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On the geometry of graphs associated to infinite-type surfaces

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Abstract

Consider a connected orientable surface S of infinite topological type, i.e. with infinitely-generated fundamental group. Our main purpose is to give a description of the geometric structure of an arbitrary subgraph of the arc graph of S, subject to some rather general conditions. As special cases, we recover results of Bavard (Geom Topol 20, 2016) and Aramayona–Fossas–Parlier (Arc and curve graphs for infinite-type surfaces. Preprint, 2015). In the second part of the paper, we obtain a number of results on the geometry of connected, \(\mathrm{Mod}(S)\)-invariant subgraphs of the curve graph of S, in the case when the space of ends of S is homeomorphic to a Cantor set.

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Notes

  1. This definition is due to Schleimer [17], who referred to witnesses as holes. The word “witness” has been suggested to us by S. Schleimer.

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Acknowledgements

This project stemmed out of discussions with Juliette Bavard, and the authors are indebted to her for sharing her ideas and enthusiasm. We want to thank LAISLA and CONACYT’s Red temática Matemáticas y Desarrollo for its support. This work started with a visit of the first named author to the UNAM (Morelia), and he would like to thank the Centro de Ciencias Matemáticas for its warm hospitality. He also thanks Brian Bowditch, Hugo Parlier, and Saul Schleimer for conversations. The authors are grateful to Federica Fanoni and Nick Vlamis for discussions and for pointing out several errors in an earlier version of this draft. Finally, thanks to the referee for comments and suggestions that helps improve the exposition.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

    Javier Aramayona

  2. Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico

    Ferrán Valdez

  3. Instituto de Ciencias Matemáticas, CSIC, Madrid, Spain

    Javier Aramayona

Authors
  1. Javier Aramayona
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  2. Ferrán Valdez
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Corresponding author

Correspondence to Javier Aramayona.

Additional information

J. Aramayona was partially funded by Grants RYC-2013-13008 and MTM2015-67781. F. Valdez was supported by PAPIIT Projects IN100115, IN103411 and IB100212.

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Aramayona, J., Valdez, F. On the geometry of graphs associated to infinite-type surfaces. Math. Z. 289, 309–322 (2018). https://doi.org/10.1007/s00209-017-1952-6

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  • DOI: https://doi.org/10.1007/s00209-017-1952-6

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