Geometry of the Homology Curve Complex

TL;DR

This paper explores the geometry of the homology multicurve complex for surfaces of genus at least two, providing algorithms for quasi-geodesic construction, analyzing distance bounds, and establishing non-hyperbolicity for higher genus.

Contribution

It introduces a simple algorithm for constructing quasi-geodesics and minimal genus surfaces, and proves linear bounds on distances and non-hyperbolicity for the homology multicurve complex.

Findings

Distance bounds are linear in intersection number for genus ≥ 3.

The complex is not δ-hyperbolic for genus ≥ 4.

An algorithm for constructing quasi-geodesics in the complex.

Abstract

Suppose $S$ is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, $\mathcal{HC}(S,\alpha)$, of $S$; a complex closely related to complexes studied by Bestvina-Bux-Margalit and Hatcher. A path in $\mathcal{HC}(S,\alpha)$ corresponds to a homotopy class of immersed surfaces in $S\times I$. This observation is used to devise a simple algorithm for constructing quasi-geodesics connecting any two vertices in $\mathcal{HC}(S,\alpha)$, and for constructing minimal genus surfaces in $S\times I$. It is proven that for $g \geq 3$ the best possible bound on the distance between two vertices in $\mathcal{HC}(S, \alpha)$ depends linearly on their intersection number, in contrast to the logarithmic bound obtained in the complex of curves. For $g \geq 4$ it is shown that $\mathcal{HC}(S, \alpha)$ is not $\delta$-hyperbolic.