Arc and curve graphs for infinite-type surfaces

Aramayona, Javier; Fossas, Ariadna; Parlier, Hugo

doi:10.1090/proc/13608

Arc and curve graphs for infinite-type surfaces
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by Javier Aramayona, Ariadna Fossas and Hugo Parlier

Proc. Amer. Math. Soc. 145 (2017), 4995-5006

DOI: https://doi.org/10.1090/proc/13608

Published electronically: August 7, 2017

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Abstract:

We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite diameter; this extends a recent result of J. Bavard to a large class of punctured surfaces.

We also study the subgraph of the curve graph spanned by those elements which intersect a fixed separating curve on the surface. We show that this graph has infinite diameter and geometric rank 3, and thus is not hyperbolic.

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