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.
|Number of pages||12|
|Journal||Advances in Applied Mathematics|
|State||Published - Sep 2005|
Bibliographical noteFunding 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).