Hyperbolicity, topology, and combinatorics of fine curve graphs and variants

TL;DR

This paper studies the hyperbolic properties, topological features, and combinatorial structures of various fine curve graphs associated with surfaces, revealing their universality and automorphism groups.

Contribution

It introduces the finitary curve graph as a limit object, proving its hyperbolicity, universality for countable graphs, and identifying its automorphism group with surface homeomorphisms.

Findings

Fine $k$-curve graphs are hyperbolic for all $k$.

The finitary curve graph contains all countable graphs as induced subgraphs.

The automorphism group of the finitary curve graph is the surface's homeomorphism group.

Abstract

Given a surface, the fine $k$-curve graph of the surface is a graph whose vertices are simple closed essential curves and whose edges connect curves that intersect in at most $k$ points. We note that the fine $k$-curve graph is hyperbolic for all $k$ and, for $k\geq 2,$ show that it contains as induced subgraphs all countable graphs. We also show that the direct limit of this family of graphs, which we call the finitary curve graph, has diameter 2, has a contractible flag complex, contains every countable graph as an induced subgraph, and has as its automorphism group the homeomorphism group of the surface. Finally, we explore some finite graphs that are not induced subgraphs of fine curve graphs.