WWPD elements of big mapping class groups

Abstract

We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the $loop$ $graphs$ introduced by Bavard and 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 and 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.