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