On the commutator lengths of certain classes of finitely presented groups.
On the completion of cyclically ordered groups
On the complexity of braids
We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators , which are Garside-like half-twists involving strings through , and by counting powered generators as instead of simply . The geometrical complexity is some natural measure of the amount of distortion of the times punctured disk caused by a homeomorphism. Our main...
On the Configuration Spaces of Grassmannian Manifolds
Let be the -th ordered configuration space of all distinct points in the Grassmannian of -dimensional subspaces of , whose sum is a subspace of dimension . We prove that is (when non empty) a complex submanifold of of dimension and its fundamental group is trivial if , and and equal to the braid group of the sphere
On the conformal gauge of a compact metric space
In this article we study the Ahlfors regular conformal gauge of a compact metric space , and its conformal dimension . Using a sequence of finite coverings of , we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute using the critical exponent associated to the combinatorial modulus.
On the Conjugacy Problem for Knot Groups.
On the cut number of a 3-manifold.
On the cut point conjecture.
On the definability of the group by its spectrum.
On the derived length of parasoluble groups
In this paper groups are considered inducing groups of power automorphisms on each factor of their derived series. In particular, it is proved that soluble groups with such property have derived length at most 3, and that this bound is best possible.
On the Dimensions of Automorphism Groups of Eight-Dimensional Double Loops.
On the dynamics and the fixed subgroup of a free group automorphism.
On the dynamics of (left) orderable groups
We develop dynamical methods for studying left-orderable groups as well as the spaces of orderings associated to them. We give new and elementary proofs of theorems by Linnell (if a left-orderable group has infinitely many orderings, then it has uncountably many) and McCleary (the space of orderings of the free group is a Cantor set). We show that this last result also holds for countable torsion-free nilpotent groups which are not rank-one Abelian. Finally, we apply our methods to the case of braid...
On the Embedding of Semigroups into 2-Engel Groups
On the excedance number of colored permutation groups.
On the existence of -local subgroups in a group.
On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane
We prove several results concerning the existence of universal covering spaces for separable metric spaces. To begin, we define several homotopy-theoretic conditions which we then prove are equivalent to the existence of a universal covering space. We use these equivalences to prove that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal covering space. As an application of these results, we prove the main result of this article,...
On the existence of universal rational structures for groups.
On the exponent of the product of two groups
On the Farrell cohomology of mapping class groups.