Geodesic
Generic properties of random subgroups of a free group for general distributions
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12) (2012).

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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. 
@article{DMTCS_2012_special_262_a12,
     author = {Bassino, Fr\'ed\'erique and Nicaud, Cyril and Weil, Pascal},
     title = {Generic properties of random subgroups of a free group for general distributions},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)},
     year = {2012},
     doi = {10.46298/dmtcs.2991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2991/}
}
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Bassino, Frédérique; Nicaud, Cyril; Weil, Pascal. Generic properties of random subgroups of a free group for general distributions. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12) (2012). doi : 10.46298/dmtcs.2991. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2991/

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