### Abstract

In the theory of random groups, we consider presentations with any fixed number mm of generators and many random relators of length ℓℓ, sending ℓ→∞ℓ→∞. If dd is a „density“ parameter measuring the rate of exponential growth of the number of relators compared to the length of relators, then many group-theoretic properties become generically true or generically false at different values of dd. The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for d<1/2d<1/2, random groups are a.a.s. infinite hyperbolic, while for d>1/2d>1/2, random groups are a.a.s. order one or two. We study random groups at the density threshold d=1/2d=1/2. Kozma had found that trivial groups are generic for a range of growth rates at d=1/2d=1/2; we show that infinite hyperbolic groups are generic in a different range. (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)

Original language | English |
---|---|

Pages (from-to) | 985-1005 |

Number of pages | 21 |

Journal | Groups, Geometry and Dynamics |

Volume | 10 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2016 |

### Keywords

- Random groups
- density

## Fingerprint Dive into the research topics of 'A sharper threshold for random groups at density one-half'. Together they form a unique fingerprint.

## Profiles

## Dr John M Mackay

- Probability, Analysis and Dynamics
- School of Mathematics - Lecturer in Pure Mathematics
- Pure Mathematics
- Analysis

Person: Academic , Member

## Cite this

Duchin, M., Jankiewicz, K., Klimer, S., Lelièvre, S., Mackay, J., & Sánchez, A. (2016). A sharper threshold for random groups at density one-half.

*Groups, Geometry and Dynamics*,*10*(3), 985-1005. https://doi.org/10.4171/GGD/374