## Shortcut graphs and groups

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## Abstract:

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the $1$-skeletons of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), $1$-skeletons of finite dimensional $\operatorname {CAT}(0)$ cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated Coxeter groups and the standard Cayley graph of the Baumslag-Solitar group $\operatorname {BS}(1,2)$. Most of these examples satisfy a strong form of the shortcut property.

The shortcut properties also have important geometric group theoretic consequences. We show that shortcut groups are finitely presented and have exponential isoperimetric and isodiametric functions. We show that groups satisfying the strong form of the shortcut property have polynomial isoperimetric and isodiametric functions.

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## Additional Information

**Nima Hoda**- Affiliation: Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75005 Paris, France; and Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
- MR Author ID: 1121947
- ORCID: 0000-0002-6308-7751
- Email: nima.hoda@mail.mcgill.ca
- Received by editor(s): November 12, 2018
- Received by editor(s) in revised form: April 14, 2019, April 14, 2019, and July 28, 2021
- Published electronically: December 20, 2021
- Additional Notes: This work was supported in part by an NSERC Postgraduate Scholarship, a Pelletier Fellowship, a Graduate Mobility Award, an ISM Scholarship, the NSERC grant of Daniel T. Wise and the ERC grant GroIsRan
- © Copyright 2021 by Nima Hoda
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 2417-2458 - MSC (2020): Primary 20F65, 20F67, 05C12
- DOI: https://doi.org/10.1090/tran/8555
- MathSciNet review: 4391723