The lengths of conjugators in the model filiform groups

Journal article

Abstract:: The conjugator length function of a finitely generated group Γ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius n in the Cayley graph of Γ. We prove that polynomials of arbitrary degree arise as conjugator length functions of finitely presented groups. To establish this, we analyse the geometry of conjugation in the discrete model filiform groups Γ_d = Z^d ⋊_φ Z where φ is the automorphism of Z^d that fixes the last element of a basis a₁, . . . , a_d and sends ai to a_ia_i+1 for i < d. The conjugator length function of Γ_d is polynomial of degree d.

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Copyright holder:: Bridson and Riley
Rights statement:: © The Author(s) 2026. Open Access. This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

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