A NOTE ON LOCAL RIGIDITY 7 N. BERGERON AND T. GELANDER 1 0 2 n a Abstract. Theaimofthisnoteistogiveageometricproofforclassical J local rigidity of lattices in semisimple Lie groups. We are reproving well 3 known results in a more geometric (and hopefully clearer) way. 2 ] R 1. Introduction G Let G be a semisimple Lie group, and let Γ ≤ G be an irreducible lattice. . h WedenotebyR(Γ,G)thespaceofdeformationsofΓinG,i.e. thespaceofall t a homomorphisms Γ → G with the topology of pointwise convergence. Recall m that Γ is called locally rigid if there is a neighborhood Ω of the inclusion map [ ρ : Γ → G in R(Γ,G) such that any ρ ∈ Ω is conjugate to ρ . In other 0 0 1 words, this means that if Γ is generated by the finite set Σ, then there is an v 2 identity neighborhood U ⊂ G such that if ρ : Γ → G is a homomorphism 4 such that ρ(γ) ∈ γ · U for any γ ∈ Σ, then there is some g ∈ G for which 3 ρ(γ) = gγg−1 for any γ ∈ Γ (such ρ is called a trivial deformation). 0 0 Local rigidity was first proved by Selberg [17] for uniform lattices in the 2. case G = SLn(R), n ≥ 3, and by Calabi [3] for uniform lattices in the 0 case G = PO(n,1) = Isom(Hn), n ≥ 3. Then, Weil [23] generalized these 7 results to any uniform irreducible lattice in any G, assuming that G is not 1 : locally isomorphic to SL (R) (in which case lattices have many non-trivial v 2 i deformations - the Teichmuller spaces). Later, Garland and Raghunathan X [7] proved local rigidity for non-uniform lattices Γ ≤ G when rank(G) = r a 1 1 under the necessary assumption that G is neither locally isomorphic to SL (R) nor to SL (C). For non-uniform irreducible lattices in higher 2 2 rank semisimple Lie groups, the local rigidity is a consequence of the much stronger property: super-rigidity, which was proved by Margulis [13]. Our aim here, is to present a simple proof of the following theorem: Theorem 1.1. Under the necessary conditions on G and Γ, if ρ is a defor- mation, close enough to the identity ρ , then ρ(Γ) is a lattice and ρ is an 0 isomorphism. Date: February 2, 2017. 1Unless otherwise specified rank(G) will always mean the real rank of G. 1 2 N.BERGERON AND T. GELANDER Remark 1.2. In the higher rank case, we need the lattice to be arithmetic, which is always true by Margulis’ theorem. But this is of course cheating. Anyway our method provides, for example, a simple geometric proof of the local rigidity of SL (Z) in SL (R) for n ≥ 3. Note also that Margulis’ n n arithmeticity theorem is easier in the non compact case [14] (it does not rely on super-rigidity) which is the only case we need here. Remark 1.3. In fact, we prove a “stronger” result than the statement of theorem 1.1. Namely, if X = G/K is the associated symmetric space, then for a small deformation ρ, X/ρ(Γ) is homeomorphic to X/Γ and ρ : Γ → ρ(Γ) is an isomorphism. With Theorem 1.1, local rigidity follows from Mostow rigidity. Recall that Mostow’s rigidity theorem [15] says that if G is a connected semisimple Lie group not locally isomorphic to SL (R), and Γ ,Γ ≤ G are isomorphic 2 1 2 irreducible lattices in G, then they are conjugate in G, i.e. Γ = gΓ g−1 for 1 2 some g ∈ G. Remark 1.4. In Selberg’s paper [17] a main difficulty was to prove a weaker version of 1.1 in his special case. Selberg also indicated that his proof (which is quite elementary and very elegant) would work for more general higher rank cases if the general version (1.1 above) of his lemma 9 could be proved. This supports the general feeling that it should be possible to give a complete elementary proof of local rigidity which does not rely on Mostow’s rigidity. However, the authors decided not to strain too much, in trying to avoid Mostow rigidity, since in the rank one case there are by now very elegant and straightforward proofs of Mostow rigidity (such as the proof of Gromov and Thurston (see [18]) for compact hyperbolic manifolds, and of Besson, Courtois and Gallot [2] for the general rank one case, note also that Mostow’s original proof is much simpler in the rank one case) while for higher rank spaces the much stronger Margulis’ supper-rigidity holds. Our proof of Theorem 1.1 relies on an old principle implicit in Ehres- mann’s work and to our knowledge first made explicit in Thurston’s notes. This principle implies in particular that any sufficiently small deformation of a cocompact lattice in a Lie group stays a lattice. It seems to have remained unknown for many years, as bothSelberg andWeil spent some effort in prov- ing partial cases of it. It is by now fairly well known, but the authors could not find a complete written proof of it in the literature although everyone knows that it is implicit in Ehresmann’s work. In the second section of this paper, we thus decided to write a complete proof, using Ehresmann’s beauti- ful viewpoint, of a slight generalization of this principle to the non-compact case needed for our purpose. In the third section we prove Theorem 1.1 in A NOTE ON LOCAL RIGIDITY 3 the rank one case. In the last section we prove Theorem 1.1 for arithmetic lattices of higher rank. Non-uniform lattices in the group PSL (C) ∼= Isom(H3) are not locally 2 rigid, although Mostow rigidity is valid for PSL (C). This shows that local 2 rigidity is not a straightforward consequence of Mostow rigidity. Moreover, it was shown by Thurston that if Γ ≤ PSL (C) is a non-uniform lattice then 2 there is an infinite family of small deformations ρ of Γ such that ρ (Γ) is n n again a lattice, but not isomorphic to Γ, and in fact many of the ρ (Γ)’s may n be constructed to be uniform. The proof which we present in this note is in a sense a product of our attempt to understand and formulate why similar phenomenon cannot happen in higher dimensions. In fact, we shall show that some weak version of local rigidity is valid also for non-uniform lattices in PSL (C). This would allow us to obtain a direct 2 proof of the fact that for any (non-uniform) lattice in G = SL (C) (as well 2 as in any connected G which is defined over Q as an algebraic group and is locally isomorphic to SL (C)), there is an algebraic number field K, and an 2 element g ∈ G, such that gΓg−1 ≤ G(K). This fact (which is well known in the general case for any lattice in any G which is not locally isomorphic to SL (R)) can be proved by a few line argument when Γ is locally rigid in G. 2 When G is locally isomorphic to SL (C), and Γ ≤ G is non-uniform (and 2 hence not locally rigid), this was proved in [7], section 8, by a special and longer argument. Our method allows to unify this special case to the general case. In section 2 we shall formulate and prove the Ehresmann-Thurston prin- ciple and use it in order to prove local rigidity for uniform lattices (Weil’s theorem). In section 3, which is in a sense the main contribution of this note, we shall present a complete and elementary proof for local rigidity of non-uniform lattices in groups of rank one (originally due to Garland and Raghunathan). Then, in section 4, we shall apply our method to higher-rank non-uniform lattices. In some cases, our proof remains completely elemen- tary. However, for the general case we shall rely on some deep results. 2. The Ehresmann-Thurston principle In this section X is a model manifold and G a transitive group of real ana- lytic homeomorphisms of X. Given such data one can construct all possible manifolds by choosing opensubsets of X andpasting them together using re- strictions of homeomorphisms from the group G. Following Ehresmann and Thurston we will call such a manifold a (G,X)-manifold. More precisely, a (G,X)-manifold is a manifold M together with an atlas {κ : U → X} such i i that the changes of charts are restrictions of elements of G. 4 N.BERGERON AND T. GELANDER Such a structure enables to develop the manifold M along its paths (by pasting the open subsets of X) into the model manifold X. We can do this in particular for closed paths representing generators of the fundamental group π (M). Starting from one of the open sets U ⊂ M of the given 1 atlas, the development produces a covering space M′ of M, a representation ρ π (M) → G called the holonomy and a structure preserving immersion, the 1 developing map M′ →d X, which is equivariant with respect to the action of π (M) on M′ and of ρ(cid:0)π (M)(cid:1) on X. 1 1 There is an ambiguity in the choice of one chart κ : U → X about a base point. However, deferent charts around a point are always identified on a common sub-neighborhood modulo the action of G. Ehresmann’s way of doing mathematics was to find neat definitions for his concepts so that theorems would then follow easily from definitions. We shall try to follow his way. Recall first his fiber bundle picture of a (G,X)- structure and his neat definition of the developing map. To make it simple, let us describe first the picture for one chart κ : U → X which is a topological embedding. We can associate a trivial fiber bundle E = U×X → U by assigning to any m ∈ U the model space X = X. The U m manifold U is embedded as the diagonal cross section s(U) = {(m,κ(m)) : m ∈ U} ⊂ U ×X. Its points are the points of tangency of fibers and base manifolds. For the global picture of a (G,X)-structure on a manifold M, we will restrict ourselves to compatible charts that are topological embeddings κ : U → X for open sets U ⊂ M. A point of the fiber bundle space E over M is, by definition, a triple {m,κ,x}, where m ∈ U ⊂ M, κ : U → X is a compatible chart and x ∈ X, modulo the equivalence relation given by the action of G, i.e. {m,κ,x} ∼ {m,κ′,x′} iff there is g ∈ G such that x′ = g·x and κ′ = g◦κ on some sub-neighborhood V ⊂ U ∩ U′. In E, the manifold M is embedded as the diagonal cross section s(M), whose points are represented by triples {m,κ,κ(m)}. The horizontal leaves represented by the triples {U,κ,x} give a foliation F of the total space E, which induces an n-plane field ξ in E transversal to the fibers and transversal to the cross section s(M). Given such data, Ehresmann has defined the classical notionof holonomy. The holonomy is obtained by lifting a closed curve starting and ending at m ∈ M, into all curves tangent to ξ, 0 and by this getting an element of G acting on X . For contractible closed m0 curves in the base space M, the holonomy is of course the identity map of the fiber X . m0 This leads us, following Ehresmann, to consider the general notion of flat Cartan connections. In fact (G,X)-manifolds are the flat cases of manifolds A NOTE ON LOCAL RIGIDITY 5 M with a general (G,X)-connection. A (G,X)-connection is defined in [6] as follows: (1) a fiber bundle X → E → M with fiber X over M, (2) a fixed cross section s(M), (3) an n-plane field ξ in E transversal to the fibers and transversal to the fixed cross section, such that (4) the holonomy of each closed curve starting and ending at m ∈ M is 0 obtained by lifting it to all curves tangent to ξ, and is a homeomor- phism of X which is induced by an element of G. m0 A(G,X)-connectionissaidtobeflatifcontractibleclosedcurves havetrivial holonomy. We have just seen that a (G,X)-structure on a manifold M yields a flat (G,X)-connection on M. Let us describe how we can recover the holonomy and the developing map of a given (G,X)-structure from the induced flat (G,X)-connection. Let M be a manifold with a flat (G,X)-connection. For general closed curves, starting and ending at m , the holonomy gives a 0 representation of π (M) into the group G acting on X . When the (G,X)- 1 m0 connection is induced by a (G,X)-structure on M, this map is the usual holonomy map of the (G,X)-structure. E can be described as the quotient π (M)\(M′ ×X) 1 where the action is the diagonal one, with π (M) acting on M′ by covering 1 transformations and on X via the holonomy map through the action of G on X. The development of a curve ending at m ∈ M, is obtained by dragging 0 along ξ the corresponding points until they arrive in the fiber X . As m0 soon as the connection ξ is flat, homotopic curves with common initial and end points give the same image of the initial point in the end fiber and the development map M′ → X is well defined. We thus conclude that any flat m0 (G,X)-connection on a manifold M yields a (G,X)-structure. It is then an easy matter to prove the following useful result, which to our knowledge first appeared in a slightly different setup in Thurston’s notes [18], but which, as we will see, and seems to be a common knowledge, is elementary using Ehresmann’s viewpoint of (G,X)-structures. For a slightly different proof, from which we have borrowed some ideas, see [4] which is a very nice reference for all basic material on (G,X)-structures. Theorem 2.1 (TheEhresmann-Thurston Principle). Let N be asmooth compact manifold possibly with boundary. Let N be the reunion of N with Th a small collar ∂N × [0,1) of its boundary. Assume N is equipped with a Th (G,X)-structure M whose holonomy is ρ : π (N) → G. Then, for any suffi- 0 1 ciently small deformation ρ of ρ in R(π (N),G), there is a (G,X)-structure 0 1 on the interior of N whose holonomy is ρ. 6 N.BERGERON AND T. GELANDER Proof. Let ρ ∈ R(π (N),G) be a deformation of ρ . We define the following 1 0 two fiber bundles over N with fibers X Th E = π (N)\(N′ ×X) ρ0 1 Th and E = π (N)\(N′ ×X) ρ 1 Th where the action of π (N) on X is respectively induced via ρ and ρ by the 1 0 natural action of G on X. We denote by (1,ρ ) and (1,ρ) the two diagonal 0 actions of π (N) on N ×X considered above. 1 Th Note that since N is compact, π (N) is finitely generated and hence 1 R(cid:0)π (N),G(cid:1) has the structure of an analytic manifold and, in particular, it 1 is locally arcwise connected. Theorem 2.1 is a consequence of the following easy claim. Claim 2.2. If ρ is a deformation of ρ in the same connected component of 0 R(cid:0)π (N),G(cid:1) then there exists a fiber bundle map F : E → E such that 1 ρ0 ρ (1) F restricts to a diffeomorphism Φ above N, (2) as ρ gets closer to ρ , the lifted diffeomorphism Φ˜ : N′×X → N′×X 0 gets closer to the identity in the compact-open topology. We temporarily postpone the proof of the claim and conclude the proof of Theorem 2.1. When ρ is in the same connected component as ρ , we have, by Claim 2.2, 0 a diffeomorphism Φ between the compact fiber bundles Ec and Ec which ρ0 ρ are defined over N. Thus, if F denotes the horizontal foliation in E , the ρ0 restriction of F to Ec induces via Φ an n-plane field ξ on Ec. When ρ is ρ0 ρ close enough to ρ , the lifted diffeomorphism Φ˜ is close to the identity so 0 that the n-plane field ξ is transversal to both the fibers and the image by Φ of the diagonal cross section in Ec . Moreover, the holonomy which is ρ0 obtained by lifting a closed curve starting and ending at m ∈ N, to all 0 curves tangent to ξ, is equal to the image by ρ in G of the homotopy class of the curve. Thus the restriction of the bundle Ec to a bundle over the ρ interior of N yields a (G,X)-connection. This connection is obviously flat and hence gives a (G,X)-structure on the interior of N whose holonomy is ρ. This concludes the proof of Theorem 2.1 modulo Claim 2.2. Let us now prove Claim 2.2. Let {U } be a finite open covering of i 0≤i≤k N such that Th • U = N −N, 0 Th • for each 1 ≤ i ≤ k, U is simply connected and the fiber bundles E i ρ and E are both trivial above U . ρ0 i ∼ For each integer 1 ≤ i ≤ k, we fix a trivialization (E ) = U ×X (resp. ρ |Ui i ∼ (E ) = U × X) of E (resp. E ) above U . For each pair of integers ρ0 |Ui i ρ ρ0 i A NOTE ON LOCAL RIGIDITY 7 1 ≤ i 6= j ≤ k, we then denote by gi,j : (U ∩ U ) × X → (U ∩ U ) × X ρ i j i j (resp. gi,j : (U ∩U )×X → (U ∩U )×X) the diffeomorphism, which is the ρ0 i j i j product of the identity on the first factor with the corresponding change of charts on the second factor, between the trivializations (E ) and (E ) ρ |Ui ρ |Uj (resp. (E ) and (E ) ). ρ0 |Ui ρ0 |Uj Let U1 = U (1 ≤ i ≤ k) and for each integer r > 0, let {Ur+1} be i i i 1≤i≤k a shrinking of {Ur} (i.e. for each integer i, Ur+1 is an open set whose i 1≤i≤k i closure is included in Ur) such that {U }∪{Ur+1} is still a covering of i 0 i 1≤i≤k N . Th AsbothE andE aretrivialaboveU , theidentitymapU ×X → U ×X ρ ρ0 1 1 1 induces a diffeomorphism F1 : (Eρ0)U11 → (Eρ)U11. We will define by induction on s, a diffeomorphism F between E and s ρ0 E above Us∪...∪Us which is equal to F above Us∪...∪Us . But for ρ 1 s s−1 1 s−1 the sake of clarity let us first define the diffeomorphism F . 2 Let’s first describe the s = 2 step. First note that, as ρ is a deformation of ρ in the same connected component, the diffeomorphism g1,2 ◦ (g1,2)−1 0 ρ ρ0 of (U ∩ U ) × X is isotopic to the identity, by an isotopy which preserves 1 2 each of the fiber {∗}×X. We denote by (m,x) 7→ (m,ϕ (x)), t ∈ [0,1], this t isotopy. We want to define F above U2, recall it will be equal to F above U2∩U2. 2 2 1 2 1 Let W be an open set such that U2 ⊂ W ⊂ W ⊂ U1. 2 2 We define F above U2 as follows: 2 2 • Above U1 − W ⊂ U , both E and E are trivial and the identity 2 2 ρ0 ρ map (U1 − W) × X → (U1 − W) × X induces a diffeomorphism 2 2 f : (E ) → (E ) . ρ0 |(U21−W) ρ |(U21−W) • We define f = F above U2∩U2. (In the coordinates (U2∩U2)×X 1 1 2 1 2 induced bythetrivializationsaboveU , thediffeomorphism f isequal 2 to the restriction of g1,2 ◦(g1,2)−1.) ρ ρ0 • We extend f above a small collar neighborhood B × [0,ε) of the boundary B of U2 ∩ U2 in W − (U2 ∩ U2) by the map from (cid:0)B × 1 2 1 2 [0,ε)(cid:1) × X to itself, given by (cid:0)(b,t),x(cid:1) 7→ (cid:0)(b,t),ϕ (x)(cid:1) where g(t) g : [0,ǫ] → [0,1] is the classical smooth bump function. • Finallywecanextendf (bytheidentity) aboveallU1 togetasmooth 2 diffeomorphism. • We then define F above U2 to be the restriction of f to that set. 2 2 If we let F to be equal to F above U2, we get a well defined diffeomorphism 2 1 1 F from E to E above U2 ∪U2. 2 ρ0 ρ 1 2 To follow the induction we need to define F above Us+1. First note s+1 s+1 that above Us+1, the fiber bundles E and E are both trivial. s ρ0 ρ 8 N.BERGERON AND T. GELANDER Let W be an open set such that Us+1 ⊂ W ⊂ W ⊂ Us . s+1 s+1 We define F above Us+1 as follows: s+1 s+1 • Above Us −W, bothE andE aretrivial and we define a function s+1 ρ0 ρ f to be the canonical diffeomorphism between their trivializations. • Above Us+1 ∩(Us+1 ∪...∪Us+1) we define f to be equal to F . s+1 1 s s • Weextendf asinthefirstinductivestep, byusinganisotopybetween changes of charts composed with a smooth bump function, to get a smooth diffeomorphism between E and E above all Us . ρ0 ρ s+1 • We define F above Us+1 to be equal to the restriction of f above s+1 s+1 Us+1. s+1 If we let F to be equal to F above Us+1∪...∪Us+1, we get a well defined s+1 s 1 s diffeomorphism F from E to E above Us+1 ∪...∪Us+1. s+1 ρ0 ρ 1 s+1 The local trivialization of the fiber bundle E depends continuously (in ρ the C∞-topology) on ρ, and hence F depends continuously on F and ρ. s+1 s We continue the induction until s = k and get a diffeomorphism between E and E above Uk ∪...∪Uk which restricts to a fiber bundle diffeomor- ρ0 ρ 1 k phism between the corresponding bundles above N. Moreover we have seen that therestriction of F above N depends continuously (inthe C∞ topology) on ρ. This concludes the proof of the claim. (cid:3) Recall now Ehresmann’s definition [6] of a complete (G,X)-structure on a manifold M. Let M bea (G,X)-manifold. Any curve c starting from a point m in M can be developed to a curve c˜in X. The (G,X)-manifold is said to 0 be complete if, conversely, any curve C˜ extending c˜in X is a development of a curve in M extending c. When X is a Riemannian homogeneous G-space, for a Lie group G of isometries of X, it is a classical theorem of Hopf and Rinow that this notion of completeness coincides with the usual notion of metric completeness. The Ehresmann-Thurston principle above is especially useful when com- bined with the following. Theorem 2.3 (Ehresmann [5]). If M is a complete (G,X)-manifold, then the developing map induces an isomorphism between its universal cover and the universal cover of X. As a corollary of Theorem 2.1 and Theorem 2.3 we get the following clas- sical result of Weil [22] (see also [23]). Corollary 2.4. Let Γ be a cocompact lattice in a connected center free semi- simple Lie group without compact factors G. If ρ is a deformation of Γ in A NOTE ON LOCAL RIGIDITY 9 G, close enough to the identity, then ρ(Γ) is still a cocompact lattice and is isomorphic to Γ. Proof of Corollary 2.4. The group Γ is finitely generated and by Selberg’s lemma it has a torsion-free finite index subgroup. It is easy to see that if 2.4 holds for a finite index subgroup of Γ then it also holds for Γ. Hence, we shall assume that Γ itself is torsion-free. Let X be the symmetric space associated to G. It is simply connected. Let M be the (G,X)-manifold Γ\X. It is compact and its fundamental group is Γ. According to Theorem 2.1, if ρ is a sufficiently small deformation of the inclusion Γ ⊂ G, then there exists a new (G,X)-structure M′ on the manifold M, whose holonomy is ρ : Γ → G. But asM iscompact andX isRiemannian, thisnew(G,X)-structureiscomplete and, by Theorem 2.3, the developing map is a Γ-equivariant isomorphism between its universal cover and X. This implies that ρ(Γ) acts properly discontinuously on X and that M′ = ρ(Γ)\X. As M′ is homeomorphic to M this concludes the proof. (cid:3) We shall conclude this section by stating the following natural generaliza- tion of Corollary 2.4 which is also due to Weil [22]. Proposition 2.5. Let G be a connected Lie group and Γ ≤ G a uniform lattice. Then for any sufficiently small deformation ρ of Γ in G, ρ(Γ) is again a uniform lattice and ρ : Γ → ρ(Γ) is an isomorphism. Proof. Assume first that G is simply connected. Then we can chose a left invariant Riemannian metric m on G, and argue verbatim as in the proof Corollary 2.4 letting (G,m) to stand for X, and G to act isometrically by left multiplication. ˜ Now consider the general case. Let G be the universal covering of G. ˜ ˜ Let Γ ≤ G be a uniform lattice and let Γ be its pre-image in G. Then ˜ ˜ ˜ ˜ Γ is a uniform lattice in G, and G/Γ is homeomorphic to G/Γ. Since Γ is finitely generated, the deformation space R(Γ,G) has the structure of a manifold and in particular it is locally arcwise connected. Therefore if ρ is sufficiently small deformation of Γ in G then there is a curve of deformations ρ ∈ R(Γ,G) with ρ = the identity, and ρ = ρ. Now for any γ ∈ Γ and any t 0 1 γ˜ ∈ Γ˜ aboveγ, thecurve ρ (γ)liftsuniquely toacurve ρ˜(γ˜). Theuniqueness t t guaranties that ρ˜ is again a homomorphism for any t. Hence ρ lifts to a t t curve ρ˜ ∈ R(Γ˜,G˜). In particular ρ = ρ lifts to a deformation ρ˜ = ρ˜ of Γ˜ t 1 1 in G˜. It is also easy to see that the lifting ρ 7→ ρ˜ on a neighborhood of the inclusion ρ is continuous as a map from an open set in R(Γ,G) to R(Γ˜,G˜). 0 (cid:3) Thus, the general case follows from the simply connected one. The proof of Theorem 1.1 will follow the same lines, we just need to understand the non-compact parts, namely the ends. 10 N.BERGERON AND T. GELANDER 3. The rank one case Let G be a connected center free simple Lie group of rank one, with as- sociated symmetric space X = G/K. Let Γ be a non-uniform lattice in G. We are going to investigate the possible small deformations of Γ, and to prove local rigidity when n = dim(X) ≥ 4, and some weak version of it for dimensions 2 and 3. BySelberg’slemma Γisalmosttorsionfree. Notethatasmalldeformation which stabilizesafiniteindexsubgroupmustbetrivial. Inthesequel weshall assume that Γ is torsion free. We keep the notation ρ : Γ → G for the inclusion, and ρ for a small 0 deformation of ρ . We denote by M = Γ\X the corresponding locally sym- 0 metric manifold. We shall show that under some conditions on ρ, ρ(Γ) is again a lattice in G and M′ = ρ(Γ)\X is homeomorphic to M, and that these conditions are fulfilled when dimX ≥ 4. Our strategy is to use Ehresmann-Thurston’s principle. As M is not com- pact, we shall decompose it to a compact part M which “exhausts most of 0 M” and to cusps. More precisely, let M ⊂ M be a fixed compact subman- 0 ifold with boundary, which is obtained by cutting all the cusps of M along horospheres. Each such cusp is contained in a connected component of the thin partofthethick-thin decomposition ofM corresponding tothe constant ǫ of the Margulis lemma (see [19] or [1]), and is homeomorphic to T ×R≥0 n for some (n−1)-dimensional compact manifold T. We shall call such cusps canonical. More precisely, a canonical cusp is a quotient of a horoball by a discrete group of parabolic isometries which preserves the horoball and acts freely and cocompactly on its boundary horosphere. Notice that a canonical cusp has finite volume. To see this, one can look at the Iwasawa decomposition G = KAN which corresponds to a lifting of the given cusp, express the Haar measure of G in terms of the Haar measures of K,A and N, and estimate the volume of the cusp in the same way as one shows that a Siegel set has a finite volume. In the argument below we shall assume, for the sake of clearness, that M has only one cusp, instead of finitely many. It is easy to see that the proof works equally well in the general case. We choose another horosphere which is contained in M and parallel to the boundary horosphere, and denote the 0 collar between them by T ×[0,1] (see figure 1). By the Ehresmann-Thurston principle, there is a (G,X)-structure M′ on 0 M whose holonomy is ρ. 0 Wewill showthatunder someconditions, the(G,X)-structureonT×[0,1] which is induced from M′ coincides with a (G,X)-structure which is induced 0 from some canonical cusp C (whose holonomy coincides with ρ(cid:0)π (T)(cid:1)) to 1 a subset of it, which is homeomorphic to T × [0,1]. In such case we can