The deformation space of non-orientable hyperbolic 3-manifolds

We consider non-orientable hyperbolic 3-manifolds of finite volume $M^3$. When $M^3$ has an ideal triangulation $\Delta$, we compute the deformation space of the pair $(M^3, \Delta)$ (its Neumann Zagier parameter space). We also determine the variety of representations of $\pi_1(M^3)$ in $\mathrm{Isom}(\mathbb{H}^3)$ in a neighborhood of the holonomy. As a consequence, when some ends are non-orientable, there are deformations from the variety of representations that cannot be realized as deformations of the pair $(M^3, \Delta)$. We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.


Introduction
Let M 3 be a complete, non-compact, hyperbolic three-manifold of finite volume.Assume first that M 3 is orientable.Assume also that M 3 has a geometric ideal triangulation ∆ (see [16] for the definition).Following Thurston's construction for the figure eight knot exterior in [18], Neumann and Zagier defined in [16] a deformation space of the pair (M 3 , ∆), by considering the set of parameters of the ideal simplices of ∆ subject to compatibility equations.We denote the Neumann-Zagier parameter space by Def(M 3 , ∆).It is proved in [16] that it is homeomorphic to an open subset of C l , where l is the number of ends of M 3 .
Another approach to deformations is based on R(π 1 (M 3 ), Isom(H 3 )), the variety of conjugacy classes of representations of π 1 (M 3 ) in Isom(H 3 ).It is proved for instance by Kapovich in [14] that a neighborhood of the holonomy of M 3 is bi-analytic to an open subset of C l .
Both approaches to deformations can be used to prove the hyperbolic Dehn filling theorem (even if it is still an open question whether M 3 orientable admits a geometric ideal triangulation or not).Among other things, one has to take into account that Def(M 3 , ∆) is a 2 l to 1 branched covering of the neighborhood in R(π 1 (M 3 ), Isom(H 3 )).When M 3 is orientable, both approaches yield the same deformation space.
In this paper we investigate the non-orientable setting, that is, M 3 is a connected, nonorientable, hyperbolic 3-manifold of finite volume.When it has an ideal triangulation ∆, we define a deformation space of the pair Def(M 3 , ∆) à la Neumann-Zagier.Here is the main result of the paper (for simplicity, we assume that M 3 has a single end, that is non-orientable): Theorem 1.1.Let M 3 be a complete non-orientable hyperbolic 3-manifold of finite volume with a single end, that is non-orientable.
The version with several cusps of this theorem is stated in Theorem 3.11.For M 3 as in the theorem, structures in the subinterval [0, 1) ⊂ (−1, 1) in the variety of representations are realized by Def(M 3 , ∆), but structures in (−1, 0) are not.This corresponds to two different kinds of representations of the Klein bottle, that we denote of type I when realized, and II when not.Those are described in Section 4.
Deformations of the complete structure are non-complete, and therefore for a deformation of the holonomy the hyperbolic structure is not unique.Deformations of type I can be realized by ideal triangulations, hence there is a natural choice of structure and we prove in Theorem 5.15 that its metric completion consist in adding a singular geodesic, so that it is the core of a solid Klein bottle and it has a singularity of cone angle in a neighborhood of zero.For deformations of type II, we prove that there is a natural choice of structure (radial), and the metric completion consists in adding a singular interval, also in Theorem 5. 15.This singular interval is the soul of a twisted disc orbi-bundle over an interval with mirror boundary.Topologically, a neighborhood of this interval is the disc sum of two cones on a projective plane.Metrically, it is a conifold, with cone angle at the interior of the interval in a neighborhood of zero.
The paper is organized as follows.In Section 2 we describe Def(M 3 , ∆) and in Section 3, R(π 1 (M 3 ), Isom(H 3 )).Section 4 is devoted to representations of the Klein bottle.Metric completions are described in Section 5, and finally in Section 6 we describe in detail the deformation space(s) of the Gieseking manifold.
Acknowledgement We thank the anonymous referee for useful suggestions.The first author is supported by doctoral grant BES-2016-079278.Both authors are partially supported by the Spanish State Research Agency through grant PGC2018-095998-B-I00, as well as the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M).

Deformation space from ideal triangulations
Before discussing non-orientable manifolds, we recall first the orientable case.The first example was constructed by Thurston in his notes [18,Chapter 4] for the figure eight knot exterior, and the general case was constructed by Neumann and Zagier in [16].We refer the reader to these references for the upcoming exposition.
From the point of view of a triangulation, the deformation of the hyperbolic structure on a manifold with a given geometric ideal triangulation is the space of parameters of ideal tetrahedra, subject to compatibility equations.
A geometric ideal tetrahedron is a geodesic tetrahedron of H 3 with all of its vertices in the ideal sphere ∂ ∞ H 3 .We say that a hyperbolic 3-manifold admits a geometric ideal triangulation if it is the union of such tetrahedra, along the geodesic faces.Though it has been established in many cases, it is still an open problem to decide whether every orientable hyperbolic threemanifold of finite volume admits a geometric ideal triangulation.
Given an ideal tetrahedron in H 3 , up to isometry we may assume that its ideal vertices in ∂ ∞ H 3 ∼ = C ∪ {∞} are 0, 1, ∞ and z ∈ C. The idea of Thurston is to equip the (unoriented) edge between 0 and ∞ with the complex number z, called edge invariant.The edge invariant determines the isometry class of the tetrahedron, and for different edges the corresponding invariants satisfy some relations, called tetrahedron relations: • Opposite edges have the same invariant.
• Given 3 edges with a common end-point and invariants z 1 , z 2 , z 3 , indexed following the right hand rule towards the common ideal vertex, they are related to z 1 by z 2 = 1 1−z 1 and z Let M 3 be a possibly non-orientable complete hyperbolic 3-manifold of finite volume, which admits a geometric ideal triangulation, ∆ = {A 1 , • • • , A n } .As we have stated before, up to (oriented) isometry, the hyperbolic structure of each tetrahedron can be determined by a single edge invariant, thus the usual parameterization of the triangulation goes as follows: we fix an edge e i in each tetrahedron A i , and consider its edge invariant, z i .Hence, the hyperbolic structure of M 3 can be parametrized by n parameters (one for each tetrahedron) and we will denote the parameters of the complete triangulation {z 0 1 , • • • , z 0 n }.The deformation space of M 3 with respect to ∆, Def(M 3 , ∆), is defined as the set of parameters {z 1 , • • • , z n } in a small enough neighborhood of the complete structure for which the gluing bestows a hyperbolic structure on M 3 .However, we find that the equations defining the deformation space are easier to work with if we use 3n parameters (one for each edge after taking into account the duplicity in opposite edges) and ask them to satisfy the second tetrahedron relation too.
When M 3 is orientable, in order for the gluing to be geometric it is necessary and sufficient that around each edge cycle [e] = {e i 1 ,j 1 , • • • , e in,jn } the following two compatibility conditions are satisfied: The geometric meaning of these equations can be seen as follows: if we try to realize in H 3 the tetrahedra around the edge cycle [e], Equation (1) means that the triangulation must 'close up' and Equation (2), that the angle around [e] must be precisely 2π (instead of a multiple).The parameters of the complete hyperbolic structure are denoted by {z 0 (e 1,1 ), • • • , z 0 (e n,3 )}.In a small enough neighborhood of {z 0 (e 1,1 ), • • • , z 0 (e n,3 )}, fulfillment of (1) implies (2).We end the overview of the orientable case with the theorem we want to extend to the non-orientable case: Theorem 2.1 (Neumann-Zagier [16]).Let M 3 be connected, oriented, hyperbolic, of finite volume with l cusps.Then Def(M 3 , ∆) is bi-holomorphic to an open set of C l .
When we deal with non-orientable manifolds, again the problem of the gluing being geometric lives within a neighborhood of the edges.The compatibility equations in this case carry the same geometric meaning as in Equations ( 1) and ( 2), but accounting for the possible change of orientation of the tetrahedra.Proposition 2.2.Let M 3 be a non-orientable manifold triangulated by a finite number of ideal tetrahedra A i .The triangulation bestows a hyperbolic structure around the edge cycle [e] = {e i 1 ,j 1 , • • • , e in,jn } if and only if the following compatibility equations are satisfied: where z(e i l ,j l ) is the edge invariant of e i l ,j l , and l = 0, 1 in such a way that, in the gluing around the edge cycle [e], a coherent orientation of the tetrahedra is obtained by gluing a copy of A i l with its orientation reversed if l = 0, (or kept the original one if l = 1), and with the initial condition that the orientation of the tetrahedron A i 1 is kept as given.
Proof.When we follow a cycle of side identifications around an edge, can always reorient the tetrahedra (maybe more than once) so that the gluing is done by orientable isometries.The compatibility equations for the orientable case can be then applied and, hence, for the neighborhood of the edge cycle to inherit a hyperbolic structure, (1) must be satisfied, with the corresponding edge invariants.Now, let us consider an edge e i,j ∈ A i with parameter z(e i j ).To see how the edge invariant changes under a non-orientable isometry, we can assume that A i has vertices 0, 1, z(e i,j ) and ∞ in the upper-space model, and consider the isometry is c, the Poincaré extension of the complex conjugation in Thus, the proposition follows with ease after changing the orientation of some tetrahedra.
Definition 2.3.Let M 3 be a connected, complete, non-orientable, hyperbolic 3-manifold of finite volume.Let ∆ be an ideal triangulation of M 3 .The deformation space of M 3 related to the triangulation ∆ is satisfying the compatibility equations (3) and (4) and the tetrahedron relations}, where U is a small enough neighborhood of the parameters (z 0 i,j ) of the complete structure.
Let M 3 + be the orientation covering of M 3 .The ideal triangulation on M 3 , ∆, can be lifted to an ideal triangulation ∆ + on M 3 + .There is an orientation reversing homeomorphism, ι, acting on M 3 + such that M 3 = M 3 + /ι and ι 2 = Id.The triangulation on M 3 + is constructed in the usual way: for every tetrahedron A i we take another tetrahedron with the opposite orientation, ι(A i ), and glue them so that the orientation is coherent.For every edge, e i,j ∈ A i , let z(e i,j ) or z i,j denote its edge invariant.Analogously, w(ι(e i,j )) or w i,j will denote the edge invariant of ι(e i,j ) ∈ ι(A i ).
Remark 2.4.The compatibility equations (3) and (4) around [e] ∈ M 3 are precisely the (orientable) compatibility equations in any lift of [e] to the orientation covering.
The orientation reversing homeomorphism acts on Def(M 3 + , ∆ + ) by pulling-back (equivalently, pushing-forward) the associated hyperbolic metric on each tetrahedron.Combinatorially, the action is described in the following lemma: Lemma 2.5.Let M 3 = M 3 + /ι, where ι is an orientation reversing homeomorphism.Let M 3 admit an ideal triangulation ∆.Then, ι acts on Def(M 3 + , ∆ + ) as Proof.The proof follows easily from the fact that ι permutes the edges and, for e i,j ∈ A i with invariant z(e i,j ), the edge invariant of c(e i,j , where c is the Poincaré extension of the complex conjugation.
Remark 2.6.Metrics on tetrahedra are considered up to isotopy.
Corollary 2.7.The map Proof.It follows from Remark 2.4 and Lemma 2.5.
Our goal is to use Corollary 2.7 and Theorem 2.1 in order to identify the deformation space of M 3 with the fixed points under an action on C k .Let us suppose for the time being that M 3 has only one cusp which is non-orientable.The section of this cusp must be a Klein bottle.In order to define the bi-holomorphism through generalized Dehn filling coefficients we must first fix a longitude-meridian pair in the peripheral torus in the orientation covering M 3 + .As we will see, there is a canonical choice.Afterwards, following Thurston, we will compute the derivative of the holonomy, hol , and translate the action of ι over there and finally, to the generalized Dehn filling coefficients.
Fixing a longitude-meridian pair.Let K 2 be Klein bottle, its fundamental group admits a presentation The elements a 2 , b in the orientation covering T 2 are generators of π 1 (T 2 ) and are represented by the unique homotopy classes of loops in the orientation covering that are invariant by the deck transformation (as unoriented curves).From now on, we will choose as longitude-meridian pair the elements: Definition 2.8.The previous generators of π 1 (T 2 ) are called distinguished elements.
Lemma 2.9.Let [α] ∈ π 1 (T ), let ι be the involution in the orientation covering M 3 + , that is, We also denote by ι the restriction of ι to the peripheral torus T .If When we compute the derivative of the holonomy of an element, hol (γ), we assume that hol(γ) fixes ∞.
Proof.We use Thurston's method for computing the holonomy through the developing of triangles in C (see [18]).Thus, the factor that each piece of path adds to the derivative of the holonomy changes as in Figure 1 under the action of ι.Proposition 2.10.For the chosen longitude-meridian pair, the action of ι on Im(hol ) where L = hol (l), M = hol (m).Let us denote by u := log hol (l), v := log hol (m), the generalized Dehn coefficients are the solutions in R 2 ∪ {∞} to Thurston's equation Indeed, Neumann-Zagier Theorem 2.1 (see also [18]) states that, for M 3 orientable, the map (z i,j ) ∈ Def(M 3 , ∆) → (p k , q k ) is a bi-holomorphism, and the image is a neighborhood of (∞, • • • , ∞) ∈ C l , where l is the number of cusps of M 3 .
Proposition 2.12.The action of ι on (p, q) ∈ U ∩ R 2 ∪ {∞}, where (p, q) are the generalized Dehn coefficients, is Proof.The action of ι can be translated through the logarithm to (u, v) from the action on the holonomy (6) as ι * (u, v) = (u, −v).Then, to find the action on generalized Dehn coefficients, we have to solve Thurston's equation ( 9) with u and −v, that is, where ι * (p, q) = (p , q ) ∈ R 2 ∪ {∞}.It is straightforward to check that (p , q ) = (−p, q) is the solution.
Corollary 2.13.The fixed points under ι, which are in correspondence with Def(M 3 , ∆), are those whose generalized Dehn filling coefficients are of type (0, q).Theorem 2.14.Let M 3 be a connected, complete, non-orientable, hyperbolic 3-manifold of finite volume.Let M 3 have k non-orientable cusps and l orientable ones and let it admit an ideal triangulation ∆.Then Def(M 3 , ∆) is real bi-analytic to an open set of R k+2l .
Finally, in general the action of ι on Im(hol ) ⊂ C k+2l can be understood as a product of as in the case for a Klein bottle cusp, and the subsequent l, ι j , j = k + 1, • • • , k + l, acting on C 2 as in the case for a peripheral torus.

Varieties of representations
The group of isometries of hyperbolic space is denoted by G, and we have the following well known isomorphisms: which we will often consider in order to identify elements of G with elements of PSL(2, C) Z 2 .The group G has two connected components, according to whether the isometries preserve or reverse the orientation.
For a finitely generated group Γ, the variety of representations of Γ in G is denoted by hom(Γ, G).
As G is algebraic, it has a natural structure of algebraic set, cf Johnson-Millson [13], but we consider only its topological structure.We are interested in the set of conjugacy classes of representations: To understand deformations, we analyze a neighborhood of the holonomy in R(Γ, G).The main result of this section is: Theorem 3.1.Let M 3 be a hyperbolic manifold of finite volume.Assume that it has k nonorientable cusps and l orientable cusps.Then there exists a neighborhood of When M 3 is orientable this result is well known, see for instance Boileau-Porti or Kapovich [4,14], hence we assume that M 3 is non-orientable.We will prove a more precise result in Theorem 3.10, as for our purposes it is relevant to describe local coordinates in terms of the geometry of holonomy structures at the ends.
Before starting the proof, we need a lemma on varieties of representations.The projection to the quotient π : hom(Γ, G) → R(Γ, G) can have quite bad properties, for instance even if hom(Γ, G) is Hausdorff, in general R(Γ, G) is not.But in a neighborhood of the holonomy we have: To describe the neighborhood of the holonomy in R(Γ, G) we use the orientation covering.

Orientation covering and the involution on representations
As mentioned, we assume M 3 non-orientable.Let denote the orientation covering, with fundamental group Γ + = π 1 (M 3 + ).In particular we have a short exact sequence: 1 The automorphism σ * depends on the choice of ζ ∈ Γ \ Γ + : automorphisms corresponding to different choices of ζ differ by composition (or pre-composition) with an inner automorphism of Γ + ; furthermore σ 2 * is an inner automorphism because ζ 2 ∈ Γ + .This automorphism σ * is the map induced by the deck transformation of the orientation covering M 3 + → M 3 .The map induced by σ * in the variety of representations is denoted by and σ * does not depend on the choice of ζ, because σ * is well defined up to inner automorphism.Furthermore σ * is an involution, (σ * ) 2 = Id.Consider the restriction map: that maps the conjugacy class of a representation of Γ to the conjugacy class of its restriction to Γ + .

Lemma 3.4. There exist
, by construction there exists A ∈ G that conjugates ρ + and ρ + •σ * .We chose the neighborhood V so that Lemma 3.2 applies, hence such an A ∈ G is unique.From uniqueness (of A and A 2 ), it follows easily that, if Finally we establish continuity of res −1 using a slice.The existence of a slice S ⊂ R(Γ + , G) at res(hol) is proved by Johnson and Millson in [13,Theorem 1.2], who point to Borel-Wallach [5, IX.5.3] for a definition of slice.From the properties of the slice, and as the stabilizer of hol| Γ + is trivial, the natural map G × S → R(Γ + , G), that maps (g, s) ∈ G × S to gsg −1 , yields a homeomorphism between G × S and a neighborhood of the orbit of res(hol), and the projection induces a homeomorphism S ∼ = V .It follows from the product structure that the A ∈ G that conjugates ρ + and ρ + • σ * is continuous on ρ + , so the extension of ρ + to a representation of the whole Γ is continuous on ρ + .Then continuity of res −1 follows by composing the homeomorphism V ∼ = S (restricted to the fixed point set of σ * ) with the extension from Γ + to Γ, and projecting to U ⊂ R(Γ, G).
As Γ + preserves the orientation, next we use the complex structure of the identity component

Representations in PSL(2, C)
The holonomy of the orientation covering M 3 + is contained in PSL(2, C), and it is well defined up to the action of G = PSL(2, C) Z 2 by conjugation.If we furthermore choose an orientation on M 3 + , then the holonomy is unique up to the action by conjugacy of G 0 = Isom + (H 3 ) ∼ = PSL(2, C), and complex conjugation corresponds to changing the orientation.We call the conjugacy class in PSL(2, C) of the holonomy of M 3 + the oriented holonomy.We consider Its local structure is well known: The fact that it is C-analytic follows for instance from [13] or [14].In Theorem 3.9 we precise C-analytic coordinates, for the moment this is sufficient for our purposes.Lemma 3.6.Let hol + be the oriented holonomy of M 3 + .Then Namely, the oriented holonomy and its complex conjugate are not conjugate by a matrix in PSL(2, C).
Proof.By contradiction, assume that hol + and hol + are conjugate by a matrix in PSL(2, C): there exists an orientation-preserving isometry A ∈ PSL(2, C) such that Consider the orientation reversing isometry B = c • A, where c is the isometry with Möbius transformation the complex conjugation, z → z.The previous equation is equivalent to Brouwer's fixed point theorem yields that the fixed point set of B in the ball compactification By (12) hol + (Γ + ) preserves Fix(B).Thus, by minimality of the limit set of a Kleinian group, since Fix(B) = ∅ is closed and hol + (Γ + )-invariant, it contains the whole ideal boundary: Hence B is the identity, contradicting that B reverses the orientation.
From Lemma 3.6 and Theorem 3.5, we have: There exists a neighborhood W ⊂ R(Γ + , PSL(2, C)) of the conjugacy class of the oriented holonomy of M + that is disjoint from its complex conjugate: By choosing the neighborhood W ⊂ R(Γ + , PSL(2, C)) sufficiently small, we may assume that its projection to R(Γ + , G) is in contained in V as in Lemma 3.4.The neighborhood V can also be chosen smaller, to be equal to the projection of W , as this map is open.Namely the neighborhoods can be chosen so that R(Γ + , PSL(2, C)) → R(Γ + , G) restricts to a homeomorphism between W (or W ) and V .In particular we can lift to W the restriction map from U to V : as above, the lift of the restriction map yields an homeomorphism: This lemma has same proof as Lemma 3.4, just taking into account that ρ(ζ) ∈ G reverses the orientation, for [ρ] ∈ U and ζ ∈ Γ \ Γ + .

Local coordinates
Here we give the local coordinates of Theorem 3.5 and we prove a stronger version of Theorem 3.1.
Theorem 3.9.Let M 3 + be as above and assume that it has n cusps.Chose γ 1 , . . ., γ n ∈ Γ + a non-trivial element for each peripheral subgroup.Then, for a neighborhood W ⊂ R(Γ + , PSL(2, C)) of the oriented holonomy, (I γ 1 , . . ., I γn ) : W → C n defines a bi-analytic map between W and a neighborhood of the origin.
This theorem holds for any orientable hyperbolic manifold of finite volume, though we only use it for the orientation covering.Again, see [4,14] for a proof.As explained in these references, this is the algebraic part of the proof of Thurston's hyperbolic Dehn filling theorem using varieties of representations.
For a Klein bottle K 2 , in Definition 2.8 we considered the presentation of its fundamental group: The elements a 2 and b are called distinguished elements.Recall that, in terms of paths, those are represented by the unique homotopy classes of loops in the orientation covering that are invariant by the deck transformation (as unoriented curves).
There exists a neighborhood U ⊂ R(Γ, G) of the holonomy of M 3 such that the map defines a homeomorphism between U and a neighborhood of the origin in R k × C l .
In addition, by construction I commutes with complex conjugation.Hence, by Lemma 3.8 the image (I • res)(U ) is the subset of a neighborhood of the origin in C k+2l defined by x i = x i , ∀i = 1, . . ., k, and z j = y j , ∀j = 1, . . ., l.
Finally, by combining Theorem 3.9 and Lemma 3.8, the map I • res is a homeomorphism between U and its image.
We can now state the generalization of Theorem 1.1 to several cusps.Here D(1) ⊂ C denotes a disk of radius 1.
Proof.Assertion (a) is Theorem 2.14, and assertion (b) is Theorem 3.10.To describe the holonomy map in coordinates, for each cusp (orientable or not) choose an orientation preserving peripheral element m and let v be the logarithm of the holonomy of m defined as in (9) in a neighborhood of the origin in C (with v ∈ iR in the non-orientable case).In particular v is a component of the local coordinates of Def(M 3 , ∆).Furthermore, the holonomy of m is conjugate to therefore it has trace ±2 cosh v 2 , and this trace is a component of the local coordinates of R(π 1 (M 3 ), Isom(H 3 )).Then the assertion follows from applying a suitable coordinate change.

Representations of the Klein bottle
Let π 1 (K 2 ) = a, b|aba −1 = b −1 be a presentation of the fundamental group of the Klein bottle, and Topologically, we can expect to have, at least, 4 (possibly empty) connected components according to the orientable nature of A and B. We are interested in studying one of them.
Definition 4.1.A representation ρ ∈ hom(π 1 (K 2 ), G) is said to preserve the orientation type if, for every γ ∈ π 1 (K 2 ), ρ(γ) is an orientation-preserving isometry if and only if γ is represented by and orientation-preserving loop of K 2 .We denote this subspace of representations by Let T 2 → K 2 be the orientation covering.The restriction map on the varieties of representations (without quotienting by conjugation) is: G be the connected component of the identity.The variety of representations hom(π For ρ 0 = res(ρ), let A = ρ(a), B = ρ(b) where a, b are generators of π 1 (K 2 ), and L = ρ(l), M = ρ(m).The following is satisfied: (Restriction of a representation to the torus) (Klein bottle relation) In fact, in order for ρ 0 to be a restriction, there must be A and B satisfying the previous conditions.We prove the theorem using these equations.
If [ρ] is in the parabolic case, by hypothesis τ = 0.Then, the solution is unique and, On the other hand, for [ρ] hyperbolic, either L corresponds to a real dilation and M to a rotation, or the other way around.In the case L(z) = e 2l z, M (z) = e αi z, l ∈ R, α ∈ (−π, π] the representation can be written as the restriction of several representations of the Klein bottle, but all of them are conjugated to A(z) = e l z, B(z) = e αi z.A similar situation happens when L(z) = e 2αi z, M (z) = e l z, l ∈ R, α ∈ (−π, π], obtaining A(z) = e αi /z, B(z) = e l z.However, in the last case, we should note down that for every such representation [ρ], we get two nonconjugated representations Thus, we obtain a classification of representations in hom(π 1 (K 2 ), G)/G 0 .To get the classification quotienting by the whole group, hom(π 1 (K 2 ), G)/G, we only have to see how the • parabolic non-degenerate in case a) and parabolic degenerate in case a'), • type I in case b), and Furthermore, type I or II are called non-degenerate if l = 0 or α = 0 respectively, and degenerate otherwise.
Remark 4.4.The holonomy of a non-orientable cusp restricts to a representation of the Klein bottle that preserves the orientation type and is parabolic non-degenerate.
Furthermore, deformations of this representation still preserve the orientation type and are non-degenerate (possibly of type I or II), by continuity.
Proof.It is a straightforward computation from Theorem 4.2.

Metric completion
As we deform non-compact manifolds, the deformations into non-complete manifolds are not unique (eg one can consider proper open subset of a non-complete manifold).We are not discussing the different issues related to this non-uniqueness, just the existence of a deformation into a metric that can be complete as a conifold (see below).
The main result of this section is Theorem 5.15.In the orientable case, the metric completion after deforming an orientable cusp is a singular space with a singularity called of Dehn type (that include non-singular manifolds), see Hodgson's thesis [11] and Boileau-Porti [4, Appendix B].In the non-orientable case, the singularity is more specific, a so called conifold.

Conifolds and cylindrical coordinates
A conifold is a metric length space locally isometric to the metric cone of constant curvature on a spherical conifold of dimension one less, see for instance [3].When, as topological space, a conifold is homeomorphic to a manifold, it is called a cone manifold, but in general it is only a pseudo-manifold.In dimension 2 conifolds are also cone manifolds, but in dimension three there may be points with a neighborhood homeomorphic to the cone on a projective plane P 2 .
We are interested in three local models of singular spaces, that as conifolds are: • The hyperbolic cone over a round sphere S 2 .This corresponds to a point with a nonsingular hyperbolic metric.
• The hyperbolic cone over S 2 (α, α), the sphere with two cone points of angle α, that is the spherical suspension of a circle of perimeter α.It corresponds to a singular axis of angle α.
• The hyperbolic cone over P 2 (α), the projective plane with a cone point of angle α.This is the quotient of the previous one by a metric involution, which is the antipodal map on each concentric sphere.
Next we describe metrically those local models, by using cylindrical coordinates in the hyperbolic space.These coordinates are defined from a geodesic line g in H 3 , and we fix a point in the unit normal bundle to g, ie a vector u of norm 1 and perpendicular to g. Cylindrical coordinates give a diffeomorphism: where r is the distance between g and p, θ is the angle parameter (the angle between the parallel transport of u and the tangent vector to the orthogonal geodesic from g to p) and h is the arc parameter of g, the signed distance between the base point of u and the orthogonal projection from p to g, Figure 2. In the upper-half space model of H 3 , if g is the geodesic from 0 and ∞, then there exists a choice of coordinates (a choice of u) so that the projection from g to the ideal boundary ∂ ∞ H 3 maps a point with cylindrical coordinates (r, θ, h) to e h+iθ ∈ C, Figure 3.A different choice of u would yield instead λe h+iθ ∈ C, for some λ ∈ C \ {0}.
Remark 5.2.The metric models are: • For the non-singular case (the cone on the round sphere) it is H 3 .
• For the cone on P 2 (α), it is the quotient

Conifolds bounded by a Klein bottle
We keep the notation of Subsection 5.1, with cylindrical coordinates.Before discussing conifolds bounded by a Klein bottle, we describe a cone manifold bounded by a torus.Definition 5.3.A solid torus with singular soul is H 3 (α)/∼, where ∼ is the relation induced by the isometric action of Z generated by The space H 3 (α)/∼ is a solid torus of infinite radius with singular soul of cone angle α, length of the singularity L > 0 and torsion parameter τ ∈ R/2πZ (the rotation angle induced by parallel transport along the singular geodesic is α 2π τ ∈ R/αZ).By considering the metric neighborhood of radius r 0 > 0 on the singular soul, we get a compact solid torus, bounded by a 2-torus.This compact solid torus depicts a tubular neighborhood of a component of the singular locus of a cone manifold (compare Hodgson-Kerckhoff [12] and Hodgson's thesis [11]).
We describe two conifolds bounded by a Klein bottle, that are a quotient of this solid torus by an involution.The space H 3 (α)/∼ is a solid Klein bottle of infinite radius with singular soul of cone angle α, and length of the singularity L > 0. We may consider a metric tubular neighborhood of radius r 0 , bounded by a Klein bottle.Its orientation cover is a solid torus with singular soul, cone angle α, length of the singularity 2L and torsion parameter τ = 0. Definition 5.5.The disc orbi-bundle with singular soul is H 3 (α)/∼, where ∼ is the relation induced by two isometric involutions: To describe this space, it is useful first to look at the action on the preserved geodesic, corresponding to r = 0.These involutions map h ∈ R to −h and to 2L − h respectively.Thus it is the action of the infinite dihedral group Z 2 * Z 2 on a line generated by two reflections.Its orientation preserving subgroup is Z acting by translations on R. Thus R/Z is a circle, and R/(Z 2 * Z 2 ) is an orbifold.The solid torus is a disc bundle over the circle, and our space is an orbifold-bundle over R/(Z 2 * Z 2 ) with fibre a disc.
This space is the quotient of an involution on the solid torus.View the solid torus as two 3-balls joined by two 1-handles, Figure 4. On each 3-ball apply the antipodal involution (on each concentric sphere of given radius), and extend this involution by permuting the 1-handles.The quotient of each ball is the (topological) cone on P 2 , hence our space is the result of joining two cones on P 2 by a 1-handle.Its boundary is the connected sum P 2 #P 2 ∼ = K 2 .Figure 4: A solid torus as two 3-balls joined by two 1-handles.
The singular locus of the disc orbi-bundle H 3 (α)/∼ is an interval (the underlying space of the orbifold bundle) of length L. The interior points of the singular locus have cone angle α, and the boundary points of the interval are precisely the points where it is not a topological manifold.
Again H 3 (α)/ ∼ has ∞ radius, and the metric tubular neighborhood of radius r of the singularity is bounded by a Klein bottle.It is the quotient of a solid torus of length 2L and torsion parameter τ = 0 by an isometric involution with two fixed points (thus, as an orbifold, its orientation orbi-covering is a solid torus).
Remark 5.6.The boundary of both, the solid Klein bottle and the disc orbi-bundle, is a Klein bottle.In both cases the holonomy preserves the orientation type, but the type of the presentation as in Definition 4.3 is different: a) The holonomy of the boundary of a solid Klein bottle with singular soul is a representation of type I.
b) The holonomy of the boundary of a disc orbi-bundle over a singular interval is of type II.
For a non-orientable cusp, the holonomy of the peripheral torus is either parabolic nondegenerate, of type I or of type II, also nondegenerate (Remark 4.4).The aim of next section is to prove that the deformations can be defined so that the metric completion is either solid Klein bottle with singular soul or a disc orbi-bundle with singular soul, according to the type.This is the content of Theorem 5.15, that we prove at the end of the section.

The radial structure
Let M 3 be a non-compact hyperbolic 3-manifold of finite volume.We deform its holonomy representation and accordingly we deform its hyperbolic metric.Nonetheless, incomplete metrics are not unique, so here we give a statement about the existence of a maximal structure, which corresponds to the one completed in Theorem 5. 15.
Let [ρ] ∈ R(π 1 (M 3 ), G) be a deformation of its complete structure.There is some nuance in associating to [ρ] a hyperbolic structure which is made explicit by Canary-Epstein-Green in [6].
Here, the authors conclude that every deformation with a given holonomy representation are related by an isotopy of the inclusion of M 3 in some fixed thickening (M 3 ) * , where a thickening is just another hyperbolic 3-manifold containing ours.
We will start by making clear what we mean by a maximal structure.
Definition 5.7.Let M be a manifold with an analytic (G, X)-structure.We say that M * is an isotopic thickening of M if it is a thickening and there is a isotopy, i , of the inclusion, i : Given two isotopic thickenings of M we say that 1 to some subset of M * 2 extending the identity on M .Hence, we say that an isotopic thickening is maximal if it is with respect the partial order relation we have just defined.
In general, it is not clear whether maximal isotopic thickenings exist, nor under which circumstances they do exist.However, we will construct in our situation an explicit maximal thickening.
Lemma 5.8.Let inj M 3 (x) denote the injectivity radius at a point x ∈ M 3 .Then, a necessary condition for a non-trivial thickening of M 3 to exist is that there must exist a sequence {x n } ⊂ M 3 with inj M 3 (x n ) → 0.
Proof.Let us suppose a thickening (M 3 ) * exists.Then, take a point The purpose of Lemma 5.8 is two-fold: first, it gives a condition for a thickening to be maximal (in the sense of the partial order relation we just defined), and second, it shows where a manifold could possibly be thickened.Taking into account a thick-thin decomposition of the manifold, the thickening can only be done in the deformed cusps.
Each cusp of M 3 is diffeomorphic to either an end, for some λ > 0, and let us denote by D ρ the developing map of a structure with holonomy ρ in the equivalence class [ρ] ∈ R(Γ, G).
Lemma 5.9.The image of the proper product subset by the developing map, Moreover, for every geodesic ray exiting orthogonally from γ, the intersection of the ray with D ρ ( K 2 ×[0, λ]) is non-empty and transverse to any section D ρ ( K 2 × {µ}), µ ∈ [0, λ] and, analogously for an orientable end.
Proof.We use a modified argument of Thurston (see his notes [18,Chapters 4 and 5]) to prove the lemma for a non-orientable end (the same idea goes for an orientable one).The original argument of Thurston shows that in an ideal triangulated manifold, the image of the universal cover of the end under the developing map is the whole tubular neighborhood but the geodesic.Let [ρ 0 ] be the parabolic representation corresponding to the complete structure, then The domain K can be taken so that D ρ 0 (K) is a rectangular prism between two horospheres.
We want to deform D ρ 0 (K) as we deform ρ 0 to ρ.We do that by deforming the horosphere centered at p ∞ to surfaces equidistant to the geodesics γ ρ invariant by the holonomy of the peripheral subgroup ρ(π 1 (K 2 )).The deformation of the horosphere to equidistant surfaces is described in [18, §4.4] in the half-space model of H 3 , see also Benedetti-Petronio [2, §E.6.iv].Alternatively, we can view the deformation of the horosphere to the equidistant surfaces as follows.Consider Z 2 < π 1 (K 2 ) the orientation preserving subgroup of index 2, ρ(Z 2 ) is contained in a unique one-complex parameter subgroup U ρ ⊂ PSL(2, C) (ie U ρ is the exponential image of a C-line in the Lie algebra sl(2, C)).This U ρ depends continuously on ρ, and given x ∈ H 3 the orbit U ρ (x) = {g(x) | g ∈ U ρ } is a surface containing x such that: when ρ = ρ 0 then U ρ (x) is a horosphere centered at p ∞ , and when when ρ = ρ 0 , then U ρ (x) is a surface equidistant to the geodesic γ ρ .Using this construction, the image of the domain D ρ 0 (K) deforms to D ρ (K) with the required properties, by following the equidistant surfaces for the factor K 2 and the geodesics orthogonal to these surfaces for the factor [0, λ].
Definition 5.10.The geodesic of Lemma 5.9 is called the soul of the end.
Remark 5.11.The face of the section of proper product subset the cusp that is glued to the thick part of the manifold is the section of the cusp which is further away from the geodesic.Hence, we will only consider thickenings "towards" the soul.
Let x be a point in a cusp of the manifold and consider its image under the developing y = D ρ (x) of any lift x.There is only one geodesic segment in H 3 such that γ(0) = y and goes towards the soul orthogonally.In cylindrical coordinates, if y = (r, θ, h), the image of the geodesic consists of {(t, θ, h) | t ∈ [0, r]}.Let us denote by γ x the corresponding geodesic in M 3 .Theorem 5.12.There exists a maximal thickening It is characterized by the following property: for every point x ∈ M , the geodesic γ x can be extended in M * so that D ρ ( γx ) is the geodesic whose cylindrical coordinates with respect to the soul are {(t, θ, h) | t ∈ (0, r]}. Proof.Given a cusp section S := K 2 or T 2 , a product subset of the end, K := S × [0, λ], a fixed fundamental domain K 0 of K and a small neighborhood of K 0 , N (K 0 ), the set T := {t ∈ Deck( K/K) | tN (K 0 ) ∩ N (K 0 ) = ∅} is finite, where Deck( K/K) denotes the group of covering transformations of the universal cover.Hence, we can suppose that D ρ|(T K 0 ) is an embedding.
Let U be an open cover of K by simply connected charts.For each U , take a lift U 0 ∈ Ũ such that U 0 ∩ K 0 = ∅ and consider D ρ (U 0 ).Given such a lift U 0 , the other possible lift that could have non-empty intersection with K 0 are tU , for t ∈ T .Furthermore, we can always assume that the chart U coincides with the image of U 0 under the developing map, D ρ (U 0 ).Thus, we can identify where the equivalence relation is by the action of hol(t), for t ∈ T .
Each U ∈ U can be thickened by identifying U with D ρ (U 0 ) and considering, in cylindrical coordinates, the set of rays R(U ) = {(t, θ, h) ∈ H 3 \ {soul} | ∃(t 0 , θ, h) ∈ U, t < t 0 }.Given two lifts of two thickened charts R(U 1 ) and R(U 2 ) with non-empty intersection with K 0 , we glue them together in the points corresponding to hol(t)(R(U 1 )) ∩ R(U 2 ), where t ∈ T .This defines a thickening of the cusp K * .
We have yet to show that it is isotopic to the original (half-open) product subset.Let us consider the section S × {0} of the cusp, the radial geodesics γ x for x ∈ S × {0} define a foliation of K * of finite length.Moreover, due to Lemma 5.9, the foliation is transversal to S × {0}.By re-parameterizing the foliation and considering its flow, we obtain a trivialization of the cusp, K * ∼ = S × [0, µ).Similarly, K * \ K is also a product.This let us construct an isotopy from K * to K.
This thickening clearly satisfies the property that γ x ⊂ K * can be extended so that D ρ ( γx ) = {(t, θ, h) | t ∈ (0, r]}.By taking geodesics γ x to geodesics through the developing map, it is clear our thickening can be mapped into every other thickening satisfying this property.Furthermore, if we consider the thickenings to be isotopic, we obtain an embedding. Regarding the maximality, we will differentiate between an orientable end and a nonorientable one.The general idea will be the same one, for another isotopic thickening (K) * * to include ours, the developing map should map some open set V into a ball W around a point y 0 in the soul, what will led to some kind of contradiction.
If K is non-orientable, let us denote the distinguishable generators of π 1 (K 2 ), a, b, with aba −1 = b −1 .If [ρ] is type I, y 0 is fixed by ρ(b).Let y ∈ W \ {soul} and x ∈ V be its preimage.W is invariant by ρ(b) and, in addition, both x and b • x belong to V .Now take the geodesic γ : I → (K) * * from x to x 0 which corresponds to the geodesic from y to y 0 .By equivariance and continuity, x 0 = lim γ(t) = lim bγ(t) = bx 0 .This contradicts b being a covering transformation.If [ρ] is type II, the previous argument with a 2 holds.
If K is orientable, we will follow the same arguments leading to the completion of the cusp (for more details see, for instance, [2]).the deformation [ρ] is characterized in terms of its generalized Dehn filling coefficients ±(p, q).The case p = 0 or q = 0 are solved as in the nonorientable cusp, so we have the 2 usual cases, p/q ∈ Q or p/q ∈ I.For p/q ∈ Q, there exists k > 0 such that k(p, q) ∈ Q 2 and (kp)a + (kq)b is a trivial loop in the new thickening.If p/q ∈ I, then y 0 is dense in {soul} ∩ V , which is a contradiction.Definition 5.13.We call the previous thickening, the radial thickening of the cusp.
Remark 5.14.If the manifold M 3 admits an ideal triangulation, the canonical structure coming from the triangulation is precisely the radial thickening of the cusp.
Theorem 5.15.For a deformation of the holonomy M 3 , the corresponding deformation of the metric can be chosen so that on a non-orientable end: • It is a cusp, a metrically complete end, if the peripheral holonomy is parabolic.
• The metric completion is a solid Klein bottle with singular soul if the peripheral holonomy is of type I.
• The metric completion is a disc orbi-bundle with singular soul if it is of type II.
Furthermore, the cone angle α and the length L of the singular locus is described by the peripheral boundary, so that those parameters start from α = L = 0 for the complete structure and grow continuously when deforming in either direction.Proof of Theorem 5.15.The proof uses the orientation covering and equivariance.More precisely, the deformation is constructed in the complete case for the orientation covering and it can be made equivariant.The holonomy of a torus restricted from a Klein bottle is either parabolic or the holonomy of a solid torus with singular soul (and τ = 0).In particular the holonomy of a Klein bottle is parabolic iff its restriction to the orientable covering is parabolic.Furthermore, by using the description in cylindrical coordinates (and using Figure 3) and as τ = 0, the solid torus is equivariant by the action of 6 Example: The Gieseking manifold We use the Gieseking manifold to illustrate the results of this paper.In particular the difference between deformation spaces obtained from ideal triangulations and from the variety of representations.
The Gieseking manifold M is a non-orientable hyperbolic 3-manifold with finite volume and one cusp, with horospherical section a Klein bottle.It has an ideal triangulation with a single tetrahedron.The orientation cover of the Gieseking manifold is the figure eight knot exterior, and the ideal triangulation with one simplex lifts to Thurston's ideal triangulation with two ideal simplices in Thurston's notes [18].
This manifold M was constructed by Gieseking in his thesis in 1912, here we follow the description of Magnus in [15], using the notation of Alperin-Dicks-Porti [1].Start with the regular ideal vertex ∆ in H 3 , with vertices {0, 1, ∞, 1−i √ 3 2 }, Figure 6.The side identifications are the non-orientable isometries defined by the Möbius transformations The identifications of the faces are defined by their action on vertices: By applying Poincaré's fundamental theorem The relation V U = U 2 V 2 corresponds to a cycle of length 6 around the edge.

The deformation space Def(M, ∆)
We compute the deformation space of the triangulation with a single tetrahedron, as in Section 2.
For any ideal tetrahedron in H 3 , we set its ideal vertices in 0, 1, ∞ and −ω, where ω is in C + , the upper half-space of C. The role played by −ω will be the one of 1−i √ 3 2 in the complete structure.For any such ω it is possible to glue the faces of the tetrahedron in the same pattern as in the Gieseking manifold via two orientation-reversing hyperbolic isometries, which we will call likewise U and V .
For the gluing to follow the same pattern, it must map U : (−ω, 0, ∞) → (−ω, 1, 0) and V : (1, 0, ∞) → (−ω, 1, ∞).The orientation-reversing isometries U and V satisfying this are: Although it is always possible to glue the faces in the same pattern as in the Gieseking manifold, not for all them the gluing will have a hyperbolic structure.
Let us label the edges as in Figure 6.For the topological manifold to be geometric, we only have to check that the pairing is proper (see [17]).In this case, the only condition which we need to satisfy is that the isometry that goes through the only edge cycle is the identity.This is given by: a and, therefore, we will have a hyperbolic structure if and only if U −1 V −1 U 2 V 2 = Id.Doing this computation, we obtain the equation Let us show that this equation matches the one obtained from Definition 2.3.If we denote by z(a) the edge invariant of a and analogously for the rest of the edges, we have that the equation describing the deformation space of the manifold in terms of this triangulation is Writing down all of the edge invariants in terms of z(a) by means of the tetrahedron relations results in the equation If we substitute z(a) = −1/ω, we obtain which is equivalent to Equation (15).We justify the remark and Figure 7. Firstly, to prove that set of algebraic solutions is homeomorphic to a circle, we write the defining equation |w(1 Thus w + 1 2 2 lies in the circle of center 1 4 and radius 1.As this circle separates 0 from ∞, the equation defines a connected covering of degree two of the circle.Secondly, the set of algebraic solutions is invariant by the involutions w → w and w → −1 − w (hence symmetric with respect to the real line and the line defined by real part equal to − 1 2 ).Furthermore it intersects the real line at w = −1± √ 5 2 and the line with real part − Let us construct the link of the cusp.We denote the link of each cusp point as in Figure 8 and glue them to obtain the link as in Figure 9, which is a Klein bottle.Now we take two tetrahedra and construct the orientation covering of M (the figure eight knot exterior).For the first tetrahedron, we will denote by z 1 := z(a), and z 2 , z 3 so that they follow the cyclic order described in the tetrahedron relations.Afterwards, in the second tetrahedron, we denote by w i the edge invariant of the corresponding edge after applying an orientation reversing isometry to the tetrahedron, that is, w i = 1 z i .We consider the link of the orientation covering.The derivative of the holonomy of the two loops in the link of the orientation covering, l 1 , l 2 , depicted in Figure 10 (which are free For the manifold to be complete, hol (l i ) = 1 for i = 1, 2, which happens if and only if 2 i.This corresponds to the regular ideal tetrahedron, which, as expected, is the manifold originally given by Gieseking.Notice that the upper loop (the one going through the side ) can be taken as a distinguished longitude.A suitable meridian is drawn in Figure 11.Let us check that both the longitude and the meridian satisfy the conditions we stated for their holonomy in remark 2.11, that is hol (l) ∈ R, |hol (m)| = 1.We have already shown it for the longitude.Regarding the meridian, hol (m) = z 2 z 3 w 2 w 3 w 2 z 1 z 2 w 1 = z 1 z 3 therefore |hol (m)| = 1.This leads to the result that the generalized Dehn filling coefficients of a lifted structure have the form (0, q), after an appropriate choice of longitude-meridian pair.
The last result could also had been obtained from Thurston's triangulation.By rotating the tetrahedra, our triangulation could be related with his, and the parameters identified.We can then check that in his choice of longitude and meridian, the holonomy has the same features if the structure is a lift from Gieseking manifold.

The Gieseking manifold as a punctured torus bundle
The Gieseking manifold M is fibered over the circle with fibre a punctured torus T 2 \ { * }.We use this structure to compute the variety of representations.The monodromy of the fibration is an automorphism The map φ is the restriction of a map of the compact torus T 2 ∼ = R 2 /Z 2 that lifts to the linear map of R 2 with matrix 0 1 1 1 .
This matrix also describes the action on the first homology group is the algebraic monodromy, the map induced by φ on the fundamental group.
The relationship with the presentation ( 14) of π 1 (M ) from the triangulation is given by Furthermore, a peripheral group is given by rsr −1 s −1 , t , which is the group of the Klein bottle.denote the space of irreducible representations (ie that have no invariant line in C 2 ).As we are interested in deformations, we restrict to representations ρ that preserve the orientation type: ρ(γ) is an orientation preserving isometry iff γ ∈ π 1 (M ) is represented by a loop that preserves the orientation of M , ∀γ ∈ π 1 (M ).We denote the subspace of representations that preserve the orientation type by hom irr + (π 1 (M ), G).Let hom irr + (π 1 (M ), G)/G be their the space of their conjugacy classes.Proposition 6.2.We have an homeomorphism, via the trace of ρ(s): where ∼ is the relation by complex conjugation.
Proof.Let ρ : π 1 (M ) → G be an irreducible representation.The fibre T 2 \ { * } is orientable, so the restriction of ρ to the free group r, s | ∼ = F 2 is contained in PSL(2, C).Furthermore, as r, s | is the commutator subgroup, we may assume that the image ρ( r, s | ) ⊂ SL(2, C), cf Heusener-Porti [9].We consider the variety of characters X(F 2 , SL(2, C)) and the action of the algebraic monodromy φ * on the variety of characters: Hence if ρ 0 denotes the restriction of ρ to F 2 , it satisfies that ρ 0 and ρ 0 • φ * are conjugate, hence they have the same character and the lemma follows.
preserves the orientation of H 3 if and only if γ is represented by a loop that preserves the orientation of M 3 Assertions (a) and (b) are proved for instance by Johnson and Millson in [13]: they define the property of good representation, that is open in R(Γ, G), it implies Assertions (a) and (b), and it is satisfied by the conjugacy class of the holonomy.Assertion (c) is clear by continuity and the decomposition of G in two components, according to the orientation.

Corollary 4 . 6 .
a) The holonomy of a representation in Def(M, ∆) is of type I b) Representations in a neighborhood of [hol] in R(M 3 , G) are or both, type I and II.c) In particular, the holonomy map Def(M, ∆) → R(M 3 , G) is not surjective in a neighborhood of the holonomy.Proof.Assertion a) follows from Remark 2.11 and Assertion b) from Theorem 3.10, in both cases using Lemma 4.5.

Figure 3 :
Figure 3: Orthogonal projection to ∂ ∞ H 3 with g the geodesic with ideal end-points 0 and ∞.

Figure 7 :
Figure 7: The set of solutions of the compatibility equations and Def(M, ∆) (the top half).

1 Figure 11 :
Figure 11: Meridian in the link of the cover.