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ABSTRACT
V. B. Styshnev showed in Citation[9] that the existence of n -th roots for a braid is decidable. Garside groups have been introduced in Citation[2] and Citation[3] as a natural proper generalization of Artin groups of finite type. We have to construct a new proof to extend Styshnev's decidability result to Garside groups, as several specific properties of braids used in Citation[9] fail in our case. We show that, under the assumption of a finiteness property of conjugacy, the problem is decidable.
ACKNOWLEDGMENTS
The author wishes to thank Patrick Dehornoy for his help and comments during the preparation of this work, and Matthieu Picantin for fruitful discussions.