Infinite-type surfaces and big mapping class groups

The purpose of this talk is to discuss the construction of Hooper-Thurston-Veech for infinite type surfaces. This construction is the generalization to the infinite context of the Thurston-Veech construction. Finally, I will give an application of this construction to the non-trivial quasimorphism space of big mapping class groups. This is joint work with Professor Ferrán Valdez.

This talk is for graduate students or anyone who wants a crash-course on translation surfaces of infinite type. It is especially aimed at people who have the definitions of topological surfaces of infinite type and are curious about translation surfaces. After the definition, we discuss singularities, look at some examples, introduce the Veech group, translation flow and mention some directions of research.

For those wanting to read more after the talk, the main reference for this talk is found here (https://www.labri.fr/perso/vdelecro/infinite-translation-surfaces-in-the-wild.html).

Abstract: Big mapping class groups are the mapping class groups of infinite-type surfaces. For a (possibly once-punctured) closed surface S minus a Cantor subset K, we show that the pure mapping class (sub)group consisting of maps fixing K pointwise is the unique maximal uncountable index normal subgroup, and the kernel of the forgetful map Map(S-K)->Map(S) is the unique minimal countable index normal subgroup. As applications, we compute the first homology of such big mapping class groups. We also show that these mapping class groups are generated by torsion and actually normally generated by a "generic" torsion element. This is joint work with Danny Calegari.

In the world of finite-type surfaces, one of the key tools to studying the mapping class group is to study its action on the curve graph. The curve graph is a combinatorial object intrinsic to the surface, and its appeal lies in the fact that it is infinite-diameter and $\delta$-hyperbolic. For infinite-type surfaces, the curve graph disappointingly has diameter 2. However, all hope is not lost! In this talk I will introduce the omnipresent arc graph and we will see that for a large collection of infinite-type surfaces, the graph is infinite-diameter and $\delta$-hyperbolic. The talk will feature a new characterization of infinite-type surfaces, which provided the impetus for this project.

This is joint work with Federica Fanoni and Alan McLeay.

For the first talk: Geometric group theory is traditionally limited to the study of finitely generated groups. But big mapping class groups are uncountable, so we need new techniques. This video introduces coarse boundedness, a concept promoted by Rosendal, which allows us to extend some traditional ideas of geometric group theory to a much larger class of groups.

For the second talk: Building on the work of Mann and Rafi, this video gives examples of big mapping class groups that are quasi-isometric to graphs of curves and arcs. In particular, the mapping class group of the punctured Cantor tree is shown to be quasi-isometric to that surface's loop graph, and the mapping class groups of "translatable surfaces" are shown to be quasi-isometric to the "translatable curve graph". This latter result is a classification, in the sense that only translatable surfaces are quasi-isometric to a curve graph of any kind.

We will introduce a definition of "big Out(Fn)" given by Algom-Kfir-Bestvina. After introducing the definition and giving some connections to the study of mapping class groups of big surfaces we investigate the coarse geometry of these groups. We classify for which locally-finite connected graphs the pure mapping class groups are globally coarsely bounded in the sense of Rosendal. This is joint work with H. Hoganson and S. Kwak.

Abstract: Given a finite subgroup G of the mapping class group of a surface S, the Nielsen realization problem asks whether G can be realized as a finite group of homeomorphisms of S. In 1983, Kerckhoff showed that for S a finite-type surface, any finite subgroup G may be realized as a group of isometries of some hyperbolic metric on S. We extend Kerckhoff’s result to orientable, infinite-type surfaces. We use this to show that topological groups containing sequences of torsion elements limiting to the identity do not embed continuously into the mapping class group of S. Finally, we show that compact subgroups of the mapping class group of S are finite, and locally compact subgroups are discrete.

We will see how to identify ribbon Higman-Thompson groups with oriented asymptotic mapping class groups and how this geometric model can be used to prove homological stability.