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

TL;DR

The paper studies fine curve graphs ${\mathcal{C}^\dagger_k}(S)$, where vertices are essential simple closed curves and edges connect curves with at most $k$ intersections, establishing hyperbolicity for all $k$. It introduces the direct limit finitary curve graph ${\mathcal{C}^\dagger_{<\infty}}(S)$, proving its diameter is $2$ and that its flag complex is contractible, while showing every countable graph embeds as an induced subgraph in suitable ambient surfaces. It further demonstrates that finite graphs embed in fine curve graphs of high genus, and that countable graphs embed in the finitary and $k$-curve variants (with $k\ge2$), including the Erdős–Rényi graph as a subgraph, while also identifying inadmissible subgraphs and surface-dependent obstructions. Finally, the automorphism group of the finitary curve graph is shown to be isomorphic to the homeomorphism group ${\mathrm{Homeo}}(S_g)$, highlighting a strong rigidity phenomenon.

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.

Figures (10)

• Figure 1: Left: a crossing intersection. Right: a touching intersection.
• Figure 2: Any pair of crossing curves in ${\mathcal{C}^\dagger_1}(S_g)$ is, up to homeomorphism of $S_g$, equivalent to the two blue curves on the left. We may find a curve disjoint from both (such as the red curve on the right) outside the torus they fill.
• Figure 3: We have an example of a curve $u$ that crosses the annulus bounded by green curves $v$ and $v'$ five times and forms two loops. The crossing strands are purple while the loops are burgundy.
• Figure 4: Top: the horizontal lines are, top to bottom, $v',\ v",$ and $v.$ Pictured as well are $u_1\subset u$ (red) and a banana neighborhood $B$ of $u_1$ that is disjoint from $v\cup v'$ except at the endpoints of $u_1.$ Bottom: the banana neighborhood $B$ of $u_1\subset u$. To ensure that we have a curve that intersects $u_1$ finitely many times, we surger $v"$ with the purple arcs, preserving the pictured orientation. In this image, we have not yet completed the final isotopy to remove the touching intersection.
• Figure 5: Left: Four curves on a torus. Center: Handles (annuli) attached connecting curves that are not directly next to each other. Right: An example of four curves that induce a subgraph of ${\mathcal{C}^\dagger}(S_{1,0})$ isomorphic to the graph on four vertices with no edges.

Theorems & Definitions (49)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7: Construction of inadmissible graphs
• Theorem 1.8
• Proposition 2.1: Base Case
• proof
• Proposition 2.2: Inductive Case