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PreprintCC BY 4.0arXiv

Conjugator Length in Iterated Cyclic Amalgams of Free Groups

  • Conan Gillis
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Abstract

Let π’œ0 be the set of finitely generated free groups on some countable alphabet and and define π’œr inductively as the union of π’œrβˆ’1 with the set of amalgamated free products G1βˆ—ZZG2, where G1,G2∈ π’œrβˆ’1. We show, for all r, that every element of π’œr has linear conjugator length, quantifying a result of Larsen and extending a result of Kharlampovich and Myasnikov. This is achieved by a linear conjugator length bound on a family of iterated HNN extensions of free groups. Also, in an appendix, we adapt a proof of A. Weiß to show that all Generalized Baumslag-Solitar Groups have linear conjugator length. As a consequence, we obtain that one-relator groups with non-trivial center have linear conjugator length, giving evidence for a conjecture of Bridson, Riley, and Sale.

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Cite

BibTeXRIS
BibTeX
@misc{gillis2026conjugator,
  title = {{Conjugator Length in Iterated Cyclic Amalgams of Free Groups}},
  author = {Conan Gillis},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21116},
  eprint = {2609.21116},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21116},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Conjugator Length in Iterated Cyclic Amalgams of Free Groups
AU  - Conan Gillis
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.21116
UR  - https://arxiv.org/abs/2609.21116
AB  - Let π’œ_0 be the set of finitely generated free groups on some countable alphabet and and define π’œ_r inductively as the union of π’œ_rβˆ’1 with the set of amalgamated free products G_1βˆ—_ZZG_2, where G_1,G_2∈ π’œ_rβˆ’1. We show, for all r, that every element of π’œ_r has linear conjugator length, quantifying a result of Larsen and extending a result of Kharlampovich and Myasnikov. This is achieved by a linear conjugator length bound on a family of iterated HNN extensions of free groups. Also, in an appendix, we adapt a proof of A. Weiß to show that all Generalized Baumslag-Solitar Groups have linear conjugator length. As a consequence, we obtain that one-relator groups with non-trivial center have linear conjugator length, giving evidence for a conjecture of Bridson, Riley, and Sale.
ER  -