Luisa Paoluzzi

On a Class of Hyperbolic 3-Manifolds and Groups with One Defining Relation

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, arbitrarily many of the same volume. The fundamental groups of these 3-manifolds are groups with one defining relation. Our main result is a classification of these manifolds up to homeomorphism, resp. isometry.

A note on the Lawrence-Krammer-Bigelow representation

A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group Bn. In their papers, Bigelow and Kram- mer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this paper is to understand this mon- odromy representation using standard tools from the theory of hyperplane arrangements. In particular, we prove that the representation of Bigelow and Krammer is a sub-representation of the monodromy representation which we consider, but that it cannot be the whole representation. AMS Classification 20F36; 52C35, 52C30, 32S22

COMMENSURABILITY CLASSIFICATION OF A FAMILY OF RIGHT-ANGLED COXETER GROUPS

We classify the members of an infinite family of right-angled Coxeter groups up to abstract commensurability.

Hyperbolic knots and cyclic branched covers

We collect several results on the determination of hyperbolic knots by means of their cyclic branched covers. We construct examples of knots having two common cyclic branched covers. Finally, we briey discuss the problem of determination of hyperbolic links.

On cyclic branched coverings of prime knots

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K nonequivalent to K. To prove the main theorem, a result concerning symmetries of knots is also obtained. This latter result can be interpreted as a characterisation of the trivial knot.

Psl(2,q) quotients of some hyperbolic tetrahedral and coxeter groups

We classify quotients of type PSL(2,q) and PGL(2,q) with torsion-free kernel for four of the nine hyperbolic tetrahedral groups. Using this result, we give a classification of the quotients with torsion-free kernel of type PSL(2q) ×Z2 of the associated Coxeter or reflection groups. These do not admit quotients of type PSL(2,q),PGL(2,q). We also study quotients of type PSL(2,q) and PGL(2,q) of the fundamental group of the hyperbolic 3-orbifold of minimal known volume.

THREE CYCLIC BRANCHED COVERS SUFFICE TO DETERMINE HYPERBOLIC KNOTS

Let n > m > 2 be two fixed coprime integers. We prove that two Conway reducible, hyperbolic knots sharing the 2-fold, m-fold and n-fold cyclic branched covers are equivalent. Using previous results by Zimmermann we prove that this implies that a hyperbolic knot is determined by any three of its cyclic branched covers.

Non-equivalent hyperbolic knots

We construct, for each integer n 3, pairs of non-equivalent hyperbolic knots with the same 2fold and n-fold cyclic branched covers. We also discuss necessary conditions for such pairs of knots to exist. uf6d9 2001 Elsevier Science B.V. All rights reserved. MSC: primary 57M25; secondary 57M12, 57M50

Non-standard components of the character variety for a family of Montesinos knots

We show that there are Montesinos knots with

A characterisation of S 3 among homology spheres

n+1geq 4