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Antolín Pichel, Yago and Ciobanu, Laura and Viles, Noelia
(2012)
*On the asymptotics of visible elements and homogeneous
equations in surface groups.*
Groups, geometry and dynamics, 6
.
619 -638.
ISSN 1661-7207

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Official URL: https://doi.org/10.4171/GGD/167

## Abstract

Let F be a group whose abelianization is Zk, k 2. An element of F is called

visible if its image in the abelianization is visible, that is, the greatest common divisor of its coordinates is 1. In this paper we compute three types of densities, annular, even and odd spherical, of visible elements in surface groups. We then use our results to show that the probability of a homogeneous equation in a surface group to have solutions is neither 0 nor 1, as the lengths of the right- and left-hand side of the equation go to infinity.

Item Type: | Article |
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Uncontrolled Keywords: | Free groups; Surface groups; Equations; Visible elements; Asymptotic behavior. |

Subjects: | Sciences > Mathematics > Algebra Sciences > Mathematics > Group Theory |

ID Code: | 76831 |

Deposited On: | 28 Feb 2023 12:57 |

Last Modified: | 28 Feb 2023 13:00 |

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