The Twist Conjecture and the Isomorphism Problem for Coxeter groups

Abstract

We prove Mühlherr's Twist Conjecture: any two angle-compatible Coxeter generating sets of a Coxeter group differ by a finite sequence of elementary twists and a conjugation. Combined with earlier work of Howlett-Mühlherr and Marquis-Mühlherr, this completes the resolution of the Isomorphism Problem for Coxeter groups. A further consequence is that Aut(W) is finitely generated for every Coxeter group W, and there is an algorithm producing a finite set of generators for Aut(W) starting from any Coxeter matrix. Of the vast literature on the Twist Conjecture, we utilise only two results in an essential way: strong rigidity of $2$--spherical Coxeter systems, due to Caprace and Mühlherr, and the framework of markings and hierarchies developed by Caprace and Przytycki for the twist-rigid case. We also exploit in a fundamental way some soft ideas from JSJ theory and an observation of Mihalik-Tschantz on splittings of Coxeter groups. No form of AI was used in the writing of this manuscript, nor in the research that it presents.

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BibTeX

@misc{fioravanti2026twist,
  title = {{The Twist Conjecture and the Isomorphism Problem for Coxeter groups}},
  author = {Elia Fioravanti},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28348},
  eprint = {2608.28348},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28348},
}

RIS

TY  - UNPB
TI  - The Twist Conjecture and the Isomorphism Problem for Coxeter groups
AU  - Elia Fioravanti
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28348
UR  - https://arxiv.org/abs/2608.28348
AB  - We prove Mühlherr's Twist Conjecture: any two angle-compatible Coxeter generating sets of a Coxeter group differ by a finite sequence of elementary twists and a conjugation. Combined with earlier work of Howlett-Mühlherr and Marquis-Mühlherr, this completes the resolution of the Isomorphism Problem for Coxeter groups. A further consequence is that Aut(W) is finitely generated for every Coxeter group W, and there is an algorithm producing a finite set of generators for Aut(W) starting from any Coxeter matrix. Of the vast literature on the Twist Conjecture, we utilise only two results in an essential way: strong rigidity of $2$--spherical Coxeter systems, due to Caprace and Mühlherr, and the framework of markings and hierarchies developed by Caprace and Przytycki for the twist-rigid case. We also exploit in a fundamental way some soft ideas from JSJ theory and an observation of Mihalik-Tschantz on splittings of Coxeter groups. No form of AI was used in the writing of this manuscript, nor in the research that it presents.
ER  -

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