Relative hyperbolicity and relative quasiconvexity for countable groups.

*(English)*Zbl 1202.20046This rather useful paper has two objectives: 1) It proves the equivalence of various definitions of relative hyperbolicity. The notion was introduced by M. Gromov [in: Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] and worked out from various perspectives by Farb, Bowditch, Osin and others. 2) To define several equivalent notions of relative quasiconvexity, unifying work of Dahmani and Osin.

\(G\) is a countable group and \(\mathbb{P}\) a collection of subgroups. The author proves the equivalence of the following definitions of relative hyperbolicity of \(G\) with respect to \(\mathbb{P}\).

1) \((G;\mathbb{P})\) has a geometrically finite convergence group action on a compact, metrizable space \(M\).

2) \( G\) has a properly discontinuous action on a proper hyperbolic space \(X\) such that the induced convergence group action on \(\partial X\) is geometrically finite. \(\mathbb{P}\) is a set of representatives of the conjugacy classes of maximal parabolic subgroups.

3) \( G\) has a properly discontinuous action on a proper hyperbolic space \(X\), and \(\mathbb{P}\) is a set of representatives of the conjugacy classes of maximal parabolic subgroups. There is a \(G\)-equivariant collection of disjoint horoballs centered at the parabolic points of \(G\), with union \(U\) open in \(X\), such that the quotient of \(X\setminus U\) by the action of \(G\) is compact.

4) A graph \(K\) is fine if each edge of \(K\) is contained in only finitely many circuits of length \(n\) for each \(n\). \(G\) acts on a fine hyperbolic graph \(K\) with finite edge stabilizers and finitely many orbits of edges. \(\mathbb{P}\) is a set of representatives of the conjugacy classes of infinite vertex stabilizers.

5) \(G\) is finitely generated relative to \(\mathbb{P}\), and each \(P_i\in\mathbb{P}\) is infinite. For some (every) finite relative generating set \(S\), the coned-off Cayley graph is hyperbolic and \((G,\mathbb{P},S)\) has bounded coset penetration.

6) \(\mathbb{P}\) is a finite collection of infinite subgroups of a countable group \(G\). \((G,\mathbb{P})\) has a finite relative presentation, and the relative Dehn function is well-defined and linear for some/every finite relative presentation.

The author next gives several definitions of relative quasiconvexity and proves their equivalence.

(i) A subgroup \(H\subset G\) is relatively quasiconvex if the following holds. Let \(M\) be some (any) compact, metrizable space on which \((G,\mathbb{P})\) acts as a geometrically finite convergence group. Then the induced convergence action of \(H\) on the limit set \(\Lambda H\subset M\) is geometrically finite.

(ii) Let \(X\) be some (any) proper hyperbolic space on which \((G,\mathbb{P})\) has a cusp uniform action. Then either \(H\) is finite, \(H\) is parabolic, or \(H\) has a cusp uniform action on a geodesic hyperbolic space \(Y\) quasi-isometric to the subspace \(\mathbf{join}(\Lambda H)\subset X\), where \(\Lambda H\) denotes the limit set of \(H\).

(iii) Let \((X,\rho)\) be some (any) proper hyperbolic space on which \((G,\mathbb{P})\) has a cusp uniform action. Let \(X\setminus U\) be some (any) truncated space for \(G\) acting on \(X\). For some (any) basepoint \(x\in X\setminus U\) there is a constant \(\mu\geq 0\) such that whenever \(c\) is a geodesic in \(X\) with endpoints in the orbit \(Hx\), we have \(c\cap X\subset\mathcal N_\mu(Hx)\) where the neighborhood is taken with respect to the metric on \(X\).

(iv) Let \((X,\rho)\) be some (any) proper \(\delta\) hyperbolic space on which \((G,\mathbb{P})\) has a cusp uniform action. Let \(X\setminus U\) be some (any) truncated space for \(G\) acting on \(X\). Then each pair of horoballs of \(U\) is separated by at least a distance \(r\), where \(r\) is a constant with a specific dependence on \(\delta\). (See Lemma 6.8 of the paper for the nature of the dependence.)

(v) Let \(S\) be some (any) finite relative generating set for \((G,\mathbb{P})\) and let \(\mathbb{P}\) be the union of all \(P_i\in\mathbb{P}\). Consider the Cayley graph \(\overline\Gamma\) with generating set \(S\cup\mathbb{P}\). Let \(d\) be some (any) proper, left invariant metric on \(G\). Then there is a constant \(\kappa=\kappa(S,d)\) such that for each geodesic \(c\) in \(\overline\Gamma\) connecting two points of \(H\), every vertex of \(c\) lies within a \(d\)-distance \(\kappa\) of \(H\).

\(G\) is a countable group and \(\mathbb{P}\) a collection of subgroups. The author proves the equivalence of the following definitions of relative hyperbolicity of \(G\) with respect to \(\mathbb{P}\).

1) \((G;\mathbb{P})\) has a geometrically finite convergence group action on a compact, metrizable space \(M\).

2) \( G\) has a properly discontinuous action on a proper hyperbolic space \(X\) such that the induced convergence group action on \(\partial X\) is geometrically finite. \(\mathbb{P}\) is a set of representatives of the conjugacy classes of maximal parabolic subgroups.

3) \( G\) has a properly discontinuous action on a proper hyperbolic space \(X\), and \(\mathbb{P}\) is a set of representatives of the conjugacy classes of maximal parabolic subgroups. There is a \(G\)-equivariant collection of disjoint horoballs centered at the parabolic points of \(G\), with union \(U\) open in \(X\), such that the quotient of \(X\setminus U\) by the action of \(G\) is compact.

4) A graph \(K\) is fine if each edge of \(K\) is contained in only finitely many circuits of length \(n\) for each \(n\). \(G\) acts on a fine hyperbolic graph \(K\) with finite edge stabilizers and finitely many orbits of edges. \(\mathbb{P}\) is a set of representatives of the conjugacy classes of infinite vertex stabilizers.

5) \(G\) is finitely generated relative to \(\mathbb{P}\), and each \(P_i\in\mathbb{P}\) is infinite. For some (every) finite relative generating set \(S\), the coned-off Cayley graph is hyperbolic and \((G,\mathbb{P},S)\) has bounded coset penetration.

6) \(\mathbb{P}\) is a finite collection of infinite subgroups of a countable group \(G\). \((G,\mathbb{P})\) has a finite relative presentation, and the relative Dehn function is well-defined and linear for some/every finite relative presentation.

The author next gives several definitions of relative quasiconvexity and proves their equivalence.

(i) A subgroup \(H\subset G\) is relatively quasiconvex if the following holds. Let \(M\) be some (any) compact, metrizable space on which \((G,\mathbb{P})\) acts as a geometrically finite convergence group. Then the induced convergence action of \(H\) on the limit set \(\Lambda H\subset M\) is geometrically finite.

(ii) Let \(X\) be some (any) proper hyperbolic space on which \((G,\mathbb{P})\) has a cusp uniform action. Then either \(H\) is finite, \(H\) is parabolic, or \(H\) has a cusp uniform action on a geodesic hyperbolic space \(Y\) quasi-isometric to the subspace \(\mathbf{join}(\Lambda H)\subset X\), where \(\Lambda H\) denotes the limit set of \(H\).

(iii) Let \((X,\rho)\) be some (any) proper hyperbolic space on which \((G,\mathbb{P})\) has a cusp uniform action. Let \(X\setminus U\) be some (any) truncated space for \(G\) acting on \(X\). For some (any) basepoint \(x\in X\setminus U\) there is a constant \(\mu\geq 0\) such that whenever \(c\) is a geodesic in \(X\) with endpoints in the orbit \(Hx\), we have \(c\cap X\subset\mathcal N_\mu(Hx)\) where the neighborhood is taken with respect to the metric on \(X\).

(iv) Let \((X,\rho)\) be some (any) proper \(\delta\) hyperbolic space on which \((G,\mathbb{P})\) has a cusp uniform action. Let \(X\setminus U\) be some (any) truncated space for \(G\) acting on \(X\). Then each pair of horoballs of \(U\) is separated by at least a distance \(r\), where \(r\) is a constant with a specific dependence on \(\delta\). (See Lemma 6.8 of the paper for the nature of the dependence.)

(v) Let \(S\) be some (any) finite relative generating set for \((G,\mathbb{P})\) and let \(\mathbb{P}\) be the union of all \(P_i\in\mathbb{P}\). Consider the Cayley graph \(\overline\Gamma\) with generating set \(S\cup\mathbb{P}\). Let \(d\) be some (any) proper, left invariant metric on \(G\). Then there is a constant \(\kappa=\kappa(S,d)\) such that for each geodesic \(c\) in \(\overline\Gamma\) connecting two points of \(H\), every vertex of \(c\) lies within a \(d\)-distance \(\kappa\) of \(H\).

Reviewer: Mahan Mj (Howrah)

##### MSC:

20F65 | Geometric group theory |

20F67 | Hyperbolic groups and nonpositively curved groups |

20F05 | Generators, relations, and presentations of groups |

##### Keywords:

relative hyperbolicity; relative quasiconvexity; relative presentations; Dehn functions; cusp uniform actions##### References:

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