Seminario Teoría de Grupos

Extensions of Higman--Thompson groups by stable mapping class groups

Ponente:  Javier Aramayona (ICMAT-CSIC)
Fecha:  martes 06 de abril de 2021 - 10:30
Lugar:  Online -

Starting with a compact n-manifold Y, one may construct a {em Cantor manifold} C(Y) by inductively gluing copies of Y in a tree-like fashion. In the first part of the talk, we will introduce the {em asymptotic mapping class group} B(Y) of this Cantor manifold, which is the group of isotopy classes of diffeomorphisms of C(Y) that are "eventually the identity". It generalizes (in several directions) a construction of Funar--Kapoudjian in the case of surfaces. As we will see, the group B(Y) surjects onto a Higman--Thompson group and contains the mapping class group of every compact n-submanifold of C(Y). Moreover, in certain situations its homology coincides with the {em stable homology} of the mapping class group. We will then give a sketch proof of the result that, in certain situations, the group B(Y) is of type F_inty. These situations include: (1) When Y is a sphere/torus, which settles two questions by Funar--Kapoudjian. (2) When Y = S^2 imes S^1. In this case, B(Y) contains Aut(F_r) for all r. (3) When Y = S^n imes S^n, with nge 3. Our arguments use results of Hatcher--Vogtmann, Hatcher--Wahl and Galatius--Randal-Williams in a fundamental way. This is joint work (in progress) with Kai-Uwe Bux, Jonas Fleschig, Nansen Petrosyan and Xiaolei Wu.