arXiv:1508.04794v3 [math.GT] 17 Dec 2017

CONVEX PROJECTIVE STRUCTURES ON
NON-HYPERBOLIC THREE-MANIFOLDS
SAMUEL A. BALLAS, JEFFREY DANCIGER, AND GYE-SEON LEE
Abstract. Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is
topologically the union along tori and Klein bottles of finitely many
sub-manifolds each of which admits a complete finite volume hyperbolic
structure on its interior. We describe some initial results in the direction
of a potential converse to Benoist’s theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to
convex projective structures with totally geodesic torus boundary. Such
structures may be convexly glued together whenever the geometry at
the boundary matches up. In particular, we prove that many doubles of
cusped hyperbolic three-manifolds admit convex projective structures.

1. Introduction
The previous decade has seen tremendous progress in the study of threedimensional manifolds. Much of that progress stems from Perelman’s proof
of Thurston’s Geometrization Conjecture which states that any closed orientable prime three-manifold admits a decomposition into geometric pieces
modeled on the eight homogeneous Thurston geometries. However, because
these geometric pieces do not glue together in any sensible geometric way,
there are some questions about three-manifolds for which a Thurston geometric decomposition of the manifold may not be useful. One example is the
question of linearity of three-manifold groups, ie whether a three-manifold
fundamental group admits a faithful linear representation and in which dimensions. While, in most cases, the Thurston geometric structure on each
piece of a geometric decomposition determines a faithful linear representation of its fundamental group, these representations can not be directly
synthesized into a representation of the fundamental group of the entire
manifold. In order to make progress on this and other problems, it is natural, given a manifold of interest, to search for a homogeneous geometry
capable of describing the entire manifold all at once.
This article studies properly convex real projective structures on threemanifolds. A domain Ω in the real projective space RPn is called properly
convex if there is an affine chart containing Ω in which Ω is convex and
bounded. A properly convex projective n–manifold is the quotient Γ\Ω of
a convex domain Ω by a discrete group Γ of projective transformations preserving Ω. Given a manifold N , a properly convex projective structure on
1

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N is a diffeomorphism of N with some properly convex projective manifold
Γ\Ω, considered up to certain equivalence. A convex projective structure
therefore induces a representation, called the holonomy representation, identifying π1 N with the discrete subgroup Γ ⊂ PGLn+1 R. Hyperbolic structures are special examples of convex real projective structures, but there
are many non-hyperbolic manifolds that admit such structures as well. See
Benoist [2, 3] or Kapovich [30] for some examples. See Benoist [4] for a
survey of the subject of convex projective structures on closed manifolds.
We mention that there are simple examples of convex projective structures
on three-manifolds coming from a convex hull construction applied to lower
dimensional domains; such structures are called decomposable and are not
of interest to us in the present article. By work of Benoist [2], the Thurston
geometric decomposition of any closed three-manifold that admits an indecomposable properly convex projective structure contains only hyperbolic
pieces glued together along tori and Klein bottles. We are concerned with
the converse problem: If a closed three-manifold N has geometric decomposition containing only hyperbolic pieces, must N admit a properly convex
projective structure? Our main theorem gives a positive answer to this
question in a special case.
Theorem 1.1. Let M be a compact, connected, orientable three-manifold
with a union of tori as boundary such that the interior of M admits a finite volume hyperbolic structure which is infinitesimally projectively rigid
rel boundary. Then the double N = 2M of M admits a properly convex
projective structure.
Using cube complex techniques, Przytycki–Wise [34] showed that any
mixed three-manifold, and therefore any manifold N as in Theorem 1.1, has
linear fundamental group. However, their methods give no control on the
dimension of the linear representation. On the other hand, it was shown
by Button [7] that there exist three-manifold groups which admit no linear
representation in dimension 4 or lower. To determine the smallest possible
dimension of a linear representation of a three-manifold group remains an
interesting open question. Since the holonomy representation of a convex
projective structure lifts to the special linear group (see Section 2.1), we
obtain:
Corollary 1.2. Let N = 2M be as in Theorem 1.1. Then π1 N admits a
discrete faithful representation into SL4 R.
The Corollary says that the property of linearity of the two hyperbolic
pieces in N may be extended to all of N . Indeed, the proof of Theorem 1.1
will show that the representation of π1 N in the Corollary may be chosen
such that the restriction to each copy of π1 M is arbitrarily close to the
holonomy representation of the finite volume hyperbolic structure on M .
We note that the assumption of infinitesimal projective rigidity rel boundary (Definition 3.1), studied by Heusener–Porti in [25], is satisfied for many

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hyperbolic manifolds, for example for infinitely many fillings of one component of the Whitehead link. On the other hand, this assumption does fail
in certain cases, for example when the hyperbolic structure on M contains
a totally geodesic surface. A related rigidity condition in closed hyperbolic three-manifolds was studied in Cooper–Long–Thistlethwaite [15, 14]
and shown experimentally to hold very often in small examples. However,
in the setting of cusped hyperbolic manifolds, it is not yet known in what
degree of generality infinitesimal projective rigidity rel boundary will hold.
Nonetheless, the Theorem gives a large new source of examples of convex
projective structures on non-hyperbolic three-manifolds. The only other
known examples come from taking covers of convex projective reflection
orbifolds. Benoist [2] constructed the first example of such an orbifold by
realizing a truncation polyhedron, ie a polyhedron obtained from a threedimensional tetrahedron by successively truncating vertices, as a reflection
polyhedron in projective space. Later, Marquis [32] completely classified the
three-dimensional convex projective orbifolds obtained from projective truncation polyhedra. Generalizing Benoist’s examples, Choi–Lee–Marquis [12]
are currently classifying convex projective reflection polyhedra and studying
their deformation theory.
1.1. Convex projective structures with totally geodesic boundary.
The proof of Theorem 1.1 is motivated by Benoist’s beautiful theory [2]
describing the geometry of convex projective structures on three-manifolds,
which we briefly review here. If N = Γ\Ω is a properly convex projective
closed three-manifold which is indecomposable, then either (i) Ω is strictly
convex and N admits a hyperbolic structure or (ii) Ω is not strictly convex
and the points on ∂Ω at which strict convexity fails form a dense set in ∂Ω
each component of which bounds a properly embedded triangle, which is the
intersection of Ω with a hyperplane. We will refer to these triangles as totally geodesic triangles. Each totally geodesic triangle descends to a totally
geodesic embedded torus or Klein bottle in N and after cutting along these
tori and Klein bottles, N is decomposed into a union of properly convex
sub-manifolds Mi with totally geodesic boundary. Each piece in this decomposition (which topologically is exactly the JSJ or geometric decomposition
of N ) admits a hyperbolic structure.
In light of the Benoist theory, in order to construct convex projective
structures on non-hyperbolic three-manifolds, we first need a source of convex projective building blocks, ie convex projective manifolds with totally
geodesic boundary. Under suitable assumptions, we are able to find such
structures by deforming the hyperbolic structure.
Theorem 1.3. Let M be a connected, orientable, finite volume, non-compact
hyperbolic three-manifold which is infinitesimally projectively rigid rel boundary. Then M admits nearby properly convex projective structures where each
cusp becomes a principal totally geodesic boundary component.

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The term principal totally geodesic boundary (see Definition 5.2, following
Goldman [23] in the setting of convex projective surfaces) refers to a totally
geodesic boundary component which admits a convex thickening. That all
totally geodesic boundary components are principal is a necessary condition
for a convex projective manifold to appear as a sub-manifold in the Benoist
decomposition of a closed convex projective manifold described above.
The deformations in the previous theorem may be understood by analogy
with the related phenomenon of the deformation of a two-dimensional finite
volume hyperbolic surface whose cusp “opens up” to a very small geodesic
circle coming in from infinity. Hyperbolic surfaces with geodesic boundary are indeed convex projective structures; the associated convex domain,
a subset of the hyperbolic plane, has in its boundary a dense collection
of segments, each of which covers the geodesic boundary circle. Although
cusp opening is not possible in three-dimensional hyperbolic geometry by
Mostow–Prasad rigidity, Theorem 1.3 shows that it is possible in the category of convex projective manifolds. Indeed, as the convex projective structures in the conclusion of the Theorem approach the hyperbolic structure,
the totally geodesic boundary tori become very small (with respect to the
Hilbert metric) and escape to infinity as the totally geodesic triangles in the
boundary of the associated convex domain converge to points; see Figure 1.

Figure 1. The principal totally geodesic triangles in the
boundaries of the convex domains from Theorem 1.3 collapse
to points as the convex domains converge to the round ball.
The proof of Theorem 1.3 boils down to a transversality argument in the
space Hom(π1 ∂M, PGL4 R) of representations of the peripheral fundamental
groups π1 ∂M . The assumption of infinitesimal projective rigidity rel boundary, also appearing in Theorem 1.1, guarantees that Hom(π1 M, PGL4 R) is
smooth at the holonomy representation ρhyp of the complete hyperbolic
structure and that (an augmented version of) the restriction map res :
Hom(π1 M, PGL4 R) → Hom(π1 ∂M, PGL4 R) submerses a neighborhood of
ρhyp onto a submanifold of Hom(π1 ∂M, PGL4 R). We prove that this submanifold transversely intersects a certain family of diagonalizable representations constructed explicitly in Lemma 4.3. This family of diagonalizable
representations is a partial slice (never tangent to conjugation orbits) whose

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construction is geometric in nature (see Section 4.1 for details). For dimensional reasons the intersection is positive dimensional and thus we can
find representations of π1 M into PGL4 R whose restriction to the fundamental group of each boundary component is diagonalizable over the reals.
Finally, we are able to conclude that the resulting representations are the
holonomy representations of convex projective structures as in the Theorem
using a “holonomy principle” that follows from recent work of Cooper–Long–
Tillmann [13] or Choi [11].
We note that the proof of Theorem 1.3 encounters immediate problems
upon removing the assumption that M is orientable. Indeed, Poincaré duality, applied to both M and its torus boundary components, is used to
determine the dimensions of relevant representation spaces.
1.2. Convex gluing. Consider a finite disjoint union M of properly convex
projective three-manifolds and two disjoint closed sub-manifolds ∂ and ∂ 0
of its boundary, each of which is a finite disjoint union of principal totally
geodesic tori. Given a homeomorphism f : ∂ → ∂ 0 , let Mf denote the
(topological) manifold obtained by identifying ∂ with ∂ 0 via f ; the topology
of Mf depends only on the isotopy class of f . By straightforward general
arguments, the projective structure on M defines a projective structure (in
fact many different structures) on Mf provided that the holonomy matching
condition is satisfied: For each component ∂j of ∂, there exists gj ∈ PGL4 R
such that
(1.1)

ρ0j (f∗ γ) = gj ρj (γ)gj−1

for all γ ∈ π1 ∂j , where ρj and ρ0j denote the holonomy representations for
the projective structures on neighborhoods of ∂j and ∂j0 = f (∂j ) respectively.
Indeed, if this condition is satisfied then f may be isotoped to a projective
map on each component of ∂ and the pieces of M may be glued together
projectively. We prove that the resulting projective structure on Mf may
in fact be taken to be properly convex. The following theorem is the threedimensional analogue of a result of Goldman [23] in the setting of convex
projective surfaces.
Theorem 1.4. Let M be a finite disjoint union of properly convex projective
three-manifolds, let ∂ and ∂ 0 be disjoint closed sub-manifolds of its boundary,
each of which is a disjoint union of principal totally geodesic tori, and let
f : ∂ → ∂ 0 be a homeomorphism. Assume that the holonomy matching
condition (1.1) is satisfied along all components of ∂. Then Mf admits a
projective structure in which all connected components are properly convex.
The natural map M → Mf is isotopic to a projective map.
When M = M t M is the disjoint union of two copies of the same properly convex projective manifold M with principal totally geodesic boundary
and f : ∂M → ∂M is the identity map, then the holonomy matching condition (1.1) is trivially satisfied. In this case Mf = 2M is the double of M .
Hence Theorem 1.3 and Theorem 1.4 imply Theorem 1.1.

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In general, any given projective manifold M and homeomorphism f : ∂ →
∂ 0 as in the hypotheses of Theorem 1.4 are unlikely to satisfy the holonomy
matching condition (1.1). To find a properly convex projective structure on
the glued manifold Mf , one may attempt to deform the properly convex
projective structures on the connected components of M with the goal of
aligning the geometry of the principal totally geodesic boundary tori of ∂ and
∂ 0 so that (1.1) is satisfied. However, global deformation theory problems
such as this are in general very difficult. Consider for example the case that
M = M1 tM2 is a disjoint union of two three-manifolds, each with boundary
homeomorphic to a torus ∂ = ∂M1 and ∂ 0 = ∂M2 , and that f : ∂ → ∂ 0 is any
fixed gluing homeomorphism. The space of representations Hom(π1 ∂j , A)
into the diagonal subgroup A ⊂ PGL4 R is six-dimensional. Furthermore,
the subset of representations which extend to π1 M1 is a half-dimensional
(Lagrangian) subvariety as is the subset of representations which extend via
f∗ to π1 M2 . Therefore, the expected intersection between these two sets is
zero-dimensional. We do not know any reason in general to expect this intersection to be non-empty. On the other hand, no example seems to be known
in which the intersection turns out to be empty. An experimental study of
some basic cases could prove enlightening; the authors hope to conduct such
experiments in future work. Of course, even if the matching problem (1.1)
is solved at the level of representations, one needs to find convex projective
structures realizing those representations as their holonomy representations
in order to apply Theorem 1.4. The following theorem shows that any deformation of the holonomy representation of a properly convex projective
manifold through representations with diagonalizable peripheral holonomy
is indeed the holonomy representation of a properly convex projective structure with totally geodesic boundary.
Theorem 1.5. Let M be a complete finite volume hyperbolic three-manifold.
Then the set of holonomy representations of properly convex projective structures on M with principal totally geodesic boundary is closed in the subspace
of representations in Hom(π1 M, PGL4 R) whose restriction to π1 ∂M is diagonalizable.
Indeed, this theorem is not difficult using our techniques. Any convex
projective structure on M with totally geodesic boundary yields a convex
projective structure on the double 2M by Theorem 1.4. By a theorem
of Benoist [1], the space of holonomy representations of convex projective
structures on this closed manifold 2M is closed in Hom(π1 2M, PGL4 R) and
any deformation of the original structure on 2M may be cut back into pieces
with principal totally geodesic boundary.
1.3. Gluing together covers of reflection orbifolds. In Section 7, we
apply the work of Benoist [2] and Marquis [32] and Theorem 1.4 to produce
many new examples of convex projective structures on non-hyperbolic threemanifolds N which, by contrast to Theorem 1.1, are not doubles. In each of

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these examples, the pieces {Mi } of the JSJ decomposition of N come from
covers of a reflection orbifold O with a cusp ∂ isomorphic to the Euclidean
(3, 3, 3)–triangle orbifold. Any convex projective structure on O induces a
convex projective structure at infinity on ∂, the space of which is well-known
to be one-dimensional. Considering only projective structures on the pieces
Mi which cover a convex projective structure on such an orbifold O greatly
simplifies the holonomy matching problem of Theorem 1.4. This technique
allows us, for example, to identify manifolds in the census of tetrahedral
cusped hyperbolic manifolds and gluing maps so that the resulting glued
up manifold admits properly convex projective structures. See Section 7 for
precise results.
Acknowledgements. We are thankful for helpful conversations with Yves
Benoist, Suhyoung Choi, Daryl Cooper, Ludovic Marquis, Maria Beatrice
Pozzetti and Anna Wienhard. We thank Joan Porti for suggesting the proofs
of Theorem 3.2 and Lemma 3.6 and for other helpful comments. We also
thank Daryl Cooper, Darren Long and Stephan Tillmann for making a preliminary version of their results [13] available to us. This work grew out of
discussions at the GEAR network retreats of 2012 and 2014. Additionally,
some of this work was done during the spring 2015 semester on Dynamics
on Moduli Spaces of Geometric Structures at MSRI and we thank MSRI for
support and for the stimulating environment there. Finally, we thank the
referees for suggesting some improvements.
S. Ballas was partially supported by the National Science Foundation under the grant DMS 1709097. J. Danciger was partially supported was partially supported by the National Science Foundation grants DMS 1103939
and DMS 1510254 and by an Alfred P. Sloan Foundation fellowship. G.-S.
Lee was supported by the DFG research grant “Higher Teichmüller Theory” and by the European Research Council under ERC-Consolidator Grant
614733. The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
2. Background
2.1. Properly convex domains. The n–dimensional real projective space
RPn is the quotient of Rn+1 \{0} by the action of R× by scaling. A point
in RPn is an equivalence class [v] of vectors v ∈ Rn+1 \{0}. The projective
general linear group PGLn+1 R, the quotient of GLn+1 R by its center, acts
faithfully on RPn .
The image, or projectivization, of a two-dimensional vector subspace under the quotient map is called a projective line and the image of an n–plane
in RPn is called a projective hyperplane. Each hyperplane is determined by a
unique projective class of linear functionals on Rn+1 , so the space of hyperplanes identifies with the dual projective space RPn∗ , which is the projective

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space associated to (Rn+1 )∗ . This correspondence is known as projective duality. The natural action of GLn+1 R on (Rn+1 )∗ descends to a faithful action
of PGLn+1 R on RPn∗ .
Let H be a projective hyperplane. The complement of H in RPn is
called an affine patch and is denoted AH . Up to change of coordinates by a
projective transformation, ie an element of PGLn+1 R, any affine patch AH
may be identified with the standard affine patch:
{[x1 , . . . , xn , 1] ∈ RPn | (x1 , . . . , xn ) ∈ Rn }.
A subset Ω of RPn is called convex if it is contained in some affine patch
(ie is disjoint from a projective hyperplane) and its intersection with every
projective line is connected. If in addition its closure Ω is contained in an
affine patch then Ω is properly convex. Equivalently, Ω is properly convex if
Ω does not contain any complete affine line. The boundary ∂Ω := Ω \ Int(Ω)
is said to be strictly convex at p ∈ ∂Ω if p is not contained in the interior
of any affine line segment in ∂Ω. If Ω is properly convex and ∂Ω is strictly
convex at every point p ∈ ∂Ω, then we say that Ω is strictly convex.
Every open properly convex domain Ω gives rise to a dual domain
Ω∗ = {[φ] ∈ RPn∗ | φ(v) 6= 0 ∀ [v] ∈ Ω}.
It is easily verified that Ω∗ is also open, non-empty, properly convex and
that (Ω∗ )∗ = Ω. For each p = [v] ∈ ∂Ω there is a (possibly non-unique)
q = [φ] ∈ ∂Ω∗ such that φ(v) = 0. The projective hyperplane dual to q is
called a supporting hyperplane at p. A point p ∈ ∂Ω has a unique supporting
hyperplane if and only if p is a C 1 point of the boundary.
Let Sn denote the n–sphere, realized as the quotient of Rn+1 \{0} by positive scaling, and let π : Sn → RPn denote the 2-to-1 covering map. The
automorphisms of Sn are given by the linear transformations SL±
n+1 R with
determinant ±1. Let [T ] ∈ PGLn+1 R be an equivalence class of linear transformations. By scaling we may arrange that T ∈ SL±
n+1 R. Additionally,
±
±
since T ∈ SLn+1 R if and only if −T ∈ SLn+1 R, there is a 2-to-1 covering
SL±
n+1 R → PGLn+1 R, which by abuse we also call π, given by π(T ) = [T ].
If Ω is a properly convex domain then we let SL± (Ω) and PGL(Ω) denote
−1
the subgroups of SL±
n+1 R and PGLn+1 R preserving π (Ω) and Ω, respectively. Then π restricts to a 2-to-1 covering homomorphism from SL± (Ω)
onto PGL(Ω). When Ω is properly convex, a homomorphic section of π
is constructed as follows. Since Ω ⊂ RPn is properly convex, the preimage of Ω under π will consist of two connected components. An element
T ∈ SL± (Ω) either preserves both of these components individually or it
interchanges them. Furthermore, T preserves both components if and only
if −T interchanges them. The desired section of π is defined by mapping
[T ] ∈ PGL(Ω) to the unique lift of [T ] to SL± (Ω) that preserves both components of π −1 (Ω). Using this section we are able to identify PGL(Ω) with
a subgroup of SL± (Ω). We may therefore regard elements of PGL(Ω) as
linear transformations when convenient.

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2.2. Projective structures on manifolds. Let M be an n–manifold.
A projective atlas on M is a collection of charts, φα : Uα → RPn , that
cover M with the property that if Uα and Uβ are charts with non-empty intersection then φα ◦ φ−1
β is locally the restriction of an element of PGLn+1 R.
Every projective atlas determines a unique maximal projective atlas and
we call a maximal projective atlas on M a projective structure on M . In
other words, a projective structure on M is a (G, X) structure on M (see
Ratcliffe [35] for an introduction to (G, X) structures) where (G, X) is real
projective geometry: G = PGLn+1 R and X = RPn . A manifold equipped
with a projective structure is called a projective manifold. Note that a projective manifold is also a smooth manifold.
If M and M 0 are projective manifolds of the same dimension, then a
continuous map f : M → M 0 is projective if for each pair of charts φ :
U → RPn of M and ψ : V → RPn of M 0 such that U ∩ f −1 (V ) 6= ∅ the
map ψ ◦ f ◦ φ−1 : φ(U ∩ f −1 (V )) → ψ(f (U ) ∩ V ) agrees with an element of
PGLn+1 R on each connected component. Such a map is necessarily smooth.
If in addition f is a diffeomorphism we say that f is a projective equivalence.
f → M the local data of a projective
After fixing a universal covering M
structure may be replaced, via analytic continuation of the charts, by global
f → RPn is a local diffeomorphism called a
data (dev, ρ), where dev : M
developing map which is equivariant with respect to a representation ρ :
π1 M → PGLn+1 R called the holonomy representation in the sense that
dev(γx) = ρ(γ)dev(x),
f. Any two developing maps dev and dev 0 for
for all γ ∈ π1 M and all x ∈ M
the same structure satisfy that dev 0 = g ◦ dev for some g ∈ PGLn+1 R; the
respective holonomy representations are related by conjugation: ρ0 = gρg −1 .
More generally, two projective structures on M are considered equivalent if
developing maps dev, dev 0 for the respective structures are related by the
f of a diffeomorphism ϕ
equation dev 0 = g ◦ dev ◦ ϕ
e where ϕ
e is the lift to M
of M that is isotopic to the identity.
Suppose that we are given a projective structure on M with development
pair (dev, ρ). If dev is a diffeomorphism onto a convex (resp. properly
convex or strictly convex) domain Ω of RPn then we say that the projective
structure is convex (resp. properly convex or strictly convex ). In this case
the holonomy representation ρ is faithful with image a discrete subgroup of
PGL(Ω). Here are some useful equivalent characterizations of convexity.
Theorem 2.1 (Goldman [23, Proposition 3.1]). Let M be a projective n–
manifold. Then the following are equivalent:
(1) M is convex.
(2) Every path in M is homotopic (rel endpoints) to a unique geodesic
segment, ie a segment that develops into a projective line.
(3) M is projectively equivalent to the natural projective structure on
f) ⊂ RPn is convex and Γ = ρ(π1 M ) ⊂
Γ\Ω where Ω = dev(M

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PGL(Ω) is a discrete group acting properly discontinuously and freely
on Ω.
Remark 2.2. If M = Γ\Ω is properly convex then M can be equipped with
the Hilbert metric (see de la Harpe [18]). Segments of projective lines, also
known as projective geodesics, are always geodesics in the Hilbert metric. If
Ω is strictly convex, then Hilbert geodesics are always projective geodesics.
Via the Klein model, the hyperbolic n–space Hn is realized as a ball in
RPn and its group of isometries is realized as PO(n, 1) ⊂ PGLn+1 R (see
Ratcliffe [35] for details). Thus a hyperbolic structure on a manifold is a
projective structure and a complete hyperbolic structure is a strictly convex
projective structure. Incomplete hyperbolic structures could be convex, e.g.
the interior of the convex core of a convex cocompact hyperbolic structure,
or could fail to be convex. For example, given a cusped hyperbolic manifold
M , Thurston’s generalized Dehn surgery space [37] consists of one complete
structure, which is strictly convex, and many incomplete structures, none of
which are convex. Some of those incomplete structures can be completed to
give hyperbolic structures on Dehn fillings of M , but even those incomplete
structures are not convex: the developing map is an infinite sheeted covering
onto the complement of a discrete countable collection of lines.
2.3. Deformation theory and projective manifolds with generalized cusps. There is a natural correspondence between (G, X) structures
and their holonomy representations often called the holonomy principle or
Ehresmann–Thurston principle. The correspondence, originally discovered
by Ehresmann [19] and popularized by Thurston [37], is a crucial tool in
the study of deformation spaces of geometric structures. We describe this
holonomy principle in the projective setting as well as some relevant generalizations in the setting of convex projective structures. For this discussion
we denote G = PGLn+1 R. Let M be the interior of a compact n–manifold
possibly with or without boundary, let Γ = π1 M , and let D(M ) be the space
of equivalence classes of marked projective structures on M . Let Hom(Γ, G)
denote the space of representations of Γ into G and let χ(Γ, G) denote its
quotient by the G action by conjugation. Let hol : D(M ) → χ(Γ, G) denote
the map that associates to each equivalence class of projective structure
the conjugacy class of its holonomy representation. Each space is equipped
with a natural topology; we refer the reader to Goldman [22] for details.
When M is closed, the holonomy principle states simply that the map
hol : D(M ) → χ(Γ, G) is a local homeomorphism; in other words, the small
deformations of a projective structure are, up to equivalence, parameterized
by small deformations of the conjugacy class of its holonomy representation.
In the case that ∂M is non-empty, the principle holds as stated only once
the definition of equivalence of projective structures is relaxed so that two
projective structures are considered equivalent if there is a diffeomorphism
of M which is projective, with respect to the one structure in the domain

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and the other in the target, away from some collar neighborhood of ∂M .
In general, the holonomy principle does not guarantee any control of the
geometry at the boundary; this is an important issue in many studies of
deformations of geometric structures (e.g. in the context of cone-manifold
structures, see Hodgson–Kerckhoff [27] or Danciger [17]), including this one.
Proofs of the holonomy principle are found in Canary–Epstein–Green [9] or
Goldman [22].
In the context of this paper, we need a more powerful holonomy principle
that allows for control of more refined geometric properties, specifically that
of convexity. Koszul [31] proved that when M is closed, proper convexity
is an open condition: a small deformation of the holonomy representation
of a properly convex projective structure is the holonomy representation of
a nearby properly convex projective structure. However, when ∂M 6= ∅,
the same statement fails. A simple example, that of incomplete hyperbolic structures nearby a complete hyperbolic structure, was already given
in the previous subsection. We now describe recent work of Cooper–Long–
Tillman [13] which, given a properly convex projective structure satisfying
certain assumptions, determines which deformations of the holonomy representation are the holonomy representation of a nearby properly convex
projective structure. Further, the result allows for some control over the
geometry at the boundary. We note that related results of Choi [10, 11]
on projective structures with radial ends also imply the holonomy principle
needed in our context.
A generalized cusp is a properly convex manifold B (with boundary) such
that B ∼
= ∂B×[0, ∞), ∂B is compact and strictly convex (ie locally the graph
of a strictly convex function), and π1 B is virtually nilpotent. The manifold
∂B is called the cusp cross section. We now discuss some motivating examples for this definition. The first is an end of a finite volume hyperbolic
manifold. In this case the strictly convex boundary ∂B of the generalized
cusp is the quotient of a horosphere by a virtually abelian group. The second
example, which is more relevant for our purposes, is a regular neighborhood
of a principal totally geodesic torus boundary component. The following
holonomy principle follows immediately from [13, Theorem 0.1].

Theorem 2.3. Let M = M c ∪ V be a connected n–manifold such that M c
is compact and connected, V ∼
= ∂V × [0, ∞), and ∂M c = ∂V. Suppose that
V1 , . . . , Vk are the components of V and that ρ is the holonomy of a properly
convex projective structure on M in which each Vi is a generalized cusp. If
ρ0 is sufficiently close to ρ in Hom(π1 M, G) and for each i, ρ0 |π1 Vi is the
holonomy of a generalized cusp structure on Vi , then ρ0 is the holonomy of
a properly convex structure on M in which each Vi is a generalized cusp.

We will use this theorem in Section 5 to prove Theorem 1.3.

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

12

3. Smoothness
We begin by setting some notation that will be used for the remainder of
the article. Henceforth, we let M denote an orientable finite volume noncompact hyperbolic three-manifold with k torus cusps, which we denote by
∂i for 1 ≤ i ≤ k. Let Γ = π1 M denote the fundamental group of M , and for
1 ≤ i ≤ k let ∆i denote a representative of the conjugacy class of peripheral
subgroups of the ith cusp. Each ∆i is isomorphic to Z × Z. We denote the
groups PGL4 R, SL±
4 R and SL4 R by G, G̃ and G̃0 , respectively. All three of
these Lie groups are locally isomorphic and thus have the same Lie algebra
which we will denote by g.
Let ρhyp be a representative of the unique conjugacy class of discrete
faithful representations of Γ into the isometry group of hyperbolic threespace. Via the Klein model, the isometry group of hyperbolic three-space
is realized as the subgroup PO(3, 1) of G that preserves the standard round
ball H3 in RP3 . Hence we may regard ρhyp as a point in the representation
space Hom(Γ, G). We also, by abuse, regard ρhyp as a point in the quotient
χ(Γ, G) = Hom(Γ, G)/G of the representation space by the action of G by
conjugation. The sense in which the quotient is taken may be left ambiguous;
if one desires χ(Γ, G) to have the structure of an algebraic variety, then the
Mumford geometric invariant theory quotient is needed rather than the naive
topological quotient. It is a standard fact that the two quotients agree (as
topological spaces) locally near any irreducible representation such as ρhyp .
We will also need to study the representation space Hom(∆i , G) of a
peripheral subgroup ∆i , which has no irreducible representations. In this
case the quotient by conjugation is not well-behaved. In the main sections
of the paper, we will avoid such issues by working exclusively in Hom rather
than in the quotient χ.
This section is dedicated to some basic results about the local structure
of Hom(Γ, G) and χ(Γ, G) near ρhyp . These results are straightforward and
many of them seem to be well-known to experts although we are not aware of
their existence anywhere in the current literature. Some analogous results
from the context of deformations in SL(2, C) and SL(n, C) can be found
in Boden–Friedl [5], Heusener–Medjerab [24], and Heusener–Porti–Suárez
Peiró [26]. The main result of this section shows that if a certain cohomological condition is satisfied (Definition 3.1), then ρhyp is a smooth point of
Hom(Γ, G) and χ(Γ, G). Since we are only interested in local behavior, it
is equivalent and will be marginally less cumbersome to work in the representation space Hom(Γ, G̃) for the matrix group G̃. Note that, since ρhyp
is the holonomy representation of a convex projective structure, it admits a
unique lift, denoted by abuse again by ρhyp , to G̃ and so does every nearby
representation (see Section 2.1).
3.1. Infinitesimal deformations and cohomology. Let ρt : Γ → G̃ be
a smooth path of representations into the matrix group G̃. Near t = 0, we

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

13

may express the path ρt as:
ρt (γ) = exp(tu(γ) + O(t2 ))ρ(γ),
d
where u : Γ → g, defined by γ 7→ dt
ρ (γ)ρ(γ)−1 , is a group cocycle with
t=0 t
coefficients in g twisted by the adjoint action of ρ = ρ0 . We denote the set
of such cocycles, defined by the condition u(γ1 γ2 ) = u(γ1 ) + Adρ(γ1 ) u(γ2 ),
by Zρ1 (Γ, g) and refer to its elements as infinitesimal deformations of ρ; it
is the Zariski tangent space of Hom(Γ, G̃) at ρ. The representation ρ is a
smooth point of Hom(Γ, G̃) if and only if each infinitesimal deformation u is
integrable, ie u is tangent to some path ρt as above. The space of coboundaries, denoted by Bρ1 (Γ, g), is the subspace of those cocycles b satisfying
the infinitesimal conjugacy condition: that there exists v ∈ g such that
b(γ) = v − Adρ(γ) v. Each such coboundary b is tangent to the conjugation
path ρt = ct ρc−1
at t = 0, where ct = exp(tv). The first cohomology group
t
Hρ1 (Γ, g) = Zρ1 (Γ, g)/Bρ1 (Γ, g) with coefficients in g twisted by the adjoint
action of ρ describes the infinitesimal deformations of ρ up to infinitesimal
conjugacy. In the case that ρ determines a smooth point of χ(Γ, G̃), this
cohomology group describes its Zariski tangent space.
The higher cohomology groups, with twisted coefficients in g, will not be
of use to us except in the following subsection. Given an infinitesimal deformation u, there is an infinite sequence of obstructions to the integrability of
u, each of which is an element of the second cohomology group Hρ2 (Γ, g).
It will be important to understand the relationship between the deformations of the representations of Γ with the deformations of representations of
the peripheral subgroups ∆1 , . . . , ∆k . The restriction map

res : Hom(Γ, G̃) → Hom(∆1 , G̃) × . . . × Hom(∆k , G̃)
is the product res = res1 × · · · × resk , where resi : Hom(Γ, G̃) → Hom(∆i , G̃)
denotes the restriction map induced by the inclusion ∆i ,→ Γ of the ith
peripheral subgroup into Γ. Each such map resi defines a restriction map
1
(∆i , g) and it is convenient
on group cohomology, (resi )∗ : Hρ1 (Γ, g) → Hres
iρ
to synthesize these into one linear map
res∗ : Hρ1 (Γ, g)

→

k
M

1
Hres
(∆i , g)
iρ

i=1

defined by
res∗ = (res1 )∗ ⊕ · · · ⊕ (resk )∗ .
When clear, we will abuse notation using res and resi to denote both the
restriction map on representations and on cohomology. We will also conserve
∗
space using Hρ∗ (∆i , g) to mean Hres
(∆i , g).
iρ
We note that, since M is aspherical, the group cohomology groups Hρ∗ (Γ, g)
coincide with the de Rham cohomology groups Hρ∗ (M, g) with coefficients in
the flat g-bundle over M associated to ρ. Similarly, there is a natural identification between Hρ∗ (∆i , g) and Hρ∗ (∂i , g) for each 1 ≤ i ≤ k and between

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

14

Lk

∗
i=1 Hρ (∆i , g)

and Hρ∗ (∂M, g). Although it will be more convenient for us
to work with group cohomology, this identification makes available the tools
commonly used in the study of cohomology of manifolds, e.g. the long exact
sequence of a relative pair and Poincaré duality (see e.g. Heusener–Porti [25]
for details).
3.2. Infinitesimal rigidity implies smoothness. The following property,
introduced and studied by Huesener–Porti [25], is the critical assumption in
Theorems 1.1 and 1.3.

Definition 3.1. Let ρhyp : Γ → SO(3, 1) ⊂ G̃ denote the holonomy representation of the complete finite volume hyperbolic structure on M . Then
M is called infinitesimally projectively rigid rel ∂M if the restriction map
L
res : Hρ1hyp (Γ, g) → ki=1 Hρ1hyp (∆i , g) is an injection.
The main theorem of this section is:
Theorem 3.2. Let M be an orientable complete finite volume hyperbolic
manifold with fundamental group Γ, and let ρhyp : Γ → SO(3, 1) ⊂ G̃ be
the holonomy representation of the complete hyperbolic structure. If M
is infinitesimally projectively rigid rel ∂M , then ρhyp is a smooth point of
Hom(Γ, G̃) and its conjugacy class is a smooth point of χ(Γ, G̃).
Remark 3.3. Huesener–Porti [25] prove that the condition of infinitesimal
projective rigidity rel boundary persists under infinitely many Dehn fillings.
They then show that there are infinitely many examples of one-cusped hyperbolic 3–manifolds that satsify the condition by studying fillings of the
Whitehead link, a two-cusped manifold that satisfies the condition. For example, the figure-eight knot complement as well as all but finitely many twist
knots are infinitesimally projective rigid rel boundary. In future work, we
hope to determine exactly which manifolds of the Hodgson–Weeks cusped
census are infinitesimally projectively rigid rel boundary. A related rigidity
condition in closed hyperbolic three-manifolds was studied in Cooper–Long–
Thistlethwaite [15, 14] and shown experimentally to hold very often in small
examples.
The proof of Theorem 3.2 requires several lemmas. Note that Lemma 3.4
does not require M to be orientable, but that Lemma 3.6 does.
Lemma 3.4. For each 1 ≤ i ≤ k, the restriction ρhyp i = resi (ρhyp ) of
ρhyp to the ith peripheral subgroup is a smooth point of Hom(∆i , G̃). Hence,
res(ρhyp ) is a smooth point of Hom(∆1 , G̃) × . . . × Hom(∆k , G̃).
Proof. We may work in G̃0 = SL4 R in place of G̃, since ρhyp i has image
in this smaller group. The variety Hom(∆i , G̃0 ) is the set of real points
of a complex affine variety Hom(∆i , SL4 C), which is defined over R. It
therefore suffices to prove that ρhyp i is a smooth point of Hom(∆i , SL4 C).
By work of Richardson [36, Theorem C], Hom(∆i , SL4 C) is an irreducible

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

15

(complex) affine variety and contains a dense set of representations whose
images consist of diagonalizable representations. Thus Hom(∆i , SL4 C) is an
18–dimensional complex variety. Heuristically this can be seen as follows:
pick two generators γ1 , γ2 for ∆i . An element ρ ∈ Hom(∆i , SL4 C) can map
γ1 arbitrarily and thus contributes 15 degrees of freedom. The only condition
on ρ(γ2 ) is that it commute with ρ(γ1 ). Generically, the image of γ1 will
be diagonalizable with distinct eigenvalues, so the centralizer Z(ρ(γ1 )) is
conjugate to the diagonal subgroup, which is three-dimensional.
To prove the Lemma, we must therefore show that the Zariski tangent
space to Hom(∆i , SL4 C) is 18–dimensional at ρhyp i . The image of ∆i under
ρhyp i is conjugate to a lattice in the Lie group of matrices of the form:


1 u v 21 (u2 + v 2 )
0 1 0

u


0 0 1

v
0 0 0
1
A simple calculation shows that the infinitesimal centralizer Hρ0hyp (∆i , g) is
i
3–dimensional. By Poincaré duality (see e.g. Hodgson–Kerckhoff [27]),
dim Hρ0hyp (∆i , g) = dim Hρ2hyp (∆i , g),
i

i

and therefore since the Euler characteristic of ∂M is zero, we have that
Hρ1hyp (∆i , g) has dimension six. Furthermore,
i

dim Hρ0hyp (∆i , g) + dim Bρ1hyp (∆i , g) = dim g = 15,
i

i

Bρ1hyp (∆i , g)
i

so the coboundaries
are 12–dimensional. Hence the Zariski
1
tangent space Zρhyp (∆i , g) has dimension 18 as desired.

i

Remark 3.5. Despite Lemma 3.4, the representation ρhyp i does not determine a smooth point of χ(∆i , G̃). Indeed, the conjugacy class of ρhyp i
contains the trivial representation in its closure. Hence, the naive quotient
of Hom(∆i , G̃) by conjugation is not Hausdorff at this point. The Mumford
GIT quotient avoids failure of Hausdorff-ness by identifying ρhyp i with the
trivial representation, which is not even a smooth point of Hom(∆i , G̃).
Next, we derive some relevant information about second cohomology
groups from the infinitesimal projective rigidity condition.
Lemma 3.6. If M is infinitesimally rigid rel boundary then the map
res∗ : Hρ2hyp (M, g) → Hρ2hyp (∂M, g)
is an injection.
Proof. Consider the long exact sequence of the pair (M, ∂M ), where the first
injection is by assumption:
res∗

0 → Hρ1hyp (M, g) ,→ Hρ1hyp (∂M, g)
res

→Hρ2hyp (M, ∂M, g) → Hρ2hyp (M, g) →∗ Hρ2hyp (∂M, g)

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

16

We showed above that dim Hρ1hyp (∂i , g) = 6 for each i and so we have
i
that dim Hρ1hyp (∂M, g) = 6k. By a standard Poincaré duality argument,
known as “half lives, half dies” (see e.g. Hodgson–Kerckhoff [27]), the image of Hρ1hyp (M, g) under the restriction map res∗ is 3k–dimensional and so
Hρ1hyp (M, g) is itself 3k–dimensional. By Poincaré duality, we have that the
group Hρ2hyp (M, ∂M, g) is also 3k–dimensional. We conclude that the map
Hρ1hyp (∂M, g) → Hρ2hyp (M, ∂M, g) must be a surjection and by exactness
that res∗ : Hρ2hyp (M, g) → Hρ2hyp (∂M, g) is also injective.

We now prove Theorem 3.2.
Proof of Theorem 3.2. To prove Theorem 3.2 we must show that each infinitesimal deformation u ∈ Zρ1hyp (Γ, g) is integrable. It is well-known that
integrability of u follows if we can show that the infinitely many obstructions all vanish (see e.g. Heusener–Porti [25, Section 8.2.4]). The obstructions to integrability are cohomology classes in Hρ2hyp (Γ, g). However, if one
of these obstruction classes is non-zero it would, by Lemma 3.6, map to a
L
non-zero class in ki=1 Hρ2hyp (∆i , g) obstructing the integrability of res∗ u in
Lk
1
i=1 Hρhyp (∆i , g). By Lemma 3.4, this is impossible. Hence all the obstructions vanish and u is an integrable infinitesimal deformation. This proves
that Hom(Γ, G̃) is smooth at ρhyp . Since ρhyp is an irreducible representation, the orbits of ρhyp and all nearby representations are closed. Since the
centralizer of all representations nearby ρhyp is constant (equal to ±I), we
conclude that χ(Γ, G̃) is a manifold near the conjugacy class of ρhyp .

3.3. The augmented restriction map. In this subsection, we formulate
and prove some results that are needed for the main transversality argument
in the next section, where the basic goal will be to find deformations of the
discrete faithful SO(3, 1) representation with certain desired behavior along
each of the peripheral subgroups. More specifically, we will construct a submanifold S ⊂ Hom(Z × Z, G̃) consisting of representations with the desired
behavior and then look for representations in Hom(Γ, G̃) whose restriction
to each peripheral subgroup are conjugate into S . For technical reasons, in
the execution of this strategy it will be more convenient to work with the
following augmented restriction map.
Definition 3.7. Let M be a finite volume hyperbolic 3–manifold with k
torus cusps and with fundamental group Γ. Then we define the augmented
restriction map of M denoted by rf
es : Hom(Γ, G̃) × G̃k−1 → Hom(∆1 , G̃) ×
· · · × Hom(∆k , G̃) by the formula
(ρ, g2 , . . . gk ) 7→ (res1 (ρ), c(g2 ) · res2 (ρ), . . . , c(gk ) · resk (ρ)) ,
where c(g) denotes the conjugation action by g.
The main result about rf
es that we will need for the transversality argument in Section 4 is:

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

17

Theorem 3.8. Let M be an orientable finite volume hyperbolic 3–manifold
with k cusps and with fundamental group Γ, and let ρhyp be the holonomy
representation of the complete hyperbolic structure. Assume that M satisfies
Theorem 3.2 so that Hom(Γ, G̃) is smooth at ρhyp . Then the augmented
restriction map
rf
es : Hom(Γ, G̃) × G̃k−1 → Hom(∆1 , G̃) × · · · × Hom(∆k , G̃)
is a local submersion onto a submanifold of codimension 3k at the point
(ρhyp , g2 , . . . , gk ) where g2 , . . . , gk are any elements of G̃.
Proof. Let U be a smooth neighborhood of ρhyp in Hom(Γ, G̃) whose elements are all irreducible representations, and let W = U × G̃k−1 . The proof
proceeds by showing that the rank of the augmented restriction map rf
es is
constant and equal to 15k for the points in W .
We begin by identifying the relevant tangent spaces. The tangent space
Tρ Hom(Γ, G̃) is Zρ1 (Γ, g) and the tangent space Tρ Hom(∆i , G̃) is Zρ1 (∆i , g).
1
Since any ρ ∈ U is irreducible we may identify Tg G̃ = Bc(g)ρ
(Γ, g) for any
g ∈ G̃. As a result, for p = (ρ, g2 , . . . , gk ) ∈ W , we may identify:
k


M
1
Bc(g
(Γ, g).
Tp Hom(Γ, G̃) × G̃k−1 = Zρ1 (Γ, g) ⊕
i )ρ
i=2

The following diagram commutes:
L
1
(Γ, g)
Zρ1 (Γ, g) ⊕ ki=2 Bc(g
i )ρ

rf
es∗

/ Z 1 (∆1 , g) ⊕
ρ

$1



Hρ1 (Γ, g)

res∗

/

Lk

Lk

1
i=2 Zc(gi )ρ (∆i , g)



$2

1
i=1 Hρ (∆i , g)

where $1 is projection to cohomology and $2 is projection to cohomology
1
in each factor followed by the identification of Hρ1 (∆i , g) and Hc(g
(∆i , g)
i )ρ
L
k
1
1
induced by c(gi ). Then ker($1 ) is given by Bρ (Γ, g) ⊕ i=2 Bc(gi )ρ (Γ, g)
L
1
and ker($2 ) is given by Bρ1 (∆1 , g) ⊕ ki=2 Bc(g
(∆i , g) so it is clear that
i )ρ
rf
es∗ (ker($1 )) = ker($2 ). After possibly shrinking U we may assume for any
ρ ∈ U that for each i, the dimension of Hρ1 (∆i , g) is 6, that Hom(∆i , G̃) is
smooth and 18–dimensional at c(gi )resi (ρ), and therefore that the dimension
of ker($2 ) is 12k. Furthermore, the “half lives, half dies” argument from
L
Lemma 3.6 shows that the rank of res∗ : Hρ1 (Γ, g) → ki=1 Hρ1 (∆i , g), and
hence of $2 ◦ rf
es∗ , is 3k. Combining these facts we see that the rank of rf
es∗
must be 15k.

Remark 3.9. When k = 1, res = rf
es are the same. However, for k ≥ 2,
while the local image of res is still a smooth submanifold, the codimension
is larger than that of rf
es.

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

18

4. Diagonalizable peripheral holonomy
Recall the notation G = PGL4 R, G̃ = SL±
4 R and G̃0 = SL4 R from the
previous section. Also recall that M is an orientable finite volume hyperbolic
three-manifold with k torus cusps, Γ = π1 M , the ith cusp is denoted ∂i , and
∆i is a peripheral subgroup for ∂i . In this section we prove:
Theorem 4.1. Assume that M is infinitesimally projectively rigid rel ∂M .
Then there exists a smooth path of representations ρt ∈ Hom(Γ, G̃) such that
ρ0 = ρhyp is the holonomy representation of the complete hyperbolic structure
and ρt (∆i ) is diagonalizable over the reals for all t 6= 0 and i ∈ {1, . . . , k}.
The proof is a transversality argument in the product of the representation spaces of the boundary tori. Let us give the rough idea in the simpler case that there is only one cusp, whose peripheral subgroup we denote
by ∆: Theorem 3.8 gives that the restriction map res maps a neighborhood
of ρhyp in Hom(Γ, G̃) onto a submanifold of Hom(∆, G̃). This submanifold has codimension three and is smoothly foliated by conjugation orbits.
Now, the key ingredient for the proof is the construction of a smooth fourdimensional partial slice S in Hom(∆, G̃) which is transverse to res at ρhyp
and all of whose representations are either diagonalizable with real eigenvalues or lie in a unipotent (parabolic) subgroup of SO(3, 1). The transverse
intersection of S with the image of res gives a one-dimensional family in
Hom(∆, G̃) through ρhyp , in which all representations, except the restriction
of ρhyp , are diagonalizable. This one-dimensional family is the image of a
one-dimensional path in Hom(Γ, G̃) as desired.
Let us begin the proof by describing the four-dimensional partial slice
in the following general setting: Let ∆ ∼
= Z × Z and let ρhyp be a representation taking ∆ to a lattice in a unipotent subgroup of SO(3, 1). The
four-dimensional partial slice S is the image of the following map Φ : R4 →
Hom(∆, G̃0 ). We use coordinates (a, b, x, y) on R4 and generators γ1 , γ2 for
∆∼
= Z × Z. Define:


0
1
0 0
0
a
b 1

Φ(a, b, x, y)(γ1 ) := exp 
0
b
−a 0
0 2(a2 + b2 ) 0 0


0
x
y
0
0
ax + by
bx − ay
x

Φ(a, b, x, y)(γ2 ) := exp 
0
bx − ay
−ax − by y 
0 2(a2 + b2 )x 2(a2 + b2 )y 0
A simple computation checks that Φ(a, b, x, y)(γ1 ) and Φ(a, b, x, y)(γ2 ) commute. We may assume using conjugacy that ρhyp coincides with Φ(0, 0, u, v)
where u + iv is the cusp shape of the cusp ρhyp (∆)\H3 with respect to
the generators γ1 and γ2 . When a = b = 0 and x, y are allowed to vary,

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

19

Φ gives a global slice for the discrete faithful unipotent SO0 (3, 1) representations of ∆; the restriction of Φ to the xy–coordinate plane is well-known
to be transverse to conjugation. Geometrically, Φ(0, 0, x, y) parameterizes
the conjugacy classes of holonomy representations of all possible torus cusps
of hyperbolic three-manifolds.
Definition 4.2. We refer to the collection of representations Φ(0, 0, x, y),
for all y 6= 0, as the cusp shape locus.
The following Lemma is the most important ingredient in the proof of
Theorem 4.1. Its proof will be given in the following subsection.
Lemma 4.3. Let u, v ∈ R with v 6= 0. Then near the point (a, b, x, y) =
(0, 0, u, v), the map Φ is a local immersion which is never tangent to conjugation orbits. Further, for each (a, b) 6= (0, 0), the representation Φ(a, b, x, y)
is diagonalizable over the reals.
The eigenvalues of the generators Φ(a, b, x, y)(γi ) may be computed explicitly. They are most naturally described using certain branched polar coordinates around the cusp shape locus: (a, b, x, y) = (t cos 3θ, t sin 3θ, x, y).
In these coordinates the eigenvalues of Φ(t cos 3θ, t sin 3θ, x, y)(γ1 ) are
√
(1, exp(2t cos θ), exp(−t(cos θ+ 3 sin θ)),
√
exp(t(− cos θ + 3 sin θ))),
and the eigenvalues of Φ(t cos 3θ, t sin 3θ, x, y)(γ2 ) (listed with respect to the
same eigenbasis) are
√
√
(1, exp(2t(x cos θ + y sin θ)), exp(−t((x − 3y) cos θ+( 3x + y) sin θ)),
√
√
exp(−t((x + 3y) cos θ + (− 3x + y) sin θ))).
Observe that when moving away from the cusp shape locus (ie increasing
t from zero) in any direction (ie for any value of 3θ), the four eigenvalues
of Φ(t cos 3θ, t sin 3θ, x, y)(γ) vary as smooth real-valued functions of t with
distinct first derivative for at least some (generic) γ ∈ ∆. However, more
is required to show that Φ is not tangent to the conjugation orbit (see
Remark 4.9). The complete proof of Lemma 4.3 will be given in the following
subsection, together with a more geometric description of the representations
Φ(a, b, x, y).
Let us now return to the context of Theorem 4.1. In order to state the next
lemma, let us introduce a useful splitting of g (see Johnson–Millson [28] for
more details). Let g = so(3, 1) ⊕ v be the Killing-orthogonal splitting of the
Lie algebra g; the splitting is invariant under the adjoint action of O(3, 1).
Since the representation ρhyp has image in the subgroup O(3, 1) ⊂ G̃, all
relevant cohomology groups split and the restriction map res : Hρ1hyp (Γ, g) →

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

Lk

1
i=1 Hρhyp (∆i , g)

20

splits into the direct sum of the two maps:

resso(3,1) :Hρ1hyp (Γ, so(3, 1)) →

k
M

Hρ1hyp (∆i , so(3, 1)),

i=1

resv :Hρ1hyp (Γ, v) →

k
M

Hρ1hyp (∆i , v).

i=1

Note that for each i ∈ {1, . . . , k}, dim Hρ1hyp (∆i , v) = 2 (see Heusener–
Porti [25, Section 5.1]).
Lemma 4.4. There exists a cohomology class of infinitesimal deformations
[z] ∈ Hρ1hyp (Γ, v) whose restriction resi ([z]) ∈ Hρ1hyp (∆i , v) is non-trivial for
all i ∈ {1, . . . , k}.
Proof. The image L = res(Hρ1hyp (Γ, v)) is a Lagrangian subspace (see [25,
L
Section 5.1]) for the cup product pairing in ki=1 Hρ1hyp (∆i , v) and thus has
dimension k. Note that under the cup product pairing, the direct sum is
orthogonal. The projection onto any single factor is not zero, or else L would
be a Lagrangian subspace of the direct sum of the complementary k − 1
factors, which is impossible since dim L = k. Hence for each j ∈ {1, . . . , k},
there exists [zj ] ∈ Hρ1hyp (Γ, v) such that resj ([zj ]) 6= 0. Then some linear
combination [z] of {[zj ]}kj=1 satisfies the conclusion of the Lemma.

The next Lemma is a basic consequence of Calabi–Weil rigidity.
Lemma 4.5. Let i ∈ {1, . . . , k}, let µi be a non-trivial element of ∆i ,
and let [v] ∈ Hρ1hyp (Γ, so(3, 1)) have non-trivial restriction to µi . Then the
eigenvalues of at least one element of ∆i (although possibly not µi ) become
complex along any path in Hom(Γ, G̃) tangent to any cocycle representative
v ∈ Zρ1hyp (Γ, g) of the class [v].
Proof. Let ρt be any path of representations which is tangent to v. Let
us first assume that the representations ρt remain in SO(3, 1). Then the
restriction resi (ρt ) lies in SO0 (3, 1) ∼
= PSL2 C and we may regard the image
under ρt of any individual element as either a 4 × 4 real matrix or a 2 × 2
complex matrix. It follows easily from Calabi–Weil rigidity [8, 40, 41] (or
see Kapovich [29]) that the PSL2 C traces of resi ρt become complex to first
order. Further, there exists an element νi ∈ ∆i such that the derivative of the
PSL2 C trace of ρt (νi ) has imaginary part larger than its real part. It follows
that the SL4 R trace of ρt (νi ) has strictly negative derivative. This first order
trace behavior holds for any path of representations ρt into G̃ = SL±
4 R that
is tangent to v. Therefore, for any sufficiently small time t > 0, the trace
of ρt (νi ) is strictly smaller than four and so ρt (νi ) has at least one pair of
complex eigenvalues (by the arithmetic mean vs geometric mean inequality
applied to the eigenvalues).


CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

21

For each i ∈ {1, . . . , k}, let Si ⊂ Hom(∆i , G̃) be a copy of the fourdimensional partial slice S described above with ∆ = ∆i and let gi ∈ G̃
be such that c(gi ) · resi (ρhyp ) ∈ Si . Without loss of generality we assume
g1 = 1. Let S = S1 × · · · × Sk ⊂ Hom(∆1 , G̃) × · · · × Hom(∆k , G̃), let VSi ⊂
Hρ1hyp (∆i , g) denote the subspace of cohomology classes of all infinitesimal
deformations tangent to Si , and let VS = VS1 ⊕ · · · ⊕ VSk . We now prove:
Lemma 4.6. The augmented restriction map rf
es is transverse to S at
(ρhyp , g2 , . . . , gk ) with k–dimensional local intersection.
Proof. By Lemma 4.5, the intersection VS ∩res∗ (Hρ1hyp (Γ, so(3, 1))) is trivial,
since none of the representations in S have complex eigenvalues. Hence,
since dim VS = 4k by Lemma 4.3, and Hρ1hyp (Γ, so(3, 1)) and its image in
Lk
1
i=1 Hρhyp (∆i , so(3, 1)) have dimension 2k, we have that
k
M

Hρ1hyp (∆i , g) = VS ⊕ res∗ (Hρ1hyp (Γ, so(3, 1))

i=1

and it follows that the subspaces VS and res∗ (Hρ1hyp (Γ, g)) intersect transversely in a k–dimensional subspace. Therefore the 4k–dimensional tangent
L
1
(∆i , g) intersects the codimension 3k image of
space to S in ki=1 Zc(g
i )ρhyp
the augmented restriction map transversely in a k–dimensional subspace.
The result follows.

Finally we prove Theorem 4.1.
Proof of Theorem 4.1. Lemma 4.4 guarantees the existence of a cohomology class of infinitesimal deformation [z] ∈ Hρ1hyp (Γ, v) whose restrictions
(resi )∗ [z] are, for each i ∈ {1, . . . , k}, non-trivial in Hρ1hyp (∆i , v). The
span of [z] and Hρ1hyp (Γ, so(3, 1)) is a (2k + 1)–dimensional subspace of
Hρ1hyp (Γ, g) whose restriction, also (2k + 1)–dimensional by the assumption that res∗ is injective, must intersect the codimension 2k subspace VS
non-trivially (and indeed, transversally). Let res∗ [u] = αres∗ [z] + res∗ [w]
be a non-trivial element of the intersection, where [w] ∈ Hρ1hyp (Γ, so(3, 1)).
Since VS ∩ res∗ (Hρ1hyp (Γ, so(3, 1))) = 0, we must have α 6= 0. In particular, for each i ∈ {1, . . . , k}, (resi )∗ [u] does not lie in Hρ1hyp (∆i , so(3, 1)).
Hence (resi )∗ [u] ∈ VSi is not tangent to the cusp shape locus. We may
therefore find a representative cocycle u ∈ Zρ1hyp (Γ, g) and coboundaries
bi ∈ Bρ1hyp (∆i , g) for i ∈ {2, . . . , k} such that res1∗ u is tangent to S1 and
for each i ∈ {2, . . . , k}, resi∗ u + bi is tangent to Si . By Lemma 4.6 there
exists a path ρt ∈ Hom(Γ, G̃) based at ρ0 = ρhyp with tangent u at t = 0 and
paths g1,t = 1 (constant), g2,t , . . . , gk,t in G̃ with g2,0 = g2 , . . . , gk,0 = gk such
that for each i ∈ {1, . . . , k}, c(gi,t ) · resi ρt ∈ Hom(∆i , G̃) lies in Si . Since
(resi )∗ [u] is not tangent to the cusp shape locus, for sufficiently small t > 0,

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

22

each of the representations c(gi,t ) · resi ρt does not lie in the cusp-shape locus
and is therefore diagonalizable by Lemma 4.3. The theorem is proved. 

Remark 4.7. The properties characterizing the infinitesimal deformation
[z] from Lemma 4.4 are stable under perturbation. Therefore there is an
open k–dimensional cone of [z] ∈ Hρ1hyp (Γ, v) satisfying the conclusion of
Lemma 4.4. The proof of Theorem 4.1 implies that this cone parametrizes a
k–dimensional family of representations satisfying the conclusion of theorem.

4.1. More on the four-dimensional partial slice S . In this section we
give the proof of Lemma 4.3, which describes the essential properties of the
four-dimensional partial slice S = Im(Φ) used in the transversality argument for Theorem 4.1. Along the way, we will give a geometric description
of the representations in S and indicate some of the intuition behind its
construction.
Let C denote the two-dimensional abelian subgroup of SO(3, 1) consisting
of unipotent matrices fixing a point p∞ on the ideal boundary of hyperbolic
space. Let us work in the paraboloid model of H3 , with the ideal boundary
∂∞ H3 described by the paraboloid
∂∞ H3 = {[(x21 + x22 )/2, x1 , x2 , 1] ∈ RPn | x1 , x2 ∈ R} ∪ [1, 0, 0, 0],
and let us take p∞ = [1, 0, 0, 0]. Then each of the cusp-shape representations
Φ(0, 0, x, y), where y 6= 0, is a lattice inside C. Note that C is contained in
its centralizer Z(C) in SL4 R, a three-dimensional abelian subgroup, maximal with respect to inclusion. Let a denote the Cartan subalgebra of sl4 (R)
consisting of traceless diagonal matrices. Let A = exp a denote the corresponding Cartan subgroup. In order to find representations nearby res(ρhyp )
which are diagonalizable, we must study the space of maximal (ie threedimensional) abelian subgroups of SL4 R near Z(C) and attempt to locate
(at least some of) those that are conjugates of A.
We now construct a smooth two-dimensional family of three-dimensional
abelian subgroups Aa,b , which are conjugate to A for all (a, b) 6= (0, 0),
and such that A0,0 = Z(C). We work in the affine chart with coordinates
[x3 , x1 , x2 , 1]. For each t > 0, consider the intersection, St , of the paraboloid
∂∞ H3 with the affine plane Pt parallel to the x1 x2 –plane at height x3 =
1/(2t2 ). In these coordinates, St is a round circle contained in Pt , invariant
under the rotation R(θ) by any angle θ in the x1 x2 –plane about the x3 –
axis. Let p(t) = [1/(2t2 ), 1/t, 0, 1] and let p1 (t, θ) = R(θ)p(t), let p2 (t, θ) =
R(θ+2π/3)p(t) and p3 (t, θ) = R(θ+4π/3)p(t). Then p1 (t, θ), p2 (t, θ), p3 (t, θ)
are the vertices of an equilateral triangle inscribed in St .
Let At,θ denote the subgroup of projective transformations which fix the
vertices p1 (t, θ), p2 (t, θ), p3 (t, θ), p∞ . Then At,θ is given by the explicit

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

p∞

p2
p1

p2
Pt
x3

23

p1

p3
St

St
p3

x2
x1

Figure 2. Left panel: The points p1 (t, θ), p2 (t, θ) and
p3 (t, θ) form an equilateral triangle on the circle St at height
1/(2t2 ) on ∂H3 in the paraboloid model. Right panel: The
same picture shown in the affine chart in which ∂H3 is the
round sphere.
formula At,θ = R(θ)Q(t)AQ(t)−1 R(θ)−1 , where



Q(t) =

1
 2t
0
2t2

1
−t
√
3t
2t2

1
−t
√
− 3t
2t2

1
0
,
0
0

R(θ) =

1
0
0
0

0
cos θ
sin θ
0

0
− sin θ
cos θ
0



0
0
0
1

Note that At,θ = At,θ+2π/3 . The Hausdorff limit as t → 0 of At,θ is exactly
Z(C), independent of θ. Indeed, (t, θ) are branched polar coordinates for
a smooth two-dimensional family of three-dimensional abelian subgroups of
SL4 R. To see this, consider the three families of traceless diagonal matrices
xt,θ , yt,θ , zt,θ ∈ a = Lie(A):


2t cos θ

xt,θ

2t cos(θ + 2π/3)


=


,

2t cos(θ + 4π/3)
0





2t sin θ
2t sin(θ + 2π/3)


yt,θ = 


,

2t sin(θ + 4π/3)
0



t

zt,θ


=



2

t2


.

t2
−3t2

For all t 6= 0 and any θ, these three vectors form a basis of a. A simple
computation shows that


0

x0t,θ

:= R(θ)Q(t)xt,θ Q(t)

−1

−1

R(θ)

1

0

0

0 t cos 3θ t sin 3θ 1
=  t sin 3θ −t cos 3θ 0
2t2

0

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS



0
0

0
yt,θ
:= R(θ)Q(t)yt,θ Q(t)−1 R(θ)−1 = 

0
t sin 3θ
−t cos 3θ
0

1
−t cos 3θ
−t sin 3θ
2t2



−3t2

2
t2


0
zt,θ
:= R(θ)Q(t)zt,θ Q(t)−1 R(θ)−1 = 

24



0
0
.
1
0





t2
t2

These three vectors form a basis for at,θ = Lie(At,θ ). We now set a = t cos 3θ
and b = t sin 3θ and rewrite in these coordinates:




0

x0a,b

0
= 0
0



1
a
b
2(a2 + b2 )
2

0
b
−a
0

0
1
,
0
0

0

0
ya,b

0

2

−3(a + b )

0
za,b

2
a2 + b2


=

0

0 b
= 0 −a
0

1
−a
−b
2(a2 + b2 )

0
0
,
1
0



.

a2 + b2
a2 + b2

0 , z0
In these coordinates it is transparent that the Lie algebra elements x0a,b , ya,b
a,b
span a smooth (in fact algebraic) family of three-dimensional abelian subalgebras aa,b = at,θ with a0,0 = Lie(Z(C)). More relevant for Lemma 4.3, for
0 span a two-dimensional
each a, b ∈ R, the Lie algebra elements x0a,b and ya,b
Lie subalgebra ca,b which generates a two-dimensional abelian subgroup Ca,b
of SL4 R. Indeed both ca,b and Ca,b are smooth families and, of course,
C0,0 = C.

Proposition 4.8. The maps (a, b) 7→ aa,b and (a, b) 7→ ca,b are both transverse to conjugation at (a, b) = (0, 0).
Proof. Since ca,b ⊂ aa,b it is enough to show that the map (a, b) 7→ ca,b is
transverse to conjugation. To do so, we must simply show that the two-
0 ] of bivectors in P ∧2 sl R
parameter family of projective classes [x0a,b ∧ ya,b
4
0 ]. This is a straightis never tangent to the conjugation orbit at [x00,0 ∧ y0,0
forward calculation. First, compute:

 

0

∂a (x0a,b

∧

0
ya,b
) (a,b)=(0,0)

0

1
0


=

0
−1

 
∧

0
0
0

0



0
0

+

1
0
0

0
0
0





0
0
1 
∧
0 
0

0
0
−1

0
−1
0

1
0
0

0
0
1
0



0
0
0
0

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS


0
∂b (x0a,b ∧ ya,b
)

(a,b)=(0,0)



0
0
1


=

1
0



0

 
∧



0
0
0

1
0
0

0
0
1
0

0
1
0

0
0
−1

0



0
0

1
0
0

+

0
0
0





0
0
1 
∧
0 
0

25



0
0
0
0

Next, the infinitesimal action by conjugation of an arbitrary element v =
0
(vij ) ∈ g on x00,0 ∧ y0,0
is given by:
0
0
ad(v)x00,0 ∧ y0,0
+x00,0 ∧ ad(v)y0,0
=


−v21
−v41
 0
0

v11 − v22
v21 − v42
v31
v41



0
0
0

0
0

+

1
0
0



−v23
−v43
0
0



0
−v31
1  0
∧
0 −v41
0
0





v12 − v24
0
v22 − v44  
∧
v32  
v42
−v32
0
−v42
0

v11 − v33
v21
v31 − v43
v41

0
0
0

1
0
0



0
0
1
0



v13 − v34
v23 
.
v33 − v44 
v43

Now, let (eij ) be the usual basis for gl4 R, thought of as the space of 4 × 4
V
V
real matrices, and work in the basis for 2 gl4 R ⊃ 2 sl4 R consisting of all
eij ∧ emn such that either i < m or i = m and j < n. Suppose that some
0 ] is tangent to the conjugation orbit
tangent vector to the family [x0a,b ∧ ya,b
0 ]. This is equivalent to the equation:
at [x00,0 ∧ y0,0
0
0
0
0
0
) = ad(v)x00,0 ∧y0,0
+x00,0 ∧ad(v)y0,0
+γx00,0 ∧y0,0
.
α∂a (x0a,b ∧ya,b
)+β∂b (x0a,b ∧ya,b

The coefficient of e13 ∧e33 on the left-hand side is α while the same coefficient
on the right-hand side is zero. The coefficient of e12 ∧ e22 on the left-hand
side is β while the same coefficient on the right-hand side is again zero. It
0 ]
follows that α = β = 0: any vector tangent to both the family [x0a,b ∧ ya,b
and the conjugation orbit is trivial. The Proposition is proved.


Proof of Lemma 4.3. First, let us rewrite Φ in the notation of this section:
Φ(a, b, x, y)(γ1 ) = exp(x0a,b )
0
Φ(a, b, x, y)(γ2 ) = exp(xx0a,b + yya,b
)

That the representations Φ(a, b, x, y) are diagonalizable with real eigenvalues whenever (a, b) 6= (0, 0) has already been demonstrated. We must
show that Φ is not tangent to the conjugation orbit in any direction at any
point (0, 0, x, y).
Consider the tangent vector
w = ȧ∂a Φ + ḃ∂b Φ + ẋ∂x Φ + ẏ∂y Φ

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

26

1
at the point Φ(0, 0, x, y) and suppose that w ∈ BΦ(0,0,x,y)
(∆, sl4 R). By
Proposition 4.8, the two-dimensional subgroup Ca,b generated by Φ(a, b, x, y)
is changing to first order in the direction of w if and only if (ȧ, ḃ) 6= (0, 0).
So we must have (ȧ, ḃ) = (0, 0). Hence w is tangent to the cusp shape locus.
Since w is a coboundary, the cusp shape must not change to first order. It
follows that ẋ = 0 and ẏ = 0. This completes the proof of Lemma 4.3. 

Remark 4.9. For fixed (a0 , b0 ) 6= (0, 0), the eigenvalues of Φ(ta0 , tb0 , u, v)
are smooth functions of t with non-zero derivative at t = 0. It is tempting to
conclude that dΦ(0,0,u,v) (a0 , b0 , 0, 0) is not an infinitesimal conjugation. Although this turns out to be the case, it is fallacious to conclude so purely from
the given behavior of eigenvalues; the above proof of Lemma 4.3 is needed.
We give an example in the simpler
setting
of 2 × 2 matrices.
Consider
 t


 two
1+t
1
e
1
paths of matrices M1 (t) =
and M2 (t) =
. Note
0 e−t
−t2 1 − t
that M1 (0) = M2 (0) and that M10 (0) = M20 (0). However, the eigenvalues of
M1 (t), which are et = 1 + t + O(t2 ) and e−t = 1 − t + O(t2 ), vary to first
order in t, while the eigenvalues of M2 (t) are both constant equal to one for
all t. Indeed M2 (t) is a conjugation path.
5. Geometry of manifolds with totally geodesic boundary
This section is dedicated to the geometry of convex projective threemanifolds with diagonalizable peripheral holonomy. We will complete the
proof of Theorem 1.3, the algebraic side of which was given in the previous
section. Recall that we denote the Lie groups PGL4 R, SL±
4 R and SL4 R by
G, G̃ and G̃0 , respectively.
Let M be an open three-manifold which is the interior of a compact
boundary incompressible manifold with k torus boundary components ∂M =
'
f−
∂1 t · · · t ∂k . Let dev : M
→ Ω ⊂ RP3 be the developing map of an indecomposable properly convex projective structure on M and denote the
holonomy representation ρ : Γ → G̃ where Γ = π1 M denotes the fundamental group. As usual denote the peripheral subgroups by ∆i = π1 ∂i .
We will assume that ρ(∆i ) is diagonalizable over the reals with eigenvectors
(i) (i) (i) (i)
(i)
p1 , p2 , p3 , p4 ∈ RP3 and further that p4 is never an attracting fixed
point (not even weakly attracting) for any non-trivial element of ρ(∆i ).
Then ρ is said to satisfy the middle eigenvalue condition, namely that for
(i)
any nontrivial γ ∈ ∆i , the eigenvalue of ρ(γ) associated to eigenvector p4
is never the largest nor the smallest. The middle eigenvalue condition is a
slight weakening of the uniform middle eigenvalue condition defined by Choi
(see [10] for more details).
The holonomy representation ρ for this structure is also the holonomy
representation for other (related) convex projective structures defined by
different domains. Let us begin by constructing a minimal convex domain
for the action of ρ(Γ). Any strongly attracting fixed point of an element

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

27

of ρ(Γ) must lie on ∂Ω. The closure of the set of strongly attracting fixed
points is called the limit set of ρ(Γ) and we define Ωmin to be the interior
of the closed convex hull of the limit set. Then Ωmin is contained in Ω,
is non-empty, open, ρ–invariant, convex, and is minimal with respect to
these properties. Pulling back Ωmin via dev determines a convex projective
structure on a sub-manifold Mmin ∼
= ρ(Γ)\Ωmin whose inclusion into M
is a homotopy equivalence. We now investigate the geometry of Mmin , in
particular of its ends.
(i)

(i)

(i)

Lemma 5.1. For any i ∈ {1, . . . , k}, each of the three fixed points p1 , p2 , p3
is a strongly attracting fixed point for some element of ρ(∆i ). Therefore
(i) (i) (i)
p1 , p2 , p3 lie on ∂Ω and ∂Ωmin .
Proof. The peripheral holonomy group ρ(∆i ) is obtained by exponentiating a
lattice, Λ, inside a two-dimensional Lie subalgebra, c(i) , of the Cartan subal(i) (i) (i) (i)
(i)
gebra a(i) ⊂ g corresponding to the basis p1 , p2 , p3 , p4 ∈ RP3 . Let A123 =
(i)
(i) (i) (i)
exp a123 be the (2–dimensional) subgroup of SL± (span{p1 , p2 , p3 }) ∼
=
(i) (i) (i)
R
of
elements
that
fix
each
of
p
,
p
,
p
.
Consider
the
natural
proSL±
3
1
2
3
(i)
jection $123 : a(i) → a123 . It follows from the middle eigenvalue condition
that the restriction of $123 to c(i) is injective. To see this, observe that if
A ∈ ker($123 ) ∩ c(i) then the eigenvalues of A corresponding to the eigen(i)
(i)
(i)
(i)
vectors p1 , p2 , p3 and p4 are λ, λ, λ and µ, respectively. As a result we
see that µ is either the smallest or largest eigenvalue of A which contradicts
the middle eigenvalue condition. Thus by dimensional considerations we
(i)
conclude that $123 (c(i) ) = a123 . Furthermore, if A ∈ c(i) , 1 ≤ j ≤ 3, and the
(i)
(i)
pj –eigenvalue of $123 (A) is the largest eigenvalue then the pj –eigenvalue
of A is also the largest eigenvalue for A.
(i)
Let 1 ≤ j ≤ 3 and let Dj be the subset of a123 consisting of elements where
(i)
the pj –eigenvalue is the largest. It is easy to see that Dj is a non-empty
open cone, which implies that Dj has non-trivial intersection with $123 (Λ).
(i)
As a result we can find an element of ρ(∆i ) such that the pj –eigenvalue
(i)

is the largest. Such an element has pj as an attracting fixed point, which
completes the proof of the lemma.

We make the following definition, following Goldman [23] in the twodimensional setting.
Definition 5.2. For each i, there is a unique (open) triangle T (i) ⊂ Ωmin
(i) (i) (i)
spanned by the points p1 , p2 , p3 . Any ρ(Γ) translate of T (i) will be called
a principal totally geodesic triangle. The group ρ(∆i ) acts properly on T (i)
and the quotient is called a principal totally geodesic torus.
We will show that Mmin admits a natural compactification whose boundary consists of principal totally geodesic tori. First, we introduce a third

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

28

convex domain Ωmax defined as follows. Let Ω∗ denote the convex domain
dual to Ω. Then Γ acts properly discontinuously on Ω∗ via ρ with diagonalizable peripheral holonomy, and we may perform the same construction as
above: Let (Ω∗ )min denote the interior of the closed convex hull of the limit
set for the ρ(Γ) action on RP3∗ . We define Ωmax to be the convex domain
dual to (Ω∗ )min . It is the maximal properly convex, ρ–invariant domain
because its dual is minimal. Next, observe that the fixed points of the dual
(i)
(i)
(i)
(i)
action of ρ(∆i ) are the hyperplanes P1 , P2 , P3 , P4 spanned respectively
(i) (i) (i)
(i) (i) (i)
(i) (i) (i)
(i) (i) (i)
by {p2 , p3 , p4 }, {p1 , p3 , p4 }, {p1 , p2 , p4 } and {p1 , p2 , p3 }. The
(i)
(i)
(i)
hyperplanes P1 , P2 and P3 are the attracting fixed points of ρ(∆i ) in
(i)
(i)
(i)
RP3∗ . Hence P1 , P2 , P3 are points on the boundary of any convex do3∗
main in RP preserved by ρ, including on ∂(Ω∗ )min . Dually, they are three
support hyperplanes for any convex domain in RP3 preserved by ρ, in particular for Ω, Ωmin and Ωmax . They bound a convex (but not properly convex)
open triangular prism U (i) which is separated by T (i) into two components
(i)
(i)
T+ , T− , each of which is an open tetrahedron.
Lemma 5.3.
(1) Each principal totally geodesic triangle is contained in ∂Ωmin .
(2) Ωmin is the interior of the intersection of the positive half-spaces
(i)
bounded by the planes ρ(Γ)P4 containing principal totally geodesic
triangles.
(3) Ωmax \ Ωmin is the disjoint union over all i ∈ {1, . . . , k} and all
(i)
(i)
γ ∈ Γ/∆i , of open tetrahedra ρ(γ)T− , where T− is the tetrahedron
lying on the opposite side from Ωmin of the principal triangle T (i) .
Proof. Let S (1) , . . . , S (k) be a pairwise disjoint collection of embedded tori in
M which are parallel to the k boundary components ∂1 , . . . , ∂k respectively.
f = Ω of S (i) that is invariant
Let D(i) denote the lift to the universal cover M
under ∆i . Then D(i) is (Γ, ∆i ) precisely invariant, meaning that ρ(γ)D(i) ∩
D(i) 6= ∅ if and only if γ ∈ ∆i . It follows from Lemma 5.1 that ∂D(i) = ∂T (i)
and therefore T (i) is also (Γ, ∆i ) precisely invariant. Next, D(i) divides Ω
into two components and one of these, the one which covers the end of M
bounded by S (i) , is also (Γ, ∆i ) precisely invariant. As a result, the limit
set of ρ(Γ) lies on one side of ∂D(i) = ∂T (i) on ∂Ω. Hence the limit set
(i)

(i)

lies entirely in one of the two closed tetrahedra T+ or T− and we take the
(i)

(i)

(i)

labeling convention that it is T+ . Let Ω+ = Ω ∩ T+ . It then follows that
the intersection
k
\ \
(i)
ρ(γ)Ω+
γ∈Γ i=1

is a closed convex set containing the limit set of ρ(Γ). We denote its interior,
which must be non-empty, by Ω0 . In fact, we will show soon that Ω0 is the

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

29

T+
T−
T+
Ωmin

T

Ω

Ωmax

Figure 3. A schematic of the configuration Ωmin ⊂ Ω ⊂
Ωmax in Lemma 5.3. Here Ωmin is red, Ω is the union of red
and purple, and Ωmax is the union of red, purple and blue.
minimal convex domain Ωmin . Until then we conclude the simple fact that
T (i) is contained in ∂Ωmin because the vertices of T (i) must be contained
in the boundary of any invariant convex domain and because T (i) lies in a
support plane for ∂Ω0 .
(i)
Since the hyperplane P4 containing T (i) is a support plane for Ωmin , it is
also a point in the boundary of the dual convex domain (Ωmin )∗ = (Ω∗ )max .
By applying the above to the dual convex domain Ω∗ in place of Ω, we see
(i)
therefore that the point p4 belongs to the boundary of Ωmax . In particular,
(i)
the entire tetrahedron T− is contained in Ωmax ; see Figure 3. Since the
action of ρ(Γ) is properly discontinuous on Ωmax (or on any invariant open
properly convex domain), we may now conclude that ρ(Γ) acts properly
on the subdomain Ω0∂ consisting of the union of Ω0 with all of the principal totally geodesic (open) triangles, and similarly on the union Ωmin,∂ of
Ωmin with all of the principal totally geodesic triangles. The quotient of
either set by ρ(Γ) is a submanifold with boundary of Mmax := ρ(Γ)\Ωmax .
The boundary of either is the collection of principal totally geodesic tori
{ρ(∆i )\T (i) }ki=1 . We conclude that Ωmin = Ω0 . It follows that the construction of Ω0 is independent of the domain Ω. The proof of (2) is thus
completed by applying the above argument in the case that Ω is the interior
of the intersection of the positive half-spaces bounded by the hyperplanes

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

30

containing principal triangles. The third statement of the Lemma then follows immediately.


(a)

(b)

(c)

(d)

Figure 4. The principal triangles on the boundary of Ωmin
as the representation ρ deforms away from the holonomy representation of the complete hyperbolic structure.
Definition 5.4. The manifold M min = ρ(Γ)\Ωmin,∂ is called a properly
convex projective manifold with totally geodesic torus boundary. By abuse,
Mmin = ρ(Γ)\Ωmin will also be said to have totally geodesic boundary; see
Figure 4.
Definition 5.5. The manifold Mmax = ρ(Γ)\Ωmax is the maximal thicken(i)
ing of M . The tetrahedron T− (or any of its orbits) is a principal tetrahedron and its quotient by ρ(∆i ) is a principal collar of Mmax ; see Figure 5.
Proof of Theorem 1.3. By assumption, the complete hyperbolic structure
on M is infinitesimally rigid rel ∂M . By Theorem 4.1 there is a path ρt of
representations through ρ0 = ρhyp such that ρt (∆i ) is diagonalizable over
the reals for all t 6= 0 and i ∈ {1, . . . , k}. The holonomy principle of Cooper–
Long–Tillmann (Theorem 2.3) or Choi [11, Corollary 1.1] guarantees that
there is some ε > 0, such that for t ∈ (0, ε), the representation ρt is the

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

(a)

(b)

(c)

(d)

31

Figure 5. The principal tetrahedra of Ωmax as the representation ρ deforms away from the holonomy representation
of the complete hyperbolic structure.
holonomy representation of a properly convex projective structure, nearby
the complete hyperbolic structure. Further, the ends of this structure have
generalized cusps, although we will not use this fact here.
By construction, resi (ρt ) belongs to Si . Hence for t > 0, ρt (∆i ) is a
lattice in a two-dimensional diagonalizable subgroup conjugate to Ca,b , as
(i) (i)
defined in Section 4.1, where (a, b) = (at , bt ) 6= (0, 0) depend on i and
continuously on t. Since all elements of Ca,b have a common fixed point
with eigenvalue one, the middle eigenvalue condition is satisfied for ρt (∆i )
for all t > 0 and i ∈ {1, . . . , k}. For each t, the minimal convex sub-manifold
Mmin,t ⊂ M is a properly convex projective manifold with totally geodesic
boundary. Further, the inclusion Mmin,t ,→ M is a homotopy equivalence
which is isotopic to a homeomorphism. Such an isotopy may be constructed
explicitly in the universal cover by flowing the points lying in a principal
f (realized as the convex domain Ωt ) radially toward the
tetrahedron of M
principal triangle bounding that tetrahedron.

Remark 5.6. In the context of the proof of Theorem 1.3, we note that
for each end, the continuous deformation of the principal totally geodesic

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

32

triangles, which open up from the parabolic fixed point of the hyperbolic
structure, may be observed explicitly in terms of the parameters (a, b) of the
last paragraph of the proof. Indeed, the eigenvectors of Ca,b were described
explicitly in Section 4.1.
We close this subsection with one more lemma which will be needed in
(i)
(i)
the next section. The region U (i) = T+ ∪ T (i) ∪ T− is a triangular prism.
Its boundary ∂U (i) is the union of three totally geodesic bigons.
(i)

Lemma 5.7. The intersection ∂Ωmax ∩ ∂U (i) is equal to ∂T− \ T (i) . In
(i)
particular, ∂Ωmin ∩∂T+ contains T (i) but does not contain any point of any
(i)
(i)
other (open) face of T+ of positive codimension that is adjacent to p4 .
(i)

Proof. Suppose p ∈ ∂Ωmax lies on a face F of T+ of positive codimension
(i)
(i)
adjacent to p4 . Although p4 is never an attractor for the action of any
(i)
ρ(γ) on T+ , it is, by dimension count, an attractor for the action of some
(i)
ρ(γ) on F . Then by convexity Ωmax contains T+ and therefore all of U (i) .
However this is impossible since U (i) is not properly convex and Ωmax is
properly convex.

6. Gluing convex projective structures
In this section we prove Theorem 1.4, which states that convex projective
structures may be glued along principal totally geodesic torus boundary
components whenever the holonomy matching condition (1.1) is satisfied.
This will complete the proof of Theorem 1.1. Theorem 1.4 is proved by
induction using the following two Lemmas:
Lemma 6.1. Let M1 = Γ1 \Ω1 , and M2 = Γ2 \Ω2 be two properly convex
projective three-manifolds. Let f : ∂1 → ∂2 be a homeomorphism between
principal totally geodesic torus boundary components ∂1 and ∂2 of M1 and
M2 , respectively, satisfying the holonomy matching condition: There exists
g ∈ G̃ such that
(6.1)

f∗ γ = gγg −1

for all γ ∈ ∆1 , where f∗ : ∆1 → ∆2 is the group homomorphism induced by
f on the fundamental groups ∆1 and ∆2 of ∂1 and ∂2 respectively. Then
the topological manifold M1 ∪f M2 admits a properly convex structure which
restricts to the given properly convex projective structures on M1 and M2 .
Lemma 6.2. Let M = Γ\Ω be a properly convex projective three-manifold
and let f : ∂1 → ∂2 be a homeomorphism between principal totally geodesic
boundary components ∂1 , ∂2 of M that satisfies the holonomy matching condition: There exists g ∈ G̃ such that
(6.2)

f∗ γ = gγg −1

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

33

for all γ ∈ ∆1 , where f∗ : ∆1 → ∆2 is the group homomorphism induced by
f on the fundamental groups ∆1 and ∆2 of ∂1 and ∂2 respectively. Then the
topological manifold Mf admits a properly convex structure which restricts
to the given properly convex projective structure on M .
We now give a detailed proof of Lemma 6.1. The proof of Lemma 6.2 is
nearly identical.
For convenience, throughout this subsection, we will identify the universal
cover of Mi with Ωi and the fundamental group π1 Mi with Γi ⊂ G̃, for both
i ∈ {1, 2}. For i ∈ {1, 2}, let ∂i = ∆i \T (i) be the given principal totally
geodesic boundary torus of Mi , where T (i) is a totally geodesic triangle
(i)
(i)
contained in ∂Ωi . For each i, let T+ and T− be defined as in the previous
(i)
(i)
section, so that Ωi ⊂ T+ and T− is a principal tetrahedron for Ωi . By
translating Ω1 by g and possibly a reflection in the centralizer Z(∆2 ), we
may assume henceforth that g = 1 and that Ω1 and Ω2 are positioned so
(2)
(1)
(1)
(2)
that T (1) = T (2) =: T , T− = T+ =: T (1) and T− = T+ =: T (2) . Then,
by (6.1), the identity projective transformation g = 1 descends to a projective
gluing map in the isotopy class of f that glues M1 to M2 along their totally
geodesic boundary components. Any neighborhood in Ω1 ∪ T ∪ Ω2 of a point
p ∈ T serves as a chart defining a projective structure on a neighborhood of
∂1 = ∂2 in the glued up manifold N = M1 ∪f M2 . The charts on M1 and M2
are compatible with these new charts, hence N is endowed with a projective
structure in which ∂1 = ∂2 is an embedded totally geodesic torus. To prove
Theorem 1.4, we now show that this projective structure is properly convex.
The fundamental group of N naturally identifies with the free product
Γ = Γ1 ∗f∗ Γ2 amalgamated over the identification f∗ : ∆1 → ∆2 . Let ∆
denote the inclusions of ∆1 and ∆2 , which are identified, in Γ. We will
denote the product of two elements α, β in the abstract group Γ by the
notation α ? β in order to avoid confusion with matrix multiplication in G̃.
e is described combinatorially as:
The universal cover N
e = Γ × Ω1,∂ / ∼1 ∪ Γ × Ω2,∂ / ∼2
N
where Ω1,∂ denotes the union of Ω1 with the Γ1 orbit of T , and ∼1 is the
equivalence relation generated by (γ, p) ∼1 (γγ1−1 , γ1 p) for all γ1 ∈ Γ1 , and
similarly for Ω2,∂ and ∼2 . We refer to each {γ} × Ωi,∂ as a tile. If p ∈
T = Ω1,∂ ∩ Ω2,∂ , then we consider the points (γ, p) ∈ Γ × Ω1,∂ and (γ, p) ∈
Γ × Ω2,∂ to be identified. The developing map dev for the natural projective
structure on N is defined by the formula dev([γ, p]) = ρ(γ)p, for any γ ∈ Γ
and p ∈ Ω1,∂ ∪ Ω2,∂ , where ρ : Γ → G̃, the holonomy representation, is
defined by the property that its restriction to Γi is the inclusion map for
each i ∈ {1, 2}. In other words if γ1 , . . . , γm are elements of Γ1 ∪ Γ2 , then
ρ(γ1 ? · · · ? γm ) = γ1 · · · γm . We will also consider an augmented version of
e , defined by
N
eaug = Γ × Ω1 / ∼1 ∪ Γ × Ω2 / ∼2
N

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

34

which includes the full boundaries of the convex tiles. The developing map
eaug → RP3 .
dev extends naturally to an augmented developing map dev : N
It is a local embedding, even at points of the tile boundaries [γ, ∂Ωi ].
We show that the developing map is injective with image contained in an
affine chart via a ping-pong lemma.
Lemma 6.3. Let γ1 ∈ Γ1 \ ∆ and γ2 ∈ Γ2 \ ∆. Then
(1) γ1 T (2) ⊂ T (1) \ Ω1 .
(2) γ2 T (1) ⊂ T (2) \ Ω2 .
Proof. We prove only the first statement as the second follows by symmetry.
(1)
Note that T (2) = T− is contained in (Ω1 )max (Definition 5.5). Hence
γ1 T (2) ⊂ (Ω1 )max as well. (Ω1 )max is contained in the triangular prism
U = T (1) ∪ T ∪ T (2) , which is convex (but not properly convex) and bounded
by three bigons. Since γ1 ∈
/ ∆, γ1 T (2) is not equal to T (2) and therefore
the two tetrahedra do not intersect. Hence γ1 T (2) must lie in T (1) . By
Lemma 5.7, we also have that γ1 T (2) = γ1 T (2) lies in T (1) , that is no point
of γ1 T (2) intersects ∂T (1) . Of course γ1 preserves Ω1 , so γ1 T (2) does not
intersect Ω1 because T (2) does not intersect Ω1 .

eaug → RP3 is injective
Lemma 6.4. The augmented developing map dev : N
and its image is contained in (Ω1 )max .
Proof. We already know that dev is an embedding when restricted to the
union of any two adjacent closed tiles. To show that dev is a global embedding, it suffices to show that dev([{γ} × Ω1 ]) = ρ(γ)Ω1 does not intersect Ω2
/ Γ1 ∪ Γ2 .
nor Ω1 as long as γ ∈
Assume that γ ∈
/ Γ1 ∪ Γ2 . Then γ may be expressed as an alternating
product γ = γ1 ? · · · ? γm of m ≥ 2 elements γi ∈ Γ1 ∪ Γ2 \ ∆ such that for
i = 1, . . . , m − 1, γi ∈ Γ1 if and only if γi+1 ∈ Γ2 . There are two possibilities
to consider. First, assume γm lies in Γ2 \ ∆. Then by Lemma 6.3,
ρ(γ)Ω1 = γ1 · · · γm−1 γm Ω1 ⊂ γ1 · · · γm−1 (T (2) \ Ω2 )
⊂ γ1 · · · γm−2 (T (1) \ Ω1 )
..
.
⊂ (T (j) \ Ωj )
where j = 1 if γ1 ∈ Γ1 (equivalently if m is even) or j = 2 if γ1 ∈ Γ2 .
Hence ρ(γ)Ω1 does not intersect either Ω1 or Ω2 . The other possibility is
that γm ∈ Γ1 \ ∆. In this case, ρ(γ)Ω1 = γ1 · · · γm−1 Ω1 , with γm−1 ∈ Γ2 \ ∆
and we proceed as in the previous case replacing γ with γ1 ? · · · ? γm−1 . This
eaug . Indeed dev is a closed
completes the proof that dev is injective on N
map, so dev is an embedding.
We may also see from the above ping-pong argument that the image
of dev is contained in (Ω1 )max . For any γ ∈ Γ1 , we have that ρ(γ)Ω1 =

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

35

Ω1 ⊂ (Ω1 )max and ρ(γ)Ω2 ⊂ ρ(γ)T (2) ⊂ (Ω1 )max . If γ ∈ Γ2 \ ∆, then
ρ(γ)Ω2 = Ω2 ⊂ (Ω1 )max and ρ(γ)Ω1 ⊂ ρ(γ)T (1) ⊂ T (2) ⊂ (Ω1 )max . Finally,
let γ ∈
/ Γ1 ∪ Γ2 . Then as above γ = γ1 ? γ2 ? · · · ? γm and ρ(γ)Ω1 ⊂ T (2) ⊂
(Ω1 )max if γ1 ∈ Γ2 . If γ1 ∈ Γ1 , then ρ(γ2 ? · · · ? γm )Ω1 is contained in T (2) , so
ρ(γ1 )ρ(γ2 ? · · · ? γm )Ω1 = ρ(γ)Ω1 is contained in (Ω1 )max . It follows similarly
that if γ ∈
/ Γ1 ∪ Γ2 then ρ(γ)Ω2 ⊂ (Ω1 )max .

Finally, we prove:
Lemma 6.5. The projective manifold N is properly convex.
To prove Lemma 6.5, we will need the following basic result about convex
sets in Euclidean space. It is similar to a well-known theorem of Nakajima [33] and Tietze [38]. We include a proof for convenience.
Lemma 6.6. Suppose A, B ⊂ Rd are closed convex subsets with non-empty
interior and non-empty intersection. Suppose that A ∪ B satisfies the following local convexity condition along C = ∂(A ∪ B) ∩ (A ∩ B): At each
point z ∈ C there is a local support plane, i.e. a hyperplane containing z
and bounding a closed half-space that contains a neighborhood of z in A ∪ B.
Suppose further there is a point of A ∩ B in the interior of A ∪ B. Then
A ∪ B is convex.
Proof. We consider first the interior Int(A ∪ B). Fix a point x ∈ Int(A)
and let S be the set of all y ∈ Int(A ∪ B) such that [x, y] ⊂ Int(A ∪ B).
S is clearly an open subset of Int(A ∪ B). We show S is also closed in
Int(A ∪ B). Suppose yn ∈ S converges to y ∈ Int(A ∪ B) but y ∈
/ S. The
open interval (x, y) is contained in A ∪ B and intersects ∂(A ∪ B) in at
least one point z. Let H be a hyperplane supporting A ∪ B locally near z,
guaranteed to exist by the local convexity assumption if z ∈ C or by the
convexity of A (resp. of B) if z ∈ ∂A \ C (resp. if z ∈ ∂B \ C). Since (x, y)
does not cross H transversely at z, we must have [x, y] ⊂ H. However, this
is a contradiction: if z ∈ A, the local support plane H does not intersect
the convex set Int(A) so it can not contain x, and if z ∈ B, then H does not
intersect Int(B) so it does not contain y. Hence S is closed in Int(A ∪ B).
We note that Int(A ∪ B) is connected, since by assumption a point of A ∩ B
is contained in Int(A ∪ B). Hence S = Int(A ∪ B). Hence for all x ∈ Int(A)
and y ∈ Int(A ∪ B), [x, y] ⊂ Int(A ∪ B) and similarly for x ∈ Int(B) by
symmetry. It now follows by taking limits that A ∪ B is convex.

Proof of Lemma 6.5. By Lemma 6.4, the augmented developing map is an
eaug is
embedding into an affine chart A of RP3 . The image of each tile of N
convex. The union of two adjacent tiles is locally convex at the boundary
of the interface between them. To check this, it suffices to examine Ω1 ∪ Ω2 .
(1)

(1)

We may assume that U (1) = T+ ∪ T− intersects the affine chart A in an
infinite triangular prism (with vertex at infinity). Since Ω1 ∪ Ω2 is contained
in U (1) and since ∂(Ω1 ∪Ω2 )∩(Ω1 ∩Ω2 ) = ∂T is contained in ∂U (1) , it follows

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

36

that Ω1 ∪ Ω2 is locally convex at the boundary of the interface T between
the two tiles: one of the three planes bounding U (1) supports Ω1 ∪ Ω2 at
each point of ∂T . The union Ω1 ∪ Ω2 is convex by Lemma 6.6. Next, since
eaug meet non-trivially, it follows by induction
no three of the closed tiles of N
and Lemma 6.6 that the image under the augmented developing map of any
eaug is convex. The image of N
eaug is an
finite connected union of tiles of N
increasing union of such convex sets and hence is convex. The image of
e , which is the interior of the image of N
eaug , is therefore also convex, and
N
indeed properly convex because it is contained in (Ω1 )max .

This completes the proof Lemma 6.1. The proof of Lemma 6.2 is nearly
the same and so we include only the following sketch which highlights the
required modifications.
(1) Possibly after modifying g by a reflection in the centralizer Z(∆2 ),
(1)
(2)
we may assume that Ω and gΩ are positioned so that gT∓ = T± ,
(i)
(i)
where for both i ∈ {1, 2}, T+ ⊃ Ω and T− is the principal tetrahedron for Ω preserved by ∆i as above.
(2) The fundamental group of Mf is the HNN extension Γ∗f∗ . The
ff = Γ ∗f
universal cover of Mf is combinatorially a union of tiles M
∗
×Ω∂ / ∼, where Ω∂ is the union of Ω and the principal triangles
covering the boundary component ∂1 and ∂2 and the equivalence
relation is generated by (γ, p) ∼ (γγ1−1 , γ1 p), for all γ1 ∈ Γ.
(3) The developing map dev for the natural projective structure on Mf
is defined by the formula dev([γ, p]) = ρ(γ)p, for any γ ∈ Γ∗f∗ and
p ∈ Ω∂ , where ρ : Γ∗f∗ → G̃, the holonomy representation, is defined
by the property that its restriction to Γ is the inclusion map and ρ
applied the stable letter of Γ∗f∗ is the gluing transformation g. The
ff,aug =
developing map dev extends to the augmented domain M
Γ ∗f∗ ×Ω / ∼.
(4) The following analogue of Lemma 6.3 holds:
(1)
(2)
• gT∓ = T± (already arranged above)
(i)

(i)

• For any i ∈ {1, 2} and γ ∈ Γ \ ∆i , γT− ⊂ T+ \Ω (follows
(i)
immediately because Ωmax contains γT− and is contained in
(i)
(i)
the union U (i) = T+ ∪ T (i) ∪ T− .)
(5) Plugging (4) into the argument from Lemma 6.4 shows that dev is an
embedding with image contained in Ωmax . Finally, as in Lemma 6.5,
ff,aug under dev is locally convex showing that the
the image of M
projective structure on Mf is properly convex.
Proof of Theorem 1.4. The theorem follows by induction by applying Lemmas 6.1 and 6.2 in succession to glue together the given collection of convex
projective manifolds along various pairs of boundary components using the
given gluing maps in any order.


CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

37

Suppose we are given a collection of cusped hyperbolic manifolds, along
with gluing homeomorphisms between some of their torus boundary components. In general, it is not known whether projective structures with totally
geodesic boundary satisfying the appropriate holonomy matching hypotheses (1.1) of Theorem 1.4 can be found. To find such glueable structures
would seem to require a description of the global deformation space of properly convex projective structures with principal totally geodesic boundary
on a given manifold. As of the writing of this article, there is no cusped
hyperbolic manifold for which this global deformation space has been computed, abstractly nor computationally (although there are certain Coxeter
orbifolds for which the deformation space has been computed, see Section 7).
Nonetheless, the matching condition is automatically satisfied in the case of
doubling a convex projective manifold with principal totally geodesic boundary. Indeed:
Proof of Theorem 1.1. Since M is assumed to be infinitesimally projectively
rigid rel boundary, Theorem 1.3 tells us that we can deform the complete hyperbolic structure on M to a convex projective structure where each boundary component is a principal totally geodesic torus. The gluing condition
(1.1) is trivially satisfied for the identity gluing, and thus Theorem 1.4 ensures that the double 2M admits a properly convex projective structure. 
7. Gluing covering manifolds of reflection 3–orbifolds
We conclude the paper by giving examples of properly convex projective
structures on non-hyperbolic manifolds which are not obtained by doubling.
To do this, we apply the Convex Gluing Theorem 1.4 to a collection of highly
symmetric properly convex projective structures with principal totally geodesic boundary. The symmetry of these structures, which are covers of
certain convex projective reflection orbifolds, greatly restricts their boundary holonomy groups. This allows us to determine which combinations of
these manifolds admit structures satisfying the matching hypothesis (1.1) of
Theorem 1.4.
7.1. Euclidean hex tori. We will be interested in three-dimensional hyperbolic reflection orbifolds with cusps isomorphic to the Euclidean (3, 3, 3)–
triangle orbifold. We now recall the geometric features of this two-orbifold
and describe the geometry of its torus covers.
Let ∆3,3,3 denote the subgroup of the isometries Isom E2 of the Euclidean
plane generated by reflections in the sides of an equilateral triangle T in
the Euclidean plane. We denote the resulting quotient orbifold by S3,3,3 =
∆3,3,3 \E2 .
If ∆H denotes the subgroup of all translations in ∆3,3,3 , then ∆H is the
maximal torsion-free subgroup of ∆3,3,3 and all other torsion-free subgroups
∆ of ∆3,3,3 are contained in ∆H . In other words, the torus SH = ∆H \E2
is the minimal torus cover of S3,3,3 , and every orientable surface S = ∆\E2

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

(a)

(b)

(c)

(d)

38

Figure 6. A collection of properly embedded triangles in
the convex domains defining convex projective structures on
the double of the figure eight knot complement. Going from
(A) to (D), the projective structure on each of the two pieces
deforms away from the complete hyperbolic structure.
covering S3,3,3 is a Euclidean torus which may be decomposed into regular
hexagons each made up of six copies of the equilateral triangle T . We call
such a torus a Euclidean regular hex torus, or just hex torus for short. The
fundamental group ∆ of a Euclidean regular hex torus S is called a hex torus
group.
Let k be a natural number and let ∆0 be a hex torus group. Then the
group ∆ = k∆0 of all k th powers of elements of ∆0 is a subgroup of ∆0 . If
S 0 and S are the hex tori associated to ∆0 and ∆, then there is a natural
homeomorphism, denoted by k : S 0 → S which, in the universal cover E2 ,
simply scales by k.
Definition 7.1. Let S1 , S2 be hex tori corresponding to the subgroups
∆1 , ∆2 ⊂ ∆H . Then a homeomorphism f : S1 → S2 is said to be a lattice
Q–isometry or a lattice kk12 –isometry if there exists a hex torus S 0 with associated group ∆0 and natural numbers k1 , k2 such that ∆1 = k1 ∆0 , ∆2 = k2 ∆0 ,
and so that the map f 0 = k2−1 f k1 : S 0 → S 0 is induced by conjugation by an
element of ∆3,3,3 . When k1 = k2 = 1, we call f a lattice isometry.

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

39

We now define an invariant, called the hex shape equivalence class which
may be used to determine when a lattice Q–isometry exists between two
hex tori. Let v1 , v2 be a pair of vectors of minimal length such that the
translations tv1 and tv2 generate ∆H . Assume further that the angle between
v1 and v2 is equal to 2π/3. Then v1 , v2 are unique up to the action
 ofthe
v1
dihedral group D6 of order 12, generated by the order six rotation
7→
v2

 
 

 
0 −1
v1
v1
1
0
v1
and the reflection
7→
. Let w1 =
1 1
v2
v2
−1 −1
v2
p11 v1 + p12 v2 and w2 = p21 v1 + p22 v2 be an ordered pair of vectors such that
the translations tw1 and tw2 generate a hex torus group ∆ with corresponding
hex torus denoted S. Then we can encode the shape of ∆ with the 2 × 2
integer matrix A∆ appearing in the equation
  
 
w1
p11 p12
v1
=
.
w2
p21 p22
v2
The matrix A∆ is only well-defined up to multiplication on the left by SL±
2Z
and multiplication on the right by the matrices of the dihedral group D6
above. We call the equivalence class of matrices A∆ the hex cusp shape of
S (or of ∆). The following is elementary.
Lemma 7.2. Let ∆1 , ∆2 ⊂ ∆3,3,3 be the fundamental groups of two regular
Euclidean hex tori S1 , S2 . Let A∆1 and A∆2 be matrices representing the
respective hex cusp shapes of S1 and S2 . Then there exists a lattice kk12 –
isometry f : S1 → S2 if and only if k2 [A∆1 ] = k1 [A∆2 ], ie if there exists
B ∈ SL±
2 Z and C ∈ D6 such that k2 A∆1 = k1 BA∆2 C.
For example, if ∆1 = h5v1 − 5v2 , 5v1 + 5v2 i and ∆2 = h8v1 , 4v2 i, then
representatives of the hex
hex tori S1 and
 cuspshape of the corresponding


5 −5
8 0
S2 are given by A∆1 =
and A∆2 =
. There exists a lattice
5 5
0 4
4
5 –isometry S1 → S2 because




1 1
0 −1
4A∆1 = 5
A∆2
.
0 1
1 1
Remark 7.3. Given a hex torus S, the area a of S, defined as the index
of the covering S → SH , is simply the absolute value of the determinant of
any matrix in the hex cusp shape. It follows from the Lemma that if S1
and S2 are lattice kk12 –isometric, then their respective areas a1 and a2 satisfy
k22 a1 = k12 a2 . Therefore the ratio a2 /a1pis a square, and the conformal
factor kk12 of the Q–isometry is given by a2 /a1 . Further, there are only
finitely many equivalence classes of hex tori whose area is at most a. Hence,
it is quite easy in practice to determine whether a lattice Q–isomorphism
S1 → S2 exists as long as the areas of S1 and S2 are small. If one exists,
then at most 12 = |D6 | exist.

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

40

7.2. Projective hex tori. The deformation space C(S3,3,3 ) of marked convex real projective structures on the triangle orbifold S3,3,3 is homeomorphic
to R and we now briefly describe this correspondence (see Goldman [21] and
Vinberg [39] for more details). Let r1 , r2 , r3 denote the reflections in the
three sides of the triangle T , generating ∆3,3,3 . We identify the triangle
T with the positive octant in the projective 2–sphere S2 = (R3 \ {0})/R+ ,
which is the interior of the convex hull of three directions e1 = (1, 0, 0),
e2 = (0, 1, 0), e3 = (0, 0, 1) of S2 . For each τ ∈ R, let s = eτ /3 , and define
the representation ζτ : ∆3,3,3 → SL±
3 R by:






−1 0 0
1 0 s
1 1s 0
ζτ (r1 ) =  s 1 0 , ζτ (r2 ) = 0 −1 0 , ζτ (r3 ) = 0 1 1s  .
1
0 0 −1
0 1
0 s 1
s
For each τ , these three elements are projective reflections in the sides of T
defining a representation ζτ which is discrete and injective. The union of
tiles, Ωτ = ∪γ∈∆3,3,3 ζτ (γ)T , is a convex open domain in S2 . When τ = 0,
the convex domain Ω0 is the affine chart defined by x1 + x2 + x3 > 0,
where (x1 , x2 , x3 ) are coordinates with respect to the standard basis, and
the representation ζ0 preserves a Euclidean metric on this affine chart; we
identify Ω0 with the Euclidean plane E2 and think of ζ0 as the inclusion into
Isom E2 ⊂ SL±
3 R. For τ 6= 0, Ωτ is the interior of the convex hull of the
directions of the three vectors
l1 (τ ) = τ (−1, s, 0),

l2 (τ ) = τ (0, −1, s)

and l3 (τ ) = τ (s, 0, −1),

where τ = τ /|τ |; see Figure 7. Any ζτ –equivariant homeomorphism of the
universal cover Se3,3,3 = E2 with Ωτ is a developing map for the unique convex
projective structure on S3,3,3 associated to the representation ζτ . Of course,
for any regular hex torus S covering S3,3,3 and for any τ ∈ R, we obtain a
convex projective structures on S by pullback. If ∆ ⊂ ∆3,3,3 denotes the
fundamental group of S, then the convex projective torus ζτ (∆)\Ωτ is called
a convex projective hex torus; indeed, such a torus may be decomposed
into “regular” hexagons, each of which is the union of six copies of the
fundamental triangle T .
The maximal torus subgroup ∆H of ∆3,3,3 is generated by three elements
(which are translations in the Euclidean structure):
g1 = r3 r2 r3 r1 ,

g2 = r1 r3 r1 r2 ,

g3 = r2 r1 r2 r3 .

For τ 6= 0, let hτ denote the change of basis taking (l1 (τ ), l2 (τ ), l3 (τ )) to the
standard basis. Then for i ∈ {1, 2, 3}, hτ ζτ (gi )h−1
τ = exp(τ αi ), where






−1 0 0
1 0 0
0 0 0
α1 =  0 0 0 , α2 = 0 −1 0 , α3 = 0 1 0  .
0 0 0
0 0 −1
0 0 1
It is then clear that for any γ ∈ ∆H and natural number k,
(7.1)

−1
hτ ζτ (kγ)h−1
τ = hkτ ζkτ (γ)hkτ .

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS
l2

l1

e2

l3

e3

41

e2

e1

e3

e1

l3

l1 l2

(a) τ = −3/4

(b) τ = 3/4

Figure 7. Fundamental domains of dual minimal hex tori
Therefore, if ∆ is any hex torus group, the path of representations ζτ restricted to k∆ looks the same, up to conjugation, as the k–times faster path
of representations ζkτ restricted to ∆. Using this we prove:
Lemma 7.4. Let S1 , S2 be Euclidean regular hex tori with fundamental
groups ∆1 , ∆2 ⊂ ∆3,3,3 and assume that f : S1 → S2 is a lattice kk21 –
isometry, with f∗ : ∆1 → ∆2 denoting the isomorphism at the level of fundamental group. Then for any τ 6= 0, there exists g ∈ SL±
3 R such that
gζτ /k1 (γ)g −1 = ζτ /k2 (f∗ (γ)) for all γ ∈ ∆1 .
Proof. By assumption, there exists ∆0 such that ∆1 = k1 ∆0 and ∆2 = k2 ∆0
and k2−1 f∗ k1 : ∆0 → ∆0 is conjugation by an element δ ∈ ∆3,3,3 . Then for
any γ ∈ ∆1 , write γ = k1 γ 0 , where γ 0 ∈ ∆0 and observe:
k2−1 f∗ k1 γ 0 = δγ 0 δ −1
f∗ (γ) = k2 δγ 0 δ −1


k2
0 −1
=⇒ ζτ /k2 (f∗ (γ)) = ζτ /k2
k1 δγ δ
k1
=⇒

= hζτ /k1 (δk1 γ 0 δ −1 )h−1
= gζτ /k1 (γ)g −1
where h ∈ SL±
3 R is determined by (7.1) and g = hζτ /k1 (δ).



7.3. Gluing 3–manifolds which cover reflection orbifolds. Let O be
a finite volume hyperbolic three-orbifold. Each cusp of O is naturally the
product ∂ × R of a Euclidean two-orbifold ∂ with an interval. We think
of O as the interior of an “orbifold with boundary”, where the boundary
includes a copy of each cusp cross-section ∂ at infinity. Note that if O
is a reflection orbifold, then any convex projective structure on O induces

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

42

a convex projective structure on its boundary components ∂. Indeed, the
fundamental group ∆ of ∂ fixes a unique point with eigenvalue one under the
holonomy representation of any convex projective structure on O; the convex
projective structure induced on ∂ may be seen in the link of this point.
Now, let ∂ ∼
= S3,3,3 be a boundary component isomorphic to the Euclidean
(3, 3, 3)–triangle orbifold. Then, for any manifold covering M → O, the
torus boundary components ∂e1 , . . . , ∂en of M which cover ∂ are each naturally
endowed with a regular Euclidean hex structure.
Theorem 7.5. For i ∈ {1, . . . , m}, let Mi be a hyperbolic three-manifold of
finite volume which covers a reflection orbifold Oi . For α ∈ {1, . . . , ni }, let
(∂ei )α be a torus component of ∂Mi which covers a component ∂i ∼
= S3,3,3 of
α,β
∂Oi . Let F be a collection of homeomorphisms of the form fi,j
: (∂ei )α →
(∂ej )β which identify the boundary components {(∂ei )α }i,α in disjoint pairs.
Assume that:
α,β
(1) There are natural numbers k1 , . . . , km such that any fi,j
∈ F is
k

a lattice kji –isometry of the induced regular hex torus structure on
(∂ei )α , and (∂ej )β .
(2) For each i ∈ {1, . . . , m}, there exists a non-trivial continuous path
ci : (−i , i ) → C(Oi ) passing through the finite volume complete
hyperbolic structure on Oi at ci (0) such that the convex projective
structure on ∂i induced by ci (t) is not constant.
Then there exists a properly convex projective structure on the manifold
obtained by gluing together the building blocks M1 , . . . , Mm using the gluing
maps in F .
Proof. By assumption, the convex projective structure on ∂i ∼
= S3,3,3 is not
constant, so for some ti 6= 0, the structure is isomorphic to the non-Euclidean
convex projective structure associated to some τi 6= 0 described in the previous section. It follows easily from the fact that the fundamental group of
Oi is generated by reflections that the restriction of the holonomy representation ρti for ci (ti ) to the fundamental group (∆3,3,3 )i of ∂i has a fixed point
pi with eigenvalue one and that in a suitable basis ρti (∆3,3,3 ) = ζτi ⊕ id1 ,
i
where id1 denotes the one-dimensional trivial representation. Hence, for
each α ∈ {1, . . . , ni } the restrictions of ρti to the fundamental group (∆i )α of
(∂ei )α is diagonalizable and satisfies the middle eigenvalue condition. Therefore the pull-back of ci (ti ) to Mi induces a convex projective structure which
after removing a collar neighborhood, as in Section 5, has principal totally
geodesic boundary along its (relevant) components.
Further, by continuity, τi may be varied continuously in a small neighborhood [0, δi ). Hence, we may choose t1 , . . . , tm so that (τ1 , . . . , τm ) =
µ(1/k1 , . . . , 1/km ) for some small µ ∈ R+ . It now follows from the assumpα,β
tion (1) and Lemma 7.4 that the maps fi,j
of F satisfy the holonomy
matching condition (1.1) and Theorem 1.4 implies the result.


CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

43

7.4. Application: Two-colorable tetrahedral hyperbolic manifolds.
A finite volume hyperbolic three-manifold is tetrahedral if it admits an ideal
triangulation consisting of regular ideal tetrahedra. A triangulated threemanifold is two-colorable if the tetrahedra can be colored using two colors
so that no two adjacent tetrahedra have the same color. Two-colorability
may be easily checked by Regina [6]. For example, the figure eight knot
complement and its sister manifold are tetrahedral and two-colorable. Recently Fominykh–Garoufalidis–Goerner–Tarkaev–Vesnin [20] gave a census
of all orientable tetrahedral manifolds with at most 25 tetrahedra.
It is well-known (see e.g. [20, Remark 5.5]) that every two-colorable tetrahedral manifold M is a cover of the Bianchi√Orbifold O3 = PSL(2, Z[ζ])\H3
of discriminant D = −3, where ζ = (1 + −3)/2. Further O3 covers the
2,2,3
reflection orbifold O3,3,3
determined by reflections in the faces of the partially ideal tetrahedron with dihedral angles π/2, π/2, π/3 at the three
edges bounding one face and π/3 along the other edges, incident on the
2,2,3
unique ideal vertex. The reflection orbifold O3,3,3
has one cusp and its
cross section is the Euclidean (3, 3, 3)–triangle orbifold. Further, it follows
2,2,3
from Benoist’s work [2] that the hyperbolic structure on the orbifold O3,3,3
may be deformed non-trivially to nearby convex projective structures. So
the deformation hypothesis (2) of Theorem 7.5 is satisfied for tetrahedral
two-colorable manifolds, and therefore the theorem may be applied given
appropriate gluing maps between torus boundary components that respect
the hex structure as in hypothesis (1). Indeed, such gluing maps seem to be
easy to find. We now give some examples.
Of the 29 orientable tetrahedral manifolds with at most 8 tetrahedra,
20 are two-colorable, and each of those has at most 2 cusps. In Table
1 below, these 20 manifolds are listed along with the hex cusp shapes of
the cusps. The hex torus structure at the cusps of tetrahedral manifolds
can be calculated easily using SnapPy [16]. In Figure 8(A), a fundamental
2,2,3
is drawn in yellow, while the cusp pattern of
triangle for the cusp of O3,3,3
the tessellation of H3 by regular ideal tetrahedra is drawn in black. The
hex torus structure of one boundary component ∂1 of the tetrahedral twocolorable manifold m207 is shown in Figure 8(B): ∆H (resp. the hex lattice
∆1 of ∂1 ) corresponds to vertices of triangle with black edges (resp. red
dots), and two generators of ∆1 are shown as pink arrows; we easily read of
the hex cusp shape: A∆1 = [ 31 13 ].
From Table 1, we may easily read off several new examples of closed
three-manifolds that admit properly convex projective structures.

Theorem 7.6. In each line of Table 2 below, there exist gluing maps identifying the boundary components of the given building blocks in pairs so that
the resulting closed three-manifold admits a properly convex projective structure. The gluing maps may be read off from Table 1.

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

Name Name [20]

∂1

∂2

Name

Name [20]

∂1

44

∂2

m003 otet020000 [ 20 02 ]

t12845 otet080001 [ 31 25 ] [ 10 03 ]

m004 otet020001 [ 10 04 ]

t12840 otet080002 [ 32 04 ] [ 10 04 ]

m202 otet040000 [ 32 13 ] [ 10 01 ] t12842 otet080003 [ 32 04 ] [ 20 12 ]
m203 otet040001 [ 20 13 ] [ 10 02 ] t12843 otet080004 [ 32 26 ] [ 10 02 ]
m206 otet040002 [ 20 04 ]

t12844 otet080005 [ 32 26 ] [ 10 02 ]

m207 otet040003 [ 31 13 ]

t12837 otet080006 [ 32 16 ]

s959 otet060002 [ 30 03 ] [ 21 12 ] t12839 otet080007 [ 40 24 ]
s961 otet060003 [ 32 04 ]

t12838 otet080008 [ 42 25 ]

s960 otet060004 [ 42 24 ]

t12836 otet080009 [ 30 23 ] [ 21 14 ]

s958 otet060006 [ 30 24 ]
t12841 otet080010 [ 42 24 ] [ 20 02 ]
Table 1. Hex torus cusps of orientable, two-colorable tetrahedral manifolds with at most 8 tetrahedra

v3

v1
v2

v1
v2

(a)

(b)

Figure 8. (A) Hex torus cusps of orientable two-colorable
tetrahedral manifolds, and (B) A hex torus cusp ∂1 of m207
Proof. Examination of Table 1 verifies that each given identification of boundary tori satisfies the hypotheses of Lemma 7.2 and therefore a lattice Q–
isometry exists between the given boundary components. For example,
∂1 (m003) is lattice 23 –isometric to ∂1 (s959) and ∂2 (s959) is lattice 21 –isometric
to ∂1 (s960), and so on. The result follows from Theorem 7.5.

Remark 7.7. As in Remark 7.3, there may be more than one (but no more
than 12) gluing map identifying a given pair of boundary components in
Theorem 7.6. Different gluing maps might or might not produce homeomorphic manifolds. We do not check carefully here the number of different

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

45

Building blocks
m003, s959, s960

Gluing maps
∂1 m003 → ∂1 s959
∂2 s959 → ∂1 s960
m003, t12841, s960
∂1 m003 → ∂2 t12841
∂1 t12841 → ∂1 s960
m004, t12840, s961
∂1 m004 → ∂2 t12840
∂1 t12840 → ∂1 s961
t12843, t12844
∂1 t12843 → ∂1 t12844
∂2 t12843 → ∂2 t12844
t12842, t12839, s961 ∂1 t12842 → ∂1 s961
∂2 t12842 → ∂1 t12839
Table 2. Building blocks and boundary pairings for Theorem 7.6

homeomorphism types of closed three-manifolds for which Theorem 7.6 gives
properly convex projective structures. It is at least five, but seems likely to
be more.
7.5. Application: The cusp covering conjecture. Finally, we briefly
mention an application of the Cusp Covering Conjecture, proven by Wise
(see [42] and [43, Corollary 16.15]).
Theorem 7.8 (Cusp Covering Conjecture, Wise [42, 43]). Let M be a finite
volume hyperbolic 3–manifold and let ∂M = ∂1 t ∂2 t · · · t ∂r . There exist
finite covers ∂i◦ → ∂i such that for any further finite covers ∂ic → ∂i◦ , there
c → M such that for each i, each cover of ∂i appearing on
is a finite cover M
c is isomorphic to ∂ c → ∂i .
the boundary of M
i
Let O be a reflection orbifold with one cusp ∂ ∼
= S3,3,3 satisfying the
hypothesis (2) of Theorem 7.5 and let M be a manifold cover. It follows
c of M
easily from the Cusp Covering Conjecture that there is a finite cover M
(in fact infinitely many such) so that the inclusion of the fundamental group
∆ci of each boundary component ∂ic into the fundamental group ∆3,3,3 of ∂
has the same image. Hence, we may glue together in pairs the boundary
c via a lattice isometry of the hex torus structure. The
components of M
resulting manifolds admit convex projective structures by Theorem 7.5. We
may also arrange that for each i, the image of some ∆ci ,→ ∆3,3,3 is a
multiple k∆0 of the fundamental group of any given hex torus subgroup
∆0 . Let ∆0 to be hex lattice isomorphic to, for example, the hex torus
fundamental group of the boundary torus ∂M 0 of some one-cusped manifold
c of
M 0 in Table 1. Then any boundary component of the resulting cover M
0
O may be glued to the boundary of a copy of M by a lattice k–isometry.
Let N be a manifold obtained by gluing together some pairs of boundary
c via hex lattices isometries and by gluing on copies of M 0
components of M
with hex lattice k–isometries to the remaining boundary components. Then

CONVEX PROJECTIVE STRUCTURES ON NON-HYPERBOLIC 3–MANIFOLDS

46

N admits a properly convex projective structure by Theorem 7.5. This
gives one way to obtain many interesting examples of closed non-hyperbolic
three-manifolds N which admits convex projective structures.

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48

Department of Mathematics, Florida State University, Tallahassee, FL
32306, USA
E-mail address: ballas@math.fsu.edu
URL: https://www.math.fsu.edu/~ballas/
Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA
E-mail address: jdanciger@math.utexas.edu
URL: https://www.ma.utexas.edu/users/jdanciger/
Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, D-69120
Heidelberg, Germany
E-mail address: lee@mathi.uni-heidelberg.de
URL: https://www.mathi.uni-heidelberg.de/~lee/

