Let W be a compact manifold and let ρbe a representation of its fundamental group into PSL(2,C). The volume of ρis defined by taking any ρ-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a non-compact (cusped) manifold M, but he did not prove the volume is well-defined in all cases. We prove here that the volume of a representation is always well-defined and depends only on the representation. We show that this volume can be easily computed by straightening any ideal triangulation of M. We show that the volume of a representation is bounded from above by the relative simplicial volume of M. Finally, we prove a rigidity theorem for representations of the fundamental group of a hyperbolic manifold. Namely, we prove that if M is hyperbolic and vol(ρ)=vol(M) then ρis discrete and faithful.
We introduce a generalization of the Dijkgraaf-Witten invariants for cusped or compact oriented 3-manifolds. We show that the generalized DW invariants distinguish some pairs of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same Turaev-Viro invariants. We also present an example of a pair of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same homology groups, whereas with distinct generalized DW invariants.
Combinatorial Ricci flow on a cusped $3$-manifold is an analogue of Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact $3$-manifolds with boundary for finding complete hyperbolic metrics on cusped $3$-manifolds. Dual to Casson and Rivin's program of maximizing the volume of angle structures, combinatorial Ricci flow finds the complete hyperbolic metric on a cusped $3$-manifold by minimizing the co-volume of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may develop singularities. We overcome this difficulty by extending the flow…
Previous work of the authors with Bus Jaco determined a lower bound on the complexity of cusped hyperbolic 3-manifolds and showed that it is attained by the monodromy ideal triangulations of once-punctured torus bundles. This paper exhibits an infinite family of minimal ideal triangulations of Dehn fillings on the link $8^3_9$ that also attain this lower bound on complexity.
For any oriented cusped hyperbolic $3$-manifold $M$, we study its $(R,ε)$-panted cobordism group, which is the abelian group generated by $(R,ε)$-good curves in $M$ modulo the oriented boundaries of $(R,ε)$-good pants. In particular, we prove that for sufficiently small $ε>0$ and sufficiently large $R>0$, some modified version of the $(R,ε)$-panted cobordism group of $M$ is isomorphic to $H_1(\text{SO}(M);\mathbb{Z})$.
Previous work of the authors studies minimal triangulations of closed 3-manifolds using a characterisation of low degree edges, embedded layered solid torus subcomplexes and 1-dimensional $\mathbb{Z}_2$-cohomology. The underlying blueprint is now used in the study of minimal ideal triangulations. As an application, it is shown that the monodromy ideal triangulations of the hyperbolic once-punctured torus bundles are minimal.
The space $\mathrm{GC} (Σ)$ of geodesic currents on a hyperbolic surface $Σ$ can be considered as a completion of the set of weighted closed geodesics on $Σ$ when $Σ$ is compact, since the set of rational geodesic currents on $Σ$, which correspond to weighted closed geodesics, is a dense subset of $\mathrm{GC}(Σ)$. We prove that even when $Σ$ is a cusped hyperbolic surface with finite area, $\mathrm{GC}(Σ)$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $Σ$ but also to weighted geodesics connecting two cusps. In addition, we …
We show that the problem of showing that a cusped 3-manifold M is not hyperbolic is in NP, assuming $S^3$-RECOGNITION is in coNP. To this end, we show that IRREDUCIBLE TOROIDAL RECOGNITION lies in NP. Along the way we unconditionally recover SATELLITE KNOT RECOGNITION lying in NP. This was previously known only assuming the Generalized Riemann Hypothesis. Our key contribution is to certify closed essential normal surfaces as essential in polynomial time in compact orientable irreducible $\partial$-irreducible triangulations. Our work is made possible by recent work of Lackenby showing several …
We show that the 1-cusped quotient of the hyperbolic space $\mathbb{H}^3$ by the tetrahedral Coxeter group $Γ_*=[5,3,6]$ has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined. Furthermore, the lattice $Γ_*$ is incommensurable to any Gromov-Piatetski-Shapiro type lattice. Our methods have their origin in the work of C. Adams. We extend considerably this approach via the geometry of the underlying horoball configuration induced by a cusp.
We show the manifolds at infinity of the complex hyperbolic triangle groups $Δ_{3,4,4;\infty}$ and $Δ_{3,4,6;\infty}$,are one-cusped hyperbolic 3-manifolds $m038$ and $s090$ in the Snappy Census respectively.That is,these two manifolds admit spherical CR uniformizations. Moreover, these two hyperbolic 3-manifolds above can be obtained by Dehn surgeries on the first cusp of the two-cusped hyperbolic 3-manifold $m295$ in the Snappy Census with slopes $2$ and $4$ respectively. In general,the main result in this paper allow us to conjecture that the manifold at infinity of the complex hyperbolic t…
This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of minimal length. The main result is that, if the surface has at least one cusp, their self-intersection numbers are exactly $k$ for large enough $k$.
In this note, we show that there exist cusped hyperbolic $3$-manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic $4$-manifolds, and by Kolpakov, Reid and Slavich on embedding arithmetic hyperbolic manifolds.
We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings; to show that the figure-8 knot complement and its sister are the unique hyperbolic 3-manifolds with minimal volume maximal cusps; and to extend results on determining low volume closed and cusped hyperbolic 3-manifolds.
We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.
Volume and homology of one-cusped hyperbolic 3-manifolds
设M是完备的、有限体积的、可定向的双曲流形，它恰好有一个尖点。如果我们假设pi_1（M）没有子群同构于亏格2曲面群，并且（a）H_1（M;Z_p）对于某个素数p至少有5维，或者（b）H_1（M;Z_2）至少有4维，那么H^2（M; Z_2）的维数最多为1，则vol M > 5.06如果我们假设H_1（M; Z_2）的维数至少为7，且M的致密核不含亏格2闭不可压缩曲面，则vol M > 5.06。
We study how the spin structures on finite-volume hyperbolic n-manifolds restrict to cusps. When a cusp cross-section is a (n-1)-torus, there are essentially two possible behaviours: the spin structure is either bounding or Lie. We show that in every dimension n there are examples where at least one cusp is Lie, and in every dimension n <= 8 there are examples where all the cusps are bounding. By work of C. Bar, this implies that the spectrum of the Dirac operator is R in the first case, and discrete in the second. We therefore deduce that there are cusped hyperbolic manifolds whose spectrum o…
In this paper, we adopt combinatorial Ricci curvature flow methods to study the existence of cusped hyperbolic structure on 3-manifolds with torus boundary. For general pseudo 3-manifolds, we prove the long-time existence and the uniqueness for the extended Ricci flow for decorated hyperbolic polyhedral metrics. We prove that the extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature. If it is the case, the flow converges exponentially fast. These results apply for cusped hyperbolic …
Any two geometric ideal triangulations of a cusped complete hyperbolic $3$-manifold $M$ are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on dihedral angles. This leads to a naive but effective algorithm to check if two hyperbolic knots are equivalent, given geometric ideal triangulations of their complements. Given a geometric ideal triangulation of $M$, we also give a lower bound on the systole length of $M$ in terms of the number of tetrahedra and a lower …
A group $Γ$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $Γ$ admits a cusp-uniform action on a proper $δ$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric, provided the spaces have "constant horospherical distortion," a condition satisfied by Groves--Manning's cusped Cayley graph and by all negatively curved symmetric spaces. Consequently the Bowditch boundary admits a canonical quasisymmetric structure, which coincides with the "naturally occurring" quasisymmetric structure of the symmetric space when considering lattices in…
We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice $Γ$, applying a classical result of Macbeath to a suitable $Γ$-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ for $d=1,3,7$ and the quaternion hyperbolic lattice ${\rm PU}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$. The implementation of the method for these groups is computer-assisted.
A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic 3-manifold with torus boundaries. This family of 3-manifolds includes the knot complements. This computation of a hyperbolic structure requires the resolution of gluing equations on a triangulation of the space, but not all triangulations admit a solution to the equations. In this paper, we propose a new method to find a triangulation that admits a solution to the …
We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichmüller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theories and such that their traces define invariants of mapping tori.
