A note on hyperbolic 3-manifolds of the same volume

We describe a method for constructing an arbitrary number of closed hyperbolic 3-manifolds of the same volume. In fact we prove that many hyperbolic 3-manifolds of finite volume have an arbitrary number of non-homeomorphic finite convering spaces of the same degree and hence the same volume. This applies, for example, to all hyperbolic 3-manifolds whose universal covering group is a subgroup of finite index in a Coxeter group generated by the reflections in the faces of a hyperbolic Coxeter polyhedron. It also applies to all hyperbolic 3-manifolds of finite volume with at least one cusp.     

The Chern-Simons Invariants of Hyperbolic Manifolds Via Covering Spaces

One important invariant of a closed Riemannian 3-manifold isthe Chern–Simons invariant [1]. The concept was generalizedto hyperbolic 3-manifolds with cusps in [11], and to geometric(spherical, euclidean or hyperbolic) 3-orbifolds, as particularcases of geometric cone-manifolds, in [7]. In this paper, westudy the behaviour of this generalized invariant under changeof orientation, and we give a method to compute it for hyperbolic3-manifolds using virtually regular coverings [10]. We confineourselves to virtually regular coverings because a coveringof a geometric orbifold is a geometric manifold if and onlyif the covering is a virtually regular covering of the underlyingspace of the orbifold, branched over the singular locus. Thereforeour work is the most general for the applications in mind; namely,computing volumes and Chern–Simons invariants of hyperbolicmanifolds, using the computations for cone-manifolds for whicha convenient Schläfli formula holds (see [7]). Among otherresults, we prove that every hyperbolic manifold obtained asa virtually regular covering of a figure-eight knot hyperbolicorbifold has rational Chern–Simons invariant. We giveexplicit examples with computations of volumes and Chern–Simonsinvariants for some hyperbolic 3-manifolds, to show the efficiencyof our method. We also give examples of different hyperbolicmanifolds with the same volume, whose Chern–Simons invariants(mod ) differ by a rational number, as well as pairs of differenthyperbolic manifolds with the same volume and the same Chern–Simonsinvariant (mod ). (Examples of this type were also obtainedin [12] and [9], but using mutation and surgery techniques,respectively, instead of coverings as we do here.) 1991 MathematicsSubject Classification 57M50, 51M10, 51M25.     

Hyper-Kähler geometry and invariants of three-manifolds

We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kähler manifold X. A Feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3-manifolds which is similar in structure to those appearing in the perturbative calculation of the Chern-Simons partition function. The sigma-model suggests a new system of weights for finite type invariants of 3-manifolds, described by trivalent graphs. The Riemann curvature of X plays the role of Lie algebra structure constants in Chern-Simons theory, and the Bianchi identity plays the role of the Jacobi identity in guaranteeing the so-called IHX relation among the weights. We argue that, for special choices of X, the partition function of the sigma-model yields the Casson-Walker invariant and its generalizations. We also derive Walker's surgery formula from the SL(2, Z) action on the finite-dimensional Hilbert space obtained by quantizing the sigma-model on a two-dimensional torus.     

The Chern-Simons invariants of cone-manifolds with Whitehead link singular set

In the present article, we obtain some explicit integral formulas for the generalized Chern-Simons function I(W(α,β)) for Whitehead link cone-manifolds in the hyperbolic and spherical cases. We also give the Chern-Simons invariant for the Whitehead link orbifolds. We find a formula for the Chern-Simons invariant of n-fold coverings of the three-sphere branched over the Whitehead link.     

Cyclically symmetric hyperbolic spatial graphs in 3-manifolds

The aims of this paper is to prove that every closed connected orientable 3-manifold with an orientation-preserving periodic diffeomorphism contains infinitely many, setwise invariant, spatial graphs whose exteriors are hyperbolic 3-manifolds.     

Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center To the Memory of Professor Shantao Liao

In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center.     

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any $k\ge 2$, we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.     

Mutative hyperbolic homology 3-spheres with the same Floer homology

We construct a large class of finitely many hyperbolic homology 3-spheres making the following invariants equal, simultaneously, the integral homology, the quantum SU(2) invariants, the hyperbolic volume, the hyperbolic isometry group, the -invariant, the Chern-Simons invariant, and the Floer homology.     

Cluster Partition Function and Invariants of 3-Manifolds

The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G.The author focuses on the case of G =SL(N,C) and M being a knot complement:M =S3 \ k.The main result presented in this note is the cluster partition function,a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G =SL(N,C).He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory.     

Some curious Kleinian groups and hyperbolic 5-manifolds

We present a new family of discrete subgroups ofSO (5, 1) isomorphic to lattices inSO (3, 1). In some of the examples the limit sets are wildly knotted 2-spheres. As an application we produce complete hyperbolic 5-manifolds that are nontrivial plane bundles over closed hyperbolic 3-manifolds and conformally flat 4-manifolds that are nontrivial circle bundles over closed hyperbolic 3-manifolds.     

Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron

It is known that the volume function for hyperbolic manifolds of dimension 3 is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even integral multiples of 4π²/3. This is “half” the set of possible values for volumes, which is the integral multiples of 4π²/3 due to the Gauss-Bonnet formula Vol(M) = 4π²/3 · χ(M).     

A remark on renormalized volume and Euler characteristic for ache 4-manifolds

This note computes a renormalized volume and a renormalized Gauss-Bonnet-Chern formula for asymptotically complex hyperbolic Einstein (so-called ache) 4-manifolds.     

A volume formula for hyperbolic tetrahedra in terms of edge lengths

We give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths. The cue of our formula is by the volume conjecture for the Turaev-Viro invariant of closed 3-manifolds, which is defined from the quantum 6j -symbols. This formula contains the dilogarithm functions, and we specify the adequate branch to get the actual value of the volumes.     

Sectional-Anosov flows

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.     

Generalised Fibonacci manifolds

Fibonacci manifolds have a hyperbolic structure which may be defined via Fibonacci numbers. Using related sequences of Lucas numbers, other 3-manifolds are constructed, their geometric structures determined, and a curious relationship between the homology and the invariant trace-field examined.Supported by the Royal Society.     

Sectional-Anosov flows

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.     

Incompressible surfaces in punctured-torus bundles

We derive in this paper the classification up to isotopy of the incompressible surfaces in hyperbolic 3-manifolds which fiber over the circle with fiber a once-punctured torus. From this classification it follows that most of the 3-manifolds obtained by compactifying these bundles via a circle at infinity are closed hyperbolic 3-manifolds which contain 1.0 incompressible surfaces, i.e., are not Haken manifolds.     

Lower bounds on volumes of hyperbolic Haken 3-manifolds

We prove a volume inequality for 3-manifolds having metrics ``bent' along a surface and satisfying certain curvature conditions. The result makes use of Perelman's work on the Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of convex cores of Kleinian groups, improved volume estimates for certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume of orientable hyperbolic 3-manifolds. An appendix compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed embedded geodesic with experimental data.

     

On a class of hyperbolic 3-manifolds and groups with one defining relation

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, arbitrarily many of the same volume. The fundamental groups of these 3-manifolds are groups with one defining relation. Our main result is a classification of these manifolds up to homeomorphism, resp. isometry.     

Admissible transverse surgery does not preserve tightness

We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots—namely, the range of slopes on which admissible transverse surgery preserves tightness—and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universally tight contact 3-manifolds whose Heegaard Floer contact invariants vanish (and which are not weakly fillable); and the existence of open books with arbitrarily high fractional Dehn twist coefficients whose compatible contact structures are not deformations of co-orientable taut foliations.