Graphs of curves on infinite-type surfaces with mapping class group actions

TL;DR

This paper investigates the actions of mapping class groups of infinite-type surfaces on graphs of curves, introducing a topological invariant to determine when such actions with unbounded orbits exist, revealing complex coarse geometry.

Contribution

It introduces a topological invariant for infinite-type surfaces that predicts the existence of unbounded orbit actions of their mapping class groups.

Findings

Many big mapping class groups have nontrivial coarse geometry.

The invariant helps identify when the group admits unbounded orbit actions.

The work connects topological properties of surfaces with geometric group actions.

Abstract

We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.