Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds

In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.

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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.

We prove that the number variance for the spectrum of an arithmetic surface is highly nonrigid in part of the universal range. In fact it is close to having a Poisson behavior. This fact was discovered numerically by Schmit, Bogomolny, Georgeot and Giannoni. It has its origin in the high degeneracy of the length spectrum, first observed by Selberg.

It was shown by Chinburg and Reid that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert showed that almost all quasi-Fuchsian 3-manifolds have all closed geodesics simple and disjoint. The natural conjecture arose that the Chinburg-Reid examples also had disjoint geodesics. Here we show that this conjecture is both almost true (they have no geodesics that intersect except at right angles) and spectacularly false (any pair of closed geodesics admits infinitely many closed geodesics which intersects both geodesics of the pair perpendicularly). The latter statement is shown to be true for all closed arithmetic hyperbolic 3-manifolds. Section 0 Introduction By a hyperbolic n-manifold we shall mean a complete orientable n-dimensional Riemannian manifold all of whose sectional curvatures are −1. If M is a hyperbolic 3-manifold, the universal cover of M can be identified with H, the upper half-space model of hyperbolic 3-space, andM is realized asH/Γ for some Γ a discrete torsionfree subgroup of Isom(H3). Now Isom(H3) can be identified with PSL(2,C) ( which in turn is isomorphic to PGL(2,C)), and Γ is called a Kleinian group. In the sequel we will only be interested in the case when M is closed, in which case Γ is referred to as cocompact. Of interest to us is the structure of the set of closed geodesics in closed hyperbolic 3-manifolds. The motivation comes from the following. A closed geodesic in a (closed) hyperbolic n-manifold is simple if it has no self-intersections, and nonsimple otherwise. In dimension 2 every closed hyperbolic manifold has a non-simple closed geodesic. However in dimension 3 the situation is much more complex. Many closed hyperbolic 3-manifolds contain immersions of totally geodesic surfaces and so there are non-simple closed geodesics. In [JR] examples were given of closed hyperbolic 3-manifolds containing a non-simple closed geodesic but having no immersed totally geodesic surface. However it was shown in [CR] that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert [BW] showed that almost all 3-manifolds arising as the quotient of H 1991 Mathematics Subject Classification. Primary: 57M50, Secondary: 30F40.

Computing shortest words via shortest loops on hyperbolic surfaces

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces through a new interpretation of the Nielsen realization problem for the mapping class groups.

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of “Bolza twins” (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface. We exploit random sampling combined with the Reidemeister–Schreier algorithm as implemented in magma to generate these surfaces.

On arithmetic surfaces over local rings of integers we examine whether any geometric point has a basis of \'etale neighborhoods whose $\mathfrak{c}$-completed \'etale homotopy types are of type $K(\pi,1)$ for a given full class~$\mathfrak{c}$ of finite groups.

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a "stable neighborhood" whose width only depends on the length of the geodesic. As an application, we show that there is a lower bound for the length of a closed geodesic having crossing number k on a hyperbolic surface. This lower bound tends to infinity with k . Our second generalization is to totally geodesic hypersurfaces of hyperbolic manifolds. Namely, we construct a tubular neighborhood function and show that an embedded closed totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface (and hence not on the geometry of the ambient manifold). The implications of this result for volumes of hyperbolic manifolds is discussed. We also derive a (hyperbolic) quantitative version of the Klein-Maskit combination theorem (in all dimensions) for free products of fuchsian groups. Using this last theorem, we construct examples to illustrate the qualitative sharpness of the tubular neighborhood function.

We introduce and study trace fields and invariant trace fields of SL2(C) and PSL2(C) representations of 3-manifold groups. We give conditions on such fields which imply that the underlying 3-manifold is virtually Haken or has virtually positive or 1 first Betti number. In particular, we define the notion of an algebraically trace-proper surface subgroup in a 3-manifold group, and show that any closed orientable irreducible 3-manifold with such a surface subgroup is virtually Haken. We give infinitely many families of closed orientable hyperbolic non-Haken 3-manifolds with algebraically trace-proper surface subgroups. In this paper a 3-manifold is always assumed to be connected and orientable, without loss of generality for the problems to be considered. A 3-manifold is said to be Haken if it is compact, irreducible, and contains a properly embedded incompressible surface. A 3-manifold is said to be virtually Haken if it has a finite cover which is Haken. The well-known virtually Haken conjecture states that every closed and irreducible 3-manifold with infinite fundamental group is virtually Haken. It is also conjectured that every closed Haken 3-manifold has virtually positive first Betti number, that is, the manifold has a finite cover which has positive first Betti number. If M is a closed hyperbolic Haken 3-manifold, then it is further conjectured that M has virtually 1 first Betti number, which means that for any positive integer n, there is a finite cover Mn of M such that the first Betti number of Mn is larger than n. These conjectures are fundamental and difficult issues in 3-manifold topology. In this paper we provide some information about these conjectures by studying traces of representations of 3-manifold groups into SL2ðCÞ or PSL2ðCÞ: For a group G and a representation r : G ! SL2ðCÞ; we define the trace field Kr of r to be the field generated by the traces of all the matrices r (g), g [ G; over the base field Q of rational numbers, that is, Kr ¼ Qðtrðr ðgÞÞ; g [ G Þ:

We define an extended Bloch group for an arbitrary field F, and show that this group is naturally isomorphic to K 3 ind ( F ) $K_3^{\operatorname{ind}}(F)$ if F is a number field. This gives an explicit description of K 3 ind ( F ) $K_3^{\operatorname{ind}}(F)$ in terms of generators and relations. We give a concrete formula for the regulator, and derive concrete symbol expressions generating the torsion. As an application, we show that a hyperbolic 3-manifold with finite volume and invariant trace field k has a fundamental class [ M ] $[M]$ in K 3 ind ( k ) ⊗ ℤ [ 1 2 ] $K_3^{\operatorname{ind}}(k)\otimes \mathbb {Z}[\frac{1}{2}]$ . If M is closed, 2 [ M ] $2[M]$ is well defined in K 3 ind ( k ) $K_3^{\operatorname{ind}}(k)$ , and if M has cusps, 8 [ M ] $8[M]$ is well defined in K 3 ind ( k ) $K_3^{\operatorname{ind}}(k)$ .

For finite volume, noncompact Riemann surfaces with their canonical hyperbolic metrics, there is a notion of pseudo-Laplace operators which include all embedded eigenvalues $(> \frac {1}{4})$ of the Laplacian as a part of their eigenvalues. Similarly, we define pseudo-Laplace operators for compact hyperbolic Riemann surfaces with short geodesics. Then, for any degenerating family of hyperbolic Riemann surfaces ${S_l} (l \geq 0)$, we show that normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_l}$ converge to normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_0}$ as $l \to 0$. In particular, normalized embedded eigenfunctions and their embedded eigenvalues of ${S_0}$ can be approximated by normalized pseudoeigenfunctions and pseudoeigenvalues of ${S_l}$ and $l \to 0$.

For finite volume, noncompact Riemann surfaces with their canonical hyperbolic metrics, there is a notion of pseudo-Laplace operators which include all embedded eigenvalues ( > 1 4 ) (> \frac {1}{4}) of the Laplacian as a part of their eigenvalues. Similarly, we define pseudo-Laplace operators for compact hyperbolic Riemann surfaces with short geodesics. Then, for any degenerating family of hyperbolic Riemann surfaces S l ( l ≥ 0 ) {S_l} (l \geq 0) , we show that normalized pseudoeigenfunctions and pseudoeigenvalues of S l {S_l} converge to normalized pseudoeigenfunctions and pseudoeigenvalues of S 0 {S_0} as l → 0 l \to 0 . In particular, normalized embedded eigenfunctions and their embedded eigenvalues of S 0 {S_0} can be approximated by normalized pseudoeigenfunctions and pseudoeigenvalues of S l {S_l} and l → 0 l \to 0 .

We prove that hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance from the boundary of the convex core for points in the thick part. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves. We use this inflexibility to give a new proof of the convergence of pseudo-Anosov double iteration on the quasi-Fuchsian space of a closed surface, and the resulting hyperbolization theorem for closed 3-manifolds that fiber over the circle with pseudo-Anosov monodromy.