deformations of non-compact, projective manifolds€¦ · deformations of non-compact, projective...

24
Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate that the complete, hyperbolic representation of vari- ous two-bridge knot and link groups enjoy a certain local rigidity property inside of the PGL 4 (R) character variety. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-trivial deformations near the complete, hyperbolic representation. 1 Introduction Mostow rigidity for hyperbolic manifolds is a crucial tool for understanding the deformation theory of lattices in Isom(H n ). Specifically, it tells us that the fundamental groups of hy- perbolic manifolds of dimension n 3 admit a unique conjugacy class of discrete, faithful representations into Isom(H n ). Recent work of [3–5,10] has revealed several parallels between the geometry of hy- perbolic n-space and the geometry of strictly convex domains in RP n . For example, the classification and interaction of isometries of strictly convex domains is analogous to the situation in hyperbolic geometry. Additionally, if the isometry group of the domain is suffi- ciently large then the strictly convex domain is known to be δ -hyperbolic. Despite the many parallels between these two types of geometry, there is no analogue of Mostow rigidity for strictly convex domains. This observation prompts the following question: is it possible to deform a finite volume, strictly convex structure on a fixed manifold? Currently, the answer is known only in certain special cases. For example, when the manifold contains a totally geodesic, hypersurface there exist non-trivial deformations at the level of representations coming from the bending construction of Johnson and Millson [15]. In the closed case, work of Koszul [17] shows that these new projective structures arising from bending remain properly convex. Further work of Benoist [4] shows that these structures are actually strictly convex. In the non-compact case recent work of Marquis [19] has shown that the projective structures arising from bending remain properly convex in this setting as well. In contrast to the previous results, there are examples of closed 3-manifolds for which no such deformations exist (see [9]). Additionally, there exist 3-manifolds that contain no 1

Upload: others

Post on 30-Jun-2020

15 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

Deformations of Non-Compact, Projective Manifolds

Samuel A. Ballas

November 1, 2012

Abstract

In this paper, we demonstrate that the complete, hyperbolic representation of vari-ous two-bridge knot and link groups enjoy a certain local rigidity property inside of thePGL4(R) character variety. We also prove a complementary result showing that undercertain rigidity hypotheses, branched covers of amphicheiral knots admit non-trivialdeformations near the complete, hyperbolic representation.

1 Introduction

Mostow rigidity for hyperbolic manifolds is a crucial tool for understanding the deformationtheory of lattices in Isom(Hn). Specifically, it tells us that the fundamental groups of hy-perbolic manifolds of dimension n ≥ 3 admit a unique conjugacy class of discrete, faithfulrepresentations into Isom(Hn).

Recent work of [3–5, 10] has revealed several parallels between the geometry of hy-perbolic n-space and the geometry of strictly convex domains in RPn. For example, theclassification and interaction of isometries of strictly convex domains is analogous to thesituation in hyperbolic geometry. Additionally, if the isometry group of the domain is suffi-ciently large then the strictly convex domain is known to be δ-hyperbolic. Despite the manyparallels between these two types of geometry, there is no analogue of Mostow rigidity forstrictly convex domains. This observation prompts the following question: is it possible todeform a finite volume, strictly convex structure on a fixed manifold?

Currently, the answer is known only in certain special cases. For example, when themanifold contains a totally geodesic, hypersurface there exist non-trivial deformations at thelevel of representations coming from the bending construction of Johnson and Millson [15].In the closed case, work of Koszul [17] shows that these new projective structures arising frombending remain properly convex. Further work of Benoist [4] shows that these structuresare actually strictly convex. In the non-compact case recent work of Marquis [19] has shownthat the projective structures arising from bending remain properly convex in this settingas well.

In contrast to the previous results, there are examples of closed 3-manifolds for whichno such deformations exist (see [9]). Additionally, there exist 3-manifolds that contain no

1

Page 2: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

totally geodesic surfaces, yet still admit deformations (see [8]). Henceforth, we will refer tothese deformations that do not arise from bending as flexing deformations.

While Mostow rigidity guarantees the uniqueness of complete structures on finite vol-ume 3-manifolds, work of Thurston [25] shows that if we remove the completeness hypothesisthen there is an interesting deformation theory for cusped, hyperbolic 3-manifolds. In orderto obtain a complete hyperbolic structure we must insist that the holonomy of the peripheralsubgroup be parabolic. Thus, if would like any sort of rigidity result for projective structureson non-compact manifolds it will be necessary to constrain how our representations restrictto the peripheral subgroups. As shown in [10], there is a notion of an automorphism ofa strictly convex domain being parabolic, and in light of [10, Thm 11.6] insisting that theholonomy of the peripheral subgroup be parabolic seems to be a natural restriction.

Prompted by these results a natural question to ask is whether or not there existflexing deformations for non-compact, finite volume manifolds. Two bridge knots and linksprovide a good place to begin exploring because they have particularly simple presentationsfor their fundamental groups. Additionally, work of [12] has shown that they contain noclosed, totally geodesic, embedded surfaces. Using normal form techniques developed insection 4 we are able to prove that several two bridge knot and link complements enjoya certain rigidity property. See section 3 for definitions related to local and infinitesimalrigidity.

Theorem 1.1. The two bridge links with rational number 52

(figure-eight) , 73, 9

5, and 8

3

(Whitehead link) are locally projectively rigid relative to the boundary near their geometricrepresentations.

Remark 1.2. The knots and links mentioned in Theorem 1.1 correspond to the 41, 52, 61,and 52

1 in Rolfsen’s table of knots and links [24]

It should be mentioned that an infinitesimal analogue of this theorem is proven in[13] for the figure-eight knot and the Whitehead link. In light of Theorem 1.1 we ask thefollowing:

Question 1. Is it possible to deform the geometric representation of any two-bridge knot orlink relative to the boundary?

In [13] it is shown that there is a strong relationship between deformations of a cuspedhyperbolic 3-manifold and deformations of surgeries on that manifold. In particular they areable to use the fact that the figure-eight knot is infinitesimally projectively rigid relative tothe boundary to deduce that there are deformations of certain orbifold surgeries of the figure-eight knot. We are able to extend these results to other amphicheiral knot complements thatenjoy a certain rigidity property in the following theorem. See section 6 for the definition ofrigid slope.

Theorem 1.3. Let M be the complement of a hyperbolic, amphicheiral knot, and supposethat M is infinitesimally projectively rigid relative to the boundary and the longitude is arigid slope. Then for sufficiently large n, M(n/0) has a one dimensional deformation space.

2

Page 3: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

Here M(n/0) is orbifold obtained by surgering a solid torus with longitudinal singularlocus of cone angle 2π/n along the meridian of M . Other than the figure-eight knot, wecannot yet prove that there exist other knots satisfying the hypotheses of Theorem 1.3.However, there is numerical evidence that the two bridge knot with rational number 13

5

satisfies these conditions. Also, in light of Theorem 1.1 and the rarity of deformations inthe closed examples computed in [9], it seems that two bridge knots that are infinitesimallyrigid relative to the cusps may be quite abundant. Additionally, there are infinitely manyamphicheiral two-bridge knots, and so there is hope that there are many situations in whichTheorem 1.3 applies.

The organization of the paper is as follows. Section 2 discusses some basics of pro-jective geometry and projective isometries while Section 3 discusses local and infinitesimaldeformations with a focus on bending and its effects on peripheral subgroups. Section 4discusses some normal forms into which parabolic isometries can be placed. In Section 5 weuse the techniques of the previous section to give a thorough discussion of deformations ofthe figure-eight knot (Section 5.1) and Whitehead link (Section 5.2), which along with com-putations in [2] proves Theorem 1.1. Finally, in Section 6 we discuss some special propertiesof amphicheiral knots and their representations and prove Theorem 1.3.

Acknowledgements

This project started at the MRC on real projective geometry in June, 2011. I would like tothank Daryl Cooper who suggested looking at projective deformations of knot complements.I would also like to thank Tarik Aougab for several useful conversations and his help withsome of the early computations related to the figure-eight knot. Neil Hoffman also offereduseful suggestions on an earlier draft of this paper. Finally, I would like to thank my thesisadvisor, Alan Reid for his encouragement and helpful feedback throughout this project.

2 Projective Geometry and Convex Projective Struc-

tures

Let V be a finite dimensional real vector space. We form the projectivization of V , denotedP (V ), by quotienting V \0 by the action of R×. When V = Rn we denote P (Rn) by RPn.If we take GL(V ) and quotient it by the action of R× acting by multiplication by scalarmatrices we get PGL(V). It is easy to verify that the action of GL(V ) on V descends to anaction of PGL(V) on P (V ). If W ⊂ V is a subspace, then we call P (W ) a projective subspaceof P (V ). Note that the codimension of P (W ) in P (V ) is the same as the codimension of Win V and the dimension of P (W ) is one less than the dimension of W . A projective line is a1-dimensional projective subspace.

A projective space P (V ) admits a double cover π : S(V ) → P (V ), where S(V ) isthe quotient of V by the action of R+. In the case where V = Rn we denote S(V ) by Sn.The automorphisms of S(V ) are SL±(V ), which consists of matrices with determinant ±1.

3

Page 4: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

Similarly, there is a 2-1 map from SL±(V ) to PGL(V).A subset of Ω of a projective space Y is convex if its intersection with any projective

line is connected. An affine patch is the complement in Y of a codimension 1 subspace. Aconvex subset Ω ⊂ Y is properly convex if Ω is contained in some affine patch. A point in∂Ω is strictly convex if it is not contained in a line segment of positive length in ∂Ω and a Ωis strictly convex if it is properly convex and every point in its boundary is strictly convex.Next, we discuss how properly convex sets sit inside of Sn. Observe that if Ω ⊂ RPn isproperly convex, then its preimage of Ω in Sn. has two components (this is easily seen byassuming that the codimension 1 subspace defining the affine patch projects to the equatorof Sn). If we let SL±(Ω) and PGL(Ω) be subsets of SL±n+1(R) and PGLn+1(R) preservingΩ, respectively, then there is a nice map between PGL(Ω) and SL±(Ω) that eliminates theneed to deal with equivalence classes of matrices. An element T ∈ PGL(Ω) has two lifts toSL±(Ω), one of which preserves the components of π−1(Ω) and another which interchangesthem. Sending T to the lift that preserves the components gives the desired map.

Let Ω be properly convex and open, then given a matrix T ∈ SL±(Ω) representingan element T ∈ PGL(Ω), the fixed points of T in RPn correspond to eigenvectors of T withreal eigenvalues. We can now classify elements of SL±(Ω) according to their fixed points asfollows: if A fixes a point in Ω, then A is called elliptic. If A acts freely on Ω, and all of itseigenvalues have modulus 1 then A is parabolic. Otherwise A is hyperbolic. It is shown in[10] that parabolic elements of SL±(Ω) are subject to certain linear algebraic constraints.

Theorem 2.1. Suppose that Ω is a properly convex domain and that A ∈ SL±(Ω) is parabolic.Then one of largest Jordan blocks of A has eigenvalue 1. Additionally, the size of this Jordanblock is odd and at least 3. If Ω is strictly convex then this is the only Jordan block of thissize.

This theorem is particularly useful in small dimensions. For example, in dimensions2 and 3 any parabolic that preserves a properly convex subset is conjugate to

1 1 00 1 10 0 1

or

1 0 0 00 1 1 00 0 1 10 0 0 1

,

respectively. This theorem also tells us that in these small dimensional cases, 1 is the onlypossible eigenvalue of a parabolic that preserves a properly convex domain.

When Ω is properly convex we can also gain more insight into the structure of SL±(Ω)by putting an invariant metric on Ω. Given x1, x2 ∈ Ω we define the Hilbert metric as follows:let ` be the line segment between x1 and x2. Proper convexity tells us that ` intersects theboundary in two points y and z, and we define dH(x1, x2) = |log([y : x1 : x2 : z])|, where[y : x1 : x2 : z] is the cross ratio, defines a metric on Ω. This metric is clearly invariant underSL±(Ω) because the cross ratio is invariant under projective automorphisms. This metricgives rise to a Finsler metric on Ω, and in the case that Ω is an ellipsoid it coincides withtwice the standard hyperbolic metric.

4

Page 5: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

We now use this metric to understand certain subsets of SL±(Ω).

Lemma 2.2. Let Ω be a properly convex domain and let x be a point in the interior of Ω,then the set ΩK

x = T ∈ SL±(Ω) | dH(x, Tx) ≤ K is compact.

Proof. Let B = x0, . . . , xn) be a projective basis (that is a set of n+ 1 point no n of whichlive in a common hyperplane) that are contained in Ω and such that x0 = x. Next, let γibe a sequence of elements of ΩK

x . the elements γix0 all live in the compact ball of radius Kcentered at x and so by passing to subsequence we can assume that γix0 → x∞0 ∈ Ω. Next,we claim that the by passing to subsequence that γixj → x∞j ∈ Ω for 1 ≤ j ≤ n. To see thisobserve that

dH(x∞0 , γixj) ≤ dH(x∞0 , γix0) + dH(γix0, γixj) = dH(x∞0 , γix0) + dH(x0, xj),

and so all of the γixj live in a compact ball centered at x∞0 . Elements ofPGLn+1(R) areuniquely determined by their action on a projective basis, and convergence of elements ofPGLn+1(R) is the same as convergence bases as subsets of (RPn)n+1. Therefore the proof willbe complete if we can show that the set x∞0 . . . , x∞n is a projective basis. Suppose that thisset is not a basis, then without loss of generality we can assume that the set v∞0 , . . . , v∞n−1is linearly dependent, where xi = [vi]. Thus v∞0 = c1v

∞1 + . . . + cn−1v

∞n−1 is a non-trivial

linear combination, and we find that γi[c1v1 + . . .+ cn−1vn−1]→ [v∞0 ]. However,

dH([v0], [c1v1 + . . .+ cn−1vn−1]) = dH(γi[v0], γi[c1v1 + . . .+ cn−1vn−1])

and

dH(γi[v0], γi[c1v1 + . . .+ cn−1vn−1])→ dH([v∞0 ], [c1v∞1 + . . .+ cn−1v

∞n−1]) = 0,

which contradicts the fact that B is a basis.

As we mentioned previously, isometries of a strictly convex domain interact in wayssimilar to hyperbolic isometries. As an example of this, recall that if φ, ψ ∈ SO(n, 1), withφ hyperbolic, then φ and ψ cannot generate a discrete group if they share exactly one fixedpoint. In particular parabolics and hyperbolics cannot share fixed points in a discrete group.A similar phenomenon takes place inside of SL±(Ω), when Ω is strictly convex.

Proposition 2.3. Let Ω be a strictly convex domain and φ, ψ ∈ SL±(Ω) with φ hyperbolic.If φ and ψ have exactly one fixed point in common, then the subgroup generated by φ and ψis not discrete.

Proof. Notice the similarity between this proof and the proof in [22, Thm 5.5.4]. Supposefor contradiction that the subgroup generated by φ and ψ is discrete. Since Ω is strictlyconvex, φ has exactly two fixed points (see [10, Prop 2.8]), x1 and x2 and without loss ofgenerality we can assume that they correspond to the eigenvalues of smallest and largestmodulus, respectively, and that ψ fixes x1 but not x2. From [10, Prop 4.6] we know that x1

5

Page 6: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

is a C1 point, and hence there is a unique supporting hyperplane to Ω at x1 and both φ andψ preserve it. This means that there are coordinates with respect to which both φ and ψare affine. In particular we can assume that

φ(x) = Ax, ψ(x) = Bx+ c,

with c 6= 0. We now examine the result of conjugating ψ by powers of φ.

φnψφ−n(x) = AnBA−n + Anc

The fixed point x2 (which had the largest eigenvalue) has now been moved to the origin andhas been projectively scaled to have eigenvalue 1. Since Ω is strictly convex, x2 is the uniqueattracting fixed point, and so after possibly passing to a subsequence we can assume thatAnc is a sequence of distinct vectors that converge to the origin. Since Anc = φnψφ−n(0)we see that φnψφ−n is also a distinct sequence. The elements of φnψφ−n all move pointson line between the fixed points of φ a fixed bounded distance, and so by Lemma 2.2 thissequence has a convergent subsequence, which by construction, is not eventually constant.The existence of such a sequence contradicts discreteness.

To close this section we briefly describe convex real projective structures on manifolds.For more details about real projective structures and more generally (G,X) structures see [22,25]. A real projective structure on an n-manifold M is an atlas of charts U → RPn such thatthe transition functions live in PGLn(R). We can globalize the data of an atlas by selectinga chart and constructing a local diffeomorphism D : M → RPn using analytic continuation.This construction also yields a representation ρ : π1(M)→ PGLn(R) that is equivariant withrespect to D. The map and representation are known as a developing map and holonomyrepresentation, respectively. Only the initial choice of chart results in ambiguity in thedeveloping construction, and different choices of initial charts will result in developing mapswhich differ by post composing by an element of g ∈ PGLn(R). Additionally, the holonomyrepresentations will differ by conjugation by g.

When the map D is a diffeomorphism onto a properly (resp. strictly) convex set wesay that the real projective structure is properly (resp. strictly) convex. A key example tokeep in mind are hyperbolic structures. Projectivizing the cone x2

1 + . . . x2n < x2

n+1 gives riseto the Klein model of hyperbolic space. The Klein model can be embedded in affine space asthe unit disk and so we realize hyperbolic structures as specific instances of strictly convexreal projective structures.

3 Local and Infinitesimal Deformations

Unless explicitly mentioned, Γ will henceforth denote the fundamental group of a complete,finite volume, hyperbolic 3-manifold. By Mostow rigidity, there is a unique class of repre-sentations of Γ that is faithful and has discrete image in SO(3, 1). We call this class the

6

Page 7: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

geometric representation and denote it [ρgeo]. From the previous section we know that realprojective structures on a manifold give rise to conjugacy classes of representations of its fun-damental group into PGLn(R). Therefore, if we want to study real projective structures nearthe complete hyperbolic structure it suffices to study conjugacy classes of representationsnear [ρgeo].

We now set some notation. LetR(Γ,PGL4(R)) be the PGL4(R) representation varietyof Γ. The group PGL4(R) acts on R(Γ,PGL4(R)) by conjugation, and the quotient by thisaction is the character variety, which we denote R(Γ,PGL4(R)). The character variety is notglobally a variety because of pathologies of the action by conjugation, however at ρgeo theaction is nice enough to guarantee that R(Γ,PGL4(R)) has the local structure of a variety.

Next, we define a refinement of R(Γ,PGL4(R)) that better controls the representa-tions on the boundary. Recall, that by Theorem 2.1 then only conjugacy class of parabolicelement that is capable of preserving a properly convex domain is

1 0 0 00 1 1 00 0 1 10 0 0 1

.

Because this is the conjugacy class of a parabolic element of SO(3, 1) we will refer to parabol-ics of this type as SO(3, 1) parabolics. LetR(Γ,PGL4(R))p be the elements ofR(Γ,PGL4(R))such that peripheral elements of Γ are mapped to SO(3, 1) parabolics, and let the relativecharacter variety, denoted R(Γ,PGL4(R))p, be the corresponding quotient by the actionPGL4(R).

If ρ is a representation, then a deformation of ρ is a smooth map, σ(t) : (−ε, ε) →R(Γ,PGL4(R)) such that σ(0) = ρ. Often times we will denote σ(t) by σt. If a class [σ] isan isolated point of the R(Γ,PGL4(R)) then we say that Γ is locally projectively rigid at σ.Similarly, we say that [σ] is locally rigid relative relative to the boundary when it is isolatedin R(Γ,PGL4(R))p.

3.1 Twisted Cohomology and Infinitesimal Deformations

We now review how the cohomology of Γ with a certain system of local coefficients helps toinfinitesimally parameterize conjugacy classes of deformations of representations. For moredetails on cohomology see [7, Chap III]

Let σ0 : Γ → PGL4(R) be a representation, then we can define a cochain com-plex, Cn(Γ, sl(4)σ0) as function from Γn to sl(4), with differential dn : Cn(Γ, sl(4)σ0) →Cn+1(Γ, sl(4)ρ0) where dnφ(γ1, . . . , γn+1) is given by

γ1 · φ(γ2, . . . , γn+1) +n∑i=1

(−1)iφ(γ1, . . . , γi−1, γiγi+1, . . . , γn+1) + (−1)n+1φ(γ1, . . . , γn),

where γ ∈ Γ acts on sl(4) by Ad(σ0(γ)). Letting Zn(Γ, sl(4)σ0) and Bn(Γ, sl(4)σ0) be the ker-nel of dn and image of dn−1, respectively, we can form the cohomology groups H∗(Γ, sl(4)σ0).

7

Page 8: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

To see how this construction is related to deformations, let σt be a deformation of σ0,then since PGL4(R) is a Lie group we can use a series expansion and write σt(γ) as

σt(γ) = (I + z(γ)t+O(t2))σ0(γ), (3.1)

where γ ∈ Γ and z is a map from Γ into sl(4), which we call an infinitesimal deformation.The above construction would have worked for any smooth function from R to PGL4(R),but the fact that σt is a homomorphism for each t puts strong restrictions on z. Let γ andγ′ be elements of Γ, then

σt(γγ′) = (I + z(γγ′)t+O(t2))σ0(γγ′) (3.2)

andσt(γ)σt(γ

′) = (I + z(γ)t+O(t2))σ0(γ)(I + z(γ′) +O(t2))σ0(γ′)

= (I + (z(γ) + γ · z(γ′))t+O(t2))σ0(γγ′),

where the action is the adjoint action of σ0, given by γ ·M = σ0(γ)Mσ0(γ)−1. By focusingon the linear terms of the two power series in (3.2), we find that z(γγ′) = z(γ) + γ · z(γ′),and so z ∈ Z1(Γ, sl(4)σ0), the set of 1-cocycles twisted by the action of σ0. Since we thinkof deformations coming from conjugation as uninteresting, we now analyze which elementsin Z1(Γ, sl(4)σ0) arise from these types of deformations. Let ct be a smooth curve in sl(4)

such that c0 = I, and consider the deformation σt(γ) = c−1t σ0(γ)ct. Again we write σt(γ) as

a power series we find that

c−1t σ0(γ)ct = (I− zct+O(t2))σ0(γ)(I + zct+O(t2)) = (I + (zc−γ · zc)t+O(t2))σ0(γ), (3.3)

and again by looking at linear terms we learn z(γ) = zc − γ · zc and so z is a 1-coboundary.We therefore conclude that studying infinitesimal deformations near σ0 up to conju-

gacy boils down to studying H1(Γ, sl(4)σ0). In the case where H1(Γ, sl(4)σ0) = 0 we saythat Γ is infinitesimally projectively rigid at σ0, and when the map ι∗ : H1(Γ, sl(4)σ0) →H1(π1(M), sl(4)σ0) induce by the inclusion ι : ∂M → M is injective we say that Γ is in-finitesimally projectively rigid relative to the boundary at σ0. The following theorem of Weil[26] shows the strong relationship between infinitesimal and local rigidity.

Theorem 3.1. If Γ is infinitesimally projectively rigid at σ then Γ is locally projectivelyrigid at σ.

More generally the dimension of H1(Γ, sl(4)ρgeo) is an upper bound for the dimension

of R(Γ,PGL4(R)) at [ρgeo] (see [15, Sec 2]). However, it is important to remember that ingeneral the character variety need not be smooth. When this occurs this bound is not sharpand so the converse to Theorem 3.1 is false.

8

Page 9: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

3.2 Decomposing H1(Γ, sl(4)ρ0)

In order to simplify the study of H1(Γ, sl(4)ρ0) we will decompose it into two factors usinga decomposition of sl(4). Consider the symmetric matrix

J =

1 0 0 00 1 0 00 0 1 00 0 0 −1

Following [13, 15] we see that this gives rise to the following decomposition of PSO(3, 1)modules.

sl(4) = so(3, 1)⊕ v, (3.4)

where so(3, 1) = a ∈ sl(4) | atJ = −Ja and v = a ∈ sl(4) | atJ = Ja. These two spacescan be thought of as the ±1-eigenspaces of the involution a 7→ −JatJ . This splitting of sl(4)gives rise to a splitting of the cohomology groups, namely

H∗(Γ, sl(4)ρ0) = H∗(Γ, so(3, 1)ρ0)⊕H∗(Γ, vρ0). (3.5)

If ρgeo is the geometric representation then the first factor of (3.5) is well understood. By workof Garland [11], H1(M, so(3, 1)ρgeo) injects into H1(∂M, so(3, 1)ρgeo) and work of Thurston

[25] shows that dimH1(M, so(3, 1)ρgeo) = 2k, where k is the number of cusps of M .

Now that we understand the structure of H1(Γ, so(3, 1)ρ0) we turn our attention to

the other factor of (3.5). The inclusion ι : ∂M →M induces a map ι∗ : H1(M, sl(4)ρgeo)→H1(∂M, sl(4)ρgeo) on cohomology. We will refer to the kernel of this map as the sl(4)ρgeo-

cuspidal cohomology. In [13], Porti and Heusener analyze the image of this map. The portionthat we will use can be summarized by the following theorem, which can be thought of as atwisted cohomology analogue of the classical half lives/half dies theorem.

Theorem 3.2. Let ρgeo be the geometric representation of a finite volume hyperbolic 3-manifold with k cusps, then dim(ι∗(H1(Γ, vρgeo)))=k. Furthermore, if ∂M = tki=1∂Mi, then

there exists γ = tki=1γi with γi ⊂ ∂Mi, such that ι∗(H1(Γ, vρgeo)) injects into⊕k

i=1 H1(γi, vρgeo)

3.3 Bending

Now that we have some upper bounds on the dimensions of our deformations spaces wewant to begin to understand what types of deformations exist near the discrete faithfulrepresentation. The most well know construction of such deformations is bending along atotally geodesic hypersurface. The goal of this section is to explain the bending constructionand prove the following theorem on the effects of bending on the peripheral subgroups

Theorem 3.3. Let S be a totally geodesic hypersurface and let [ρt] ∈ R(Γ,PGLn+1(R)),obtained by bending along S. Then [ρt] is contained in R(Γ,PGLn+1(R))p if and only if S isclosed or each curve dual to the intersection of ∂M and S has zero signed intersection withS.

9

Page 10: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

Following [15], let M be a finite volume hyperbolic manifold of dimension ≥ 3 andρgeo be its geometric representation. Next, let S be a properly embedded, totally geodesic,hypersurface. Such a hypersurface gives rise to a curve of representations in R(Γ,PGL4(R))passing through ρ0 as follows. We begin with a lemma showing that the hypersurface S givesrise to an element of sl(n+ 1) that is invariant under ∆ = π1(S) but not all of Γ = π1(Γ).

Lemma 3.4. Let M and S as above, then there exists a unique 1-dimensional subspace ofsl(n + 1) that is invariant under ∆. Furthermore this subspace is generated by a conjugatein PGLn+1(R) of (

−n 00 I

),

where I is the n× n identity matrix.

Proof. Γ is a subgroup of PO(n, 1) (the projective orthogonal group of the form x21 + x2

2 +. . . x2

n − x2n+1). Since S is totally geodesic we can assume that after conjugation that ∆ pre-

serves both the hyperplane where x1 = 0 and its orthogonal complement which is generatedby (1, 0, . . . , 0). Hence if A ∈ ∆ then

A =

(1 0T

0 A

),

where A ∈ PO(n−1, 1) (the projective orthogonal group of the form x22+x2

3+. . .+x2n−x2

n+1)and 0 ∈ Rn. If x ∈ sl(n + 1) is invariant under ∆ then we know that B(t) = exp (tx)

commutes with every A ∈ ∆. If we write B(t) =

(b11 b12

b21 b22

), where b ∈ R, bT12, b21 ∈ Rn,

and b22 ∈ SLn(R), then(b11 b12

Ab21 Ab22

)=

(1 0

0 A

)(b11 b12

b21 b22

)=

(b11 b12

b21 b22

)(1 0

0 A

)=

(b11 b12A

b21 b22A

).

From this computation we learn that b12 and b21 are invariants of ∆ and that b22 is inthe commutator of ∆ in SLn(R). However, the representation of ∆ into PO(n − 1, 1) isirreducible, and so the only matrices that commute with every element of ∆ are scalarmatrices and the only invariant vector of ∆ is 0, and so

B =

(e−nλt 0

0 eλtI

),

where I is the identity matrix. Differentiating B(t) at t = 0 we find that

x =

(−nλ 0

0 λI

),

and the result follows.

10

Page 11: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

The vector from Lemma 3.4 will be called a bending cocycle, and we will denote it xS.We can now define a family of deformations of ρ. The construction breaks into two casesdepending on whether or not S is separating.

If S is separating then Γ splits as the following amalgamated free product:

Γ ∼= Γ1 ∗∆ Γ2,

where Γi are the fundamental groups of the components of the complement of S in M , andwe can define a family of representations ρt as follows. Since ρ is irreducible we know thatxS is not invariant under all of Γ and so we can assume without loss of generality that it isnot invariant under Γ2. So let ρt|Γ1 = ρ and ρt|Γ2 = Ad(exp (txS)) · ρ. Since these two mapsagree on ∆ they give a well defined family of homomorphisms of Γ, such that ρ0 = ρ.

If S is nonseparating, then Γ is realized as the following HNN extension:

Γ ∼= Γ′∗∆,

where Γ′ is the fundamental group of M\S. If we let α be a curve dual to S then we candefine a family of homomorphisms through ρ by ρt|Γ′ = ρ and ρt(α) = exp (txS)ρ(α). SincexS is invariant under ∆ the values of ρt(ι1(∆)) do not depend on t, where ι1 is the inclusionof the positive boundary component of a regular neighborhood of S into M\S, and so wehave well defined homomorphisms of the HNN extension.

In both cases ρt gives rise to a non-trivial curve of representations and by examiningthe class of ρt ∈ H1(Γ, sl(n+1)ρ0) Johnson and Millson [15] showed that [ρt] actually definesa non-trivial path in R(Γ,PGLn+1(R)).

We can now prove Theorem 3.3.

Proof of Theorem 3.3. In the case that S is closed and separating a peripheral element,γ ∈ Γ, is contained in either Γ1 or Γ2 since it is disjoint from S. In this case ρt(γ) is eitherρ(γ) or some conjugate of ρ(γ). In either case we have not changed the conjugacy classof any peripheral elements and so [ρt] is a curve of representations in R(Γ,PGLn+1(R))p.Similarly if S is closed and non-separating then if γ ∈ Γ is peripheral, then γ ∈ π1(M\S),and so its conjugacy class does not depend on t.

In the case that S is non-compact we need to analyze its intersection with the bound-ary of M more carefully. First, let M ′ be a finite sheeted cover of M and S ′ a lift of Sin M ′. If bending along S preserves the peripheral structure of M then bending along S ′

will preserve the peripheral structure of M ′. Therefore, without loss of generality we maypass to a finite sheeted cover of M such that ∂M ′ = tki=1Ti, where each of the Ti is a torus(this is possible by work of [20]). By looking in the universal cover, it is easy to see thatsince S is properly embedded and totally geodesic that for any boundary component Ti,Ti ∩ S = tlj=1tj, where the tj are parallel n− 2 dimensional tori. The result of the bendingdeformation on the boundary will be to simultaneously bend along all of these parallel tori,however some of the parallel tori may bend in opposite directions and can sometimes cancelwith one another. In fact, if we look at a curve α ∈ Ti dual to one (hence all) of the parallel

11

Page 12: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

tori, then intersection points of α with S are in bijective correspondence with the tj, andthe direction of the bending corresponds to the signed intersection number of α with S.

When S is separating the signed intersection number of α with S is always zero,and so bending along S has no effect on the boundary. When S is non-separating then thesigned intersection can be either zero of non-zero. In the case where the intersection numberis non-zero the boundary becomes non-parabolic after bending. This can be seen easily bylooking at the eigenvalues of peripheral elements after they have been bent.

Remark 3.5. The Whitehead link is an example where bending along a non-separating totallygeodesic surface has no effect on the peripheral subgroup of one of the components (See section5.2)

Corollary 3.6. Let M be a finite volume, hyperbolic 3-manifold. If M is locally projectivelyrigid relative the boundary near the geometric representation then M contains no closed,embedded totally geodesic surfaces or finite volume, embedded, separating, totally geodesicsurfaces.

The Menasco-Reid conjecture [21] asserts that hyperbolic knot complements do notcontain closed, totally geodesic surfaces. As a consequence of Corollary 3.6 we see that anyhyperbolic knot complement that is locally projectively rigid relative to the boundary nearthe geometric representation will satisfy the conclusion of the Menasco-Reid conjecture.

4 Some Normal Forms

The goal of this section will be to examine various normal forms into which we can put twonon commuting SO(3, 1) parabolics. In [23], Riley shows how two non commuting parabolicelements a and b inside of SL2(C) can be simultaneously conjugated into the following form:

a =

(1 10 1

), b =

(1 0ω 1

). (4.1)

In this same spirit we would like to take two SO(3, 1) parabolics A and B that are sufficientlyclose to the SO(3, 1) parabolics A0 and B0 in SO(3, 1) and show that A and B can besimultaneously conjugated into the following normal form, similar to (4.1).

A =

1 0 2 1 + a14

0 1 2 10 0 1 10 0 0 1

, B =

1 0 0 0b21 1 0 0

b31 + b21b32 2b32 1 0b21 + b41 2 0 1

. (4.2)

At first, this may appear to be an odd normal for, however, it provides a nice symmetrybetween A and B and their inverses. To start we need to discuss how to build homomorphismfrom SL2(C) to PGL4(R) that takes our old favorite normal forms from SL2(C) to our new

12

Page 13: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

favorite normal forms in PGL4(R). More precisely if a and b are two parabolics in PSL2(C),then our homomorphism should take a and b to matrices of the form (4.2)

Let

X =

1 x12 x13 x14

x12 x22 x23 x24

x13 x23 x33 x34

x14 x24 x34 x44

be a symmetric matrix. If detX < 0, then X has signature (3, 1) (up to ±I). We wantto discover the restrictions placed on the coefficients of A, B, and X by knowing that thequadratic form determined by X is preserved by matrices of type (4.2). The restriction thatATXA = X and BTXB = X tell us that

a14 = −2b32, b31 = 0, b41 = −2, x12 = −1, x13 = 2b32, x14 = −b21/2, (4.3)

x22 = 1, x23 = −2b32, x24 = 2b232, x33 = b21, x34 = −b21b32, x44 = b21b

232

With these restrictions we see that our matrix X looks like

X =

1 −1 2b32 −b21/2−1 1 −2b32 2b2

32

2b32 −2b32 b21 −b21b32

−b21/2 2b232 −b21b32 b21b

232

For the time being we will assume that the entries of A and B satisfy (4.3).

If we let x, y, z, t be coordinates for R4 then we see that the quadratic form on R4 byX is

x2 − 2xy + y2 − b21xt+ 4b232yt− 2b21b32zt+ 4b32xz − 4b32yz + b21z

2 + b21b232t

2 (4.4)

Let d = b21 − 4b232, then a simple calculation shows that detX < 0 if and only if d > 0.

Next, consider the map of quadratic spaces from (R4, X) to (Herm2,−Det), whereHerm2 is the space of 2× 2 hermitian matrices, that takes (x, y, z, t) to

(x x− y + 2b32z − 2b232t+ i(

√dz − b32

√dt)

x− y + 2b32z − 2b232t− i(√dz − b32

√dt) dt

)(4.5)

Let M be a 2× 2 matrix and let N ∈ SL2(C) act on M by N ·M = NMN∗, where∗ denotes the conjugate transpose operator on matrices. This action is linear and clearlypreserves determinants and therefore gives us a map, φ′, from SL2(C) to SO(3, 1). Since−I acts trivially this map descends to another map that we also call φ′ from PSL2(C) toSO(3, 1). A simple calculation shows that

φ′((

1 i/√d

0 1

))=

1 0 2 1− 2b32

0 1 2 10 0 1 10 0 0 1

.

13

Page 14: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

Let g ∈ PSL2(C) be a matrix that fixes 0 and ∞ and sends

(1 10 1

)to

(1 i/

√d

0 1

),

then if we let φ = φ′ cg, where cg is conjugation by g, then

φ

((1 10 1

))=

1 0 2 1− 2b32

0 1 2 10 0 1 10 0 0 1

.

A simple, yet tedious, computation shows that if

b21 = |ω|2 and b32 = Re(ω)/2 (4.6)

then φ will take elements of the form (4.1) to elements of the form (4.2). With theseassumptions on b21 and b32 we see that

d = b21 − 4b232 = |ω|2 − Re(ω)2 = Im(ω)2,

and so as long as Im(ω) 6= 0, φ(a) and φ(b) will preserve a form of signature (3, 1). One ofthe utilities of the previous computation is that it can help us identify the discrete faithfulrepresentation in PGL4(R) in the normal form (4.2), provided that we have found the discretefaithful representation in the normal form (4.1) inside of SL2(C). To see this observe that ifwe have the discrete, faithful representation in the form (4.1), then combining the value ofω with equations (4.3) and (4.6) we can construct the entries of our matrices in the normalform (4.2).

Let F2 = 〈α, β〉 be the free group on two letters and let R(F2,PGL4(R))p be the setof homomorphisms, f , from F2 to PGL4(R) such that f(α) and f(β) are SO(3, 1) parabolics.The topology of R(F2,PGL4(R))p is induced by convergence of the generators. There is anatural action of PGL4(R) on R(F2,PGL4(R))p by conjugation, and we denote the quotientof this action by R(F2,PGL4(R))p. We have the following lemma about the local structureof R(F2,PGL4(R))p.

Lemma 4.1. Let f0 ∈ R(F2,PGL4(R))p satisfy the following conditions

1. 〈f0(α), f0(β)〉 is irreducible and conjugate into SO(3, 1).

2. 〈f0(α), f0(β)〉 is not conjugate into SO(2, 1).

Then for f ∈ R(F2,PGL4(R))p sufficiently close to f0 there exists a unique (up to ±I)element G ∈ SL4(R) such that

G−1f(α)G =

1 0 2 1 + a14

0 1 2 10 0 1 10 0 0 1

, G−1f(β)G =

1 0 0 0b21 1 0 0

b31 + b21b32 2b32 1 0b21 + b41 2 0 1

Additionally, the map taking f to its normal form is continuous.

14

Page 15: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

Proof. The previous argument combined with properties 1 and 2 ensure that f0(α) and f0(β)can be put into the form (4.2). Let A = f(α) and B = f(β). Let EA and EB be the 1-eigenspaces of A and B, respectively. Since both A and B are SO(3, 1) parabolics both ofthese spaces are 2-dimensional. Irreducibility is an open condition we can assume that f isalso irreducible and so EA and EB have trivial intersection. Therefore R4 = EA⊕EB. If weselect a basis with respect to this decomposition then our matrices will be of the followingblock form. (

I AU0 AL

),

(BU 0BL I

).

Observe that 1 is the only eigenvalue of AL (resp. BU) and that neither of these matrices isdiagonalizable (otherwise (A − I)2 = 0 (resp. (B − I)2 = 0) and so A (resp. B) would nothave the right Jordan form). Thus we can further conjugate EA and EB so that

AL =

(1 a34

0 1

), BU =

(1 0b21 1

),

where a34 6= 0 6= b21. Conjugacies that preserve this form are all of the formu11 0 0 0u21 u22 0 00 0 u33 u34

0 0 0 u44

.

Finally, a tedious computation1 allows us to determine that there exist unique values of theuijs that will finish putting our matrices in the desired normal form. Note that the existenceof solutions depends on the fact that the entries of A and B are close to the entries of f0(α)and f0(β), which live in SO(3, 1).

Lemma 4.1 gives us the following information about the dimension of R(F2,PGL4(R))p.

Corollary 4.2. The space R(F2,PGL4(R))p is 5-dimensional.

We conclude this section by introducing another normal form for SO(3, 1) parabolics.If we begin with A and B in form (4.2) then we can conjugate by the matrix

V =

1 0 0 0

2−b21−b414

2+b21+b414

0 00 0 1

22b32+b21b32+b32b41−1

2

0 0 0 2+b21+b412

and then change variables, the resulting form will be

A =

1 0 1 a14

0 1 1 a24

0 0 1 a34

0 0 0 1

, B =

1 0 0 0b21 1 0 0b31 1 1 01 1 0 1

(4.7)

1This computation is greatly expedited by using Mathematica.

15

Page 16: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

Conjugation by V makes sense because by equation (4.3) and (4.6), 2 + b21 + b41 6= 0, andso V is non-singular.

5 Two Bridge Examples

In this section we will examine the deformations of various two bridge knots and links. Twobridge knots and links always admit presentations of a particularly nice form. Given a twobridge knot or link, K, with rational number p/q where q is odd, relatively prime to p and0 < q < p there is always a presentation of the form

π1(S3\K) = 〈A,B | AW = WB〉, (5.1)

where A and B are meridians of the knot. The word W can be determined explicitly fromthe rational number, p/q, see [18] for details. We now wish to examine deformations of Γ =π1(S3\K) that give rise to elements of sl(4)ρgeo-cuspidal cohomology. Such a deformationsmust preserve the conjugacy class of both of the meridians of Γ, and so we can assume thatthroughout the deformation our meridian matrices, A and B, are of the form (4.7). Thereforewe can find deformations by solving the matrix

AW −WB = 0 (5.2)

over R.

5.1 Figure-eight knot

The figure-eight knot has rational number 5/3, and so the word W = BA−1B−1A. Solving(5.2) we find that

b31 = 2, a34 =b21

2(b21 − 2), a24 =

1

(b21 − 2), a14 =

3− b21

b21 − 2, (5.3)

and so we have found a 1 parameter family of deformations. However, this family does notpreserve the conjugacy class of any non-meridional peripheral element. A longitude of thefigure-eight is given by L = BA−1B−1A2B−1A−1B, and a simple computation shows that if

(5.3) is satisfied then tr(L) = 48+(b21−2)4

8(b21−2). This curve of representations corresponds to the

cohomology classes guaranteed by Theorem 3.2, where γ1 can be taken to be the longitude.If we want the entire boundary to be parabolic we must add the equation tr(L) = 4 to (5.2)and when we solve we find that over R the only solution is

b21 = 4, b31 = 2, a14 = −1

2, a34 = 1, a24 =

1

2. (5.4)

This proves Theorem 1.1 for the figure-eight knot.

16

Page 17: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

Figure 1: The Whitehead link and its totally geodesic, thrice punctured sphere.

5.2 The Whitehead Link

The Whitehead link has rational number 8/3, and so the word W = BAB−1A−1B−1AB. Ifwe again solve (5.2) we see that

b21 = 4, b31 = −1, a14 = 0, a24 = −2, a34 = 2, (5.5)

and again we see that the Whitehead link has no deformations preserving the conjugacyclasses of the boundary elements near the geometric representation, thus proving Theorem1.1 for the Whitehead link. This time it was not necessary to place any restriction on thetrace of a longitude in order to get a unique solution. Theorem 3.2 tells us that there are still2 dimensions of infinitesimal deformations of Γ that we have not yet accounted for. However,we can find deformations that give rise to these extra cohomology classes. To see this noticethat there is a totally geodesic surface that intersects one component of the Whitehead linkin a longitude that we can bend along. With reference to Figure 5.2, we denote the totallygeodesic surface S, and the two cusps of the Whitehead link by C1 and C2. If we bendalong this surface the effect on C1 will be to deform the meridian (or any non-longitudinal,peripheral curve) and leave the longitude fixed. This bending has no effect on C2, since C2

intersects S in two oppositely oriented copies of the meridian, and the bending along thesetwo meridians cancel one another (see Theorem 3.3). This picture is symmetric and so thereis another totally geodesic surface bounding the longitude of C2 and the same argumentshows that we can find another family of deformations.

Remark 5.1. Similar computations have been done for the two bridge knots with rationalnumber 7/3 and 9/5 and in both cases the solutions form a discrete set, however in these twoexamples it is not known if the cohomology classes coming from Theorem 3.2 are integrable.The details of these computations can be found in [2] and serve to complete the proof ofTheorem 1.1.

17

Page 18: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

6 Rigidity and Flexibility After Surgery

In this section we examine the relationship between deformations of a cusped hyperbolicmanifold and deformations of manifolds resulting from surgery. The overall idea is as follows:suppose that M is a 1-cusped hyperbolic 3-manifold of finite volume, α is a slope on ∂M , andM(α) is the manifold resulting from surgery along α. If M(α) is hyperbolic with geometricrepresentation ρgeo and ρt is a non-trivial family of deformations of ρgeo into PGL4(R). Sinceπ1(M(α)) is a quotient of π1(M) we can pull ρt back to a non-trivial family of representations,ρt, such that ρt(α) = 1 for all t (It is important to remember that ρ0 is not the geometricrepresentation for M). In terms of cohomology, we find that the image of the elementω ∈ H1(M, sl(4) ρ0) corresponding to ρt is trivial in H1(α, sl(4) ρ0). With this in mind we

will call slope α-rigid if the map H1(M, vρgeo) → H1(α, vρgeo) is non-trivial. Roughly theidea is that if a slope is rigid then we can find deformations that do not infinitesimally fixα. The calculations from section 5.2 show that either meridian of the Whitehead link is arigid slope.

6.1 The cohomology of ∂M

Before proceeding we need to understand the structure of the cohomology of the boundary.For details of the facts in this section see [13, Section 5]. Let T is a component of ∂M andlet γ1 and γ2 be generators of π1(T ) whose angle is not an integral multiple of π/3, then wehave an injection,

H1(T, vρ0)ι∗γ1⊕ι

∗γ2→ H1(γ1, vρ0)⊕H1(γ2, vρ0), (6.1)

where ι∗γi is the map induced on cohomology by the inclusion γi → T . Additionally, if ρu isthe holonomy of an incomplete, hyperbolic structure, then

H∗(∂M, vρu) ∼= H∗(∂M,R)⊗ vρu(π1(∂M)), (6.2)

where vρu(π1(∂M)) are the elements of v invariant under ρi(π1(∂M)). Additionally, for all suchrepresentations the image of H1(M, vρu) under ι∗ in H1(∂M, vρu) is 1 dimensional.

6.2 Deformations coming from symmetries

From the previous section we know that the map H1(M, vρ) → H1(∂M, vρ) has rank 1whenever ρ is the holonomy of an incomplete, hyperbolic structure, and we would like to knowhow this image sits insideH1(∂M, vρ). Additionally, we would like to know when infinitesimaldeformations coming from cohomology classes can be integrated to actual deformations.Certain symmetries of M can help us to answer this question.

Suppose that M is the complement of a hyperbolic, amphicheiral knot complement,then M admits an orientation reversing symmetry, φ, that sends the longitude to itself andthe meridian to its inverse. The existence of such a symmetry places strong restrictions on

18

Page 19: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

the shape of the cusp. Let m and l be the meridian and longitude of M , then it is alwayspossible to conjugate so that

ρ(m) =

(ea/2 1

0 e−a/2

), ρ(l) =

(eb/2 τρ0 e−b/2

).

The value τρ is easily seen to be an invariant of the conjugacy class of ρ and we will henceforthrefer to it as the τ invariant (see also [6, App B]). When ρ is the geometric representationthis coincides with the cusp shape.

Suppose that [ρ] is a representation such that the metric completion of H3/ρ(π1(M)) isthe cone manifold M(α/0) or M(0/α), where M(α/0) (resp. M(0/α)) is the cone manifoldsobtained by Dehn filling along the meridian (resp. longitude) with a solid torus with singularlongitude of cone angle 2π/α. Using the τ invariant we can show that the holonomy of thesingular locus of these cone manifolds is a pure translation.

Lemma 6.1. Let M be a hyperbolic, amphicheiral, knot complement, and let α ≥ 2. IfM(α/0) is hyperbolic then the holonomy of the longitude is a pure translation. Similarly, ifM(0/α) is hyperbolic then the holonomy around the meridian is a pure translation.

Proof. We prove the result for M(α/0). Provided that M(α/0) is hyperbolic and α ≥ 2,rigidity results for cone manifolds from [14, 16] provides the existence of an element Aφ ∈PSL2(C) such that ρ(φ(γ)) = Aφρ(γ)A−1

φ , where γ is complex conjugation of the entries ofthe matrix γ. Thus we see that τρφ is τρ. On the other hand φ preserves l and sends m toits inverse and so we see that the τρφ is also equal to −τρ, and so we see that τρ is purelyimaginary.

The fact that ρ(m) and ρ(l) commute gives the following relationship between a, band τρ:

τρ sinh(a/2) = sinh(b/2). (6.3)

Using the fact that tr(ρ(m))/2 = cosh(a/2) and the analogous relationship for tr(ρ(l)) wecan rewrite (6.3) as

τ 2ρ (tr2(ρ(m))− 4) + 4 = tr2(ρ(l)). (6.4)

For the cone manifold M(α/0), ρ(m) is elliptic and so tr2(ρ(m)) is real and between 0 and 4.This forces tr2(ρ(l)) to be real and greater than 4, and we see that ρ(l) is a pure translation.The proof for M(0/α) is identical with the roles of m and l being exchanged.

With this in mind we can prove the following generalization of [13, Lemma 8.2], whichlets us know that φ induce maps on cohomology that act as we would expect.

Lemma 6.2. Let M be the complement of a hyperbolic, amphicheiral knot with geometricrepresentation ρgeo, and let ρu be the holonomy of a incomplete, hyperbolic structure whosecompletion is M(α/0) or M(0/α), where α ≥ 2, then

1. H∗(∂M, vρu) ∼= H∗(∂M,R)⊗vρu(π1(∂M)) and φ∗u = φ∗⊗ Id, where φu is the map inducedby φ on H∗(∂M, vρu), φ∗ is the map induced by φ on H∗(∂M,R), and

19

Page 20: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

2. ι∗l φ∗0 = ι∗l and ι∗m φ∗0 = −ι∗m, where φ∗0 is the map induced by φ on H1(∂M, vρgeo).

Proof. The proof that H∗(∂M, vρu) ∼= H∗(∂M,R)⊗ vρu(π1(∂M)) is found in [13]. For the firstpart we will prove the result of M(α/0) (the other case can be treated identically). By theprevious paragraph we know that ρu(m) is elliptic and that ρu(l) is a pure translation. Oncewe have made this observation our proof is identical to the proof given in [13].

For the second part, we begin by observing that by Mostow rigidity there is a matrixA0 ∈ PO(3, 1) such that ρ0(φ(γ)) = A0 · ρ0(γ), where the action here is by conjugation. Thefact that our knot is amphicheiral tells us that the cusp shape of M is imaginary, and so inPSL2(C) we can assume that

ρ0(m) =

(1 10 1

)and ρ0(l) =

(1 ic0 1

),

where c is some positive real number. Under the standard embedding of PSL2(C) into SL4R(as the copy of SO(3, 1) that preserves the form x2

1 + x22 + x2

3 − x24) we see that

ρ0(m) = exp

0 0 0 00 0 −1 10 1 0 00 1 0 0

and ρ0(l) = exp

0 0 −c c0 0 0 0c 0 0 0c 0 0 0

.

Thus we see that A0 = T

1 0 0 00 −1 0 00 0 1 00 0 0 1

, where T is some parabolic isometry that fixing

the vector (0, 0, 1, 1), which corresponds to ∞ in the upper half space model of H3. Such aT will be of the form

T = exp

0 0 −a a0 0 −b ba b 0 0a b 0 0

,

where a and b can be thought of as the real and imaginary parts of the complex number thatdetermines this parabolic translation. Since the cusp shape is imaginary the angle betweenm and l is π/2 and we know from [13, Lemma 5.5] that the cohomology classes given by thecocycles zm and zl generate H1(∂M, vρ0). Here zm is given by zm(l) = 0 and zm(m) = al,where

al =

−1 0 0 00 3 0 00 0 −1 00 0 0 −1

,

20

Page 21: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

and zl given by zl(m) = 0 and zl(l) = am, where

am =

3 0 0 00 −1 0 00 0 −1 00 0 0 −1

.

Observe that φ∗0zl(m) = 0 = zl(m) and that

φ∗0zm(m) = A−10 · zm(m−1) = −A−1

0 · ρ0(m−1) · al.

In [13] it is shown how the cup product associated to the Killing form on v gives rise to a non-degenerate pairing. We can now use this cup product to see that ι∗mφ

∗0zm(m) and −ι∗mzm(m)

are cohomologous. The cup product yields a map H1(m, vρ0)⊗H0(m, vρ0)→ H1(m,R) ∼= Rgiven by (a ∪ b)(m) = B(a(m), b) = 8tr (a(m)b) (where we are thinking of H1(m,R) ashomomorphisms from Z to R). Observe that

(ι∗mφ∗0zm ∪ am)(m) = B(φ∗0zm(m), am) = 32 = −B(al, am) = −(ι∗mzm ∪ am)(m).

Since this pairing is non-degenerate and since [ι∗mφ∗0zm] and [−ι∗mzm] both live in the same 1

dimensional vector space we see that they must be equal. Thus we see that ι∗m φ∗0 = −ι∗m.After observing that

(ι∗l φ∗0zl ∪ al)(l) = B(φ∗0zl(l), al) = −32 = B(am, al) = (ι∗l zl ∪ ll)(l),

a similar argument shows that ι∗l φ∗0 = ι∗l

This lemma immediately helps us to answer the question of how the image ofH1(M, vρ0)sits inside of H1(∂M, vρ0). Since π1(∂M) is invariant under φ we see that the image ofH1(M, vρ0) is invariant under the involution φ∗, and so the image will be either the ±1eigenspace of φ∗. In light of Lemma 6.2 we see that these eigenspaces sit inside of H1(l, vρ0)and H1(m, vρ0), respectively. Under the hypotheses of Theorem 1.3 the previous fact isenough to show that certain infinitesimal deformations are integrable.

Proof of Theorem 1.3. In this proof O = M(n/0) and N will denote a regular neighborhoodof the singular locus of M(n/0). Hence we can realize O as M t N . By using a Taylorexpansion we see that given a family of representations ρt of π1(O) into SL4(R) such that ρ0

is the geometric representation, ρgeo, of O. We can therefore write

ρt(γ) = (I + u1(γ)t+ u2(γ)t2 + . . .)ρgeo(γ),

where the ui are 1-cochains. The ρt will be homomorphisms if and only if for each k

δuk +k∑i=1

ui ∪ uk−i = 0, (6.5)

21

Page 22: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

where a∪ b is the 2-cochain given by (a∪ b)(c, d) = a(c)c · b(d), where the action is byconjugation. Conversely, if we are given a cocycle u1 and a collection uk such that (6.5)is satisfied for each k we can apply a deep theorem of Artin [1] to show that we can findan actual deformation ρt in PGL4(R) that is infinitesimally equal to u1. Using Weil rigidityand the splitting (3.5) we see that H1(M(n/0), sl(4)ρn) ∼= H1(M(n/0), vρn), where ρn isthe holonomy of the incomplete structure whose completion is M(n/0). Therefore, givena cohomology class [u1], we can assume that [u1] ∈ H1(M(n/0), vρn), and we need to findcochains uk satisfying (6.5). In order to simplify notation H1(∗, vρn) will be denoted H1(∗).The orbifold O is finitely covered by an aspherical manifold, and so by combining a transferargument [13] with the fact that singular cohomology and group cohomology coincide foraspherical manifolds [27], we can conclude that group cohomology for π1(O) is the sameas singular cohomology for O. Therefore we can use a Mayer-Vietoris sequence to analyzecohomology. Consider the following section of the sequence.

H0(M)⊕H0(N)→ H0(∂M)→ H1(O)→ H1(M)⊕H1(N)→ H1(∂M). (6.6)

Next, we will determine the cohomology of N . Since N has the homotopy type ofS1 it will only have cohomology in dimension 0 and 1. Since ρn(∂M) = ρn(N) we see thatH0(N) ∼= H0(∂M) (both are 1-dimensional). Finally by duality we see that H1(N) is also1-dimensional. Additionally, H0(O) is trivial since ρn is irreducible. Combining these facts,we see that the first arrow is an isomorphism and thus the penultimate arrow of (6.6) isinjective. We also learn that H1(O) injects into H1(M), because if a cohomology class fromH1(O) dies in H1(M) then exactness tells us that it must also die when mapped into H1(N)(since H1(N) injects into H1(∂M)). However, this contradicts the fact that H1(O) injectsinto H1(M) ⊕ H1(N). Using the fact that the longitude is a rigid slope, Lemma 6.2, and[13, Cor 6.6] we see that φ∗ acts as the identity on H1(O), but since H1(M) and H1(N) havethe 1-eigenspace of φ∗ as their image in H1(∂M), the last arrow of (6.6) is not a surjection,and so H1(O) is 1-dimensional.

Duality tells us that H2(O) is 1-dimensional and we will now show that φ∗ act asmultiplication by -1. By duality we see that H3(O) = 0 and so the Mayer-Vietoris sequencecontains the following piece.

H1(M)⊕H1(N)→ H1(∂M)→ H2(O)→ H2(M)⊕H2(N)→ H2(∂M)→ 0. (6.7)

Since H2(O) is 1-dimensional, the second arrow of (6.7) is either trivial or surjective.If this arrow is trivial then the third arrow is an injection and thus an isomorphism fordimensional reasons, but this is a contradiction since the penultimate arrow is a surjectionand H2(∂M) is non-trivial. Since the first arrow of (6.7) has the 1-eigenspace of φ∗ as itsimage we see that the -1-eigenspace of φ∗ surjects H2(O). However, since the Mayer-Vietorissequence is natural and φ respects the splitting of O into M ∪ N we see that φ∗ acts onH2(O) as -1.

22

Page 23: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

We can now construct the ui. Let [u1] be a generator of H1(O) (we know this is onedimensional by duality). Since φ is an isometry of a finite volume hyperbolic manifold weknow that it has finite order when viewed as an element of Out(π1(M)) [25], and so thereexists a finite order map ψ that is conjugate to φ. Because the maps φ and ψ are conjugate,they act in the same way on cohomology [7]. Let L be the order of ψ. Since ψ acts as theidentity on H1(O), the cocycle

u∗1 =1

L(u1 + φ(u1) + . . .+ φL−1(ui))

is both invariant under ψ and cohomologous to u1. By replacing u1 with u∗1 we can assumethat u1 is invariant under ψ. Next observe that

−u1 ∪ u1 ∼ ψ(u1 ∪ u1) = ψ(u1) ∪ ψ(u1) = u1 ∪ u1,

and so we see that u1 ∪ u1 is cohomologous to 0, and so there exists a 1-cochain, u2, suchthat δu2 + u1 ∪ u1 = 0. Using the same averaging trick as before we can replace u2 by theψ-invariant cochain u∗2. By invariance of u1 we see that u∗2 has the same boundary as u2 sothis replacement does not affect the first part of our construction. Again we see that

−(u1 ∪ u2 + u2 ∪ u1) ∼ ψ(u1 ∪ u2 + u2 ∪ u1) = u1 ∪ u2 + u2 ∪ u1,

and so there exist u3 such that (6.5) is satisfied. Repeating this process indefinitely, we canfind the sequence ui satisfying (6.5), and thus by Artin’s theorem we can find our desireddeformation.

References

[1] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. MR0232018 (38#344)

[2] Sam Ballas, Examples of two bridge knot and link deformations, 2012. http://www.ma.utexas.edu/users/sballas/research/Examples.nb.

[3] Yves Benoist, Convexes divisibles. I, Algebraic groups and arithmetic, 2004, pp. 339–374. MR2094116(2005h:37073)

[4] , Convexes divisibles. III, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 5, 793–832. MR2195260(2007b:22011)

[5] , Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math. 164 (2006), no. 2,249–278. MR2218481 (2007g:22007)

[6] Michel Boileau and Joan Porti, Geometrization of 3-orbifolds of cyclic type, Asterisque 272 (2001), 208.Appendix A by Michael Heusener and Porti. MR1844891 (2002f:57034)

[7] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag,New York, 1982. MR1324339 (96a:20072)

23

Page 24: Deformations of Non-Compact, Projective Manifolds€¦ · Deformations of Non-Compact, Projective Manifolds Samuel A. Ballas November 1, 2012 Abstract In this paper, we demonstrate

[8] D. Cooper, D. D. Long, and M. B. Thistlethwaite, Computing varieties of representations of hyperbolic3-manifolds into SL(4,R), Experiment. Math. 15 (2006), no. 3, 291–305. MR2264468 (2007k:57032)

[9] , Flexing closed hyperbolic manifolds, Geom. Topol. 11 (2007), 2413–2440. MR2372851(2008k:57031)

[10] D. Cooper, D. D. Long, and S. Tillmann, On Convex Projective Manifolds and Cusps, ArXiv e-prints(September 2011), available at 1109.0585.

[11] Howard Garland, A rigidity theorem for discrete subgroups, Trans. Amer. Math. Soc. 129 (1967), 1–25.MR0214102 (35 #4953)

[12] A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79(1985), no. 2, 225–246. MR778125 (86g:57003)

[13] Michael Heusener and Joan Porti, Infinitesimal projective rigidity under Dehn filling, Geom. Topol. 15(2011), no. 4, 2017–2071. MR2860986

[14] Craig D. Hodgson and Steven P. Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehnsurgery, J. Differential Geom. 48 (1998), no. 1, 1–59. MR1622600 (99b:57030)

[15] Dennis Johnson and John J. Millson, Deformation spaces associated to compact hyperbolic manifolds,Discrete groups in geometry and analysis (New Haven, Conn., 1984), 1987, pp. 48–106. MR900823(88j:22010)

[16] Sadayoshi Kojima, Deformations of hyperbolic 3-cone-manifolds, J. Differential Geom. 49 (1998), no. 3,469–516. MR1669649 (2000d:57023)

[17] J.-L. Koszul, Deformations de connexions localement plates, Ann. Inst. Fourier (Grenoble) 18 (1968),no. fasc. 1, 103–114. MR0239529 (39 #886)

[18] Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts inMathematics, vol. 219, Springer-Verlag, New York, 2003. MR1937957 (2004i:57021)

[19] L. Marquis, Exemples de varietes projectives strictement convexes de volume fini en dimension quel-conque, ArXiv e-prints (April 2010), available at 1004.3706.

[20] D. B. McReynolds, A. W. Reid, and M. Stover, Collisions at infinity in hyperbolic manifolds, ArXive-prints (February 2012), available at 1202.5906.

[21] William Menasco and Alan W. Reid, Totally geodesic surfaces in hyperbolic link complements, Topology’90 (Columbus, OH, 1990), 1992, pp. 215–226. MR1184413 (94g:57016)

[22] John G. Ratcliffe, Foundations of hyperbolic manifolds, Second, Graduate Texts in Mathematics,vol. 149, Springer, New York, 2006. MR2249478 (2007d:57029)

[23] Robert Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3) 24 (1972), 217–242. MR0300267 (45 #9313)

[24] Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish Inc., Houston, TX,1990. Corrected reprint of the 1976 original. MR1277811 (95c:57018)

[25] Willam P Thurston, The geometry and topology of three-manifolds, Princeton lecture notes (UnknownMonth 1978).

[26] Andre Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157. MR0169956(30 #199)

[27] George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York, 1978. MR516508 (80b:55001)

24