Prelectura de tesis Rodrigo de Pool Alcántara

Martes 1 de abril de 2025

Sala 520

11:30 hr.

Título de la tesis: 

Rigidity results for the mapping class groups 

Director de tesis:

Javier Aramayona

Resumen:

The mapping class group of a surface S is the group of homeomorphisms of S up to homotopy.  The purpose of this thesis is to establish various results concerning the rigidity of the mapping class group and other associated spaces. Apart from its intrinsic interest in the context of geometric group theory, these results are motivated by the comparison between mapping class groups and irreducible lattices in higher-rank Lie groups. In this setting, a natural analogue of Margulis superrigidity is whether every homomorphism between mapping class groups lifts to a map at the level of diffeomorphism groups. This question leads the main discussion of the thesis.

In the first part of the thesis we characterize solutions to the braid equation for certain pairs of elements in the mapping class group. Building on this work, we then classify homomorphisms between different mapping class groups for an infinite family of surfaces. As an application, we show that, indeed, every homomorphism between these mapping class groups lifts to a map between the diffeomorphism groups. In the second part of the thesis, we study rigidity results for related objects. In joint work with Souto, we describe holomorphic maps between moduli spaces for the same range of surfaces. And, in joint work with Aramayona, Skipper, Tao, Vlamis and Wu, we extend some of the techniques to the infinite-type case, in particular, we proof that infinite-type surfaces with non-planar ends have topologically Hopfian mapping class groups.