Solving the conjugacy problem in Garside groups by cyclic sliding

We present a solution to the conjugacy decision problem and the conjugacy search problem in Garside groups, which is theoretically simpler than the usual one, with no loss of efficiency. This is done by replacing the well known cycling and decycling operations by a new one, called cyclic sliding, which appears to be a more natural choice. We give an analysis of the complexity of our algorithm in terms of fundamental operations with simple elements, so our analysis is valid for every Garside group. This paper intends to be self-contained, not requiring any previous knowledge of prior algorithms, and includes all the details for the algorithm to be implemented on a computer.Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona

A family of pseudo-Anosov braids with large conjugacy invariant sets

We show that there is a family of pseudo-Anosov braids independently parameterized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length and conclude that the conjugacy problem remains exponential in the braid index under the current knowledge.Comment: 16 pages, 6 figure

Fast nielsen-thurston classification of braids

We prove the existence of an algorithm which solves the reducibility problem in braid groups and runs in quadratic time with respect to the braid length for any fixed braid index

The development version of the CHEVIE package of GAP3

I describe the current state of the development version of the CHEVIE package, which deals with Coxeter groups, reductive algebraic groups, complex reflection groups, Hecke algebras, braid monoids, etc... Examples are given, showing the code to check some results of Lusztig.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1003.492

Dual Garside structure and reducibility of braids

Benardete, Gutierrez and Nitecki showed an important result which relates the geometrical properties of a braid, as a homeomorphism of the punctured disk, to its algebraic Garside-theoretical properties. Namely, they showed that if a braid sends a curve to another curve, then the image of this curve after each factor of the left normal form of the braid (with the classical Garside structure) is also standard. We provide a new simple, geometric proof of the result by Benardete-Gutierrez-Nitecki, which can be easily adapted to the case of the dual Garside structure of braid groups, with the appropriate definition of standard curves in the dual setting. This yields a new algorithm for determining the Nielsen-Thurston type of braids

The root extraction problem for generic braids

International audienceWe show that, generically, finding the k-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid x on n strands and canonical length l, and an integer k>1 , computes a k-th root of x, if it exists, or guarantees that such a root does not exist. The generic-case complexity of this algorithm is O(l(l+n)n3logn) . The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee. This algorithm uses the fact that the ultra summit set of a braid is, generically, very small and symmetric (through conjugation by the Garside element Δ ), consisting of either a single orbit conjugated to itself by Δ or two orbits conjugated to each other by Δ

Limits of sequences of pseudo-Anosov maps and of hyperbolic 3–manifolds

There are two objects naturally associated with a braid $\beta\in B_n$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism $\varphi_\beta\colon S^2\to S^2$; and the finite volume complete hyperbolic structure on the 3-manifold $M_\beta$ obtained by excising the braid closure of $\beta$, together with its braid axis, from $S^3$. We show the disconnect between these objects, by exhibiting a family of braids $\{\beta_q:q\in{\mathbb{Q}}\cap(0,1/3]\}$ with the properties that: on the one hand, there is a fixed homeomorphism $\varphi_0\colon S^2\to S^2$ to which the (suitably normalized) homeomorphisms $\varphi_{\beta_{q}}$ converge as $q\to 0$; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form $\lim_{k\to\infty} M_{\beta_{q_k}}$, for sequences $q_k\to 0$.Comment: Author accepted manuscrip