АННОТАЦИЯ
We study the computational complexity of the Word Problem (WP) in free solvable groups Sr,d, where r≥2 is the rank and d≥2 is the solvability class of the group. It is known that the Magnus embedding of Sr,d into matrices provides a polynomial time decision algorithm for WP in a fixed group Sr,d. Unfortunately, the degree of the polynomial grows together with d, so the uniform algorithm is not polynomial in d. In this paper we show that WP has time complexity O(rnlog2n) in Sr,2, and O(n3rd) in Sr,d for d≥3. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in Sr,2 is NP-complete. We prove also that one can compute Fox derivatives of elements from Sr,d in time O(n3rd), in particular one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as, on a relatively new geometric ideas, in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs
ЦИТАТА
The Word and Geodesic Problems in Free Solvable Groups / A. Myasnikov, V. Romankov, A. Ushakov, A. Vershik статья в открытом архиве № arXiv:0807.1032 07.07.2008 0:00:00
