Probabilistic Solutions of Equations in the Braid Group

Details Probabilistic Solutions of Equations in the Braid Group Probabilistic Solutions of Equations in the Braid Group

2004-04-05

arxiv.org/abs/math/0404076v4

Advances in Applied Mathematics 35 (2005), 323--334

Mathematics

Group Theory

Small updates

Scientific paper

10.1016/j.aam.2005.03.002

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 representation 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.

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