Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

💡 Research Summary

This paper investigates the statistical behavior of finitely generated subgroups of free groups and of finite group presentations under two distinct probabilistic models: the classical word‑based distribution and the newer graph‑based distribution introduced by Bassino, Nicaud, and Weil. In the word‑based model, a subgroup is generated by a fixed‑size k‑tuple of reduced words whose maximal length n tends to infinity. Classical results show that, with overwhelming (exponentially generic) probability, such tuples consist of long words whose short prefixes are pairwise distinct, which in turn guarantees properties such as malnormality and purity.

The graph‑based model, by contrast, represents a subgroup via its Stallings graph and selects uniformly among all admissible A‑graphs with a given number n of vertices. The authors develop combinatorial tools to count such graphs and to analyse the probability of various structural features. They prove that, as n grows, the proportion of graphs whose associated subgroups are malnormal or pure decays exponentially to zero; thus these properties are negligible in the graph‑based setting.

A striking intermediate result is the identification of a property that occurs with limiting probability e⁻ʳ (where r is the rank of the free group). This property is invisible in the word‑based model but has a non‑trivial asymptotic frequency in the graph‑based model, illustrating how the choice of distribution can dramatically alter what is considered “typical”.

The paper also extends the analysis to finite presentations ⟨A | R⟩. Under the word‑based distribution, random relators of bounded length generate a normal subgroup whose quotient is almost surely infinite and hyperbolic, a fact established by Ol’shanskiĭ and Champetier. In the graph‑based distribution, however, random relators correspond to random Stallings graphs; the resulting normal subgroup almost always coincides with the whole free group, so the quotient is generically the trivial group. This reversal underscores the sensitivity of generic‑group results to the underlying probabilistic model.

Overall, the work demonstrates that “generic” and “negligible” are not intrinsic properties of subgroups or presentations but depend heavily on how randomness is introduced. By contrasting the two natural stratifications—by word length versus by graph size—the authors reveal that many properties previously thought to be typical (e.g., malnormality, purity, hyperbolicity) become rare, while others become typical, when viewed through the lens of Stallings graphs. The findings have implications for algorithmic group theory, random group constructions, and cryptographic applications where the notion of a “random” subgroup or presentation is pivotal. Future research directions include exploring hybrid distributions, refining the asymptotic estimates, and assessing the impact of these statistical differences on computational problems in combinatorial group theory.