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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 -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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Quasiconvexity and Dehn filling
We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with recent work of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hyperbolic 3-manifolds, a result first proved by Wise. Additionally, we explain how previous results on multiplicity and height can be generalized to the relative setting to control the relative height of relatively quasiconvex subgroups under appropriate Dehn fillings.
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- Award ID(s):
- 1904913
- PAR ID:
- 10475808
- Publisher / Repository:
- Johns Hopkins University Press
- Date Published:
- Journal Name:
- American Journal of Mathematics
- Volume:
- 143
- Issue:
- 1
- ISSN:
- 1080-6377
- Page Range / eLocation ID:
- 95 to 124
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1353/ajm.2021.0007
