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Freiheitssatz and phase transition for the density model of random groups

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Abstract

Magnus’ Freiheitssatz [20] states that if a group is defined by a presentation with m generators and a single cyclically reduced relator, and this relator contains the last generating letter, then the first \(m-1\) letters freely generate a free subgroup. We study an analogue of this theorem in the Gromov density model of random groups [14], showing a phase transition phenomenon at density \(d_r = \min \{\frac{1}{2}, 1-\log _{2m-1}(2r-1)\}\) with \(1\le r\le m-1\): we prove that for a random group with m generators at density d, if \(d < d_r\) then the first r letters freely generate a free subgroup; whereas if \(d > d_r\) then the first r letters generate the whole group. This result partially answers a general problem proposed by Gromov in 2003 [15]: existence/nonexistence of non-free subgroups in a random group.

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Acknowledgements

I would like to thank my thesis advisor, Thomas Delzant, for his patience and guidance, and for many interesting and helpful discussions on the subject. I am also very grateful to the anonymous referee for his/her thorough review of the manuscript and greatly appreciate the comments and suggestions.

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  1. Institut de Recherche Mathématique Avancée, 7 Rue René Descartes, 67000, Strasbourg, France

    Tsung-Hsuan Tsai

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Tsai, TH. Freiheitssatz and phase transition for the density model of random groups. Math. Z. 303, 65 (2023). https://doi.org/10.1007/s00209-022-03186-2

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