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# An infinite genus mapping class group and stable cohomology

• ##### Christophe Kapoudjian
Communications in Mathematical Physics (Impact Factor: 1.97). 06/2005; DOI: 10.1007/s00220-009-0728-1
Source: arXiv

ABSTRACT 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.
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