47 pages, 4 figures. V4: minor modifications, mostly in the introductions. V3: improved introduction and abstract; some simplifications in Section 4. V2: This is in large part a new paper, now also including a marked length-spectrum rigidity result. All the material in Section 4 is new and this allows us to weaken hypotheses to the optimal ones
Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cubulations of arbitrary Gromov hyperbolic groups $G$. Under weak assumptions, we show that the space of cubulations of $G$ naturally injects into the space of $G$-invariant cross ratios on the Gromov boundary $\partial_{\infty}G$.
A consequence of our results is that essential, hyperplane-essential cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work in arXiv:1903.02447.
Along the way, we describe the relationship between the Roller boundary of a ${\rm CAT(0)}$ cube complex, its Gromov boundary and - in the non-hyperbolic case - the contracting boundary of Charney and Sultan.
All our results hold for cube complexes with variable edge lengths.
Agol proved that hyperbolic cubulated groups are virtually special. The aim of these notes is to make the proof accessible to a wider audience; we retain the underlying ideas and constructions of Agol, but substantially change or add to many parts of the argument to give a more transparent and detailed account.
We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol's criterion, we find that every subgroup separable closed hyperbolic 3-manifold is virtually fibered.
We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical $C'(1/6)$--small cancellation condition is cubulable. This yields a new large class of relatively hyperbolic groups that can be cubulated, and constitutes the first instance of a cubulability theorem for relatively hyperbolic groups which does not require any geometric assumption on the peripheral subgroups besides their cubulability. We do this by constructing appropriate wallspace structures for such groups, by combining walls of the free fac…
The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a very short hierarchy. Moreover, we describe a 60-sheeted cover in which the associated cubulation is special. We also describe the natural cubulation and covers of the spherical dodecahedral space (aka Poincaré homology sphere).
We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds with one cusp. More generally, we show that the number of $k$-cusped hyperbolic four-manifolds with volume smaller than V grows like $C^{V log V}$ for any fixed $k$. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.
We provide a condition on the links of polygonal complexes that is sufficient to ensure groups acting properly discontinuously and cocompactly on such complexes contain a virtually free codimension-1 subgroup. We provide stronger conditions on the links of polygonal complexes, which are sufficient to ensure groups acting properly discontinuously and cocompactly on such complexes act properly discontinuously on a CAT(0) cube complex. If the group is hyperbolic then this action is also cocompact, hence by Agol's Theorem the group is virtually special (in the sense of Haglund-Wise); in particular…
The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-metric" and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds, and integral convergence to hyperbolic for the metrics under consideration.
We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing density for links is bounded above by 3.1263.... Moreover, there is a sequence of links with cusp crossing density approaching 3. The least upper bound for cusp crossing density remains an open question. For two-component hyperbolic links, cusp crossing density is shown to be dens…
We provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to index 1, 2, 4 or 8.
We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a proper hyperbolic geodesic metric space.
In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston's hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of ``uniform'' size yield hyperbolic structures. Here the size of a…
The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action on R^3 by polynomial automorphisms preserving the cubic polynomial and an area form on the level surfaces. We describe the dynamical decomposition of this action. The domain of discontinuity of this action corresponds to geometric structures: either complete hyperbolic structures on the 2-holed …
We study maximal representations of surface groups $ρ:π_1(Σ)\to\mathrm{SO}_0(2,n+1)$ via the introduction of $ρ$-invariant pleated surfaces inside the pseudo-Riemannian space $\mathbb{H}^{2,n}$ associated to maximal geodesic laminations of $Σ$. We prove that $ρ$-invariant pleated surfaces are always embedded, acausal, and possess an intrinsic pseudo-metric and a hyperbolic structure. We describe the latter by constructing a shear cocycle from the cross ratio naturally associated to $ρ$. The process developed to this purpose applies to a wide class of cross ratios, including examples arising fr…
We compute and analyse the moduli space of those real projective structures on a hyperbolic 3-orbifold that are modelled on a single ideal tetrahedron in projective space. Parameterisations are given in terms of classical invariants, traces, and geometric invariants, cross ratios.
For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.
The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [0,v_{oct}]. We construct sequences of alternating knots whose volume and determinant densities both converge to any x in [0,v_{oct}]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these inv…
Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connection between the intrinsic geometry of a 3-manifold and its number-theoretic invariants. Further, we explore the question of choosing a minimal collection of arcs and associated parameters to generate the field. We prove that for a tunnel number $k$ manifold it is enough to choose…
We obtain an exact modularity relation for the $q$-Pochhammer symbol. Using this formula, we show that Zagier's modularity conjecture for a knot $K$ essentially reduces to the arithmeticity conjecture for $K$. In particular, we show that Zagier's conjecture holds for hyperbolic knots $K\neq 7_2$ with at most seven crossings. For $K=4_1$, we also prove a complementary reciprocity formula which allows us to prove a law of large numbers for the values of the colored Jones polynomials at roots of unity. We conjecture a similar formula holds for all knots and we show that this is the case if one as…
We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $a/b vol(M)=(-D)^{3/2}/(2π)^{2n-4} (ζ_K(2))/(2ζ(2))$ for a manifold M whose invariant trace field $K$ has a single complex place, discriminant $D$, degree $n$, and Dedekind zeta value $ζ_K(2)$. The largest numerator of the 998 invariants of Hodgson-Weeks manifolds is, astoundingly, $a=2^4\times23\times37\times691 =9,408,656$; the largest denominator is merely b=9. We also study the rational invariant a/b for single-complex-place cusped manifolds, …
