Departamento de Matemáticas UAM

Seminario Teoría de grupos

14 de marzo de 20236

11:30 Aula Roja, IFT

Speaker: Martin Palmer (Mathematical Institute of the Romanian Academy)

Title: The homology of big mapping class groups

Abstract: Big mapping class groups -- mapping class groups of 
infinite-type surfaces -- have recently become the subject of 
intensive study, having connections for example with geometric group 
theory and dynamical systems. However, their homology in degrees above 
one has so far been very little understood.

I will describe two contrasting results, from joint work with Xiaolei 
Wu, that exhibit very different behaviours of the homology of big 
mapping class groups. First, we find an uncountable family of big 
mapping class groups (including the mapping class group of the disc 
minus a Cantor set) whose integral homology vanishes in all positive 
degrees. Second, we find another uncountable family of big mapping 
class groups (including the mapping class groups of the flute surface 
and of the Loch Ness monster surface) whose integral homology is 
uncountable in each positive degree.

We also study the pure subgroups of big mapping class groups, namely 
the subgroups consisting of mapping classes that fix each end of the 
surface. These have more uniform behaviour: we prove that, for every 
infinite-type surface, its pure mapping class group has uncountable 
homology in each positive degree.