Shortcut graphs and groups

Hoda, Nima

doi:10.1090/tran/8555

Shortcut graphs and groups
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by Nima Hoda PDF

Trans. Amer. Math. Soc. 375 (2022), 2417-2458

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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