Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques

Dani, Pallavi; Levcovitz, Ivan

doi:10.1090/tran/8801

Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques
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by Pallavi Dani and Ivan Levcovitz;

Trans. Amer. Math. Soc. 376 (2023), 1427-1444

DOI: https://doi.org/10.1090/tran/8801

Published electronically: November 16, 2022

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Abstract:

We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings’ foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be $2$-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are as short as possible (length two words), finitely generated free subgroups, non-finitely presentable subgroups, and subgroups of fundamental groups of square complexes of nonpositive sectional curvature.

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