Probabilistic Solutions of Equations in the Braid Group

Given a system of equations in a “random” finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability. Moreover, with a significant probability, the solution will be the first in the list. This gives a probabilistic solution to: the conjugacy problem, the group membership problem, the shortest presentation of an element, and other combinatorial group-theoretic problems in random subgroups of the braid group. We use a memory-based extension of the standard length-based approach, which in principle can be applied to any group admitting an efficient, reasonably behaving length function.  2005 Elsevier Inc. All rights reserved. ✩ This paper is a part of the PhD thesis of the second named author at Bar-Ilan University. * Corresponding author. E-mail addresses: [email protected], [email protected] (D. Garber), [email protected] (S. Kaplan), [email protected] (M. Teicher), [email protected] (B. Tsaban), [email protected] (U. Vishne). URL: http://www.cs.biu.ac.il/~tsaban (B. Tsaban). 0196-8858/$ – see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aam.2005.03.002 324 D. Garber et al. / Advances in Applied Mathematics 35 (2005) 323–334 1. The general method 1.1. Systems of equations in a group Fix a group G. A pure equation in G with variables Xi , i ∈ N, is an expression of the form X σ1 k1 X σ2 k2 · · ·Xn kn = b, (1) where k1, . . . , kn ∈ N, σ1, . . . , σn ∈ {1,−1}, and b is given. A parametric equation is one obtained from a pure equation by substituting some of the variables with given (known) parameters. By equation we mean either a pure or a parametric one. Since any probabilistic method to solve a system of equations implies a probabilistic mean to check that a given system has a solution, we will confine attention to systems of equations which possess a solution. Given a system of equations of the form (1), it is often possible to use algebraic manipulations (taking inverses and multiplications of equations) in order to derive from it a (possibly smaller) system of equations all of which share the same leading variable, that is, such that all equations have the form