Ara Basmajian

THE ORTHOGONAL SPECTRUM OF A HYPERBOLIC MANIFOLD

Introduction. In this paper, we investigate the geometry of embedded hypersurfaces in hyperbolic manifolds of dimension greater than one. Our focus is on hypersurfaces that are either totally geodesic, horospherical (quotients of horospheres), or the boundary of a hyperbolic ball. These hypersurfaces have a natural decomposition into two pieces; one of which corresponds to base points of normal rays that satisfy a specified crossing condition, and the other piece being a disjoint union of embedded discs. We show that these embedded discs are in one to one correspondence with a certain set of lengths of orthogonals (the orthogonal spectrum) emanating from the hypersurface. In fact, the area of each disc is a monotonic function of the corresponding length in the spectrum. Thus the area (induced volume) of the hypersurface is intimitely related to the orthogonal spectrum (see theorem 1.1 for a precise statement). In special cases, (corollary 1.2) the piece of the hypersurface corresponding to the base points of normal rays is related to the limit set of the discrete isometry group that represents the manifold. As a consequence, we show that the measure of the limit set of this group is zero if and only if the area of the piece consisting of base points of normal rays is zero. The next theorem (theorem 1.3) relates the orthogonal spectrum of a closed embedded totally geodesic hypersurface with the lengths of geodesic loops based at some point of the ambient manifold. In particular, by applying a result of Sullivan, we can show that the geodesic flow acts ergodically on the unit tangent bundle of the manifold if and only if the sum of the areas of the discs embedded in the hypersurface diverges (corollary 1.4).

Hyperbolic structures for surfaces of infinite type

Our main objective is to understand the geometry of hyperbolic structures on surfaces of infinite type. In particular, we investigate the properties of surfaces called flute spaces which are constructed from infinite sequences of «pairs of pants», each glued to the next along a common boundary geodesic. Necessary and sufficient conditions are supplied for a flute space to be constructed using only «tight pants», along with sufficient conditions on when the hyperbolic structure is complete. An infinite version of the Klein-Maskit combination theorem is derived. Finally, using the above constructions a number of applications to the deformation theory of infinite type hyperbolic surfaces are examined

Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds

SummaryWe show that a closed embedded totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface. 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. In particular, we show that ifM is a hyperbolic 3-manifold containingn rank two cusps andk disjoint totally geodesic embedded closed surfaces, then the volume ofM is bigger than

Space form isometries as commutators and products of involutions

(\tfrac{{\sqrt 3 }}{4})n + (4.4)k

Large parameter spaces of quasiconformally distinct hyperbolic structures

. 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 result, we construct examples to illustrate the qualitative sharpness of our tubular neighborhood function in dimension three. As an application of our results we give an eigenvalue estimate.

Metric flows in space forms of nonpositive curvature

We derive a generalized collar lemma, called the stable neighborhood theorem, for nonsimple closed geodesics. 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 only depends on k and tends to infinity as k goes to infinity. Also, we show that the shortest nonsimple closed geodesic on a closed hyperbolic surface has (geometric) crossing number bounded above by a constant which only depends on the genus.

Generalizing the hyperbolic collar lemma

In dimensions 2 and 3 it is well known that given two orientationpreserving hyperbolic isometries that generate a non-elementary group, one can find a triple of involutions so that each isometry can be expressed as a product of two of the three involutions; in this case, we say that the isometries are linked. In this paper, we investigate the extent to which a pair of isometries in higher dimensions can be linked. This question separates naturally into two parts. In the first part, we determine the least number of involutions needed to express an isometry as a product, and give two applications of our results; the second part is devoted to the question of linking. In general, the commutator (involution) length of a group element is the least number of elements needed to express that element as a product of commutators (involutions), and the commutator (involution) length of the group is the supremum over all commutator (involution) lengths. Let Gn be the group of orientation-preserving isometries of one of the space forms, the (n − 1)sphere, Euclidean n-space, hyperbolic n-space. For n ≥ 3, we show that the commutator length of Gn is 1; i.e., every element of Gn is a commutator. We also show that every element of Gn can be written as a product of two involutions, not necessarily orientation-preserving; and, depending on the particular space and on the congruence class of n mod 4, the involution length of Gn is either 2 or 3. In the second part of the paper, we show that all pairs in SO4 are linked but that the generic pair in the orientation-preserving isometries of hyperbolic 4-space or in Gn, n ≥ 5, is not.

Hyperbolic 3-Manifolds With Nonintersecting Closed Geodesics

We show that the group of conformal homeomorphisms of the boundary of a rank one symmetric space (except the hyperbolic plane) of noncompact type acts as a maximal convergence group. Moreover, we show that any family of uniformly quasiconformal homeomorphisms has the convergence property. Our theorems generalize results of Gehring and Martin in the real hyperbolic case for Mobius groups. As a consequence, this shows that the maximal convergence subgroups of the group of self homeomorphisms of the d-sphere are not unique up to conjugacy. Finally, we discuss some implications of maximality.