Subgroup separability and virtual retractions of groups
Let \(H\subset\Gamma\) be a subgroup of a group \(\Gamma\). If for every \(g\in\Gamma\setminus H\), there is a subgroup \(K\) of finite index in \(\Gamma\) such that \(H\subset K\) but \(g\neq K\), then \(\Gamma\) is called \(H\)-separable. The group \(\Gamma\) is called subgroup separable or LERF if \(\Gamma\) is \(H\)-separable for all finitely generated subgroups \(H\). As one of the main results the authors prove that if \(\Gamma\) is LERF, \(H\) is a finitely generated subgroup and \(\theta: H\to A\) is a homomorphism onto a finite group, then there is subgroup of finite index \(V\) of \(\Gamma\) containing \(H\) and a homomorphism \(\Theta: V\to A\) with \(\Theta|_H= \theta\). Next they introduce the definition of virtual retracts, i.e. \(\Gamma\) virtually retracts onto \(H\) if there is a finite index subgroup \(V\) of \(\Gamma\) with (1) \(H\subset V\), (2) there is a homomorphism \(\theta: V\to H\) which is the identity when restricted to \(H\). \(V\) is called a retractor. Then authors introduce a new property of groups. \(\Gamma\) has Property LR (it admits Local Restriction) if for every finitely generated subgroup \(H\subset\Gamma\), there is a finite index subgroup which retracts to it. For example the authors prove the following result. If \(\Gamma\) is a finitely generated group which virtually embeds into an all right hyperbolic Coxeter subgroup of \(\text{Isom}(\mathbb{H}^n)\), then \(\Gamma\) virtually retracts to its geometrically finite subgroups. Moreover in the article examples of LR groups are given and some open problems and questions are formulated. Also many other results are proved which are formulated in the language of LR groups. Most of them are quite technical. The reader should be familiar with hyperbolic geometry.
scientific article; zbMATH DE number 5544209