Two simultaneous actions of big mapping class groups

TL;DR

This paper explores two distinct actions of big mapping class groups—one on a Gromov-hyperbolic graph and another on a circle—revealing their interplay and constructing non-trivial quasimorphisms.

Contribution

It introduces new geometric objects and tools for big mapping class groups, establishing connections between their actions and dynamics, and constructing quasimorphisms beyond acylindrical hyperbolicity.

Findings

Loxodromic elements have rational rotation numbers.

Existence of equators for infinite type surfaces.

Construction of non-trivial quasimorphisms on subgroups.

Abstract

We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of the paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the circle. The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.