The central tree property and algorithmic problems on subgroups of free groups
Mallika Roy, Enric Ventura, Pascal Weil
TL;DR
This work analyzes the average-case complexity of the uniform membership problem for subgroups of free groups and introduces the central tree property (ctp) as a generic structural condition on word tuples. By exploiting the ctp, the authors design algorithms whose expected running time is substantially smaller than the worst-case, particularly when the tuple size grows at most polynomially with input length, and they quantify the probabilistic likelihood of the ctp holding. The paper then leverages Shpilrain's constant-average-case primitivity algorithm to address the relative primitivity problem, showing that combining these approaches yields efficient average-case performance for membership and primitivity tasks in subgroups of free groups. Overall, the results provide concrete average-case bounds and practical procedures for core algorithmic problems in free groups, with implications for related decision problems and primitive testing in subgroups. The methods rely on Stallings graphs, growth modulus analysis, and probabilistic properties of random word tuples, delivering improvements over worst-case complexity in a broad range of input regimes.
Abstract
We study the average case complexity of the uniform membership problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.