The grand arc graph

TL;DR

This paper introduces the grand arc graph, a new combinatorial structure for infinite-type surfaces, demonstrating its hyperbolic geometry and the properties of the mapping class group action.

Contribution

It constructs the grand arc graph for infinite-type surfaces and analyzes its geometric and group action properties, including hyperbolicity and orbit structure.

Findings

Grand arc graph is infinite-diameter and hyperbolic under certain conditions.

Mapping class group acts by isometries on the grand arc graph.

Action has finitely many orbits when the surface has stable maximal ends.

Abstract

In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three different self-similar equivalence classes of maximal ends, then the grand arc graph is infinite-diameter and $\delta$-hyperbolic. In this case, we also show that the mapping class group acts on the grand arc graph by isometries and acts on the visible boundary continuously. When the surface has stable maximal ends, we also show that this action has finitely many orbits.