We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration of M over the circle is the modular elliptic curve E=X_0(49), which admits multiplication by the ring of integers of Q[sqrt(-7)]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K=Q[sqrt(-3)], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of level 7 of the multiplicative group of a maximal order of D. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a non-compact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.
We give the first examples of closed fibered hyperbolic 3-manifolds whose fundamental groups are distinguished from every other finitely generated, residually finite group by their finite quotients. One of the examples is also the first example of a non-orientable profinitely rigid hyperbolic 3-manifold.
We construct a hyperbolic fibered slice knot with right-veering monodromy, giving a negative answer to the question posed by Hubbard-Kawamuro-Kose-Martin-Plamenevskaya-Raoux-Truong-Turner.
Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects the combinatorics of L `at infinity'. The class of links at infinity of affine F-line arrangements is properly included in the class of links at infinity of hyperbolic F-line arrangements. Many links at infinity of (essentially non-affine) connected hyperbolic C-line arrangement…
Let $\barγ$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $γ$ in $\mathcal O.$ We prove that the complement $M_{\barγ}$ of $\barγ$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.
We complete the classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. This is the final step in understanding all exceptional surgeries on hyperbolic pretzel knots. We also present results toward similar classifications for non-pretzel Montesinos knots of length three.
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.
In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of examples of primitive/Seifert-fibered knots. By analyzing the twisted torus knots, we prove that nearly all possible triples of multiplicities of the critical fibers arise via Dehn surgery on primitive/Seifert-fibered knots.
Suppose $α$ and $R$ are disjoint simple closed curves in the boundary of a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $α$ is Seifert in $H$(i.e., a 2-handle addition $H[α]$ is a Seifert-fibered space) and not the meridian of $H[R]$. Then for a slope $γ$ of $k$ represented by $α$, $γ$-Dehn surgery $k(γ)$ is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was i…
By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic $3$-manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.
Let $p$ be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually $p$. Let $M=M^3$ be a virtually fibered 3-manifold. It is well-known that $G=π_1(M)$ is residually solvable and even residually finite solvable. We prove that $G$ is always virtually residually $p$. Using recent work of Wise, we prove that every hyperbolic 3-manifold is either closed or virtually fibered and hence has a virtually residually $p$ fundamental group. We give some generalizations to pro-$p$ completions of groups, mapping class groups, residually torsion-free ni…
Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to the circle, according as the center of the fundamental group of M is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings of…
It is proved by Sakuma and Brooks that any closed orientable $3$-manifold with a Heegaard splitting of genus $g$ admits a $2$-fold branched cover that is a hyperbolic $3$-manifold and a genus $g$ surface bundle over the circle. This paper concerns entropy of pseudo-Anosov monodromies for hyperbolic fibered $3$-manifolds. We prove that there exist infinitely many closed orientable $3$-manifolds $M$ such that the minimal entropy over all hyperbolic, genus $g$ surface bundles over the circle as $2$-fold branched covers of the $3$-manifold $M$ is comparable to $1/g$.
We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-manifolds. If a surface bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of "extremal" tight contact structures. Specifically, there is exactly one contact structure whose Euler class, when evaluated on the fiber, equals the Euler number of the fiber. This rigidity theorem is a consequence of properties of the action of pseudo-Anosov maps on the complex of curves of the fiber and a remarkable flexibility property of convex surfaces i…
Let $K\subset S^3$ be a hyperbolic fibered knot such that $S^3_{p/q}(K)$, the $\frac pq$--surgery on $K$, is non-hyperbolic. We prove that if the monodromy of $K$ is right-veering, then $0\le\frac pq\le 4g(K)$. The upper bound $4g(K)$ cannot be attained if $S^3_{p/q}(K)$ is a small Seifert fibered L-space. If the monodromy of $K$ is neither right-veering nor left-veering, then $|q|\le3$. As a corollary, for any given positive torus knot $T$, if $p/q\ge4g(T)+4$, then $p/q$ is a characterizing slope. This improves earlier bounds of Ni--Zhang and McCoy. We also prove that some finite/cyclic slope…
We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.
We study the smallest positive eigenvalue $λ_1(M)$ of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold $M$ which fibers over the circle, with fiber a closed surface of genus $g\geq 2$. We show the existence of a constant $C>0$ only depending on $g$ so that $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2, C\log {\rm vol}(M)/{\rm vol}(M)^{2^{2g-2}/(2^{2g-2}-1)}]$ and that this estimate is essentially sharp. We show that if $M$ is typical or random, then we have $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2,C/{\rm vol}(M)^2]$. This rests on a result of independent interest about reccurence properties of axe…
A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries.
In this paper we study the systoles of arithmetic hyperbolic 2- and 3-manifolds. Our first result is the construction of infinitely many arithmetic hyperbolic 2- and 3-manifolds which are pairwise noncommensurable, all have the same systole, and whose volumes are explicitly bounded. Our second result fixes a positive number x and gives an upper bound for the least volume of an arithmetic hyperbolic 2- or 3-manifold whose systole is greater than x. We conclude by determining, for certain small values of x, the least volume of a principal arithmetic hyperbolic 2-manifold over Q or 3-manifold ove…
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "s…
We discuss fibered commensurability of fibrations on a hyperbolic 3-manifold, a notion introduced by Calegari, Sun and Wang. We construct manifolds with non-symmetric but commensurable fibrations on the same fibered face. We also prove that if a given manifold M does not have any hidden symmetries, then M does not admit non-symmetric but commensu- rable fibrations. Finally, Theorem 3.1 of Calegari, Sun and Wang shows that every hyperbolic fibered commensurability class contains a unique minimal element. In this paper we provide a detailed discussion on the proof of the theorem in the cusped ca…
We prove that for any closed, connected, oriented 3-manifold M, there exists an infinite family of 2-fold branched covers of M that are hyperbolic 3-manifolds and surface bundles over the circle with arbitrarily large volume.
In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum in which volume and 1-systole are controlled, and analyze the growth of the genera of minimal surfaces across commensurability classes. These results have applications to the study of how Heegard genus grows across commensurability classes.
Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level $r$). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants $TV_r$ of hyperbolic 3-manifolds. We show that if the $r$-growth of $|TV_r(M)|$ for a hyperbolic 3-manifold $M$ that fibers over the circle is exponential, then the monodromy of the fibration of $M$ satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constru…
Given $M_φ$, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy $φ$ can be bounded above by the complexity of an essential surface with non-zero slope. Furthermore we prove that the minimal complexity of a surface with non-zero slope in $M_{φ^n}$ tends to infinity as $n\to\infty$. Additionally, we show that an infinite family of fibered hyperbolic knots has translation distance bounded above by two, satisfying a conjecture by Schleimer which postulates that this behavior should hold for all fibered knots.
Let X be a hyperbolic surface and H the fundamental group of a hyperbolic 3-manifold that fibers over the circle with fiber X. Using the Birman exact sequence, H embeds in the mapping class group Mod(Y) of the surface Y obtained by removing a point from X. We prove that a subgroup G in H is convex cocompact in Mod(Y) if and only if G is finitely generated and purely pseudo-Anosov. We also prove a generalization of this theorem with H replaced by an arbitrary Gromov hyperbolic extension of the fundamental group of X, and an additional hypothesis of quasi-convexity of G in H. Along the way, we o…
In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twisted Ruelle zeta function of a closed arithmetic hyperbolic 3-manifold and we express the leading coefficient of its Laurent expansion at the origin in terms of the orders of the torsion subgroups of the c…
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…
We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.
For a hyperbolic fibered 3-manifold M, we prove results that uniformly relate the structure of surface projections as one varies the fibrations of M. This extends our previous work from the fully-punctured to the general case.
It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds with no 2-torsion in their first homology. Then we exhibit families of examples of 3-manifolds (obtained by surgery on links), with fixed linking form and cohomology ring, that are not homology cobordant to any Seifert fibered space (as shown by their rational cohomology ring). T…
For d=2n+1 a positive odd integer, we consider sequences of arithmetic subgroups of SO_0(d,1) and Spin(d,1) yielding corresponding hyperbolic manifolds of finite volume and show that, under appropriate and natural assumptions, the torsion of the associated cohomology groups grows exponentially.
This paper initiates a systematic study of the relation of commensurability of surface automorphisms, or equivalently, fibered commensurability of 3-manifolds fibering over the circle. We show that every hyperbolic fibered commensurability class contains a unique minimal element, whereas the class of Seifert manifolds fibering over the circle consists of a single commensurability class with infinitely many minimal elements. The situation for non-geometric manifolds is more complicated, and we illustrate a range of phenomena that can occur in this context.
Algebraically fibering group is an algebraic generalization of the fibered 3-manifold group in higher dimensions. Let $M(\mathcal{P})$ and $M(\mathcal{E})$ be the cusped and compact hyperbolic real moment-angled manifolds associated to the hyperbolic right-angled 24-cell $\mathcal{P}$ and the hyperbolic right-angled 120-cell $\mathcal{E}$, respectively. Jankiewicz-Norin-Wise showed in [13] that $π_1(M(\mathcal{P}))$ and $π_1(M(\mathcal{E}))$ are algebraic fibered. Namely, there are two exact sequences $$1\rightarrow H_{\mathcal{P}}\rightarrow π_1(M(\mathcal{P}))\xrightarrow{ϕ_{\mathcal{P}}} \m…
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 set…
We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a "short geodesic conjecture", and prove its equivalence with "Lehmer's conjecture" for Salem numbers.
A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichmüller translation distance and hyperbolic volume. It relies on a key estimate which we supply here: using geometric inflexibility of hyperbolic 3-manifolds, we show that for $S$ a closed surface, and $ψ\in \text{Mod}(S)$ pseudo-Anosov, the double iteration $Q(ψ^{-n}(X),ψ^n(X))$ has convex core volume differing from $2n \text{vol}(M_ψ)$ by a uniform additive constant, where $M_ψ$ is the hyperbolic mapping torus for $ψ$. We combine this estimate with work of Schlenker, and a branched coverin…
It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurability class. It is known, however, that non-commensurable arithmetic hyperbolic surfaces may share arbitrarily large portions of their length spectra. In this paper we investigate this phenomenon and prove a number of quantitative results about the maximum cardinality of a family of …
These are mostly expository notes based on the course of lectures on arithmetic invariants of hyperbolic manifolds given at the workshop associated with the final "Volume Conference," held at Columbia University, June 2009. Some new results are included.
In this paper we provide the first examples of arithmetic hyperbolic 3-manifolds that are rational homology spheres and bound geometrically either compact or cusped hyperbolic 4-manifolds.
This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov--Pyatetski-Shapiro and Agol--Belolipetsky--Thomson non-arithmetic manifolds embed geodesically. Moreover, we show that the number of commensurability classes of hyperbolic manifolds with a representative of volume $\leq v$ that bounds geometrically is at least $v^{Cv}$, for $v$ large enough.
We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.
We prove there are exactly 16 arithmetic lattices of hyperbolic 3-space which are generated by two elements of finite orders p and q with p,q at least six. We also verify a conjecture of H.M. Hilden, M.T. Lozano, and J.M. Montesinos concerning the orders of the singular sets of arithmetic orbifold Dehn surgeries on two bridge knot and link complements.
Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with invariant trace field $k$. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic fiel…
In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type $B_n$ and $D_n$. We use a combination of techniques from algebraic groups and quadratic forms to prove several results about these spaces.
