A mathematical preprint reports a proof that resolves a central comparison problem for Coxeter groups. The paper addresses whether groups arising from two Coxeter systems are isomorphic, meaning whether they have the same abstract group structure. Its main result is a proof of Mühlherr’s Twist Conjecture. For the angle-compatible Coxeter generating sets covered by the theorem, the paper says one set can be changed into the other through finitely many elementary twists and a conjugation. In plain language, the result supplies a finite sequence of permitted transformations linking the two generating descriptions. The paper then states that the Isomorphism Problem for Coxeter groups is algorithmically decidable, so the yes-or-no question can be settled by an algorithm within the stated framework.
A precise setting for the result
The theorem is deliberately precise about which generating sets it covers. It focuses on two sets, called S and R, that are angle-compatible. The paper’s term “angle-compatible” marks the specific compatibility condition required by the theorem. The main statement says that S and R are twist-equivalent relative to a family C of compatible subsets. That qualification matters: the result is framed under the theorem’s compatibility conditions, rather than as an unrestricted comparison of any two descriptions. Twists and conjugation provide the link between the generating sets, while the corollary turns that link into a decision procedure for the groups.
A proof built from smaller cases
The proof is organized around a complete proof of Theorem A. That proof assumes Theorems 2.6 and 2.8, which are established in Sections 4 and 7. The argument is not statistical and does not use an empirical sample. Instead, it combines formal definitions and lemmas with induction, tree splittings, geometric arguments and algorithmic constructions. The paper says the proof relies essentially on strong rigidity of 2-spherical Coxeter systems, markings and hierarchies, JSJ ideas and an observation about splittings. Theorem A itself is proved by induction on the cardinality of S, making the size of the generating set the measure used to organize the argument.
One of the technical mechanisms is a shortening construction. The paper says this usually requires finitely many twists rather than a single twist. Another result, Theorem 7.8, produces a new generating set and a transformed subset under its stated hypotheses. The transformed subset is R-geometric, and it generates the same subgroup as the original subset. The replacement therefore preserves the subgroup while putting it into the geometric form named in the theorem.
From theorem to algorithm
The consequences reach beyond the original twist statement. The paper reports that the automorphism group of every Coxeter group is finitely generated. In ordinary terms, the group’s full collection of structure-preserving self-maps can be built from a finite list of generators. It also gives an algorithm that produces such a finite generating set from a Coxeter matrix. The appendix adds a related result: the relative automorphism group Aut(W; P) is finitely generated as well. These conclusions connect the twist argument to formal procedures for comparing Coxeter groups and describing their symmetries.
What the result does not establish
The main boundary is scope. The central theorem concerns angle-compatible generating sets and is stated relative to compatible subsets. The broader isomorphism conclusion relies on additional procedures and earlier work. Theorem A alone therefore does not directly address arbitrary non-angle-compatible generating sets. The paper’s stated boundaries also matter: elementary twists do not always extend to automorphisms of the Coxeter group, so the twist operation and the automorphism consequences should not be treated as interchangeable.
Practical questions remain unanswered. No runtime bounds, complexity analysis, implementation details or benchmarks are reported for the algorithms. The result establishes algorithmic decidability and an algorithm for finite automorphism generators, but it does not establish how quickly those procedures run or how they perform in concrete implementations. That leaves a clear distinction between a proof that a procedure exists and an assessment of its cost in practice.
Preprint status
The manuscript is a preprint on arXiv, listed as version 1 and dated 28 August 2026. The author reports support from Emmy Noether grant 515507199 and DFG grant 541703614. It also states that no form of AI was used in writing the manuscript or conducting the research.
Paper data and sources
Original title: The Twist Conjecture and the Isomorphism Problem for Coxeter groups
Authors: Elia Fioravanti
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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