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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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