Height in splittings of relatively hyperbolic groups (Q2039255)

Consider a relatively hyperbolic group \(G\). A subgroup \(H\subset G\) has relative height at most \(n\) if, for every intersection \(\bigcap_{i=1}^{n} g_i H g_i^{-1}\) that contains a loxodromic element, there exists \(i\neq j\) such that \(g_i H= g_jH\) (and in particular \( g_i H g_i^{-1}= g_j H g_j^{-1}\)). It is known that relatively quasiconvex subgroups have finite relative height. The main result of the paper goes in the converse direction, thus generalizing a Theorem by \textit{M. Mitra} [Proc. Indian Acad. Sci., Math. Sci. 114, No. 1, 39--54 (2004; Zbl 1059.20040)]. It can be stated as follows. Consider a graph of groups \(\Gamma\) where each vertex group \(G_v\) is relatively hyperbolic with respect to a collection of subgroups \(\mathcal{H}_v\). Assume that edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups and that the fundamental group \(G\) of \(\Gamma\) is relatively hyperbolic with respect to \(\bigcup_v \mathcal{H}_v\). If all vertex groups have finite relative height in \(G\), then all vertex groups are relatively quasiconvex in \(G\). The proof relies on the techniques in [\textit{M. Mj} and \textit{A. Pal}, Geom. Dedicata 151, 59--78 (2011; Zbl 1222.57013)].