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Anti-de Sitter strictly GHC-regular groups which are not lattices
Gye-Seon Lee, Ludovic Marquis
To cite this version:
Gye-Seon Lee, Ludovic Marquis. Anti-de Sitter strictly GHC-regular groups which are not lattices.
Transactions of the American Mathematical Society, American Mathematical Society, 2019, 372 (1), pp.153-186. �10.1090/tran/7530�. �hal-01579212v2�
WHICH ARE NOT LATTICES
GYE-SEON LEE AND LUDOVIC MARQUIS
ABSTRACT. For d =4,5,6,7,8, we exhibit examples of AdSd,1 strictly GHC-regular groups which are not quasi-isometric to the hyperbolic spaceHd, nor to any symmetric space. This provides a negative answer to Question 5.2 in [9A12] and disproves Conjecture 8.11 of Barbot–
Mérigot [BM12].
We construct those examples using the Tits representation of well-chosen Coxeter groups.
On the way, we give an alternative proof of Moussong’s hyperbolicity criterion [Mou88] for Coxeter groups built on Danciger–Guéritaud–Kassel [DGK17] and find examples of Coxeter groupsWsuch that the space of strictly GHC-regular representations ofW into POd,2(R)up to conjugation is disconnected.
CONTENTS
1. Introduction 1
2. Coxeter groups and Davis complexes 6
3. Proof of Theorem B 8
4. Tits representations and Convex cocompactness 11
5. Topological actions of reflection groups 16
6. Proof of Theorems C and D 18
7. An alternative proof of Moussong’s criterion 20
8. Proof of Theorems E and F 21
Appendix A. The tables of the Coxeter groups 25
Appendix B. The spherical, affine and Lannér diagrams 29
References 30
1. INTRODUCTION
LetRd,ebe the vector spaceRd+eendowed with a non-degenerate symmetric bilinear form
⟨⋅,⋅⟩d,eof signature(d,e), and let qbe the associated quadratic form. A coordinate represen- tation of q with respect to some basis ofRd,e is:
q(x)=x12+ ⋯ +x2d−x2d+1− ⋯ −x2d+e
The(d+e−1)-dimensionalpseudohyperbolic spaceHd,e−1is the quotient of the hyperquadric Q ={x∈Rd,e∣q(x)=−1}
2010Mathematics Subject Classification. 20F55, 20F65, 20H10, 22E40, 51F15, 53C50, 57M50, 57S30.
Key words and phrases. Anti-de Sitter spaces, Anosov representations, Quasi-Fuchsian groups, Coxeter groups, Discrete subgroups of Lie groups.
1
by the involution x↦−x and is endowed with the semi-Riemannian metric induced by the quadratic form q. It is a complete semi-Riemannian manifold of constant sectional curva- ture −1. The isometry group of Hd,e−1 is POd,e(R), and so Hd,e−1 may be identified with POd,e(R)/Od,e−1(R). It will be more convenient for us to work with the projective model of Hd,e−1, namely:
Hd,e−1={[x]∈P(Rd,e) ∣q(x)<0}
In particular, for e=2, the pseudohyperbolic space Hd,e−1 is called the(d+1)-dimensional anti-de Sitter spaceAdSd,1.
Let Γ be a finitely presented group. A representation ρ∶Γ→POd,2(R) is strictly GHC- regular if it is discrete, faithful and preserves anacausal (i.e. negative) subset1 Λρ in the boundary Eindof AdSd,1 such thatΛρ is homeomorphic to a(d−1)-dimensional sphere and the action of Γ on the convex hull2 ofΛρ in AdSd,1 is cocompact. An AdSd,1 strictly GHC- regular group (simply strictly GHC-regular group) is a subgroup of POd,2(R) which is the image of a strictly GHC-regular representation.
The first examples of strictly GHC-regular representations areFuchsian representations.
Namely, consider any uniform lattice Γ of POd,1(R), which is isomorphic to O+d,1(R), and let ρ0∶Γ→POd,2(R) be the restriction to Γ of the natural inclusion POd,1(R)≃O+d,1(R)↪ POd,2(R) (hence Od,1(R) corresponds to the stabilizer of a point p0∈AdSd,1 in POd,2(R)).
The representationρ0is strictly GHC-regular and the convex hull ofΛρ0 is a copy of the hy- perbolicd-spaceHd inside AdSd,1. Moreover, any small deformationρt∶Γ→POd,2(R)of the representationρ0is strictly GHC-regular (see Barbot–Mérigot [BM12]). Even more, Barbot [Bar15] proved that every representation ρ∶Γ→POd,2(R) in the connected component of Hom(Γ,POd,2(R))containingρ0 is strictly GHC-regular.
We will not use the following point of view, but we remark that strictly GHC-regular rep- resentations of torsion-free groups are exactly the holonomies of GHMC AdS manifolds (see Mess [Mes07] and Barbot [Bar08]). We quickly recall this terminology: an AdS manifoldM isglobally hyperbolic spatially compact(GHC) if it contains a compactCauchy hypersurface, i.e. a space-like hypersurface S such that all inextendible time-like lines intersect S ex- actly once. It isglobally hyperbolic maximal compact(GHMC) if in addition every isometric embedding of Minto another GHC AdS manifold of the same dimension is an isometry.
Recently, Barbot–Mérigot [BM12] found an alternative description of AdSd,1strictly GHC- regular groups. In the case when Γ is isomorphic to the fundamental group of a closed, negatively curved Riemanniand-manifold, a representationρ∶Γ→POd,2(R)is strictly GHC- regular if and only if it is P1d,2-Anosov, where P1d,2 is the stabilizer of an isotropic line l0∈
1We denote by∂Hd,e−1the boundary ofHd,e−1inP(Rd,e). In the casee=2, the boundary∂Hd,e−1=∂AdSd,1 is called theEinstein universe, denoted Eind. A subsetΛof∂Hd,e−1isnegativeif it lifts to a cone ofRd,e∖{0}on which all inner products⟨⋅,⋅⟩d,eof noncollinear points are negative. By aconewe mean a subset ofRd,ewhich is invariant under multiplication by positive scalars.
2An element of PGL(Rd,e)isproximalif it admits a unique attracting fixed point inP(Rd,e). The (proximal) limit setof a discrete subgroupΓ0of PGL(Rd,e)is the closure of the set of attracting fixed points of elements of Γ0which are proximal. In fact, theρ(Γ)-invariant subsetΛρmust be the limit set ofρ(Γ), and the convex hull ofΛρ is well-defined becauseΛρis acausal.
Rd,2in POd,2(R)(see Labourie [Lab06] or Guichard–Wienhard [GW12] for more background on Anosov representations) and the limit set of ρ(Γ) is a topological (d−1)-sphere in the boundary Eind of AdSd,1.
Very recently, Danciger–Guéritaud–Kassel [DGK17] introduced a notion ofconvex cocom- pactness in Hd,e−1. A discrete subgroup Γ of POd,e(R) is Hd,e−1-convex cocompact if it acts properly discontinuously and cocompactly on some properly convex3closed subsetCofHd,e−1
with non-empty interior4so thatC∖Cdoes not contain any non-trivial projective segment. A main result of [DGK17] is as follows: an irreducible discrete subgroupΓof POd,e(R)isHd,e−1- convex cocompact if and only if Γis Gromov-hyperbolic, the natural inclusion Γ↪POd,e(R) isP1d,e-Anosov, and the limit setΛΓis negative.
In particular, for any discrete faithful representation ρ∶Γ→POd,2(R), we have thatρ is strictly GHC-regular if and only ifρ(Γ)is AdSd,1-convex cocompact (which implies thatΓis Gromov-hyperbolic) and the Gromov boundary ofΓis homeomorphic to a(d−1)-dimensional sphere.
In [9A12], the authors asked the following:
Question A(Question 5.2 of [9A12]). Assume thatρ∶Γ→POd,2(R)is a strictly GHC-regular representation. IsΓisomorphic to a uniform lattice ofPOd,1(R)?
The positive answer to this question was conjectured by Barbot and Mérigot (see Conjec- ture 8.11 of [BM12]). In this paper, we give a negative answer to QuestionA.
Theorem A. For d=4,5,6,7,8, there exists anAdSd,1 strictly GHC-regular group which is not isomorphic (even not quasi-isometric) to a uniform lattice in a semi-simple Lie group.
The counterexamples are the Tits representations of well-chosen Coxeter groups.
Remark 1. For d=2, there exist no examples as in Theorem A by work of Tukia [Tuk88], Gabai [Gab92] and Casson–Jungreis [CJ94]: ifΓis a Gromov-hyperbolic group whose bound- ary is homeomorphic toS1, thenΓadmits a geometric action onH2, i.e. there exists a prop- erly discontinuous, cocompact and isometric action of Γ on H2 (for a proof see Chapter 23 of Dru¸tu–Kapovich [DK17]). In the case d=3, recall that Cannon’s conjecture claims that ifΓ is a Gromov-hyperbolic group whose boundary is homeomorphic toS2, then Γadmits a geometric action onH3. If Cannon’s conjecture is true, then any AdS3,1strictly GHC-regular group is isomorphic to a uniform lattice of PO3,1(R). For Coxeter groups, Cannon’s conjecture is in fact known to be true (see Bourdon–Kleiner [BK13]).
An important invariant of a Coxeter group W is the signature sW of W. It means the signature of the Cosine matrix of W (see Section 2 for the basic terminology of Coxeter groups). Recall that the signature of a symmetric matrix A is the triple (p,q,r) of the positive, negative and zero indices of inertia of A.
The outline of the proof of Theorem A is as follows. First, observe that if the signature of a Coxeter group W is (d,2,0), then the Tits representation ρ of W is a discrete faithful
3A subsetCofP(Rd,e)isproperly convexif its closureCis contained and convex in some affine chart.
4In the caseΓis irreducible, any non-emptyΓ-invariant convex subset ofP(Rd,e)has non-empty interior, and so “Cwith non-empty interior” may be replaced by “Cnon-empty”.
representation of W into POd,2(R). Second, we prove Theorem 4.6 implying that if in ad- dition W is Gromov-hyperbolic and the Coxeter diagram ofW has no edge of label ∞, then the representation ρ∶W→POd,2(R)is AdSd,1-convex cocompact. Remark that Theorem 4.6 is a generalization of Theorem 8.2 of [DGK17]. Third, in order to prove that ρ is strictly GHC-regular, we only need to check that the Gromov boundary ofW is a(d−1)-dimensional sphere. Coxeter groups satisfying all these properties exist:
Theorem B. Every Coxeter groupW in Tables5,6,7,8,9satisfies the following:
● the groupW is Gromov-hyperbolic;
● the Gromov boundary∂W of W is homeomorphic to a(d−1)-sphere;
● the signaturesW ofW is(d,2,0). Here, d denotes the rank of W minus2.
TheoremAis then a consequence of the following:
Theorem C. Let W be a Coxeter group in Tables 5,6,7, 8,9. Then the Tits representation ρ∶W→POd,2(R)is strictly GHC-regular and the groupW is not quasi-isometric to a uniform lattice of a semi-simple Lie group.
Remark 2. With Selberg’s lemma, TheoremCalso allows us to obtain a finite index torsion- free subgroup Γ of W so that AdSd,1/ρ(Γ) is a GHMC AdS manifold and Γ is not quasi- isometric to a uniform lattice of a semi-simple Lie group.
Applying the same method as for TheoremBand TheoremC, we can prove:
Theorem D. LetW be a Coxeter group in Table4. Then the following hold:
● the groupW is Gromov-hyperbolic;
● the Gromov boundary ofW is homeomorphic to a3-dimensional sphere;
● the signature ofW is(5,1,0);
● the Tits representationρ∶W→PO5,1(R)is quasi-Fuchsian;
● the groupW is not quasi-isometric to a uniform lattice of a semi-simple Lie group.
Here, a representation ρ∶Γ→POd+1,1(R) is called Hd+1-quasi-Fuchsian (simply quasi- Fuchsian) if it is discrete and faithful, ρ(Γ) is Hd+1-convex cocompact and the Gromov boundary ofΓis homeomorphic to a(d−1)-dimensional sphere.
Remark 3. In [Ess96], Esselmann proved that each Coxeter group W in Table 3 admits a geometric action on the hyperbolic space H4 such that a fundamental domain for the W- action on H4 is a convex polytope whose combinatorial type is the product of two triangles.
Esselmann’s examples in Table 3 are adapted to obtain the Coxeter groups in Tables 4, 5, 6,7,8,9. In particular, the nerve of each Coxeter group in Tables 3, 4, 5 is isomorphic to the boundary complex of the dual polytope of the product of two triangles. We are currently classifying Coxeter groups with nerve isomorphic to the boundary complex of the dual poly- tope of the product of two simplices of dimension ⩾2, and in the meanwhile, we found the Coxeter groups in Tables4,5,6,7,8,9.
We denote byχ(Γ,POd,2(R))the POd,2(R)-character variety of Γ and byχs(Γ,POd,2(R)) the space of strictly GHC-regular characters inχ(Γ,POd,2(R)). In [Bar15], Barbot asked the following:
Question B(Discussion after Theorem 1.5 of [Bar15]). Assume thatΓis a lattice ofPOd,1(R). Is the spaceχs(Γ,POd,2(R))connected? In other words, does every strictly GHC-regular char- acter deform to a Fuchsian character?
We can extend QuestionBto the case whenΓis Gromov-hyperbolic:
Question C. LetΓbe a Gromov-hyperbolic group. Assume thatχs(Γ,POd,2(R))is non-empty.
Is the spaceχs(Γ,POd,2(R))connected?
The following theorem provides a negative answer to QuestionC.
Theorem E. LetW be a Coxeter group in Tables12or13, and letd be the rank of W minus 3.
Then there exist two characters[ρ1],[ρ2]∈χs(W,POd,2(R))such that there isnocontinuous path from[ρ1]to[ρ2]inχ(W,POd,2(R)), let alone in the subspaceχs(W,POd,2(R)).
Similarly, we denote by χ(Γ,POd+1,1(R)) the POd+1,1(R)-character variety of Γ and by χq(Γ,POd+1,1(R))the space of quasi-Fuchsian characters inχ(Γ,POd+1,1(R)). It follows from work of Barbot–Mérigot [BM12] and Barbot [Bar15] that χs(Γ,POd,2(R))is a union of con- nected components of χ(Γ,POd,2(R)). However, χq(Γ,POd+1,1(R))is open but not closed in χ(Γ,POd+1,1(R))in general. Nevertheless, we are able to prove:
Theorem F. LetW be a Coxeter group in Tables14or15, and letd be the rank of W minus 3.
Then there exist two characters[ρ1],[ρ2]∈χq(W,POd+1,1(R))such that there isnocontinuous path from[ρ1]to[ρ2]inχ(W,POd+1,1(R)).
However, QuestionBis still an open problem.
Remark 4. Each Coxeter groupW in Tables12,13,14,15is not quasi-isometric toHd even though it is a Gromov-hyperbolic group whose boundary is homeomorphic toSd−1.
Remark5. In [Tum07], Tumarkin proved that each Coxeter groupW in Table10(resp. Table 11) admits a geometric action on the hyperbolic spaceH4(resp. H6). Tumarkin’s examples in Tables10,11are adapted to obtain the Coxeter groups in Tables12,13,14,15. In particular, the nerves of Coxeter groups in Table 10(resp. Table 11) are isomorphic to the nerves of Coxeter groups in Tables12,14(resp. Tables13,15).
In Section 7, we also recover Moussong’s hyperbolicity criterion for Coxeter groups (see Theorem3.4) built on [DGK17].
Acknowledgments. We are thankful for helpful conversations with Ryan Greene, Olivier Guichard, Fanny Kassel and Anna Wienhard. We thank Daniel Monclair for introducing Question 5.2 in [9A12] to the second author a long time ago. Finally, we would like to thank the referee for carefully reading this paper and suggesting several improvements.
G.-S. Lee was supported by the European Research Council under ERC-Consolidator Grant 614733 and by DFG grant LE 3901/1-1 within the Priority Programme SPP 2026 “Geome- try at Infinity”, and he acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures And Representation vari- eties” (the GEAR Network).
2. COXETER GROUPS ANDDAVIS COMPLEXES
Apre-Coxeter systemis a pair(W,S)of a groupW and a setS of elements ofW of order 2 which generates W. ACoxeter matrixM=(Mst)s,t∈S on a setS is a symmetric matrix with the entriesMst∈{1,2,...,m,...,∞}such that the diagonal entriesMss=1 and othersMst≠1.
For any pre-Coxeter system (W,S), we have a Coxeter matrix MW on S such that each entry (MW)st of MW is the order of st. To any Coxeter matrix M=(Mst)s,t∈S is associated a presentation for a group ˆWM: the set of generators for ˆWM is S and the set of relations is {(st)Mst =1∣ Mst≠∞}. A pre-Coxeter system (W,S) is called a Coxeter system if the surjective homomorphism ˆWMW →W, defined bys↦s, is an isomorphism. If this is the case, thenW is a Coxeter groupand S is afundamental set of generators. We denote the Coxeter group W also by WS orWS,M to indicate the fundamental set of generators or the Coxeter matrix of(W,S). TherankofWS is the cardinality♯SofS.
The Coxeter diagram of WS,M is a labeled graph GW such that (i) the set of nodes (i.e.
vertices) of GW is the setS; (ii) two nodes s,t∈S are connected by an edge st of GW if and only if Mst∈{3,...,m,...,∞}; (iii) the label of the edge stis Mst. It is customary to omit the label of the edgestifMst=3. A Coxeter groupW isirreducibleif the Coxeter diagramGW is connected. TheCosine matrixofWS,M is anS×S symmetric matrixCW whose entries are:
(CW)st=−2cos(Mπ
st) for everys,t∈S
An irreducible Coxeter groupWS isspherical(resp.affine, resp. Lannér) if for every s∈S, the (s,s)-minor of CW is positive definite and the determinant det(CW) of CW is positive (resp. zero, resp. negative). Remark that every irreducible Coxeter group W is spherical, affine or large, i.e. there is a surjective homomorphism of a finite index subgroup ofW onto a free group of rank⩾2 (see Margulis–Vinberg [MV00]). For a Coxeter groupW (not neces- sarily irreducible), each connected component of the Coxeter diagram GW corresponds to a Coxeter group, called acomponentof the Coxeter groupW. A Coxeter groupW isspherical (resp. affine) if each component ofW is spherical (resp. affine). We will often refer to Ap- pendix B for the list of all the irreducible spherical, irreducible affine and Lannér Coxeter diagrams.
Let(W,S)be a Coxeter system. For eachT⊂S, the subgroupWT ofW generated byT is called aspecial subgroup of W. It is well-known that(WT,T)is a Coxeter system. A subset T⊂S is said to be “something” if the Coxeter group WT is “something”. For example, the word “something” can be replaced by “spherical”, “affine”, “Lannér” and so on. Two subsets T,U⊂SareorthogonalifMtu=2 for everyt∈Tand everyu∈U. This relationship is denoted T⊥U.
Aposetis a partially ordered set. ThenerveNWofW is the poset of all non-empty spherical subsets of S partially ordered by inclusion. We remark that the nerve NW is an abstract simplicial complex. Recall that an abstract simplicial complexS is a pair (V,E) of a setV, which we call the vertex set ofS, and a collection E of (non-empty) finite subsets of V such that (i) for each v∈V,{v}∈E; (ii) if T∈E and if ∅≠U⊂T, thenU∈E. An element ofE is a simplexofS, and thedimensionof a simplexT is♯T−1. The vertex set of the nerve NW of
a Coxeter groupWS is the set of generators S and a non-empty subsetT ⊂S is a simplex of NW if and only ifT is spherical.
Theoppositeposet to a posetP is the posetPopwith the same underlying set and with the reversed order relation. Given a convex polytopeP, letF(∂P)denote the set of all non-empty faces of the boundary∂P ofPpartially ordered by inclusion. A polytopePissimplicial(resp.
simple) ifF(∂P)(resp. F(∂P)op) is a simplicial complex.
Remark 2.1 (The nerve of a geometric reflection group, following Example 7.1.4 of [Dav08]).
LetPbe a simple convex polytope inX=RdorHdwith dihedral angles integral submultiples of π. To the polytope P is associated a Coxeter matrix M=(Mst)s,t∈S on a set S: (i) the set S consists of all facets5of P; (ii) for each pair of distinct facets s,tof P, if s,tare adjacent6 and if the dihedral angle between s and t is mπst, then we set Mst =mst, and otherwise Mst=∞. Let WP be the reflection group generated by the set of reflections σs across the facets sof P. The homomorphism σ∶WS,M→WP defined byσ(s)=σsis an isomorphism by Poincaré’s polyhedron theorem, and so the Coxeter groupWS,M obtained in this way is called a geometric reflection group. Moreover, the nerve NW of the Coxeter group WS,M identifies with the simplicial complex F(∂P∗)= F(∂P)op, where P∗ is the dual polytope of P, hence NW is PL homeomorphic to a(d−1)-dimensional sphere.
Inspired by the previous remark, we make the following:
Definition 2.2. A Coxeter groupWis anabstract geometric reflection group of dimensiondif the geometric realization of the nerveNWofW is PL homeomorphic to a(d−1)-dimensional sphere.
Aspherical cosetof a Coxeter groupW is a coset of a spherical special subgroup ofW. We denote byWS the set of all spherical cosets ofW, i.e.
WS = ⊔
T∈NW∪{∅}
W/WT
It is partially ordered by inclusion. To any posetP is associated an abstract simplicial com- plex Flag(P)consisting of all non-empty, finite, totally ordered subsets ofP. The geometric realization of Flag(WS)is called theDavis complexΣW ofW.
Example 2.3. IfW is generated by the reflections in the sides of a square tile, thenNW is a cyclic graph of length 4, and ΣW is the complex obtained by subdividing each tile of the infinite square grid into 8 isosceles right triangles, meeting at the center.
Theorem 2.4(Moussong [Mou88], Stone [Sto76], Davis–Januszkiewicz [DJ91]). If a Coxeter groupW is an abstract geometric reflection group of dimension d, then the following hold:
● the Davis complexΣW ofW admits a natural CAT(0)-metric;
● the cell complexΣW is homeomorphic toRd;
● the CAT(0) boundary ofΣW is homeomorphic toSd−1.
5Afacetof a convex polytopeP is a face of codimension 1.
6Two facetss,tofPareadjacentif the intersection ofsandtis a face of codimension 2.
Sketch of the proof. For the reader’s convenience, we outline the important steps of the proof as in the book [Dav08] and refer to the relevant theorems in [Dav08].
Moussong showed that the Davis complex ΣW equipped with its natural piecewise Eu- clidean structure is CAT(0) (see Theorem 12.3.3), and soΣW is contractible (see also Theorem 8.2.13). Since (the geometric realization of) the nerve ofW is PL homeomorphic to a(d−1)- dimensional sphere, ΣW is a PL d-manifold (see Theorem 10.6.1). Therefore, the cell com- plexΣW is a simply connected, nonpositively curved, piecewise Euclidean, PL manifold. By Stone’s theorem [Sto76],ΣWis homeomorphic toRdand moreover by Davis–Januszkiewicz’s theorem, the boundary ofΣW is a(d−1)-sphere (see Theorem I.8.4).
3. PROOF OF THEOREM B
Proposition 3.1. Let(W,S)be a Coxeter system with Coxeter diagramGW in Tables4,5,6, 7,8,9and letd be the rank of WS minus2. ThenWS is an abstract geometric reflection group of dimension d.
Proof. Let S1 be the set of white nodes of GW and S2 the set of black nodes of GW so that S=S1∪S2. Note that the colors are just for reference and have no influence on the definition of the group. Using the tables in AppendixB, check that:
● the special subgroupsWS1 andWS2 are Lannér;
● the special subgroupWT withT⊂Sis spherical if and only ifS1⊄T andS2⊄T. It therefore follows that the nerveNW ofW is isomorphic to the nerve of the Coxeter group WS1×WS2 (as a poset). In other words, the nerveNW is isomorphic to the join ofN1and N2, where Ni denotes the nerve ofWSi for eachi∈{1,2}.
Moreover, sinceWSi is Lannér, its nerveNi is isomorphic to F(∂∆i)where ∆i is the sim- plex of dimension ni=♯Si−1, and in particular it is a simplicial complex whose geometric realization is a (ni−1)-dimensional sphere. Finally, since the join of the n-sphere and the m-sphere is the (n+m+1)−sphere, the geometric realization of the nerveNW is a(d−1)-
dimensional sphere (recall thatd=n1+n2).
Remark 3.2. The proof of Proposition 3.1 is essentially the same as in Example 12.6.8 of [Dav08] (see also Section 18 of Moussong [Mou88]).
Corollary 3.3. In the setting of Proposition 3.1, the Coxeter groupW is Gromov-hyperbolic and the Gromov boundary ofW is a(d−1)-dimensional sphere.
Proof. First, we know from Theorem2.4 that the Davis complexΣW ofW is a CAT(0) space homeomorphic to Rd and that the CAT(0) boundary of W is homeomorphic to Sd−1. Sec- ond, it is easy to verify that the Coxeter group W is Gromov-hyperbolic using Moussong’s hyperbolicity criterion (see Theorem 3.4). Finally, since the action of W on ΣW is proper and cocompact, we have that W is quasi-isometric to ΣW and the Gromov boundary of W
identifies to the CAT(0) boundary ofΣW.
Theorem 3.4 (Moussong’s hyperbolicity criterion [Mou88]). Assume that WS is a Coxeter group. Then the groupWS is Gromov-hyperbolic if and only ifS does not contain any affine subset of rank⩾3nor two orthogonal non-spherical subsets.
In order to complete TheoremB, it remains to show the following:
Proposition 3.5. In the setting of Proposition3.1, the signature ofW is(d,2,0).
Proof. As seen in the proof of Proposition 3.1, if S1 (resp. S2) denotes the set of all white (resp. black) nodes ofGW, then the special subgroupsWS1 andWS2 are Lannér, and for each (s,t)∈S1×S2, the special subgroupWS∖{s,t}is spherical. Thus the Coxeter groupW contains a spherical special subgroup of rank d and a Lannér special subgroup. For example, if the Coxeter diagram GW ofW is the diagram (A) in Figure1, then the diagram (B) corresponds to a spherical Coxeter groupB3×I2(p)(see Table16) and the diagram (C) corresponds to a Lannér Coxeter group in Table18.
WS 2
1
3
4 5 6 7
4 p⩾7
(A)
WT 2
1
3
6 7
4 p⩾7
(B)
WS1 2
1
3 4 4
(C)
FIGURE 1. A Coxeter groupWS in Table6and two special subgroupsWT and WS1 ofWS withS={1,2,3,4}∪{5,6,7},T=S∖{4,5}andS1={1,2,3,4}
Since the signature of any spherical (resp. Lannér) Coxeter group of rank r is (r,0,0) (resp. (r−1,1,0)), the signature sW of W can only be (d,2,0), (d,1,1) or(d+1,1,0), and so sW is determined by the sign of the determinant det(CW) of CW. More precisely, the signaturesW is(d,2,0),(d,1,1)and(d+1,1,0)if det(CW)>0,=0 and<0, respectively.
Now we check carefully the sign of det(CW)in the decreasing order of dimensiond.
In the case when d=8 or 7 (see Tables9or8), a simple computation shows that det(CW) is positive. For example, if the Coxeter diagramGW ofW is:
1 5 2 3 4 5 6 7 8 9 5 10
then the Cosine matrixCW is:
⎛⎜⎜
⎜⎜⎜⎜
⎜⎜⎜⎜
⎜⎜⎜⎜
⎜⎜⎜⎜
⎝
2 −c5 0 0 0 0 0 0 0 0
−c5 2 −1 0 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0 0
0 0 −1 2 −1 0 0 0 0 0
0 0 0 −1 2 −1 0 0 0 0
0 0 0 0 −1 2 −1 0 0 0
0 0 0 0 0 −1 2 −1 0 0
0 0 0 0 0 0 −1 2 −1 0
0 0 0 0 0 0 0 −1 2 −c5
0 0 0 0 0 0 0 0 −c5 2
⎞⎟⎟
⎟⎟⎟⎟
⎟⎟⎟⎟
⎟⎟⎟⎟
⎟⎟⎟⎟
⎠ where c5=2cos(π5), and the determinant ofCW is 1
2(25−11√
5)≈0.201626.
In the case whend=6,5,4 (see Tables7,6,5), there is a one-parameter family(Wp)p or a two-parameter family(Wp,q)p,q of Coxeter groups for each item of the Tables. The following Lemma3.6shows that the determinant of the Cosine matrix increases when the parameter increases. Here, the (partial) order on the set of two parameters(p,q)is given by:
(p′,q′)⩾(p′′,q′′) ⇔ p′⩾p′′ and q′⩾q′′
By an easy but long computation, we have that det(CWp)(resp. det(CWp,q)) is positive for every minimal element p(resp. (p,q)) in the set of parameters. For example, if the Coxeter diagramGWp ofWp is:
5
5 p⩾11
then the determinant ofCWp is equal to−4(3+√
5)+8(1+√
5)cos(2πp )≈0.834557 for p=11.
Remark that if p=10, then det(CWp)=0 and so Wp is the geometric reflection group of a compact hyperbolic 4-polytope with 6 facets (see Esselmann [Ess96]).
Lemma 3.6. If (Wp)p is a one-parameter family of Coxeter groups in Tables 5, 6, 7, then (det(CWp))p is an increasing sequence and its limit is a positive number. Similarly, if (Wp,q)p,q is a two-parameter family of Coxeter groups in Table 5, then (det(CWp,q))p,q is also an increasing sequence and its limit is a positive number as p and q go to infinity.
Proof. We denoteWp orWp,q by WS,M (simplyWS). LetS1 (resp. S2) be the set of all white (resp. black) nodes of the Coxeter diagramGW
S.
Assume first that there is a unique edge st between S1andS2 inGW
S such thats∈S1 and t∈S2.7 By Proposition 13 of Vinberg [Vin84], we have:
det(CW
S) det(CW
S∖{s,t})= det(CW
S1) det(CW
S1∖{s})⋅ det(CW
S2) det(CW
S2∖{t})−4cos2(Mπ
st)
which is equivalent to:
det(CWS)=det(CWS
1)det(CWS
2)−4det(CW
S1∖{s})det(CW
S2∖{t})cos2(Mπ
st)
by the following observation: for any two orthogonal subsetsT⊥UofS(such asS1∖{s}and S2∖{t}here),
det(CWT
∪U)=det(CWT)det(CWU).
Moreover, it is easy to see the following:
● the sequence p↦−det(CW
S2)is positive and increasing to a positive number;
● the sequence p↦det(CW
S2∖{t})is positive and decreasing to zero;
● in the case WS =Wp, the numbers −det(CW
S1) and det(CW
S1∖{s}) are positive; and in the case WS =Wp,q, the sequence q↦−det(CW
S1) is positive and increasing to a positive number, and the sequence q↦det(CW
S1∖{s}) is positive and decreasing to zero.
Hence, the sequence p↦det(CW
S) (or (p,q)↦det(CW
S)) is increasing and converges to a positive number.
7In practice, it means thatGW
S does not belong to the last row of Table5.
Now assume that there are two edgesrt and st between S1andS2inGWS such thatr,s∈S1 and t∈S2.8By Proposition 12 of Vinberg [Vin84], we have:
det(CW
S) det(CW
S∖{t})−2=⎛
⎜⎝
det(CW
S1∪{t}) det(CW
S1) −2⎞
⎟⎠+⎛
⎜⎝
det(CW
S2) det(CW
S2∖{t})−2⎞
⎟⎠
which is equivalent to det(CW
S)=det(CW
S1)det(CW
S2)+det(CW
S1)det(CW
S2∖{t})⎛
⎜⎝
det(CW
S1∪{t}) det(CW
S1) −2⎞
⎟⎠
by the above observation. Moreover, it is easy to see (cf. Table 2 of Esselmann [Ess96]) that (1) det(CW
S1∪{t}) det(CW
S1) =⎧⎪⎪
⎨⎪⎪⎩
5+2√ 5+3√
2+√
2 10 ifGW
S1∪{t} is the left diagram in Figure2;
3+√
5 ifGW
S1∪{t} is the right diagram in Figure2.
t 5 4
t 5 5
FIGURE 2. Two possible Coxeter diagrams ofWS1∪{t} Hence, the sequence p↦det(CW
S) is increasing and converges to a positive number, be-
cause the both values in Equation (1) are greater than 2.
Similarly, by an easy computation, we can prove:
Proposition 3.7. Let(W,S)be a Coxeter system with Coxeter diagramGW in Table4. Then the signature ofWS is(5,1,0).
4. TITS REPRESENTATIONS AND CONVEX COCOMPACTNESS
4.1. Cartan matrices and Coxeter groups. An n×n matrix A=(Ai j)i,j=1,...,n is called a Cartan matrixif the following hold:
● for all i=1,...,n,Aii=2, and for all i≠j, Ai j⩽0, and Ai j=0⇔Aji=0;
● for all i≠j, Ai jAji⩾4 orAi jAji=4cos2(mπi j)with some mi j∈N∖{0,1}.
A Cartan matrix Aisreducibleif (after a simultaneous permutation of rows and columns) Ais the direct sum of smaller matrices. Otherwise, Aisirreducible. It is obvious that every Cartan matrix A is the direct sum of irreducible matrices A1⊕⋯⊕Ak. Each irreducible matrix Ai, i=1,...,k, is called a component of A. If x=(x1,...,xn)and y=(y1,...,yn)∈Rn, we write x>yifxi>yi for everyi, andx⩾yifxi⩾yi for everyi.
Proposition 4.1(Theorem 3 of Vinberg [Vin71]). If A is an irreducible Cartan matrix of size n×n, then exactly one of the following holds:
(P) The matrix A is nonsingular; for every vector x∈Rn, if Ax⩾0, thenx>0or x=0.
8In other words,GW
Sbelongs to the last row of Table5.
(Z) The rank of A is n−1; there exists a vector y∈Rnsuch that y>0and A y=0; for every vector x∈Rn, if Ax⩾0, thenAx=0.
(N) There exists a vector y∈Rnsuch that y>0and A y<0; for every vector x∈Rn, if Ax⩾0 and x⩾0, thenx=0.
We say that A is of positive, zeroornegative typeif(P),(Z)or(N)holds, respectively.
A Cartan matrix A is of positive (resp. zero) type if every component of A is of positive (resp. zero) type.
Corollary 4.2. Let A be a Cartan matrix (not necessarily irreducible). If there exists a vector x>0such that Ax=0, then A is of zero type.
A Cartan matrix A=(Ast)s,t∈S iscompatiblewith a Coxeter groupWS,M if for every s≠t∈ S, AstAts=4cos2(Mπst) if Mst<∞, and AstAts⩾4 otherwise. For any Coxeter groupW, the Cosine matrixCW ofW is in fact compatible withW.
Proposition 4.3(Propositions 22 and 23 of Vinberg [Vin71]). Assume that a Cartan matrix A is compatible with a Coxeter group W. Then the following hold:
(1) the matrixA is of positive type if and only if W is spherical;
(2) ifA is of zero type, then W is affine;
(3) conversely, ifA=CW andW is affine, then A is of zero type.
4.2. Coxeter polytopes. Let V be a vector space over R, and let S(V) be the projective sphere, i.e. the space of half-lines in V. We denote by SL±(V)the group of automorphisms of S(V), i.e. SL±(V)={X ∈GL(V) ∣det(X)=±1}. The projective sphere S(V) and the group SL±(V) are double covers of the projective space P(V) and the group of projective transformations PGL(V), respectively.
We denote bySthe natural projection ofV∖{0}onto S(V). For any subsetU ofV,S(U) denotes S(U∖{0})for the simplicity of the notation. A subset C of S(V) isconvex if there exists a convex coneU of V such thatC =S(U), andC is properly convex if in addition its closure C does not contain a pair of antipodal points. In other words,C is properly convex if and only ifC is contained and convex in some affine chart. Note that ifCis a properly convex subset of S(V), then the subgroup SL±(C) of SL±(V)preserving C is naturally isomorphic to a subgroup of PGL(V).
Aprojective polytopeis a properly convex subset P of S(V)such that P has a non-empty interior and P =⋂ni=1S({x∈V ∣αi(x)⩽0}), where αi, i=1,...,n, are linear forms on V. We always assume that the set of linear forms is minimal, i.e. none of the half spaces S({x∈V ∣αi(x)⩽0}) contain the intersection of all the others. A projective reflection is an element of SL±(V) of order 2 which is the identity on a hyperplane. Every projective reflectionσcan be written as:
σ=Id−α⊗b
whereαis a linear form onV and bis a vector ofV such thatα(b)=2.
Apre-Coxeter polytopeis a pair of a projective polytopeP ofS(V): P=⋂
s∈S
S({x∈V∣αs(x)⩽0})
and a set of projective reflections(σs=Id−αs⊗bs)s∈S withαs(bs)=2. A pre-Coxeter polytope is called aCoxeter polytopeif
γP○∩P○=∅ for everyγ∈ΓP/{Id}
where P○ is the interior of P andΓP is the subgroup of SL±(V)generated by the reflections (σs)s∈S. If this is the case, then ΓP is a projective Coxeter group generated by reflections (σs)s∈S. In [Vin71], Vinberg showed that a pre-Coxeter polytope(P,(σs=Id−αs⊗bs)s∈S)is a Coxeter polytope if and only if the matrix A=(αs(bt))s,t∈S is a Cartan matrix. We often denote the Coxeter polytope simply by P, and we call A the Cartan matrix of the Coxeter polytopeP.
For any Coxeter polytopeP, there is a unique Coxeter group, denotedWP, compatible with the Cartan matrix of P. If q is a subset of P, then we set Sq={s∈S∣q⊂S(Kerαs)}, where Kerαsis the kernel ofαs.
Theorem 4.4 (Tits, Chapter V of [Bou68], and Theorem 2 of Vinberg [Vin71]). Let P be a Coxeter polytope of S(V) with Coxeter group WP, and letΓP be the group generated by the projective reflections(σs)s∈S. Then the following hold:
(1) the homomorphism σ∶WP →SL±(V) defined by σ(s)=σs is well-defined and is an isomorphism ontoΓP, which is a discrete subgroup ofSL±(V);
(2) theΓP-orbit of P is a convex subsetCP ofS(V);
(3) ifΩP is the interior ofCP, thenΓP acts properly discontinuously onΩP; (4) for each pointx∈P, the subgroupσ(WSx)is the stabilizer of x inΓP; (5) an open face f of P lies inΩP if and only if the Coxeter groupWSf is finite.
4.3. Tits representations. To a Cartan matrix A=(Ast)s,t∈S is associated a Coxeter sim- plex∆A ofS(RS)as follows:
● for each t∈S, we set ˜αt=e∗t, where(e∗t)t∈S is the canonical dual basis ofRS;
● for each t∈S, we take the unique vector ˜bt∈RS such that ˜αs(b˜t)=Astfor alls∈S;
● the Coxeter simplex∆Ais the pair of the projective simplex∩s∈SS({x∈RS∣α˜s(x)⩽0}) and the set of reflections(σ˜s=Id−α˜s⊗b˜s)s∈S.
We letWA denote the unique Coxeter group compatible with A. Then Theorem 4.4 pro- vides the discrete faithful representation ˜σA∶WA→SL±(RS)whose image ˜ΓA=σ˜A(WA) is the group generated by the reflections(σ˜s)s∈S.
Assume now that none of the components of A is of zero type. We letVA denote the linear span of (b˜s)s∈S. For each s∈S, we setαs=α˜s∣VA, the restriction of ˜αstoVA, andbs=b˜s. By Proposition 13 of Vinberg [Vin71], the convex subset
PA=⋂
s∈S
S({x∈VA∣αs(x)⩽0})
of S(VA) has a non-empty interior. Moreover, since dim(VA)=rank(A), the subset PA is properly convex by Proposition 18 of [Vin71]. In other words, PA is a projective polytope of S(VA) (cf. Corollary 1 of [Vin71]). The projective polytope PA of S(VA)together with the reflections (σs=Id−αs⊗bs)s∈S is a Coxeter polytope, denoted again PA, and Theorem 4.4 provides the discrete faithful representationσA∶WA→SL±(VA)whose imageΓA=σA(WA)
