Hyperbolic Manifold Of Finite Volume Research Articles

Infinitesimal rigidity for cubulated manifolds

We prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. These examples arise as infinite cyclic coverings of finite-volume hyperbolic manifolds obtained by colouring right-angled polytopes, already described in the papers (Battista in Trans Am Math Soc 375(04):2597–2625, 2022; Italiano et al. in Invent Math, 2022. https://doi.org/10.1007/s00222-022-01141-w). The 5-dimensional example is diffeomorphic to Ntimes {{mathbb {R}}} for some aspherical 4-manifold N which does not admit any hyperbolic structure. To this purpose, we develop a general strategy to study the infinitesimal rigidity of cyclic coverings of manifolds obtained by colouring right-angled polytopes.

Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups

For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.

Effective drilling and filling of tame hyperbolic 3-manifolds

We give effective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin parts of the manifold, we give effective bounds on the change in complex length of a short closed geodesic. These results quantify the filling theorem of Brock and Bromberg, and extend previous results of the authors from finite volume hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main results, we assemble tools from Kleinian group theory into a template for transferring theorems about finite-volume manifolds into theorems about infinite-volume manifolds. We also prove and apply an infinite-volume version of the 6-Theorem.

Asymptotics of twisted Alexander polynomials and hyperbolic volume

For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we prove the asymptotic behavior for cusped hyperbolic manifolds of finite volume. The proof relies on results of Muller, and Menal-Ferrer and the last author. Using the uniformity of the convergence, we also deduce a similar asymptotic result for the Mahler measures of those polynomials.

Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence

<p style='text-indent:20px;'>A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems.</p>

Quasiconvexity and Dehn filling

We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with a recent result of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hyperbolic 3-manifolds, a result first proved by Wise. Additionally, we explain how the results of [2, Appendix A] can be generalized to the relative setting to control the relative height of relatively quasiconvex subgroups under appropriate Dehn fillings.

Constructing convex projective 3‐manifolds with generalized cusps

We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling which types of cusp occur. Using these methods we produce the first known example of a 1-cusped hyperbolic 3-manifold that admits a convex projective structure with a type 2 cusp. We also use these techniques to produce new 1-cusped manifolds that admit a convex projective structure with a type 1 cusp.

Trees of manifolds as boundaries of spaces and groups

We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these spaces as boundaries of arbitrary Coxeter groups with manifold nerves, and as Gromov boundaries of the fundamental groups of singular spaces obtained from some finite volume hyperbolic manifolds by cutting off their cusps and collapsing the resulting boundary tori to points.

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.

Constructing thin subgroups of SL(n + 1, ℝ) via bending

In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it is possible to find infinitely many non-commensurable lattices in SL(n+1,R) that contain a thin subgroup isomorphic to a finite index subgroup of the original arithmetic lattice. This class of arithmetic lattices includes all non-cocompact arithmetic lattices and all cocompact arithmetic lattices when $n$ is even.

Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel–Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.

Hyperbolic Eisenstein series on [formula omitted]-dimensional hyperbolic spaces

We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.

Fat Flats in Rank One Manifolds

We study closed nonpositively curved Riemannian manifolds M that admit “fat k-flats”; that is, the universal cover M˜ contains a positive-radius neighborhood of a k-flat on which the sectional curvatures are identically zero. We investigate how the fat k-flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank 1 nonpositively curved manifolds with a fat 1-flat that corresponds to a twisted cylindrical neighborhood of a geodesic on M. As a result, M contains an embedded closed geodesic with a flat neighborhood, but M nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite volume hyperbolic manifolds. Our second main result is a proof of a closing theorem for fat flats, which implies that a manifold M with a fat k-flat contains an immersed, totally geodesic k-dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when k≥2. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.

Profinite Rigidity in the SnapPea Census

A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72,942 finite volume hyperbolic 3-manifolds have the same finite quotients.

Concordance of Seifert surfaces

This paper proves that every oriented non-disk Seifert surface $F$ for a knot $K$ in $S^3$ is smoothly concordant to a Seifert surface $F^{\prime}$ for a hyperbolic knot $K^{\prime}$ of arbitrarily large volume. This gives a new and simpler proof of the result of Friedl and of Kawauchi that every knot is $S$-equivalent to a hyperbolic knot of arbitrarily large volume. The construction also gives a new and simpler proof of the result of Silver and Whitten and of Kawauchi that for every knot $K$ there is a hyperbolic knot $K^{\prime}$ of arbitrarily large volume and a map of pairs $f:(S^3,K^{\prime})\rightarrow (S^3,K)$ which induces an epimorphism on the knot groups. An example is given which shows that knot Floer homology is not an invariant of Seifert surface concordance. The paper also proves that a set of finite volume hyperbolic 3-manifolds with unbounded Haken numbers has unbounded volumes.

The cusped hyperbolic census is complete

From its creation in 1989 through subsequent extensions, the widely-used SnapPea now aims to represent all cusped finite-volume hyperbolic 3-manifolds that can be obtained from <= 8 ideal tetrahedra. Its construction, however, has relied on inexact computations and some unproven (though reasonable) assumptions, and so its completeness was never guaranteed. For the first time, we prove here that the census meets its aim: we rigorously certify that every ideal 3-manifold triangulation with <= 8 tetrahedra is either (i) homeomorphic to one of the census manifolds, or (ii) non-hyperbolic. In addition, we extend the census to 9 tetrahedra, and likewise prove this to be complete. We also present the first list of all minimal triangulations of all census manifolds, including non-geometric as well as geometric triangulations.

An application of lattice points counting to shrinking target problems

We apply lattice points counting results to solve a shrinking target problem in the setting of discrete time geodesic flows on hyperbolic manifolds of finite volume.

Logarithm laws for unipotent flows on hyperbolic manifolds

We prove logarithm laws for unipotent flows on non-compact finite-volume hyperbolic manifolds. Our method depends on the estimate of norms of certain incomplete Eisenstein series.

A quantitative version of a theorem by Jungreis

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume $$\Vert M\Vert $$ of M is equal to $$\mathrm{Vol}(M)/v_n$$ , where $$v_n$$ is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio $$\mathrm{Vol}(M)/\Vert M\Vert $$ is strictly smaller than $$v_n$$ if M is compact with nonempty geodesic boundary. We prove here a quantitative version of Jungreis’ result for $$n\ge 4$$ , which bounds from below the ratio $$\Vert M\Vert /\mathrm{Vol}(M)$$ in terms of the ratio $$\mathrm{Vol}(\partial M)/\mathrm{Vol}(M)$$ . As a consequence, we show that, for $$n\ge 4$$ , a sequence $$\{M_i\}$$ of compact hyperbolic n-manifolds with geodesic boundary satisfies $$\lim _i \mathrm{Vol}(M_i)/\Vert M_i\Vert =v_n$$ if and only if $$\lim _i \mathrm{Vol}(\partial M_i)/\mathrm{Vol}(M_i)=0$$ . We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension 3.

Coxeter groups, quiver mutations and geometric manifolds

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every quiver or diagram of finite or affine type a CW-complex with a proper action of a finite (or affine) Coxeter group. These CW-complexes undergo mutations agreeing with mutations of quivers and diagrams. We also generalize the construction to quivers and diagrams originating from unpunctured surfaces and orbifolds.