Roberto Frigerio

Small hyperbolic 3-manifolds with geodesic boundary

We classify the orientable finite-volume hyperbolic 3-manifolds having nonempty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, describe their canonical Kojima decomposition, and discuss manifolds having cusps. The eight manifolds built from one or two tetrahedra were previously known. There are 151 different manifolds built from three tetrahedra, realizing 18 different volumes. Their Kojima decomposition always consists of tetrahedra (but occasionally requires four of them). There is a single cusped manifold, which we can show to be a knot complement in a genus-2 handlebody. Concerning manifolds built from four tetrahedra, we show that there are 5,033 different ones, with 262 different volumes. The Kojima decomposition consists either of tetrahedra (as many as eight of them in some cases), of two pyramids, or of a single octahedron. There are 30 manifolds having a single cusp and one having two cusps. Our results were obtained with the aid of a computer. The complete list of manifolds (in SnapPea format) and full details on their invariants are available on the world wide web.

Isometric embeddings in bounded cohomology

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along \pi_1-injective boundary components with amenable fundamental group.

The simplicial volume of hyperbolic manifolds with geodesic boundary

Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if the boundary of M is non-empty, then such a ratio is strictly less than v_n. We prove here that for every a>0 there exists k>0 (only depending on a and n) such that if the ratio between the volume of the boundary of M and the volume of M is less than k, then the ratio between vol(M) and ||M|| is greater than v_n-a. As a consequence we show that for every a>0 there exists a compact orientable hyperbolic n-manifold M with non-empty geodesic boundary such that the ratio between vol(M) and ||M|| is greater than v_n-a. Our argument also works in the case of empty boundary, thus providing a somewhat new proof of the proportionality principle for non-compact finite-volume hyperbolic n-manifolds without boundary.

Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling

We establish a bijective correspondence between the set \mathcal{T}n of 3-dimensional triangulations with n tetrahedra and a certain class Hn of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n + 1. We show that the manifolds in Hn are hyperbolic (with geodesic boundary, and cusps corresponding to the loops), have least possible volume, and simplest boundary loops. Mirroring the elements of Hn in their geodesic boundary we obtain a set \mathcal{D}n of cusped hyperbolic manifolds, previously considered by D. Thurston and the first named author. We show that also \mathcal{D}n corresponds bijectively to \mathcal{T}n, and we study some Dehn fillings of the manifolds in \mathcal{D}n. As consequences of our constructions, we also show that: A triangulation of a 3-manifold is uniquely determined up to isotopy by its 1-skeleton; If a 3-manifold M has an ideal triangulation with edges of valence at least 6, then M is hyperbolic and the edges are homotopically non-trivial, whence homotopic to geodesics; Every finite group G is the isometry group of a closed hyperbolic 3-manifold with volume less than const × |G|9.

The simplicial volume of 3-manifolds with boundary

We provide sharp lower bounds for the simplicial volume of compact

Levels of knotting of spatial handlebodies

3

AN INFINITE FAMILY OF HYPERBOLIC GRAPH COMPLEMENTS IN S3

-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of

On deformations of hyperbolic 3-manifolds with geodesic boundary.

3

Alexander quandle lower bounds for link genera

-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the first exact computation of the simplicial volume of a compact manifold whose boundary has positive simplicial volume. We also compute the minimal number of tetrahedra in a (loose) triangulation of the product of a surface with the interval.

A quantitative version of a theorem by Jungreis

If H is a spatial handlebody, i.e. a handlebody embedded in the 3-sphere, a spine of H is a graph Γ ⊂ S3 such that H is a regular neighbourhood of Γ. Usually, H is said to be unknotted if it admits a planar spine. This suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties. Building on this remark, we define several levels of knotting of spatial handlebodies, and we provide a complete description of the relationships between these levels, focusing our attention on the case of genus 2. We also relate the knotting level of a spatial handlebody H to classical topological properties of its complement M = S3 \H, such as its cut number. More precisely, we show that if H is not highly knotted, then M admits special cut systems for M , and we discuss the extent to which the converse implication holds. Along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies. These obstructions are based on recent quandle–coloring invariants for spatial handlebodies, on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links, on the analysis of appropriate coverings of handlebody complements, and on the study of the classical Alexander elementary ideals of their fundamental groups.