Finite height subgroups of extended admissible groups (Q6614202)

Let \(G\) be a finitely generated group and \(H \leq G\) a finitely generated subgroup of \(G\). If \(H\) is finite, the height of \(H\) is \(0\). If there is a largest \(n\) so that there are distinct cosets \(\{ g_{1}H,\ldots, g_{n} H\}\) with the intersection \(\bigcap_{i=1}^{n} H^{g_{i}}\) is infinite, then it is the height of \(H\) in \(G\). Finally, if such an integer does not exist then the height is infinite.\N \NIf \(\mathcal{S}\) and \(\mathcal{A}\) are finite generating sets of \(G\) and \(H\) respectively, then \(H\) is called undistorted in \(G\) if the inclusion map \(H \rightarrow G\) induces a quasi-isometric embedding from the Cayley graph \(\Gamma(H,\mathcal{A})\) into \(\Gamma(G,\mathcal{S})\) (undistorted subgroups are independent of the choice of finite generating sets).\N\NA first result in the paper under review is Theorem 1.3: Let \((G,\mathbb{P})\) be a finitely generated relatively hyperbolic group and \(H\) an undistorted subgroup of \(G\). Then \(H\) has finite height in \(G\) if and only if \(H \cap P^{g}\) has finite height in \(P^{g}\) for each conjugate of a peripheral subgroup \(P\) in \(\mathbb{P}\).\N\NThe subgroup \(H\) is strongly quasiconvex in \(G\) if for any \(L \geq 1\), \(C \geq 0\) there exists \(M= M(L, C)\) such that every \((L,C\))-quasi-geodesic in \(G\) with endpoints in \(H\) is contained in the \(M\)-neighborhood of \(H\) and strongly quasiconvex subgroups are independent of the choice of finite generating sets.\N\NThe following theorem generalizes the main result of the author et al. [Math. Ann. 381, No. 1--2, 405--437 (2021; Zbl 1477.57018)], but with a different proof. \N\NTheorem 1.5: Let \(G\) be an extended admissible group. Suppose that \(H\) is a finitely generated, undistorted subroup of \(G\). Then \(H\) has finite height in \(G\) if and only if \(H\) is strongly quasiconvex in \(G\).