Geometry of the Homology Curve Complex

Details Geometry of the Homology Curve Complex Geometry of the Homology Curve Complex

2011-07-18

arxiv.org/abs/1107.3547v2

Mathematics

Geometric Topology

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Geometry of the Homology Curve Complex does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Geometry of the Homology Curve Complex, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Geometry of the Homology Curve Complex will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-272591