Decomposing groups by codimension-1 subgroups
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Abstract:
The paper is concerned with Kropholler’s conjecture on splitting a finitely generated group over a codimension-$1$ subgroup. For a subgroup $H$ of a group $G$, we introduce the notion of ‘finite splitting height’ which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-$1$ subgroup $H$ in $G$, we show that the Kropholler-Roller conjecture holds when $H$ has finite splitting height in $G$. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include relatively quasiconvex subgroups of relatively hyperbolic groups with virtually polycyclic peripheral subgroups. In particular, our results extend Stallings’ theorem and generalise a theorem of Sageev on decomposing a hyperbolic group by quasiconvex subgroups.References
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Additional Information
- Nansen Petrosyan
- Affiliation: School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom
- Email: n.petrosyan@soton.ac.uk
- Received by editor(s): June 15, 2020
- Received by editor(s) in revised form: August 24, 2021
- Published electronically: August 5, 2022
- Communicated by: David Futer
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 4587-4601
- MSC (2020): Primary 20F67, 20E08
- DOI: https://doi.org/10.1090/proc/16136