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PreprintCC BY 4.0arXiv

Finiteness properties of centralisers in Out(FN) and automorphisms of free-by-cyclic groups

  • Naomi Andrew
  • Yassine Guerch
  • Camille Horbez
  • Richard D. Wade
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Abstract

Let F be a finitely generated free group. We prove that the centraliser of any element of Out(F) or Aut(F) is of type VF. We also prove that the (outer) automorphism group of any free-by-cyclic group M=Frtimesφℤ is of type VF, and so are the McCool subgroups of Out(M). This relies on the construction of a new Out(M)-invariant splitting of M, that enables us to reduce the proof of the above statements to the case where the outer class of φ has finite order.

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Cite

BibTeXRIS
BibTeX
@misc{andrew2026finiteness,
  title = {{Finiteness properties of centralisers in $\mathrmOut(F_N)$ and automorphisms of free-by-cyclic groups}},
  author = {Naomi Andrew and Yassine Guerch and Camille Horbez and Richard D. Wade},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04581},
  eprint = {2611.04581},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04581},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finiteness properties of centralisers in Out(F_N) and automorphisms of free-by-cyclic groups
AU  - Naomi Andrew
AU  - Yassine Guerch
AU  - Camille Horbez
AU  - Richard D. Wade
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04581
UR  - https://arxiv.org/abs/2611.04581
AB  - Let F be a finitely generated free group. We prove that the centraliser of any element of Out(F) or Aut(F) is of type VF. We also prove that the (outer) automorphism group of any free-by-cyclic group M=Frtimes_φℤ is of type VF, and so are the McCool subgroups of Out(M). This relies on the construction of a new Out(M)-invariant splitting of M, that enables us to reduce the proof of the above statements to the case where the outer class of φ has finite order.
ER  -