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Math-2291 · mathematics · Group theory

Parabolic subgroup intersections in Artin groups, including Euclidean type

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
In an Artin group, a standard parabolic subgroup is generated by a subset of the standard generators, and a parabolic subgroup is a conjugate of such a subgroup. Must an intersection of two parabolic subgroups again be parabolic? In particular, does this hold for Artin groups of Euclidean type?

Formulation references: [S1], [S2].

Short English statement

In an Artin group, a standard parabolic subgroup is generated by a subset of the standard generators, and a parabolic subgroup is a conjugate of such a subgroup. Must an intersection of two parabolic subgroups again be parabolic? In particular, does this hold for Artin groups of Euclidean type?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2291-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2291

The all-finite-Coxeter-matrix theorem makes intersections of arbitrary families of parabolic subgroups parabolic. Affine Artin groups are included, so its conclusion covers the Euclidean case and is stronger than the requested pairwise statement.

Parabolic subgroup intersections in Artin groups, including Euclidean type

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Every intersection, including an infinite intersection, of parabolic subgroups in a finite-rank Artin group is claimed parabolic.

Mathematical statement of the claimed result

Result 1.
Let $A_M$ be the Artin group of a finite Coxeter matrix on $S$; $A_X$ denotes its standard parabolic subgroup for $X\subseteq S$. For arbitrary $X,Y\subseteq S$ and $g,h\in A_M$, the claim is \[ \exists Z\subseteq S,\ k\in A_M:\qquad gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}. \] Every family of parabolics has the same intersection as some subfamily of size at most $|S|$, so arbitrary intersections are parabolic and every subset has a least parabolic closure. The algorithmic corollary takes $M,X$ and words $u,v$ and decides \[ v\in uA_Xu^{-1}; \] choosing $X=\varnothing$ yields a uniform word-problem algorithm. If the diagram is irreducible and has an infinite label, the additional corollary gives acylindrical hyperbolicity and, for every proper parabolic $P$, a $g$ with $P\cap gPg^{-1}=\{1\}$.

Outline of the argument

Relative categorical layers track positive divisors and caps. Two-point uniqueness and a relative frame control null layers, while harmonic heights produce connected minima and bounded positive-height representatives; these give the parabolic carrier of the intersection.

Formal verification

For family 254, the scope includes finite-rank Salvetti contractibility, the 116-generator Artin CAT$0$ counterexample, and arbitrary parabolic intersections with closure consequences. Each has its own Comparator statement. The listed Comparator files are ArtinCAT0.lean, ArtinParabolicIntersections.lean, HarmonicArtin.lean. Named declarations include OAI.ArtinCAT0.main, OAI.HarmonicArtin.salvetti_cover_contractible. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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