Bert Wiest

Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups

We prove by explicit construction that graph braid groups and most surface groups can be embedded in a natural way in right-angled Artin groups, and we point out some consequences of these embedding results. We also show that every right-angled Artin group can be embedded in a pure surface braid group. On the other hand, by generalising to right- angled Artin groups a result of Lyndon for free groups, we show that the Euler characteristic −1 surface group (given by the relation x 2 y 2 = z 2 ) never embeds in a right-angled Artin group. AMS Classification 20F36, 05C25; 05C25 Keywords Cubed complex, graph braid group, graph group, right-angled Artin group, configuration space

On the structure of the centralizer of a braid

Abstract The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on n strands, which has at most k ( k + 1 ) 2 elements if n = 2 k , and at most k ( k + 3 ) 2 elements if n = 2 k + 1 . These bounds are shown to be sharp, due to work of N.V. Ivanov and of S.J. Lee. Finally, we describe how one can explicitly compute this generating set.

Reducible braids and Garside Theory

We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it is reducible, then its reducibility is geometrically obvious: it has a round or almost round reducing curve. Moreover, for any given braid, an element of its stabilized set of sliding circuits can be found using the well-known cyclic sliding operation. This leads to a polynomial time algorithm for deciding the Nielsen-Thurston type of any braid, modulo one well-known conjecture on the speed of convergence of the cyclic sliding operation.

Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the -norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of and for all . As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the group . Examples include hyperbolic surface groups, some HNN-extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3-manifold.

Free group automorphisms, invariant orderings and topological applications

We are concerned with orderable groups and particularly those with orderings invariant not only under multiplication, but also under a given automorphism or family of automorphisms. Several applications to topology are given: we prove that the fundamental groups of hyper- bolic nonorientable surfaces, and the groups of certain fibred knots are bi- orderable. Moreover, we show that the pure braid groups associated with hyperbolic nonorientable surfaces are left-orderable. AMS Classification 6F15; 57M05

Computational Explorations in Thompson’s Group F

Here we describe the results of some computational explorations in Thompson’s group F. We describe experiments to estimate the cogrowth of F with respect to its standard finite generating set, designed to address the subtle and difficult question whether or not Thompson’s group is amenable. We also describe experiments to estimate the exponential growth rate of F and the rate of escape of symmetric random walks with respect to the standard generating set.

UPPER BOUNDS FOR THE COMPLEXITY OF TORUS KNOT COMPLEMENTS

We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.

Fast algorithmic Nielsen-Thurston classification of four-strand braids

We give an algorithm which decides the Nielsen-Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round.

On the genericity of loxodromic actions

Suppose that a finitely generated group G acts by isometries on a δ-hyperbolic space, with at least one element acting loxodromically. Suppose that the elements of G have a normal form such that the language of normal forms can be recognized by a finite state automaton. Suppose also that a certain compatibility condition linking the automatic and the δ-hyperbolic structures is satisfied. Then we prove that in the “ball” consisting of elements of G whose normal form is of length at most l, the proportion of elements which act loxodromically is bounded away from zero, as l tends to infinity. We present several applications of this result, including the genericity of pseudo-Anosov braids.

ON WORD REVERSING IN BRAID GROUPS

It has been conjectured that in a braid group, or more generally in a Garside group, applying any sequence of monotone equivalences and word reversings can increase the length of a word by at most a linear factor depending on the group presentation only. We give a counter-example to this conjecture, but, on the other hand, we establish length upper bounds for the case when only right reversing is involved. We also state a new conjecture which would, like the above one, imply that the space complexity of the handle reduction algorithm is linear.