A. Yu. Vesnin

Volumes of hyperbolic Löbell 3-manifolds

In 1931 F. Löbell constructed the first example of a closed orientable three-dimensional hyperbolic manifold. In the present paper we study properties of closed hyperbolic 3-manifolds generalizing Löbells classical example. Explicit formulas for the volumes of these manifolds in terms of the Lobachevski function are obtained.

Two-sided bounds for the complexity of hyperbolic three-manifolds with geodesic boundary

We construct an infinite family of hyperbolic three-manifolds with geodesic boundary that generalize the Thurston and Paoluzzi-Zimmermann manifolds. For the manifolds of this family, we present two-sided bounds for their complexity.

Three-Dimensional Manifolds with Poor Spines

A special spine of a 3-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev-Viro invariants, we establish that every compact 3-manifold M with connected nonempty boundary has a finite number of poor special spines. Moreover, all poor special spines of the manifold M have the same number of true vertices. We prove that the complexity of a compact hyperbolic 3-manifold with totally geodesic boundary that has a poor special spine with two 2-components and n true vertices is equal to n. Such manifolds are constructed for infinitely many values of n.

Cusped hyperbolic 3-manifolds of complexity 10 having maximum volume

We give a complete census of orientable cusped hyperbolic 3-manifolds obtained by gluing at most ten regular ideal hyperbolic tetrahedra. Although the census is exhaustive, the question of nonhomeomorphism remains open for some pairs of manifolds with one, two, and three cusps.

Two-Sided Bounds for the Volume of Right-Angled Hyperbolic Polyhedra

For a compact right-angled polyhedron R in Lobachevskii space ℍ3, let vol(R) denote its volume and vert(R), the number of its vertices. Upper and lower bounds for vol(R) were recently obtained by Atkinson in terms of vert(R). In constructing a two-parameter family of polyhedra, we show that the asymptotic upper bound 5v3/8, where v3 is the volume of the ideal regular tetrahedron in ℍ3, is a double limit point for the ratios vol(R)/ vert(R). Moreover, we improve the lower bound in the case vert(R) ≤ 56.

Spherical coxeter groups and hyperelliptic 3-manifolds

Reflection groups of Coxeter polyhedra in three-dimensional Thurston geometries are examined. For a wide class of Coxeter groups, the existence of subgroups of finite index that uniformize hyperelliptic 3-manifolds is established.