Probabilistic solutions of equations in the braid group

David Garber; Shmuel Kaplan; Mina Teicher; Boaz Tsaban; Uzi Vishne

doi:10.1016/j.aam.2005.03.002

Probabilistic solutions of equations in the braid group

David Garber, Shmuel Kaplan, Mina Teicher, Boaz Tsaban, Uzi Vishne

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

26 Scopus citations

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
Pages (from-to)	323-334
Number of pages	12
Journal	Advances in Applied Mathematics
Volume	35
Issue number	3
DOIs	https://doi.org/10.1016/j.aam.2005.03.002
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).

Funding

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).

Funders	Funder number
Minerva Foundation	8008/02-3
Israel Science Foundation	EU-network HPRN-CT-2009-00099

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10.1016/j.aam.2005.03.002

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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.",

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Probabilistic solutions of equations in the braid group

Abstract

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