We show how to count and randomly generate finitely generated subgroups of the modular group $textsf{PSL}(2,mathbb{Z})$ of a given isomorphism type. We also prove that almost malnormality and non-parabolicity are negligible properties for these subgr
oups. The combinatorial methods developed to achieve these results bring to light a natural map, which associates with any finitely generated subgroup of $textsf{PSL}(2,mathbb{Z})$ a graph which we call its silhouette, and which can be interpreted as a conjugacy class of free finite index subgroups of $textsf{PSL}(2,mathbb{Z})$.
We count the finitely generated subgroups of the modular group $textsf{PSL}(2,mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $Gamma(H)$, we consider the number of vertices of $Gamma(H)$ to be the size of
$H$ and we count the subgroups of size $n$. Since an index $n$ subgroup has size $n$, our results generalize the known results on the enumeration of the finite index subgroups of $textsf{PSL}(2,mathbb{Z})$. We give asymptotic equivalents for the number of finitely generated subgroups of $textsf{PSL}(2,mathbb{Z})$, as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size $n$ subgroup and prove a large deviations statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size $n$ subgroup (resp. finite index subgroup, free subgroup) of $textsf{PSL}(2,mathbb{Z})$.
This is a survey of results on random group presentations, and on random subgroups of certain fixed groups. Being a survey, this paper does not contain new results, but it offers a synthetic view of a part of this very active field of research.
Asymptotic properties of finitely generated subgroups of free groups, and of finite group presentations, can be considered in several fashions, depending on the way these objects are represented and on the distribution assumed on these representation
s: here we assume that they are represented by tuples of reduced words (generators of a subgroup) or of cyclically reduced words (relators). Classical models consider fixed size tuples of words (e.g. the few-generator model) or exponential size tuples (e.g. Gromovs density model), and they usually consider that equal length words are equally likely. We generalize both the few-generator and the density models with probabilistic schemes that also allow variability in the size of tuples and non-uniform distributions on words of a given length.Our first results rely on a relatively mild prefix-heaviness hypothesis on the distributions, which states essentially that the probability of a word decreases exponentially fast as its length grows. Under this hypothesis, we generalize several classical results: exponentially generically a randomly chosen tuple is a basis of the subgroup it generates, this subgroup is malnormal and the tuple satisfies a small cancellation property, even for exponential size tuples. In the special case of the uniform distribution on words of a given length, we give a phase transition theorem for the central tree property, a combinatorial property closely linked to the fact that a tuple freely generates a subgroup. We then further refine our results when the distribution is specified by a Markovian scheme, and in particular we give a phase transition theorem which generalizes the classical results on the densities up to which a tuple of cyclically reduced words chosen uniformly at random exponentially generically satisfies a small cancellation property, and beyond which it presents a trivial group.
