## Abstract

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.

Original language | English |
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Pages (from-to) | 323-334 |

Number of pages | 12 |

Journal | Advances in Applied Mathematics |

Volume | 35 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2005 |

### Bibliographical note

Funding Information:This research was partially supported by the Israel Science Foundation through an equipment grant to the School of Computer Science in Tel-Aviv University. The authors were partially supported by: Golda Meir Fellowship (first named author), EU-network HPRN-CT-2009-00099(EAGER), Emmy Noether Research Institute for Mathematics, the Minerva Foundation, and the Israel Science Foundation grant #8008/02-3 (second and third named authors).