1 · The essentials
These manuscripts study groups called Artin groups, which are specified by generators and rules relating them. The main result concerns every such group with finitely many standard generators. It establishes that a standard space built for the group captures its loop structure without additional higher-dimensional holes. A companion manuscript shows that intersecting any collection of certain distinguished subgroups gives another subgroup of the same kind. Another companion constructs a group with 116 generators that cannot act on a space of a proposed geometric kind under the required conditions. The main result and the geometric counterexample have Lean formalizations; the intersection result is presented in a manuscript without that verification status.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, think of a group as a set of instructions for moving around a building, and its associated space as a map of the possible routes. The main result says that the standard map records the intended loop structure without extra higher-dimensional holes; the geometric counterexample warns that this does not make every desired style of geometric model available.
Why it matters
The main result resolves the Artin conjecture for groups with finitely many standard generators. The intersection manuscript gives an affirmative resolution of the Parabolic Intersection Conjecture in the same finite-rank setting. The geometric counterexample provides a contrasting boundary: the CAT(0) conjecture for Artin groups is false.
2 · Going further
The main problem is whether the standard Salvetti complex of a finite-rank Artin group is aspherical. Here, finite rank means having finitely many standard generators, and asphericity means that the higher homotopy groups vanish. The main manuscript, “Harmonic heights and the Artin K(pi,1) conjecture,” establishes this for every finite-rank Artin group, resolving the conjecture in that scope. This main result has a Lean formalization.
The subgroup companion addresses a different structural question. It shows in the manuscript that every intersection of parabolic subgroups of a finite-rank Artin group is parabolic. The statement covers arbitrary intersections, not merely intersections of two subgroups, and permits arbitrary finite or infinite Coxeter labels. The supplied source does not identify this result as Lean-formalized.
The geometric companion constructs an Artin group on 116 generators admitting no proper, cocompact isometric action on any nonempty proper CAT(0) space. That result has a Lean formalization and refutes the CAT(0) conjecture for Artin groups. Together, the manuscripts concern a standard classifying-space model, closure of a distinguished subgroup family under intersections, and an obstruction to a stronger geometric model. The supplied abstracts state the conclusions but do not describe the arguments, so they do not support an account of the proof mechanisms or dependencies between the manuscripts.
Useful notions
- Groups, generators, and relations
- Subgroups and their intersections
- Loops and higher homotopy groups
- Group actions on metric spaces
- Formal verification in Lean
What was known before / what is new
Before
The source identifies the Artin conjecture, the Parabolic Intersection Conjecture, and the CAT(0) conjecture for Artin groups as the questions addressed. It supplies no account of earlier partial results or historical development.
New
The main Lean-formalized result establishes asphericity of the standard Salvetti complex for every finite-rank Artin group. The intersection manuscript shows closure of parabolic subgroups under arbitrary intersections in finite rank, including arbitrary finite or infinite Coxeter labels. The other Lean-formalized result supplies a 116-generator counterexample to the asserted existence of geometric CAT(0) actions.
Limits and assumptions
The asphericity and intersection statements are restricted to finite-rank Artin groups; the supplied source states no conclusion for infinite rank. The geometric obstruction concerns an explicit 116-generator group and excludes actions satisfying all the stated conditions: properness, cocompactness, and preservation of distances, on a nonempty proper CAT(0) space. It does not assert that every Artin group has this obstruction. The main asphericity result and the geometric counterexample have Lean formalizations, whereas the intersection result is shown in the manuscript without a supplied formalization status. The abstracts give no argument details or further list of open questions.
Mini glossary
- Finite rank —
- In this setting, the condition that an Artin group has finitely many standard generators.
- Salvetti complex —
- The standard space associated with an Artin group whose asphericity is the subject of the main result.
- Aspherical —
- A space is aspherical when its homotopy groups in dimensions greater than one vanish.
- Parabolic subgroup —
- A subgroup conjugate to one generated by a subset of the standard generators of the Artin group.
- Geometric CAT(0) action —
- In the stated obstruction, a proper, cocompact action by isometries on a nonempty proper CAT(0) space.
Keywords : Artin groups · Asphericity · CAT(0) obstruction
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT space.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Harmonic heights and the Artin K(pi,1) conjecture
23/09/2026We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin conjecture in finite rank.
An Artin group with no geometric CAT(0) action
23/09/2026We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.
Parabolic intersections in Artin groups
23/09/2026We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.