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PreprintCC0arXiv

Cosets with constant characteristic polynomial

  • Nir Avni
  • Tsachik Gelander
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Abstract

Let H be a linear group. We show that if there is an invertible matrix x such that all the elements of xH share the same characteristic polynomial then H is virtually solvable. There are plenty of applications that will be presented in future paper. Here, we discuss some applications to the generalized Weigold conjecture and present an alternative straightforward proof of the Formanek--Procesi nonlinearity theorem for Aut(F_n), n>2, over every field. When n>4 our non-linearity proof gives a stronger result than the original Formanek--Procesi theorem.

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Cite

BibTeXRIS
BibTeX
@misc{avni2026cosets,
  title = {{Cosets with constant characteristic polynomial}},
  author = {Nir Avni and Tsachik Gelander},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27204},
  eprint = {2609.27204},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27204},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Cosets with constant characteristic polynomial
AU  - Nir Avni
AU  - Tsachik Gelander
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.27204
UR  - https://arxiv.org/abs/2609.27204
AB  - Let H be a linear group. We show that if there is an invertible matrix x such that all the elements of xH share the same characteristic polynomial then H is virtually solvable. There are plenty of applications that will be presented in future paper. Here, we discuss some applications to the generalized Weigold conjecture and present an alternative straightforward proof of the Formanek--Procesi nonlinearity theorem for Aut(F_n), n>2, over every field. When n>4 our non-linearity proof gives a stronger result than the original Formanek--Procesi theorem.
ER  -