Arithmetic Hyperbolic 3-manifolds Research Articles

Improved computation of fundamental domains for arithmetic Fuchsian groups

A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.

Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds

We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in certain range, growing linearly in the degree of the invariant trace field, the volume of the $R$-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic $3$-manifold $M$ is of order at most $\mathrm{Vol}(M)^{11/12}$. As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic $3$-manifolds: We show that there are constants $A,B$ such that every such manifold $M$ is homotopy equivalent to a simplicial complex with at most $A\mathrm{Vol}(M)$ vertices, all of degrees bounded by $B$.

Absolute profinite rigidity and hyperbolic geometry

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ with $\omega^2+\omega+1 = 0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic 3-manifold of minimal volume).

Exponential growth of torsion in the cohomology of arithmetic hyperbolic manifolds

For $$d=2n+1$$ a positive odd integer, we consider sequences of arithmetic subgroups of $${\text {SO}}_0(d,1)$$ and yielding corresponding hyperbolic manifolds of finite volume and show that, under appropriate and natural assumptions, the torsion of the associated cohomology groups grows exponentially.

Many cusped hyperbolic 3-manifolds do not bound geometrically

In this note we show that there exist cusped hyperbolic 3 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 4 -manifolds and by Kolpakov, Reid, and Slavich on embedding arithmetic hyperbolic manifolds.

Special subgroups of Bianchi groups

We determine C-special subgroups of the Bianchi groups of index bounded above by 120 by effectivising the arguments of Agol-Long-Reid. These subgroups are congruence of level 2 or 4 and retract to the free group on two generators. As a consequence, we find a C-special 20-sheeted cover of the figure-eight knot complement. We also determine C-special congruence subgroups for a family of cocompact arithmetic Kleinian groups.

Systole of congruence coverings of arithmetic hyperbolic manifolds

In this paper we prove that, for any arithmetic hyperbolic $n$-manifold $M$ of the first type, the systole of most of the principal congruence coverings $M\_{I}$ satisfy $$ \mathrm{sys}(M\_{I})\geq \frac{8}{n(n+1)}\mathrm{log}(\mathrm{vol}(M\_{I}))-c, $$ where $c$ is a constant independent of $I$. This generalizes previous work of Buser and Sarnak, and Katz, Schaps, and Vishne in dimension 2 and 3. As applications, we obtain explicit estimates for systolic genus of hyperbolic manifolds studied by Belolipetsky and the distance of homological codes constructed by Guth and Lubotzky. In Appendix A together with Cayo Dória we prove that the constant $\frac{8}{n(n+1)}$ is sharp.

Non-LERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups

We will show that for any noncompact arithmetic hyperbolic m-manifold with m>3, and any compact arithmetic hyperbolic m-manifold with m>4 that is not a 7-dimensional one defined by octonions, its fundamental group is not locally extended residually finite (LERF). The main ingredient in the proof is a study on abelian amalgamations of hyperbolic 3-manifold groups. We will also show that a compact orientable irreducible 3-manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.

A torsion Jacquet-Langlands correspondence

We prove a numerical form of a Jacquet–Langlands correspondence for torsion classes on arithmetic hyperbolic 3-manifolds.

Geometrically finite amalgamations of hyperbolic 3‐manifold groups are not LERF

We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic $3$-manifold groups along abelian subgroups. A consequence of this result is that closed arithmetic hyperbolic $4$-manifolds have nonLERF fundamental groups. Along with the author's previous work, we get that, for any arithmetic hyperbolic manifold with dimension at least $4$, with possible exceptions in $7$-dimensional manifolds defined by the octonion, its fundamental group is not LERF.

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≥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.

Parametrizing Shimura subvarieties of $${\mathrm{A}_1}$$ Shimura varieties and related geometric problems

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of \({X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b}\). A special case describes all Shimura subvarieties of type \({\mathrm{A}_1}\) Shimura varieties. We produce, for any \({n\geq 1}\), examples of manifolds/Shimura varieties with precisely n commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic 3-manifolds and quaternionic Shimura surfaces, where the presence of one commensurability class of geodesic submanifolds implies the existence of infinitely many classes.

Bounds for eigenforms on arithmetic hyperbolic 3-manifolds

On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the volume. By a novel combination of diophantine and geometric arguments in a noncommutative setting, we obtain bounds as strong as the best corresponding results on arithmetic surfaces.

Correction to “L∞-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds,” Duke Math. J. 77 (1995), 799–817

Correction to “L∞-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds,” Duke Math. J. 77 (1995), 799–817

Perfectoid Shimura varieties

This note explains some of the author’s work on understanding the torsion appearing in the cohomology of locally symmetric spaces such as arithmetic hyperbolic 3-manifolds. The key technical tool was a theory of Shimura varieties with infinite level at p: As p-adic analytic spaces, they are perfectoid, and admit a new kind of period map, called the Hodge–Tate period map, towards the flag variety. Moreover, the (semisimple) automorphic vector bundles come via pullback along the Hodge–Tate period map from the flag variety. In the case of the Siegel moduli space, the situation is fully analyzed in [12]. We explain the conjectural picture for a general Shimura variety.

On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds

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 $\Gamma \subset \mathrm {SL}\_2(\mathbb C)$, provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence ${\Gamma\_i}\_i$ of finite index congruence subgroups of $\Gamma$ such that the first Betti number satisfies $$ b\_1(\Gamma\_i) \gg \[\Gamma:\Gamma\_i]^{1/2} $$ as $i$ goes to infinity.

Computing arithmetic Kleinian groups

Arithmetic Kleinian groups are arithmetic lattices in PSL_2(C). We present an algorithm which, given such a group Gamma, returns a fundamental domain and a finite presentation for Gamma with a computable isomorphism.

Injectivity radii of hyperbolic integer homology 3–spheres

We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H^3 whose normalized Ray-Singer analytic torsions do not converge to the L^2-analytic torsion of H^3. This contrasts with the work of Abert et. al. who showed that Benjamini-Schramm convergence forces convergence of normalized betti numbers. Our results shed light on a conjecture of Bergeron and Venkatesh on the growth of torsion in the homology of arithmetic hyperbolic 3-manifolds, and we give experimental results which support this and related conjectures.

Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds

Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are low density parity check codes with linear rate and distance nε. Their rate is evaluated via Euler characteristic arguments and their distance using \documentclass[12pt]{minimal}\begin{document}$\mathbb {Z}_2$\end{document}Z2-systolic geometry. This construction answers a question of Zémor [“On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction,” in Proceedings of Second International Workshop on Coding and Cryptology (IWCC), Lecture Notes in Computer Science Vol. 5557 (2009), pp. 259–273], who asked whether homological codes with such parameters could exist at all.

Triple product L-functions and quantum chaos on SL(2, ℂ)

Let Y be an arithmetic hyperbolic 3-manifold. We establish a link between quantum unique ergodicity for sections of automorphic vector bundles on Y and subconvexity for the triple product L-function, which extends a result of Watson in the case of hyperbolic 2-manifolds. The proof combines the representation theoretic microlocal lift for bundles developed by Bunke and Olbrich with the triple product formula of Ichino. A key step is determining the asymptotic behaviour of the local integrals at complex places that appear in Ichino’s formula.