Detecting trivial elements of periodic quotient of hyperbolic groups

Coulon, Rémi

doi:10.24033/bsmf2772

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Detecting trivial elements of periodic quotient of hyperbolic groups

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Coulon, Rémi
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.24033/bsmf2772
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ArXiv_1211.4267.pdf
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Citation

Coulon, R. (2018). Detecting trivial elements of periodic quotient of hyperbolic groups. Bulletin de la Société Mathématique de France, 146(4), 745-806. doi:10.24033/bsmf2772.

Cite as: https://hdl.handle.net/21.11116/0000-0003-C4F9-7

Abstract

In this article we give a sufficient and necessary condition to determine whether an element of the free group induces a nontrivial element of the free Burnside
group of sufficiently large odd exponents. Although this result is “well known” among
specialists, it has never been stated with such a level of simplicity. Moreover, our proof highlights some important differences between the Delzant-Gromov approach to the Burnside problems and others that exist. This criterion can be stated without any
knowledge regarding Burnside groups, in particular about the proof of its infiniteness.
Therefore, it also provides a useful tool to study outer automorphisms of Burnside
groups. In addition, we state an analogue result for periodic quotients of torsion-free
hyperbolic groups.