### Alexander polynomial, finite type invariants and volume of hyperbolic knots.

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In this paper, we present a new approach to the construction of Einstein metrics by a generalization of Thurston's Dehn filling. In particular in dimension 3, we will obtain an analytic proof of Thurston's result.

We study the family of closed Riemannian n-manifolds with holonomy group isomorphic to Z2n-1, which we call generalized Hantzsche-Wendt manifolds. We prove results on their structure, compute some invariants, and find relations between them, illustrated in a graph connecting the family.

We definite invariants of translation surfaces which refine Veech groups. These aid in exact determination of Veech groups. We give examples where two surfaces of isomorphic Veech group cannot even share a common tree of balanced affine coverings. We also show that there exist translation surfaces of isomorphic Veech groups which cannot affinely cover any common surface. We also extend a result of Gutkin and Judge and thereby give the first examples of noncompact Fuchsian...

It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group, is bounded by a quadratic polynomial in g (but not by a linear one). In the present paper we prove a generalization for handlebodies of arbitrary dimension d, uniformizing handlebodies...

We give a short proof of the proportionality principle for cusped hyperbolic manifolds.

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic $n$-manifolds, for $n\ge 3$. In the course of the proof of the main result,...

The focus of this paper are questions related to how various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. The paper is for the large part a survey of recent work.