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On some convex cocompact groups in real hyperbolic space. (English) Zbl 1294.20054
It is classical problem to characterize the word-hyperbolic groups that admit nice actions on a real hyperbolic space $$\mathbb H^p$$. More precisely, the question is what kind of word-hyperbolic groups can be realized as a convex-cocompact subgroup of $$\mathbb H^p$$. In the paper under review, the question is studied for word-hyperbolic Coxeter groups. It should be noticed that the characterization of word-hyperbolic groups is given in Moussong’s Thesis. Also, it is known that some word-hyperbolic Coxeter groups can not be realized as reflection convex cocompact subgroups of $$\mathbb H^p$$ [A. Felikson and P. Tumarkin, “A series of word-hyperbolic Coxeter groups”, arXiv:math/0507389] and that there are word-hyperbolic Coxeter groups that are not discrete cocompact subgroups of $$\text{Iso}(\mathbb H^p)$$ [G. Moussong, Colloq. Math. Soc. János Bolyai. 56, 535-546 (1992; Zbl 0795.53038)].
The authors prove the following result. Let $$X$$ be a simply connected even-gonal complex (roughly a two-dimensional complex with the faces being regular $$2m$$-polygons) such that the minimum number of sides of the polygons is $$\geq 4$$ and the minimum of the girth of the links of vertices is $$\geq 4$$. Let $$\Gamma$$ be a uniform lattice of $$X$$ that is virtually cubically special. Then $$\Gamma$$ admits a faithful representation in the automorphism group of the Davis complex of a Coxeter group $$W$$ and its image is virtually contained in $$W$$. If all the exponents in the Coxeter presentation of $$W$$ can be chosen to be finite, then we get a convex-cocompact representation if the minimum number of the sides of polygons is $$\geq 8$$. If the all the polygons in $$X$$ are $$2m$$-gons, then the target group can be chosen to be $$W(p,m)$$, the Coxeter group on $$p$$ generators with all the exponents in the Coxeter presentation of $$W$$ equal to $$m$$.
As an application of this result, let $$(W,S)$$ be a Coxeter system such that the weights in the Coxeter graph of the group are all equal to a number $$m\geq 4$$ and the girth of the underlying graph is $$\geq 4$$. Then the Davis complex satisfies the conditions of the above result and $$W$$ is cubically special [F. Haglund and D. T. Wise, Adv. Math. 224, No. 5, 1890-1903 (2010; Zbl 1195.53055)]. Also, it generalizes a result of Kapovich that even-gons of finite groups admit a convex-cocompact action on $$\mathbb H^p$$ [M. Kapovich, Geom. Topol. 9, 1915-1951 (2005; Zbl 1163.20029)].
As a special case, the authors prove that the groups $$W(p,m)$$ admit a convex-cocompact action on $$\mathbb H^{p-1}$$. Combining the last result with the main result, the authors show that a group $$\Gamma$$ that satisfies the conditions in the last part of the main result, then it admits a convex-cocompact action on $$\mathbb H^p$$.
There are two main remarks that should be made. For the Coxeter groups considered, the Davis complex is two dimensional. Also, the action on $$\mathbb H^p$$ is not necessarily by reflections.
The authors state the conjecture that every Coxeter group with finite exponents in the Coxeter presentation $$\geq 4$$ admits a convex-cocompact action on $$\mathbb H^p$$. This conjecture, together with Agol’s Theorem [I. Agol et al., Geom. Topol. 13, No. 2, 1043-1073 (2009; Zbl 1229.20037)] implies that any uniform lattice on an even-ton with minimum number of sides of polygons $$\geq 8$$ and minimum girth of links of vertices $$\geq 4$$ is convex-cocompact in $$\mathbb H^p$$. Also, it implies that every Coxeter group with exponents in the Coxeter presentation $$\geq 4$$ admits a convex-cocompact action on $$\mathbb H^p$$.
The second main open question that the authors state is the following. Let $$\Gamma$$ be a virtually special uniform lattice of a holonomy free $$2m$$-gonal complex $$X$$. Does the pair $$(X,\Gamma)$$ embed into $$(\Sigma(W(p,m)),W(p,m))$$? Here $$\Sigma(W(p,m))$$ is the Davis complex of $$W(p,m)$$. A positive answer to this question will show that there are word-hyperbolic Coxeter groups that admit a convex-cocompact action on $$\mathbb H^p$$ but they do not admit any faithful reflection representation in any $$\mathbb H^r$$.

MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 22E40 Discrete subgroups of Lie groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 51F15 Reflection groups, reflection geometries 57M20 Two-dimensional complexes (manifolds) (MSC2010) 20F65 Geometric group theory 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 20E26 Residual properties and generalizations; residually finite groups
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