The nth root of a braid is unique up to conjugacy

By Juan González-Meneses. 2003

70 citations recorded by the source corpus.

We prove a conjecture due to Makanin: if a and β are elements of the Artin braid group B n such that α k = β k for some nonzero integer k, then a and β are conjugate. The proof involves the Nielsen-Thurston classificationof braids.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4dde95c38243b6223370cd68e9dc28cfb0002847

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Groups, Orders, and Dynamics (2014, unknown): “derable groups satisfying the C.R.P. are not necessarily ℓ- orderable. Concrete examples are braid groups: in §1.2.5, we will see that these groups are left-orderable, the C.R.P. for them is shown in [77], and the fact that Bn is not ℓ-orderable (for n≥3) is proved in [127]. 1.3.2 Locally-invariant ”
Basic results on braid groups (2010, background): “[43] For k 6 = 0 , all k th roots of a braid are conjugate.”
A new twist on Lorenz links (2007, background): “By the results in [16] we know that periodic braids have unique roots.”
Foundations of Garside Theory (2013, background): “Gonz ález-Meneses proved in [140] that any twodth roots of an element of B n are conjugate inB + n .”
Conjugacy in Garside groups I: cyclings, powers and rigidity (2006, methodology): “It uses the structure of centralizers of PA braids, and the uniqueness of their roots [23], some particular properties of periodic braids, and also the geometric decomposition of a reducible braid along its invariant curves.”
Braids: A Survey (2004, background): “Other relevant papers are [77] and [78].”
Conjugacy in Garside groups III: Periodic braids (2006, unknown): “In Section 2, we will review what is known and explain why steps (1) and (2) of Theorem 1 follow easily from the work in [16,21,23].”