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Relative hyperbolicity and relative quasiconvexity for countable groups. (English) Zbl 1202.20046
This 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 $$H$$.
Reviewer: Mahan Mj (Howrah)

##### MSC:
 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups 20F05 Generators, relations, and presentations of groups
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