Notes on relatively hyperbolic groups and relatively quasiconvex subgroups. (Q498642)

Let \(G\) be a group and \(\mathbf H\) be a family of non-trivial subgroups of \(G\). Several equivalent conditions for relative hyperbolicity of \((G,\mathbf H)\) are given in terms of coned-off Cayley graph, Cayley graph and hyperbolic embeddabilty of \(\mathbf H\) in \(G\). It is shown that if \(\mathbf H\) is finite, then the results reduce to the known classical results. Also, the notion of quasi-convex subgroup is defined for arbitrary family \(\mathbf H\) and some basic properties of quasi-convex subgroups are investigated. It is shown that in the case when \(\mathbf H\) is finite, their definition is the same as the classical one. Relative Dehn functions of presentations are not considered up to equivalence and several times, the authors say that the relative Dehn functions of the relative hyperbolic groups are linear. This is not true because such Dehn functions are just equivalent to linear functions. However, the validity of the results is not affected by this assumption.