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Pré-Publication, Document De Travail Année : 2015

Generic properties of subgroups of free groups and finite presentations

Résumé

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 representations: here we assume, as is often done, 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. Gromov's 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 faster than its length. Under this hypothesis, we generalize the classical results on the free basis property (generically a randomly chosen tuple is a basis of the subgroup it generates), on malnormality or on small cancellation properties. We then 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 generically satisfies a small cancellation property, and beyond which it presents a trivial group.
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Dates et versions

hal-01171484 , version 1 (03-07-2015)
hal-01171484 , version 2 (12-11-2015)

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Frédérique Bassino, Cyril Nicaud, Pascal Weil. Generic properties of subgroups of free groups and finite presentations. 2015. ⟨hal-01171484v1⟩
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