Recubulating free groups.
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 Using an unpublished result of G. H. Hruska and D. T. Wise it is proved that this action is proper.
actions of groups on CAT(0) cube complexes
scientific article; zbMATH DE number 6105493
scientific article