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Arc and curve graphs for infinite-type surfaces


Authors: Javier Aramayona, Ariadna Fossas and Hugo Parlier
Journal: Proc. Amer. Math. Soc. 145 (2017), 4995-5006
MSC (2010): Primary 57M15, 57M50; Secondary 05C63
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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Additional Information

Javier Aramayona
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain
Email: aramayona@gmail.com

Ariadna Fossas
Affiliation: Department of Mathematics, University of Fribourg, Chemin du Musée 23, CH-1700 Fribourg, Switzerland
Email: ariadna.fossastenas@gmail.com

Hugo Parlier
Affiliation: Department of Mathematics, University of Fribourg, Chemun du Musée 23, CH-1700, Fribourg, Switzerland
Address at time of publication: Mathematics Research Unit, University of Luxembourg, 4364, Esch-sur-Alzette, Luxembourg
Email: hugo.parlier@uni.lu

DOI: https://doi.org/10.1090/proc/13608
Received by editor(s): April 29, 2016
Received by editor(s) in revised form: August 23, 2016, and November 4, 2016
Published electronically: August 7, 2017
Additional Notes: The first author was supported by a Ramón y Cajal grant RYC-2013-13008
The second author was supported by ERC grant agreement number 267635 - RIGIDITY
The third author was supported by Swiss National Science Foundation grants numbers PP00P2_128557 and PP00P2_153024
The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “NMS: Geometric structures and representation varieties” (the GEAR Network)
Communicated by: Ken Bromberg
Article copyright: © Copyright 2017 American Mathematical Society

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