Recubulating free groups.
type: Item
instance of: scholarly article
title: Recubulating free groups.
zbMATH Open document ID: 1277.20049
author: Daniel T. Wise
published in: Israel Journal of Mathematics
publication date: 2012-11-13T00:00:00Z
review text: The author proves the following Theorem. For each finitely generated infinite index subgroup \(H\) of a finitely generated free group \(F\), there is a \(\text{CAT}(0)\) cube complex \(X\) on which \(F\) acts properly and cocompactly, such that \(H\) is the stabilizer of a hyperplane of \(X\) and such that there is exactly one \(F\)-orbit of hyperplane. For the construction of the \(\text{CAT}(0)\) cube complex \(X\) on which \(F\) acts properly and cocompactly, the author starts with the corresponding based covering space and using the fact that the subgroup is finitely generated and of infinite index in \(F\) to obtain an \(H\)-wall. Now \(X\) is the \(\text{CAT}(0)\) cube complex dual to the system of walls associated to the translates of \(W\). By a result of \textit{M. Sageev}'s [in J. Algebra 189, No. 2, 377-389 (1997; Zbl 0873.20028)] it is obtained an action of the free group \(F\) on \(X\) which is cocompact. The stabilizer of \(W\) is equal to \(H\), therefore the stabilizers of hyperplanes of \(X\) are conjugates of \(H\). Using an unpublished result of G. H. Hruska and D. T. Wise it is proved that this action is proper.
reviewed by: Dimitrios Varsos
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 6105493
zbMATH Keywords: free groups
zbMATH Keywords: actions of groups on CAT(0) cube complexes
zbMATH Keywords: subgroups of infinite index
MaRDI profile type: Publication
full work available at URL: s11856-011-0208-0
OpenAlex ID: W1995321917
cites work: Finiteness properties of cubulated groups
cites work: Cubulating random groups at density less than $1/6$
cites work: Ends of Group Pairs and Non-Positively Curved Cube Complexes
cites work: Codimension-1 subgroups and splittings of groups
cites work: Cubulating small cancellation groups.
DOI: 10.1007/S11856-011-0208-0
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description: scientific article; zbMATH DE number 6105493