Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2,C) Isom(H3). The volume of ρ is defined by taking any ρ-equivariant map from the universal cover W̃ to H and then by integrating the pullback of the hyperbolic volume form on a fundamental domain. This volume does not depend on the choice of the equivariant map because two equivariant maps are always eq...

where a = (234) and b = (243)(12) Here, the complete classification of G in S4 is given. These calculations are mostly obtained in [6]. These representations are divided into eight non-equivalent classes. For each class a representative homomorphism is considered and the corresponding fundamental group of 4-sheeted covering space of the overhand knot has been calculated by the algorithm of Prof...

In this paper, we show that the class of all properly 3-realizable groups is closed under amalgamated free products (and HNN-extensions) over finite groups. We recall that G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π1(K) = G and whose universal cover K̃ has the proper homotopy type of a 3-manifold (with boundary).

This paper concerns the class of contractible open 3-manifolds which are “locally finite strong end sums” of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follows that such a 3-manifold cannot cover a closed 3-manifold. For each countable f...

The category of finite dimensional modules for the proalgebraic differential Galois group of the differential Galois theoretic closure of a differential field F is equivalent to the category of finite dimensional F spaces with an endomorphism extending the derivation of F . This paper presents an expository proof of this fact modeled on a similar equivalence from polynomial Galois theory, whose...

The classical Cartan-Hadamard theorem asserts that a closed Riemannian manifold M with non-positive sectional curvature has universal cover M̃ diffeomorphic to R, and a byproduct of the proof is that ∂∞M̃n is homeomorphic to Sn−1. We prove analogues of these two results in the case where M has a non-empty totally geodesic boundary. More precisely, if M 1 ,M n 2 are two negatively curved Riemannia...

Characterizing the universal coverings of smooth projective varieties is an old and hard question. Central to the subject is a conjecture of Shafarevich according to which the universal cover X̃ of a smooth projective variety is holomorphically convex, meaning that for every infinite sequence of points without limit points in X̃ there exists a holomorphic function unbounded on this sequence. In t...

We determine the L spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M whose universal covering X is a symmetric space of non-compact type with rank one. More precisely, we show that the L spectra of M and X coincide if the fundamental group of M is small and if the injectivity radius of M is bounded away from zero. In the L case, the restriction on the inje...

In this paper it is shown how the graphical methods developed by Stephen for analyzing inverse semigroup presentations may be used to study varieties of inverse semigroups. In particular, these methods may be used to solve the word problem for the free objects in the variety of inverse semigroups generated by the five-element combinatorial Brandt semigroup and in the variety of inverse semigrou...

Suppose that G is a definably connected, definable group in an ominimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism p̃ : G̃ −→ G is a locally definable covering homomorphism and π1(G) is isomorphic to the o-minimal fundamental group π(G) of G defined using locally definable covering homomorphisms. ∗With partial support from the FCT (Fundação para a Ci...