A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3-manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these immersions. This is used to develop and implement an algorithm to compute the unit ball of the Thurston norm for cusped hyperbolic 3-manifolds of finite volume. As an application of independent interest, we give an upper bound on the minimal entropy of pseudo-Anosov maps of surfaces with number of cusps bounded linearly in genus.

ln this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to M × [ 0 , ∞ ) M\times [0,\infty ) where M M is a closed Euclidean manifold. These are classified by Ballas, Cooper, and Leitner [J. Topol. 13 (2020), pp. 1455-1496]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of π 1 M \pi _1M . It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on M M , and the fiber is a closed cone in the space of cubic differentials. For 3 3 -dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.

The area of a convex projective surface of genus [Formula: see text] is at least [Formula: see text] where [Formula: see text] is the vector of triangle invariants of Bonahon–Dreyer and [Formula: see text] are the Fock–Goncharov triple ratios.

This is an expository proof that, if $M$ is a compact $n$-manifold with no boundary, then the set of holonomies of strictly-convex real-projective structures on $M$ is a subset of $\operatorname{Hom}(\pi_1M,\operatorname{PGL}(n+1,\mathbb RR))$ that is both open and closed.

There is a compactification of the space of representations of a finitely generated group into the groups of isometries of all spaces with $\Delta$-thin triangles. The ideal points are actions on $\mathbb R$-trees. It is a geometric reformulation and extension of the Culler-Morgan-Shalen theory concerning limits of representations into $\operatorname{SL}(2,{\mathbb C})$ and more generally $\operatorname{O}(n, 1)$. This paper was written and circulated in the early 90's, but never published.

Suppose G is finitely generated group and 𝒞(G) consists of all ρ:G→PGL(n+1,ℝ) for which there exists a properly convex set in ℝℙ^n that is preserved by ρ(G). Then the image of 𝒞(G) is closed in the character variety. Suppose G does not contain an infinite, normal, abelian subgroup and 𝒟(G)⊂𝒞(G) is the subset of holonomies of properly-convex n-manifolds with fundamental group G. Then the image 𝒟(G) is closed in the character variety. If M is the interior of a compact n-manifold and G=π_1M is as above, and either M is closed, or π_1M contains a subgroup of infinite index isomorphic to ℤ^n-1, then 𝒟(G) is closed. If, in addition, M is the interior of a compact manifold N such that every component of ∂ N is π_1-injective, and finitely covered by a torus, then every element of 𝒟(G) is the holonomy of a properly-convex structure on M, and 𝒟(G) is a union of connected components of a semi-algebraic set.

We study a generalized cusp C that is diffeomorphic to [ 0 , ∞ ) times a closed Euclidean manifold. Geometrically, C is the quotient of a properly convex domain in R P n by a lattice, Γ, in one of a family of affine Lie groups G ( ψ ) , parameterized by a point ψ in the (dual closed) Weyl chamber for SL ( n + 1 , R ) , and Γ determines the cusp up to equivalence. These affine groups correspond to certain fibered geometries, each of which is a bundle over an open simplex with fiber a horoball in hyperbolic space, and the lattices are classified by certain Bieberbach groups plus some auxiliary data. The cusp has finite Busemann measure if and only if G ( ψ ) contains unipotent elements. There is a natural underlying Euclidean structure on C unrelated to the Hilbert metric.

Suppose $E$ is an end of an irreducible, properly convex, real-projective $n$-manifold $M$. If $\pi_1E$ contains a subgroup of finite index isomorphic to ${\mathbb Z}^{n-1}$, and $E\hookrightarrow M$ is $\pi_1$-injective, then $E$ is a generalized cusp. We list some consequences when all ends are of this type. Under certain hypotheses we prove the holonomy of a properly convex manifold is irreducible.

This paper proves that every finite volume hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, non-asymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary of this result and a companion statement about surfaces with cusps, we recover Wise's theorem that the fundamental group of M acts freely and cocompactly on a CAT(0) cube complex.

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an open subset of the representation variety. We also give a relative version for non-compact (G,X)-manifolds of the openess of their holonomies.

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transform into another. We develop a general framework to describe transitions in the context that both geometries involved are represented as sub-geometries of a larger ambient geometry. Specializing to the setting of real projective geometry, we classify the geometric limits of any sub-geometry whose structure group is a symmetric subgroup of the projective general linear group. As an application, we classify all limits of three-dimensional hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and Sol geometry among the limits. We prove, however, that the other Thurston geometries, in particular $\mathbb{H}^2 \times \mathbb{R}$ and $\widetilde{\operatorname{SL}_2 \mathbb{R}}$, do not embed in any limit of hyperbolic geometry in this sense.

This paper gives the first example of a unipotent group that is not virtually abelian and preserves a strictly convex domain.

We show that the connected sum of two copies of real projective 3-space does not admit a real projective structure. This is the first known example of a connected 3-manifold without a real projective structure.

The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.

We give counterexamples to a version of the simple loop conjecture in which the target group is PSL(2,C). These examples answer a question of Minsky in the negative.

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space. Except in dimension 3, there are only finitely many topological types of strictly convex manifolds with bounded volume. In dimension 4 and higher, the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part. There is an algebraic characterization of strict convexity in terms of relative hyperbolicity.

The area in the Hilbert metric of a compact, properly convex, projective surface $F$ is at least $\pi^2|\chi (F)|$. The area of an ideal triangle with Fock-Goncharov parameter $t$ is at least $(\pi^2+(\log t)^2)/2$.

For a smooth, closed $n$-manifold $M$, we define an upper semi-continuous integer-valued complexity function on $H^1(M;{\mathbb R})$ using Morse theory. This measures how far an integral class is from being a fiber of a fibration. The fact complexity minimisers are open generalises Tischler's result on the openness of classes dual to fibrations. We then use this to define a complexity function on 1-dimensional cohomology of a finitely presented group, which is constant on open rays from the origin and vanishes precisely on the geometric invariant due to Bieri, Neumann and Strebel.

Any one-cusped hyperbolic manifold M with an unknotting tunnel tau is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where M is obtained by "generic" Dehn filling, we prove that tau is isotopic to a geodesic, and characterize whether tau is isotopic to an edge in the canonical decomposition of M. We also give explicit estimates (with additive error only) on the length of tau relative to a maximal cusp. These results give generic answers to three long-standing questions posed by Adams, Sakuma and Weeks.We also construct an explicit sequence of one-tunnel knots in S-3, all of whose unknotting tunnels have length approaching infinity.