}}({\cal H}_r)$ of the real Hilbert space generated by the homology classes of non-separating circles on $\su$, which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in $H^2(\M,\Z)$ is the pull-back of the Pressley-Segal class on the restricted linear group ${\rm GL_{res}}({\cal H})$ via the inclusion ${\rm Sp_{res}}({\cal H}_r)\subset {\rm GL_{res}}({\cal H})$. Comment: 14p., 8 figures, to appear in Commun.Math.Phys

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    ABSTRACT: The braided Ptolemy-Thompson group $T^*$ is an extension of the Thompson group $T$ by the full braid group $B_{\infty}$ on infinitely many strands. This group is a simplified version of the acyclic extension considered by Greenberg and Sergiescu, and can be viewed as a mapping class group of a certain infinite planar surface. In a previous paper we showed that $T^*$ is finitely presented. Our main result here is that $T^*$ (and $T$) is asynchronously combable. The method of proof is inspired by Lee Mosher's proof of automaticity of mapping class groups.
    Commentarii Mathematici Helvetici 02/2006; DOI:10.4171/CMH/239 · 0.94 Impact Factor
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    ABSTRACT: We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson's groups by infinite (spherical) braid groups. We will outline the main features of these groups and some applications to the quantization of Teichm\"uller spaces. The chapter provides an introduction to the subject with an emphasis on some of the authors results.


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