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archives-ouvertes.fr/hal-00669827 Submitted on

14 Feb

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enseignement et de recherche fran?ais ou étrangers, des laboratoires publics ou privés. On some convex cocompact groups in real hyperbolic space Marc Desgroseilliers, Frédéric Haglund To cite this version: Marc Desgroseilliers, Frédéric Haglund. On some convex cocompact groups in real hyperbolic space. 2012. ?hal-00669827? ON SOME CONVEX COCOMPACT GROUPS IN REAL HYPERBOLIC SPACE MARC DESGROSEILLIERS AND FR? ED? ERIC HAGLUND Abstract. We generalize to a wider class of hyperbolic groups a construction by Misha Kapovich yielding convex cocompact representations into real hyperbolic space. Contents 1. Introduction.

2 1.1. Background on Coxeter groups and polygonal complexes.

2 1.2. Statements of the results.

4 1.3. Concluding remarks and questions.

6 1.4. Organization of the paper.

7 2. Geometry of even-gonal complexes.

7 2.1. Non-positive curvature conditions.

7 2.2. Square subdivision.

8 2.3. Straight and rami?ed hyperplanes of even-gonal complexes.

9 2.4. Intersection and osculation of hyperplanes.

12 2.5. Two-dimensional and two-spherical Coxeter groups.

14 3. Quasi-isometric embedding of CAT(0) large-gonal complexes.

16 4. Real hyperbolic convex cocompact Coxeter groups.

17 4.1. Re?ection subgroups in Hp.

18 4.2. Convex cocompactness when there is no pair of asymptotic faces (proof of Theorem 4.7).

21 4.3. Constructions using the Witt-Tits quadratic form.

25 4.4. Computation of signatures, proof of Proposition 4.21 and new examples.

29 5. Faithfull representation of large even-gonal groups into two-dimensional Coxeter groups.

31 5.1. Various complications for the action of a group on the set of hyperplanes.

31 5.2. The Coxeter group associated to an action without ambiguous intersection.

33 5.3. The special representation is faithfull and convex cocompact when the action is special.

36 5.4. Constructing a convex cocompact 2-spherical representation.

37 6. Wall-de?ned representations and virtual specialness.

39 References

40 Date: February 14, 2012.

2000 Mathematics Subject Classi?cation. 53C23, 20F55, 20F67, 20F65, 51F15, 57M20, 20E26, 22E40, 20H10. Key words and phrases. CAT(0) Polygonal Complexes, Coxeter Groups, Separable Subgroups, Convex Cocompact Representations, Real Hyperbolic Spaces.

1 ON SOME CONVEX COCOMPACT GROUPS IN REAL HYPERBOLIC SPACE

2 1. Introduction. In this paper we study discrete cocompact isometry groups of CAT(?1) polygonal complexes, and try to represent them faithfully as convex cocompact groups of Hp (for some large integer p). The case of a (large) polygon of ?nite groups was ?rst handled by Misha Kapovich in [19]. In fact the argument of Kapovich generalizes to a much wider class of groups. We now present more precisely our results. 1.1. Background on Coxeter groups and polygonal complexes. Recall a group Γ <

Isom(Hp) is convex-cocompact provided Γ acts properly on Hp and is cocompact on the convex hull of its limit set. Equivalently, any orbital map Γ → Hp, sending γ to γx, is a quasi-isometric embedding. More generally for any geodesic metric space Y we say that a representation ρ : Γ → Isom(Y ) is convex-cocompact whenever there exists a closed convex subspace Z ? Y which is invariant under Γ and the Γ-action on Z is proper and cocompact. (Recall that a subspace Z ? Y is convex if any geodesic segment with endpoints inside Z is entirely contained inside Z.) When the metric space Y is hyperbolic in the sense of Gromov and Γ is convex cocompact in Y , it follows that Γ is word-hyperbolic (see [12] or [11] for references on hyperbolic groups). In particular when a group is convex cocompact in Hp then it is a word-hyperbolic group. The converse problem is then : What kind of word-hyperbolic groups are convex cocompact in Hp ? The above question is extremely general and we will focus on a very particular class of groups. Any convex cocompact group of Hp inherits the Haagerup property of Isom(Hp) (see [9]), so we must investigate the class of word-hyperbolic groups with this property. Recall ?rst that Coxeter groups correspond to presentations of the form W = s1,sr | (si)2 = 1, (sisj)mij =