## Arc and curve graphs for infinite-type surfaces

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- by Javier Aramayona, Ariadna Fossas and Hugo Parlier PDF
- Proc. Amer. Math. Soc.
**145**(2017), 4995-5006 Request permission

## 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
- MR Author ID: 796736
- Email: aramayona@gmail.com
**Ariadna Fossas**- Affiliation: Department of Mathematics, University of Fribourg, Chemin du Musée 23, CH-1700 Fribourg, Switzerland
- MR Author ID: 992983
- 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
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 4995-5006 - MSC (2010): Primary 57M15, 57M50; Secondary 05C63
- DOI: https://doi.org/10.1090/proc/13608
- MathSciNet review: 3692012