WWPD elements of big mapping class groups

TL;DR

This paper classifies loxodromic elements with WWPD property in big mapping class groups acting on loop graphs, and uses this to identify subgroups with infinite-dimensional second bounded cohomology.

Contribution

It provides a complete classification of WWPD elements in big mapping class groups and establishes criteria for subgroups to have infinite-dimensional second bounded cohomology.

Findings

Classified all WWPD loxodromic elements in big mapping class groups.

Provided criteria for subgroups to have infinite-dimensional second bounded cohomology.

Simplified proofs for infinite-dimensional cohomology of certain subgroups.

Abstract

We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-Fujiwara's weak proper discontinuity and is useful for constructing non-trivial quasimorphisms. We use this classification to give a sufficient criterion for subgroups of big mapping class groups to have infinite-dimensional second bounded cohomology and use this criterion to give simple proofs that certain natural subgroups of big mapping class groups have infinite-dimensional second bounded cohomology.