## Automorphisms of the loop and arc graph of an infinite-type surface

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Anschel Schaffer-Cohen
**HTML**| PDF - Proc. Amer. Math. Soc. Ser. B
**9**(2022), 230-240

## Abstract:

We show that the extended based mapping class group of an infinite-type surface is naturally isomorphic to the automorphism group of the loop graph of that surface. Additionally, we show that the extended mapping class group stabilizing a finite set of punctures is isomorphic to the arc graph relative to that finite set of punctures. This extends a known result for sufficiently complex finite-type surfaces, and provides a new angle from which to study the mapping class groups of infinite-type surfaces.## References

- Javier Aramayona, Ariadna Fossas, and Hugo Parlier,
*Arc and curve graphs for infinite-type surfaces*, Proc. Amer. Math. Soc.**145**(2017), no. 11, 4995–5006. MR**3692012**, DOI 10.1090/proc/13608 - Juliette Bavard,
*Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire*, Geom. Topol.**20**(2016), no. 1, 491–535 (French, with English and French summaries). MR**3470720**, DOI 10.2140/gt.2016.20.491 - Juliette Bavard, Spencer Dowdall, and Kasra Rafi,
*Isomorphisms between big mapping class groups*, Int. Math. Res. Not. IMRN**10**(2020), 3084–3099. MR**4098634**, DOI 10.1093/imrn/rny093 - Allen Hatcher,
*On triangulations of surfaces*, Topology Appl.**40**(1991), no. 2, 189–194. MR**1123262**, DOI 10.1016/0166-8641(91)90050-V - Jesús Hernández Hernández, Israel Morales, and Ferrán Valdez,
*Isomorphisms between curve graphs of infinite-type surfaces are geometric*, Rocky Mountain J. Math.**48**(2018), no. 6, 1887–1904. MR**3879307**, DOI 10.1216/rmj-2018-48-6-1887 - Jesús Hernández Hernández, Israel Morales, and Ferrán Valdez,
*The Alexander method for infinite-type surfaces*, Michigan Math. J.**68**(2019), no. 4, 743–753. MR**4029627**, DOI 10.1307/mmj/1561773633 - Sebastian Hensel, Piotr Przytycki, and Richard C. H. Webb,
*1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs*, J. Eur. Math. Soc. (JEMS)**17**(2015), no. 4, 755–762. MR**3336835**, DOI 10.4171/JEMS/517 - Elmas Irmak and John D. McCarthy,
*Injective simplicial maps of the arc complex*, Turkish J. Math.**34**(2010), no. 3, 339–354. MR**2681579** - Nikolai V. Ivanov,
*Automorphism of complexes of curves and of Teichmüller spaces*, Internat. Math. Res. Notices**14**(1997), 651–666. MR**1460387**, DOI 10.1155/S1073792897000433 - Mustafa Korkmaz,
*Automorphisms of complexes of curves on punctured spheres and on punctured tori*, Topology Appl.**95**(1999), no. 2, 85–111. MR**1696431**, DOI 10.1016/S0166-8641(97)00278-2 - Feng Luo,
*Automorphisms of the complex of curves*, Topology**39**(2000), no. 2, 283–298. MR**1722024**, DOI 10.1016/S0040-9383(99)00008-7 - Howard A. Masur and Yair N. Minsky,
*Geometry of the complex of curves. I. Hyperbolicity*, Invent. Math.**138**(1999), no. 1, 103–149. MR**1714338**, DOI 10.1007/s002220050343 - Howard Masur and Saul Schleimer,
*The geometry of the disk complex*, J. Amer. Math. Soc.**26**(2013), no. 1, 1–62. MR**2983005**, DOI 10.1090/S0894-0347-2012-00742-5

## Additional Information

**Anschel Schaffer-Cohen**- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
- MR Author ID: 1303454
- ORCID: 0000-0003-4885-258X
- Email: anschel@math.upenn.edu
- Received by editor(s): December 17, 2019
- Received by editor(s) in revised form: February 16, 2021
- Published electronically: April 29, 2022
- Communicated by: Kenneth Bromberg
- © Copyright 2022 by the author under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B
**9**(2022), 230-240 - MSC (2020): Primary 57K20, 20F65; Secondary 05C25
- DOI: https://doi.org/10.1090/bproc/83
- MathSciNet review: 4414904

Dedicated: For Carlos