The Word and Geodesic Problems in Free Solvable Groups

TL;DR

This paper analyzes the computational complexity of the Word and Geodesic Problems in free solvable groups, providing efficient algorithms for the Word Problem and establishing NP-completeness for the Geodesic Problem in certain cases.

Contribution

It introduces new complexity bounds for the Word Problem in free solvable groups and demonstrates NP-completeness for the Geodesic Problem in $S_{r,2}$, linking Fox derivatives to geometric flows.

Findings

Word Problem in $S_{r,2}$ can be solved in $O(r n \,\log n)$ time.

Word Problem in $S_{r,d}$ for $d\geq 3$ can be solved in $O(n^3 r d)$ time.

Computing Fox derivatives in $S_{r,d}$ is achievable in $O(n^3 r d)$ time.

Abstract

We study the computational complexity of the Word Problem (WP) in free solvable groups $S_{r,d}$, where $r \geq 2$ is the rank and $d \geq 2$ is the solvability class of the group. It is known that the Magnus embedding of $S_{r,d}$ into matrices provides a polynomial time decision algorithm for WP in a fixed group $S_{r,d}$. 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 $O(r n \log_2 n)$ in $S_{r,2}$, and $O(n^3 r d)$ in $S_{r,d}$ for $d \geq 3$. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in $S_{r,2}$ is $NP$-complete. We prove also that one can compute Fox derivatives of elements from $S_{r,d}$ in time $O(n^3 r d)$, 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.