Generic properties of random subgroups of a free group for general distributions

Abstract

We consider a generalization of the uniform word-based distribution for finitely generated subgroups of a free group. In our setting, the number of generators is not fixed, the length of each generator is determined by a random variable with some simple constraints and the distribution of words of a fixed length is specified by a Markov process. We show by probabilistic arguments that under rather relaxed assumptions, the good properties of the uniform word-based distribution are preserved: generically (but maybe not exponentially generically), the tuple we pick is a basis of the subgroup it generates, this subgroup is malnormal and the group presentation defined by this tuple satisfies a small cancellation condition.

Keywords

• [math.math-co] mathematics [math]/combinatorics [math.co]
• [math.math-gr] mathematics [math]/group theory [math.gr]
• [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds]