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@@ -1,54 +1,8 @@
================================================================
PARABOLIC INTERSECTIONS IN ARTIN GROUPS
-A gentle introduction
-================================================================
-
-This document explains a recent theorem about a family of groups
-called ARTIN GROUPS. It is written for a reader who knows what a
-group is (or is willing to learn in Section 1), likes concrete
-examples, and wants to understand what the theorem claims and
-roughly why it is true. The full research proof is 27,000 words
-of graduate-level topology; we will state the theorem precisely,
-explain the proof's big idea, and hand-verify small examples.
-
-Honesty note up front: some inner steps of the proof need linear
-algebra, topology of chain complexes, and a bit of fusion theory.
-We will be honest about which parts are accessible and which are
-"trust the specialists, here is the shape of their argument."
-The theorem itself is a yes/no statement about subgroups, and
-that statement is completely accessible.
================================================================
-1. WHAT IS A GROUP
-================================================================
-
-A GROUP is a set G together with an operation * (think "combine
-two things") satisfying:
- 1. Closure: if a, b are in G, then a*b is in G.
- 2. Associativity: (a*b)*c = a*(b*c).
- 3. Identity: there is an element 1 with 1*a = a*1 = a.
- 4. Inverse: every a has an a^{-1} with a*a^{-1} = 1.
-
-Examples:
- - The integers under addition. Identity 0. Inverse -n.
- - The integers mod 7 under addition.
- - The nonzero rationals under multiplication.
- - Rotations of a square: 0, 90, 180, 270 degrees. 4 elements.
- - All permutations of {1, 2, 3}: the symmetric group S_3, with
- 6 elements.
-
-A PRESENTATION writes a group as <generators | relations>. The
-group is "the largest group in which those relations hold". Two
-words represent the same element iff you can convert one to the
-other using the relations plus trivial bookkeeping.
-
- Z (integers) = <a | >. No relations. a^n means "+n".
- Z/7Z = <a | a^7 = 1>.
- Z^2 = <a, b | ab = ba>. Pairs of integers, added pointwise.
- F_2 (free group on 2 letters) = <a, b | >. No relations.
-
-================================================================
2. COXETER MATRICES
================================================================
@@ -198,68 +152,6 @@ That is the whole theorem. Two features make it non-trivial:
positive monoids fail to admit normal forms in the
classical way; the "positive fractional" machinery breaks.
-================================================================
-6. WHAT WAS KNOWN BEFORE 2026
-================================================================
-
-Cases:
- 1997 Paris, 1970's van der Lek: A_X ∩ A_Y = A_{X ∩ Y} for
- standards (parabolics generated by subsets). This is easy.
- 2007 Duncan-Kazachkov-Remeslennikov: arbitrary intersections of
- parabolics are parabolic, for RIGHT-ANGLED Artin groups.
- 2019 Cumplido-Gebhardt-Gonzalez-Meneses-Wiest: the conjecture
- for SPHERICAL Artin groups (using Garside theory).
- 2021 Morris-Wright: FC-type, one side spherical.
- 2022 Antolin-Foniqi: even FC-type, both sides spherical.
- 2023 Cumplido-Martin-Vaskou: LARGE type.
- 2024 Moller-Paris-Varghese: one side spherical, arbitrary
- ambient. Also extended pairwise to arbitrary intersections,
- ASSUMING pairwise.
- 2026 Character (this paper): full case, arbitrary ambient,
- arbitrary labels.
-
-================================================================
-7. THE STRATEGY: REPRESENT A ON SOMETHING
-================================================================
-
-Intersections of subgroups are hard to see from a presentation.
-The standard trick is to REPRESENT the group: find some object on
-which A acts, in such a way that subgroups correspond to "things
-that are fixed".
-
-Ordinary example: S_3 acts on a triangle. The subgroup fixing
-vertex #1 is a parabolic (a stabilizer of a point).
-
-For Artin groups, the right representation is on a CATEGORY of
-CHAIN COMPLEXES over a finite auxiliary algebra Z. The paper
-builds Z explicitly, defines an action of A on these complexes,
-and then translates:
-
- "a is in A_X" becomes "P_{f, L} is preserved by a, for
- all labels L"
-
-for a distinguished projective P_{f, L} built inside Z-complexes.
-
-Rough plan:
- STEP 1 (Section 8): Build Z and an action of A by invertible
- functors on the homotopy category of graded
- Z-complexes.
-
- STEP 2 (Section 9): Exhibit "test" objects P_{f, L} that
- satisfy: for every x in A, the set
- {a in A : B_a P_{f, L}(x) is homotopy
- equivalent to P_{f, L}(x) for all L}
- equals the stabilizer H(S, x), and
- H(S, 1) = A_X.
-
- STEP 3 (Section 10-13): Prove the STABILIZER THEOREM: for any
- subset Y and any x in A, the stabilizer
- H(Y, x) is itself a conjugate of a
- standard parabolic of A_Y.
-
- STEP 4 (Section 14): Use the case Y = S to conclude the main
- theorem.
-
================================================================
8. THE AUXILIARY ALGEBRA Z
================================================================
@@ -825,292 +717,5 @@ PROOF: all three parts reduce to "intersections of parabolics are
parabolic" plus the Moller-Paris-Varghese (2024) transfer results. QED
================================================================
-18. WORKED EXAMPLES
-================================================================
-
-We check the theorem by hand in five small cases. The whole point is
-to see the theorem in action where the calculation is short enough
-to verify line by line.
-
-------------------------------------------------------------------
-EXAMPLE 1: Z^2 (abelian, "everything commutes")
-
- S = {a, b}, m(a, b) = 2. A = Z^2. Every parabolic is standard
- (conjugation does nothing in an abelian group): {0}, Z a, Z b,
- Z^2.
-
- Intersections:
- Z a ∩ Z b = {0} = Z^0 OK
- Z a ∩ Z^2 = Z a = Z^a OK
- Z^2 ∩ Z^2 = Z^2 OK
- {0} ∩ anything = {0} OK
-
- Verdict: trivially true; holds for any abelian group.
-
-------------------------------------------------------------------
-EXAMPLE 2: F_2 (free group, right-angled)
-
- S = {a, b}, m(a, b) = infinity. A = F_2, pars are {1}, <a>, <b>,
- F_2, and conjugates. Conjugates of <a>: the groups <waw^{-1}> for
- w in F_2 (all infinite cyclic).
-
- By the classical theory of free groups, an intersection of two
- infinite cyclic subgroups of a free group is cyclic (possibly
- trivial). Everything is parabolic.
-
- Verdict: trivially true by hand.
-
-------------------------------------------------------------------
-EXAMPLE 3: Braid_3 = <a, b | aba = bab>
-
- S = {a, b}, m(a, b) = 3. A = Braid_3 (braid group on 3 strands).
- Standard parabolics: {1}, <a>, <b>, Braid_3.
-
- Take P = <a> and Q = <b a b^{-1}>. Both are infinite cyclic.
-
- Question: is P ∩ Q trivial?
-
- If x ∈ P ∩ Q is nontrivial, then a^k = x = (bab^{-1})^j for some
- nonzero integers k, j. So a and bab^{-1} are both nontrivial
- powers of the same element x, hence they COMMUTE.
-
- But: take the natural quotient Braid_3 → S_3 sending a → (12) and
- b → (23). Then bab^{-1} → (23)(12)(23) = (13). In S_3,
- (12) · (13) = (132), (13) · (12) = (123),
- which are different. So (12) and (13) do NOT commute in S_3, so
- a and bab^{-1} do NOT commute in Braid_3. Contradiction.
-
- Hence P ∩ Q = {1} = A_empty. Parabolic.
-
- Verdict: hand-checkable, correct.
-
- (Note: by the Cumplido-Gebhardt-Gonzalez-Meneses-Wiest 2019
- theorem, ALL intersections of parabolics in spherical Braid_3 are
- parabolic. Our example is a special case.)
-
-------------------------------------------------------------------
-EXAMPLE 4: right-angled Artin on the path a - b - c
-
- S = {a, b, c}, m(a, b) = m(b, c) = 2, m(a, c) = infinity.
- Presentation: A = <a, b, c | ab = ba, bc = cb>. That is: a and b
- commute, b and c commute, a and c do not interact.
-
- Structure: A = Z^2 *_<b> Z^2, an amalgamated free product of the
- two Z^2 factors <a, b> and <b, c>, merged along their common
- subgroup <b> = Z.
-
- This is nonspherical: the associated Coxeter group is
- W = (Z/2 × Z/2) *_<b> (Z/2 × Z/2),
- which is infinite.
-
- Take P = <a, b> = Z^2 and Q = <b, c> = Z^2. By the amalgam
- structure, an element of both P and Q must lie in the amalgam
- <b>:
-
- P ∩ Q = <b> = A_b.
-
- Parabolic (rank 1). GOOD
-
- Now a conjugate version. Note that b commutes with c, so
- c b c^{-1} = b in this group. Take
-
- P = <a, b>, Q = c <a, b> c^{-1} = <c a c^{-1}, b> = <u, b>,
-
- where u = c a c^{-1} is a conjugate of a.
-
- Both P and Q contain <b>. Suppose a^k b^l = u^m b^n with k, m
- both nonzero. Then a^k b^{l-n} = u^m = c a^m c^{-1}. The
- left side is in the first amalgam factor Z^2 = <a, b>; the right
- side is (c) · a^m · (c^{-1}), a normal-form expression that
- begins and ends with elements of the second factor outside the
- amalgam <b>. In reduced normal form, such an expression cannot
- equal an element of the first factor (unless m = 0, reducing it
- to 1). So m = 0, whence a^k b^{l-n} = 1 in Z^2, so k = 0 and
- l = n. Therefore P ∩ Q = <b>.
-
- Verdict: hand-checkable, correct.
-
-------------------------------------------------------------------
-EXAMPLE 5: a nonspherical case with braid relations
-
- S = {a, b, c}, m(a, b) = m(b, c) = 3, m(a, c) = infinity.
- Diagram: a - b - c path, edges labelled 3, 3; no relation between
- a and c.
-
- Associated Coxeter group:
- W = <a, b, c | a^2 = b^2 = c^2 = 1, (ab)^3 = 1, (bc)^3 = 1>.
- The subgroups <a, b> and <b, c> are each S_3 (order 6); they
- share <b> = Z/2. So
- W = S_3 *_<b> S_3,
- an infinite amalgamated free product. Nonspherical.
-
- Standard parabolics:
- {1}, <a> = Z, <b> = Z, <c> = Z,
- <a, b> = Braid_3 (spherical),
- <b, c> = Braid_3 (spherical),
- <a, c> = F_2 (free, since m(a, c) = infinity),
- <a, b, c> = A (nonspherical).
-
- Take P = <a, b> and Q = <b, c>. Both are Braid_3-type. They
- share <b>. Is the intersection bigger than <b>?
-
- By van der Lek's 1983 theorem (for standard parabolics defined
- by subdiagrams), A_{X} ∩ A_{Y} = A_{X ∩ Y} for any X, Y ⊆ S.
- Here X = {a, b}, Y = {b, c}, X ∩ Y = {b}. So:
-
- P ∩ Q = A_{a,b} ∩ A_{b,c} = A_b = <b>.
-
- Parabolic (rank 1). GOOD
-
- This is the simplest honest case where the ambient group A is
- nonspherical, both intersected parabolics are spherical, and the
- intersection is a proper subset that still sits as a standard
- parabolic. The main theorem says this works for ANY Artin group,
- not just ones where such a hand check is possible.
-
-================================================================
-19. WHAT IS AND IS NOT CLAIMED
-================================================================
-
-WHAT IS CLAIMED:
-
- - The intersection of two parabolics in any finite-rank Artin
- group is parabolic. No assumption of sphericity. No finiteness
- of labels.
-
- - Arbitrary intersections of parabolics are parabolic, and at
- most |S| members are ever needed.
-
- - The word problem and the "membership in u A_X u^{-1}" problem
- are decidable for every finite-rank Artin group.
-
- - The Parabolic Intersection Conjecture implies (combined with
- other work) acylindrical hyperbolicity of non-clique Artin
- groups, and reduces "complete parabolic" membership results
- (FA', lcH-slender) to the complete parabolics themselves.
-
-WHAT IS NOT CLAIMED:
-
- - No complexity bound is supplied for the algorithm of Corollary B.
-
- - The construction of Z is nontrivial in practice (it involves
- fusion-labelled Temperley-Lieb-Jones categories, Jones-Wenzl
- idempotents, and exact linear algebra over a specific number
- field). The paper shows it is COMPUTABLE in principle, not
- that it is EFFICIENT.
-
- - This document does not give the full inner proofs of the
- categorical and homological lemmas (Sections 8, 9, 11, 13 of
- the research paper). The shape of those arguments is described
- but not reproduced.
-
-================================================================
-20. GLOSSARY
-================================================================
-
-ARTIN GROUP (A_M): the group generated by s in S with braid
- relations of length m(s,t). Section 3.
-
-CANDIDATE: a typed coset satisfying the two-point prescription.
- Section 12.
-
-CAP (at level q on D): no generator above level q at any colour
- of D. Section 11.
-
-COCHAIN / INTERNAL DEGREE: the two indices on a generator of a
- graded Z-complex. Section 9.
-
-CONJUGATE: g P g^{-1} for some g in A. Section 4.
-
-COXETER GROUP (W_M): quotient of the Artin group by s^2 = 1.
- Section 3.
-
-ELIGIBLE COLOUR: a colour whose actual level-q count is strictly
- below the harmonic interpolation, or equal with a "+1" sign mask.
- Section 12.
-HEIGHT: the lexicographic comparison of level blocks at a vertex
- of P(Y, x). Section 12.
-
-H(Y, x) (STABILIZER): { a in A_Y : a preserves E_L(x) for all
- labels L }. Section 10.
-
-INVISIBLE: colours whose harmonic interpolation and that of their
- exterior neighbours all vanish. Section 12.
-
-LAYER (Y^d(E)): the generators at level d, with their differential.
- Section 9.
-
-LEVEL (d = c + u): cochain degree plus internal degree. Section 9.
-
-m(s, t): Coxeter matrix entry. Section 2.
-
-PARABOLIC CLOSURE (PC(E)): smallest parabolic containing E.
- Section 17.
-
-PARABOLIC SUBGROUP: a conjugate of a standard A_X. Section 4.
-
-POSITIVE WORD: a word in the generators (no inverses). Section 1.
-
-SPHERICAL (Coxeter matrix): its Coxeter group is finite. Section 2.
-
-STANDARD PARABOLIC (A_X): subgroup generated by X ⊆ S. Section 4.
-
-STABILIZER THEOREM: H(Y, x) is a conjugate of a standard parabolic
- of A_Y. Section 14.
-
-TEST (e_a E): right projective at a ⊗ E; its cohomology detects
- the layer's top and socle. Section 9.
-
-TRUNCATION TRIANGLE: 0 -> t_{<=k}E -> E -> t_{>=k+1}E -> 0.
- Section 9.
-
-VISIBLE (T_+): complement of the invisible set in the type T.
- Section 12.
-
-WEIGHTED FUSION: the identity Sum_L w_L [D ⊗ L : H] = w_H dim D.
- Section 8.
-
-Z: the auxiliary finite-dimensional graded algebra. Section 8.
-
-================================================================
-21. FURTHER READING
-================================================================
-
-Coxeter and Artin groups: Bourbaki, "Lie Groups and Lie Algebras,
-Chapters 4-6" (2002). Davis, "The Geometry and Topology of Coxeter
-Groups" (2008).
-
-Standard parabolics and their intersections: van der Lek's thesis
-(1983). Paris, "Parabolic subgroups of Artin groups", J. Algebra
-157 (1997). Charney-Paris, "Convexity of parabolic subgroups in
-Artin groups", Bull. LMS 46 (2014).
-
-Prior cases of the Parabolic Intersection Conjecture:
- Cumplido-Gebhardt-Gonzalez-Meneses-Wiest, Ann. of Math. 190 (2019)
- - spherical.
- Duncan-Kazachkov-Remeslennikov, J. Algebra 316 (2007) - right-angled.
- Morris-Wright, J. Pure Appl. Algebra 225 (2021) - FC, one side.
- Moller-Paris-Varghese, Israel J. Math (2024) - one side spherical;
- and generalization to arbitrary intersections assuming pairwise.
- Antolin-Foniqi, Proc. Edinburgh Math. Soc. 65 (2022) - even FC.
- Cumplido-Martin-Vaskou, Math. Proc. Cambridge (2023) - large type.
-
-Categorical action: Khovanov-Seidel, JAMS 15 (2002). Huerfano-
-Khovanov, J. Algebra 246 (2001). Heng-Licata, "Fusion-stable
-modules over a twisted algebra" (2024).
-
-Acylindrical hyperbolicity and non-clique behaviour: Charney-Martin-
-Morris-Wright (2026). Kato-Oguni (2025).
-
-lcH-slender and complete parabolics: Moller-Paris-Varghese (2024).
-
-The full research paper: "Parabolic Intersections in Artin Groups,"
-source files included in this repository (01_intro.tex, 02_category.tex,
-03_layers.tex, 04_caps_twopoint.tex, 05_null.tex, 06_heights.tex,
-07_spherical.tex, 08_stabilizers.tex). That paper gives every lemma
-and proof in full, at the level of graduate algebra and topology.
-
-================================================================
-END
================================================================