In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form $H^5/Γ$ where $Γ$ is a torsion-free subgroup of minimal index of the congruence two subgroup $Γ^5_2$ of the group $Γ^5$ of positive units of the Lorentzian quadratic form $x_1^2+...+x_5^2-x_6^2$. We also show that $Γ^5_2$ is a reflection group with respect to a 5-dimensional right-angled convex polytope in $H^5$. As an application, we construct a hyperbolic 5-manifold of smallest known volume $7ζ(3)/4$.
We provide an explicit lower bound for the sytole in principal congruence covers of compact quaternionic hyperbolic manifolds. We also prove the optimality of this lower bound.
Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove that for any $ε>0$ there exist infinitely many congruence covers $\{M_i\}$ such that, for any $x \in M$, $M_i$ contains an embbeded ball $B_x$ (with center $x$) satisfying $\text{vol}(B_x) > (\text{vol}(M_i))^{\tfrac{1}{4}-ε}$. We get similar results in the arithmetic non-compact case.
We give a constructive proof that the Regge symmetry is a scissors congruence in hyperbolic space. The main tool is Leibon's construction for computing the volume of a general hyperbolic tetrahedron. The proof consists of identifying the key elements in Leibon's construction and permuting them.
Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number λ_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, the number of Laplacian eigenvalues of the congruence cover Gamma(q)\ H^n smaller than lambda_0 is at most of order [Gamma:Gamma(q)]^c for some c=c(λ_0)>0.
This paper is a follow-up to our joint paper with I. Agol, P. Storm and K. Whyte "Finiteness of arithmetic hyperbolic reflection groups". The main purpose is to investigate the effective side of the method developed there and its possible application to the problem of classification of arithmetic hyperbolic reflection groups.
For any triple $(W,L,ρ)$, where W is a closed connected and oriented 3-manifold, L is a link in W and $ρ$ is a flat principal B-bundle over W (B is the Borel subgroup of $SL(2,\mc)$), one constructs a $\Dd$-scissors congruence class $\cG_{\Dd}(W,L,ρ)$ which belongs to a (pre)-Bloch group $\Pp (\Dd)$. The class $\cG_{\Dd}(W,L,ρ)$ may be represented by $\Dd$-triangulations $\Tt=(T,H,\Dd)$ of $(W,L,ρ)$. For any $\Tt$ and any odd integer $N>1$, one defines a ``quantization'' $\Tt_N$ of $\Tt$ based on the representation theory of the quantum Borel subalgebra $\Ww_N$ of $U_q(sl(2,\mc))$ specialized …
The integral Burau representation provides a map from the braid group into a group of integral matrices. This allows for a definition of congruence subgroups of the braid group as the preimage of the usual principal congruence subgroups of integral matrices. We explore the structure these congruence subgroups by examining some of the quotients that may arise in the series induced by divisibility of levels. We build on the work of Stylianakis on symmetric quotients of congruence subgroups, which itself generalizes the quotient of the braid group by the pure braid group. We accomplish this by ut…
This is a survey of our work on Quantum Hyperbolic Invariants (QHI) of 3-manifolds. We explain how the theory of scissors congruence classes is a powerful geometric framework for QHI and for a `Volume Conjecture' to make sense.
We describe a criterion for a real or complex hyperbolic lattice to admit a RFRS tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups $\mathrm{PSL}(\mathcal{O}_d)$ are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups that are not a subgroup of a product of surface groups and abelian groups.
We introduce here explicit integral formulas for linking, twisting, writhing and helicity on the 3-sphere and in hyperbolic 3-space. These formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological, in the sense that their integrands are invariant under orientation-preserving isometries of the ambient space. They are obtained by developing and then applying a steady-state version of classical electrodynamics in these two spaces, including an explicit Biot-Savart formula for the magnetic field and a corresponding Ampere's law contained in Maxwell's equation…
We show that on any hyperbolic knot in $S^3$ there is at most one non-integral Dehn surgery which yields a manifold containing an incompressible torus.
We define an invariant β(M) of a finite volume hyperbolic 3-manifold M in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M. β(M) lies in a subgroup of \B(\C) of finite \Q-rank determined by the invariant trace field of M. Moreover, the Chern-Simons invariant of M is determined modulo rationals by β(M). This leads to a simplicial formula and rationality results for the Chern Simons invariant which appear elsewhere. Generalizations of β(M) are also described, as well as several interesting examples. An appendix describes a scisso…
Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the 3-sphere S^3? It is conjectured that if r-surgery on a hyperbolic knot in S^3 yields a Seifert fiber space, then r is an integer. We show that for each integer n, there exists a tunnel number one, hyperbolic knot K_n in S^3 such that n-surgery on K_n produces a small Seifert fiber space.
Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical manifolds and give refined upper bounds of integral foliated simplicial volume in terms of stable integral simplicial volume. This allows us to compute the integral foliated simplicial volume of hyperbolic 3-manifolds. This is complemented by the calculation of the integral foliat…
We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce the following theorem: Given an arithmetically defined cocompact subgroup of SL(2,C), provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence of finite index subgroups of such that the first Betti number grows at least as fast as the squa…
For a hyperbolic knot in the 3-sphere, at most finitely many Dehn surgeries yield non-hyperbolic 3-manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed 3-manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. We show that the distance between two integral toroidal slopes for a hyperbolic knot, except the figure-eight knot, is at most four. Hence any hyperbolic knot admits at most 5 toroidal surgeries.
A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $π_1(M)$. The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and the group $G$ is approximated (in an algebraic sense) by its quotients obtain…
It is shown that a hyperbolic knot in the 3-sphere admits at most nine integral surgeries yielding 3-manifolds which are reducible or whose fundamental groups are not infinite word-hyperbolic.
We investigate the geometry of $π_1$-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any $e>0$, if the manifold $M$ has sufficiently large systole $\sys_1(M)$, the genus of any such surface in $M$ is bounded below by $\exp((1/2-e)\sys_1(M))$. Using this result we show, in particular, that for congruence covers $M_i\to M$ of a compact arithmetic hyperbolic 3-manifold we have: (a) the minimal genus of $π_1$-injective surfaces satisfies $\log \sysg(M_i) \gtrsim (1/3)\log\vol(M_i)$; (b) there exist such sequences with the ratio Heegard genus$(M_i)/\sysg(M_i) \gtrsim \v…
