#### Volume 3, issue 2 (2003)

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The $n$th root of a braid is unique up to conjugacy

### Juan Gonzalez-Meneses

Algebraic & Geometric Topology 3 (2003) 1103–1118
 arXiv: math.GT/0306070
##### Abstract

We prove a conjecture due to Makanin: if $\alpha$ and $\beta$ are elements of the Artin braid group ${B}_{n}$ such that ${\alpha }^{k}={\beta }^{k}$ for some nonzero integer $k$, then $\alpha$ and $\beta$ are conjugate. The proof involves the Nielsen–Thurston classification of braids.

##### Keywords
braid, root, conjugacy, Nielsen-Thurston theory.
##### Mathematical Subject Classification 2000
Primary: 20F36
Secondary: 20F65.
##### Publication
Revised: 16 October 2003
Accepted: 20 October 2003
Published: 1 November 2003
##### Authors
 Juan Gonzalez-Meneses Universidad de Sevilla Dep. Matemática Aplicada I ETS Arquitectura Av. Reina Mercedes 2 41012-Sevilla Spain www.personal.us.es/meneses