Finitely-presented big mapping class groups
Vendredi, 17 Novembre, 2017 - 11:00
(Joint work with Louis Funar.)
In this talk, we will define the "asymptotically rigid" mapping class group $B_g$ of genus $g$. The group $B_g$ is a subgroup of the mapping class group of a certain infinite-type surface, and contains the mapping class group of every compact surface of genus $\le g$ with non-empty boundary.
The group $B_g$ turns out to be finitely presented, and its $k$-th homology group coincides with the $k$-th stable homology of the mapping class group of genus $g$, for $k$ small enough with respect to $g$. We then proceed to prove that the family of groups so obtained enjoys a number of properties analogous to known results for finite-type mapping class groups.
More concretely, there are no (weakly) injective maps $B_h \to B_g$ if $g<h$ every automorphism of $B_g$ is geometric, and every homomorphism from a higher-rank lattice to $B_g$ has finite image.
In addition to the translation of classical results about finite-type mapping class groups, we will show that $B_g$ is not linear, and does not have Kazhdan's Property (T).
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