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PARABOLIC INTERSECTIONS IN ARTIN GROUPS
A gentle introduction
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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.
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1. WHAT IS A GROUP
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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.
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2. COXETER MATRICES
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A COXETER MATRIX on a finite set S is a symmetric table m(s,t)
of "braid lengths":
m(s, s) = 1
m(s, t) = m(t, s)
m(s, t) in {2, 3, 4, 5, ...} or the symbol "infinity"
Intuition: m(s,t) tells you how strongly s and t interact.
m = 2: s and t commute (weakest interaction)
m = 3: s t s = t s t (the "braid" relation)
m = 4: s t s t = t s t s (dihedral relation of order 4)
m = infinity: no relation at all between s and t
Draw a picture: one node per element of S, one edge labelled m(s,t)
between s and t whenever m(s,t) >= 3. Nodes with no edge commute.
Nodes joined by an edge labelled "infinity" do not interact.
Examples:
- S = {a, b}, m(a, b) = 2: two commuting nodes.
- S = {a, b}, m(a, b) = 3: a "braided" pair.
- S = {a, b}, m(a, b) = infinity: a "free" pair.
- S = {a, b, c}, m(a, b) = m(b, c) = 3, all other pairs 2:
the "A_3 diagram" (three nodes in a line with labelled edges;
a and c commute because m(a, c) = 2).
A Coxeter matrix is SPHERICAL if its associated Coxeter group
(defined in Section 3) is finite. Spherical diagrams have been
classified since the 1930's: types A_n, B_n, D_n, E_6, E_7, E_8,
F_4, G_2, H_3, H_4, I_2(m), and disjoint unions.
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3. ARTIN GROUPS AND COXETER GROUPS
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Fix a Coxeter matrix m on finite set S. The ARTIN GROUP A = A_M
is the group with generators s (for s in S) and one relation per
finite edge:
for m(s, t) < infinity:
sts... (m letters) = tst... (m letters)
The COXETER GROUP W = W_M is what you get by ALSO imposing s^2 = 1
for every s.
Examples:
All m(s, t) = 2 (zero edges):
A_M = Z^n (free abelian group on n generators).
W_M = (Z/2)^n.
All m(s, t) = 3, on n generators in a line (the "A_n" diagram):
A_M = Braid_{n+1} (braid group on n+1 strands).
W_M = S_{n+1} (symmetric group on n+1 letters).
All m(s, t) = infinity (no relations beyond nothing):
A_M = F_n (free group on n generators).
W_M = (Z/2) * (Z/2) * ... (free product of n copies of Z/2).
S = {a}, single generator:
A_M = Z, W_M = Z/2.
S = {a, b}, m(a, b) = 3:
A_M = Braid_3 = <a, b | aba = bab>.
W_M = S_3 (order 6).
S = {a, b}, m(a, b) = 4:
A_M = <a, b | abab = baba> (the "I_2(4)" Artin group).
W_M = the dihedral group of order 8 (symmetry of a square).
S = {a, b, c}, m(a, b) = m(b, c) = 2, m(a, c) = infinity:
A_M = <a, b, c | ab = ba, bc = cb> (RIGHT-ANGLED Artin group
on the path a - b - c).
W_M = (Z/2 x Z/2) *_<b> (Z/2 x Z/2), infinite.
A_M is nonspherical.
Braid group on 3 strands, Braid_3 = <a, b | aba = bab>, is the
smallest interesting spherical example. It is infinite, so an
Artin group need not be finite even when it is "spherical type".
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4. PARABOLIC SUBGROUPS
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Fix a Coxeter matrix m on S, Artin group A = A_M.
For any subset X of S, let A_X be the subgroup generated by
{s : s in X}. (It is itself an Artin group, on the restricted
matrix.) Call A_X a STANDARD PARABOLIC subgroup of rank |X|.
A PARABOLIC subgroup is any conjugate of a standard parabolic:
g A_X g^{-1}, where X is a subset of S and g is in A.
Two notable things:
- Different conjugates of the same A_X can be different
subgroups. For instance in F_2 = <a, b>, <a> and <b a b^{-1}>
are two different infinite cyclic subgroups, both conjugates
of <a>. Their intersection is {1}.
- The word "parabolic" comes from the geometry of hyperplane
arrangements. Coxeter groups act on Euclidean spaces by
reflections; the "parabolic" subgroups are stabilizers of
points where certain coordinate directions have collapsed.
The name is inherited by the group-theoretic definition.
Examples:
A = Z^2: standard parabolics are {0}, Z*a, Z*b, Z^2.
Since A is abelian, every parabolic is standard.
A = F_2: standard parabolics are {1}, <a>, <b>, F_2.
Conjugates of <a> are the infinite cyclic subgroups
<w a w^{-1}> for w in F_2.
A = Braid_3: parabolics are {1}, and conjugates of <a>, <b>,
and Braid_3 itself.
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5. THE PARABOLIC INTERSECTION CONJECTURE
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Informally: intersection of any two parabolics is again parabolic.
Precisely, the theorem of this document:
THEOREM (Main). For every Coxeter matrix M on a finite set S,
every subsets X, Y of S, and every elements g, h in A, there are
a subset Z of S and an element k in A such that
g A_X g^{-1} INTERSECT h A_Y h^{-1} = k A_Z k^{-1}.
In words: intersect two conjugates of standard parabolics; the
result is another conjugate of a standard parabolic.
That is the whole theorem. Two features make it non-trivial:
(1) The ambient Artin group A is allowed to be nonspherical.
Many classical tools (Garside structure, normal forms via
the "positive monoid", parabolic closures computed from
Garside theory) assume sphericity or at least "FC type"
(every pair of generators with finite label forms a
spherical pair). The theorem does not.
(2) The Coxeter matrix M can have infinite labels. Then
positive monoids fail to admit normal forms in the
classical way; the "positive fractional" machinery breaks.
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6. WHAT WAS KNOWN BEFORE 2026
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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.
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7. THE STRATEGY: REPRESENT A ON SOMETHING
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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.
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8. THE AUXILIARY ALGEBRA Z
================================================================
We need a finite-dimensional graded algebra Z. (Graded means: Z
splits as a direct sum of pieces Z^0, Z^1, Z^2, ...; multiplying
an element of Z^p by an element of Z^q gives an element of
Z^{p+q}. For us Z is finite-dimensional over a field, and
everything lives in a bounded range of degrees.)
Construction idea (approximate, hand-wavey):
- Attach a special vertex f to the diagram.
- The vertex f interacts with each generator s of S. The
interaction is encoded by a "fusion rule" with a small tensor
category.
- Z = the path algebra of a quiver whose vertices are "colour
and label" positions and whose arrows implement the labelling.
The important consequence is a certain TWIST MECHANISM: for each
generator s of A, there is a bimodule chain complex B_s such
that:
(i) B_s is invertible up to chain homotopy equivalence.
(ii) B_s B_t B_s = B_t B_s B_t (chain homotopy) whenever
m(s, t) = 3.
(iii) Similarly for m(s, t) = 4, 5, ..., infinity (the "no
relation" case means the twists B_s and B_t commute up to
homotopy, which is exactly the right-angled paradigm
m(s, t) = 2 generalised).
So we get an honest action of A on the homotopy category of
bounded graded Z-complexes. For a word x in A, B_x is the
corresponding functor.
The essential trick: the action of B_s on a complex is by POSITIVE
operators: differentials never decrease a certain degree called
"level" (Section 9), and B_s increases the level by at most 1.
The B_s^{-1} decreases by at most 1.
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9. LAYERS, TESTS, AND HEIGHT
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A generator of a complex has two degrees: COCHAIN degree (call it
c) and INTERNAL degree (call it u). The LEVEL is d = c + u.
This bookkeeping accounts for all shifts.
For a complex E, the LOW subcomplex t_{<=k} E has all generators
with level <= k. The HIGH quotient t_{>=k+1} E has all generators
with level >= k+1. The short exact sequence
0 -> t_{<=k} E -> E -> t_{>=k+1} E -> 0
is called the TRUNCATION TRIANGLE at level k.
The PURE LAYER at level d, written Y^d(E), is the subquotient
t_{<=d} / t_{<=d-1}: the generators at level exactly d, with their
differentials.
Because every B_s has amplitude [0, 1] (bounded level change), the
layers of B_s E are controlled by the layers of E. In particular,
for a pure input at level d, B_s E has just two layers: a "bottom"
at d and an "upper" at d+1 (upper supported only on colour s).
The KEY INSTRUMENT: for a vertex a (corresponding to a generator s
and a label L), the RIGHT TEST of E at a is:
e_a E = (right projective at a) TENSOR E
It is a complex of plain vector spaces. Its cohomology splits by
ordinary cochain degree. Two specific degrees reveal the layer:
H^d(e_a E) = socle of Y^d(E) at a
H^{d+2}(e_a E) = (top of Y^d(E) at a) <2> (internal shift 2)
So the "degree-d" and "degree-(d+2)" tests pick out the socle and
top of the pure layer. In particular a layer is nonzero at colour
s iff one of these special cohomology classes is nonzero.
HEIGHT. For a word x in A, the FAMILY at x is:
E_L(x) = B_x P_{f, L} for each label L.
The (relative) HEIGHT of a word a in A_Y relative to x is the
maximum total degree (cochain + internal) at which ANY test of ANY
member of E(x) is nonzero. Finite whenever nonempty. The height
is:
- nondecreasing under positive words in A_Y (Section 13);
- strictly increasing under any positive word that contains
every colour of Y (Lemma 7.cochain-drop of the paper).
The height being bounded along a sequence of positive words forces
strong consequences. That is the central engine of the proof.
-- Section 10 continues in the next chunk of this file --
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10. REDUCTION TO PROPER SPHERICAL PARABOLICS
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Fix Y (a subset of S, possibly nonspherical) and x in A. The
STABILIZER of the family at x inside A_Y is
H(Y, x) = { a in A_Y : the family E_L(x) is preserved by B_a
for every label L }.
Equivalently: a in A_Y lies in H(Y, x) iff every member of the
family at x is homotopy-equivalent to itself after B_a.
For Y = S and x = 1 we will have H(S, 1) = A_X (Section 16). And
the algebra X of the theorem becomes the Z we have defined.
So the main theorem has become a statement about the stabilizer
subgroups H(Y, x). Specifically:
STABILIZER THEOREM. For every Y and x, there exist a subset
R of Y and an element c of A_Y with
H(Y, x) = c^{-1} A_R c.
That is the content of Section 14. The proof splits into two
phenomena:
PHENOMENON 1 (spherical Y): a positive-word argument shows that
the stabilizer is either everything or is contained in a proper
spherical parabolic of smaller rank.
PHENOMENON 2 (nonspherical Y): a height argument shows that
the stabilizer is contained in a conjugate of a proper parabolic
whose intersection with A_Y has smaller rank.
Both phenomena reduce the rank. Induction on |Y| then finishes.
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11. CAPS AND HARMONIC INTERPOLATION
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Fix Y, x, and the family E_L(x). For a colour subset D and an
integer q, the family is CAPPED at q on D if no member has any
generator of any colour in D at level greater than q. A cap is
inclusive: level exactly q is allowed.
Lemma 11.1 (Cap accessibility). If a cap holds on a colour set
D that includes a visible colour, then it can be moved inward:
there is another representative of the same coset with a cap on
a strictly smaller set D'.
The cap is a mechanism for reducing freedom. If we can cap on a
set D that already saturates what we need, we're done: the level
counts are controlled.
Lemma 11.2 (Harmonic interpolation). For a spherical subset T
of S, the linear map "value at exterior vertices" -> "interior
entries" has a unique extension to the interior such that
(C p)_T = 0, where C is the Cartan matrix of the spherical type.
This extension is called the HARMONIC INTERPOLATION of the
exterior data on T.
The interpolation is nonnegative on nonnegative exterior data,
by a positivity property of the inverse Cartan matrix C^{-1}.
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12. HARMONIC HEIGHT AND CONNECTED MINIMA
================================================================
Fix Y (any, no sphericity), x in A. The POSET OF SPHERICAL COSETS
P(Y, x) has vertices:
(T, A_T z), where T subset of Y is spherical, z in A_Y x.
So a vertex is a right coset of a SPHERICAL parabolic, inside the
coset A_Y x. Two vertices are adjacent when their cosets are
nested. The undirected comparability graph is connected.
Each vertex v = (T, A_T z) and level k carries:
- m^k(z) : generator counts on A_T z (independent of the
representative);
- p^k(v) : harmonic interpolation of m^k(z) on T;
- eta^k(v): sign mask (+-1 or 0) depending on visibility.
The HEIGHT of v is a lexicographic comparison of a "block"
combining: total sum p^k(v), the vector (p_i^k(v))_i, and the
sign coordinates eta_i^k(v) restricted to T. Blocks are ordered
from the highest level k down.
Key facts proved by structure of the interpolation:
Lemma 12.1 (Well-order). The heights are well-ordered, and the
least block has total sum zero.
Lemma 12.2 (Descending upper). Ascending neighbours either
don't exist or belong to a spherical extension. Specifically:
a NULL VISIBLE component (a set on which the harmonic vector is
a positive null vector with all signs 0) cannot be a positive
support component; this is exactly the NULL-LAYER OBSTRUCTION of
Section 13. Hence all "active" colours form a spherical set, and
descending upper neighbours are connected.
Lemma 12.3 (Descending lower). A descending lower neighbour is
obtained by opening an ELIGIBLE colour (one whose actual count is
strictly below the harmonic prediction, or equal with a +1 sign
mask). Opening all eligible colours is a well-defined descending
move; the resulting lower neighbours are connected.
Proposition 12.4 (Connected minima). In P(Y, x), the vertices of
minimum height form a nonempty connected subgraph. Their sets K
(invisible colours) and T_+ (visible colours) are constant along
this graph.
PROPOSITION 12.5 (Containment from a minimum). Fix a minimum
vertex (T, A_T z). Let K and T_+ be as above, and D = T_+ u K.
Then T_+ and K are non-adjacent, T_+ is spherical, and
x^{-1} H(Y, x) x is contained in z^{-1} A_D z.
So we have placed H(Y, x) inside a conjugate of A_D. If D is a
proper subset of Y, we can apply the conjugation lemma and induct.
The case D = Y is the hard case. That is exactly when Y itself
is a positive null component: the harmonic vector on Y is a
positive null vector and every representative strictly exceeds it.
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13. THE NULL-LAYER OBSTRUCTION
================================================================
This is the central technical lemma. It rules out the "bad case"
D = Y of Proposition 12.5 unconditionally, using only:
(a) a NULL VECTOR hypothesis: c_I is a positive null vector of
the Cartan matrix C_I on a support I (so C_I c_I = 0 and
c_I > 0);
(b) a CONNECTED SUPPORT hypothesis: the minimal complex of any
single family member has connected color support containing
the f-frame generator.
PROPOSITION 13.1 (Relative null-layer obstruction).
Fix a representative z of a spherical coset. Suppose
- every color of I is "run" in the sense that its level-q
generator count is strictly above c_I;
- every boundary color of I is capped at level q;
- some member of the family has a level-q top at a color of I;
and the additional exterior hypothesis:
- the exterior generators of the family are fixed and satisfy
no "strict" case at level q.
Then we get a contradiction.
Proof outline (three steps):
STEP 1. By the cap projection of Section 11, reduce to a
representative that is capped on T (a set containing I).
STEP 2. The "excess" vector m^q(z) - c_I is a strictly positive
convex combination of generator dimensions of an actual
subobject M of the level-q layer (the "capped-layer hull").
STEP 3. Compute the TENSOR of the highest right divisor with
the left family. Using the weighted Euler characteristic
(a formula balancing two linear maps), we get an exact numeric
identity:
D_0 v^T C y = h_0 + h_0' - h_1
with h_j nonnegative dimensions from the linear Hom complex.
The left side is zero, so each h_j is zero. That vanishing
forces a boundary color to be occupied at level below q, but
the cap rules this out. Contradiction.
The consequence: in the harmonic height filtration, no visible
colored component can be a positive null component. Every
"visible" color is surrounded by "invisible" neighbours on the
side that matters. This is what turns the case D = Y into a
contradiction (i.e., into a proper subset D < Y).
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14. STABILIZER THEOREM
================================================================
With Sections 10-13 in hand, we can now prove the STABILIZER
THEOREM by induction on |Y|. Let us state the proof in stages.
THEOREM 14.1 (Relative stabilizers are parabolic). For every
subset Y of S and every x in A there exist R subset of Y and c in
A_Y with
H(Y, x) = c^{-1} A_R c.
LEMMA 14.2 (Conjugating a stabilizer). Let D subset of Y and
b in A_Y. If H(Y, x) is contained in b^{-1} A_D b then
b H(Y, x) b^{-1} = H(D, b x).
Proof: For c in A_D, invertibility of B_b gives the equivalence
"E_L(c b x) is homotopy-equiv to E_L(b x)" iff "E_L(b^{-1} c b x)
is homotopy-equiv to E_L(x)". Right side says precisely that
b^{-1} c b is in H(Y, x). So H(D, b x) = A_D ∩ b H(Y, x) b^{-1},
which by hypothesis is b H(Y, x) b^{-1}.
PROOF OF 14.1. Induction on |Y|.
CASE Y = empty: trivial (H is the trivial group, R = empty,
c = 1).
CASE |Y| >= 1, two subcases:
SUBCASE A: Y is spherical. If all tests on the operated family
at x are acyclic (i.e., every test cohomology vanishes), then
every generator of A_Y fixes every family member, so H(Y, x) =
A_Y. Take R = Y, c = 1.
If not all tests are acyclic, Proposition 15.3 (Section 15)
gives a containment H(Y, x) inside b^{-1} A_D b for some b in
A_Y and some PROPER D subset of Y with D SPHERICAL. Apply
Lemma 14.2 to get H(Y, x) = b^{-1} H(D, b x) b. Then induction
on D applies, giving R subset D (thus R subset Y) and c in A_D
(thus c in A_Y) with H(D, bx) = c^{-1} A_R c. Hence
H(Y, x) = (bc)^{-1} A_R (bc). Done.
SUBCASE B: Y is nonspherical. By Proposition 12.5 with
a minimum vertex, we have H(Y, x) contained in z^{-1} A_D z for
some z in A_Y and some D = T_+ u K, where T_+ is spherical and
K is the invisible set. If D is PROPER in Y, we're done by the
same conjugation-and-induct argument. If D = Y, we are in the
bad case; but Proposition 13.1 rules out D = Y being a positive
null component (spherical). So D is proper. Done.
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15. SPHERICAL POSITIVE HEIGHT
================================================================
We need to fill the gap in SUBCASE A: for spherical Y and any x,
if not all tests are acyclic, H(Y, x) is contained in a conjugate
of a proper spherical parabolic.
This is the "positive height" argument. Suppose for a
contradiction that H(Y, x) is NOT contained in any proper
parabolic. Then there is an infinite sequence a_1, a_2, ... of
positive words in A_Y such that h(a_1) <= h(a_2) <= ... where
h(a_k) is the height of any test at a_k, and all colours of Y
appear in each a_k.
But the height h is bounded above by a constant (if the operated
test cohomology is nonzero) and by -infinity (if acyclic). The
"bounded below" of h under positive words is thus the real issue.
Lemma 15.1 (Path bound, Ocneanu/Jones-Wenzl). For spherical Y,
there is an integer N = N(Y) >= 1 such that in a minimal bounded
complex whose generators on a cochain interval all sit in Y, any
composition of N successive arrow coefficient maps vanishes.
This is a nilpotence statement in the Jones-Wenzl algebra, a
finite-dimensional operator algebra. The proof uses the positive
definiteness of the Cartan matrix C_Y.
Lemma 15.2 (Cochain drop). Suppose all tests on Y in a bounded
family have ordinary cochain cohomology in degrees at most m.
Positive operations in Y preserve this bound. After a positive
word that contains every colour of Y, the bound improves to m-1.
Iterating 15.2 forces cochain degrees to drop without bound. The
path bound 15.1 converts "cochain degree dropped below every
exterior generator" into "empty cochain placement", which is
zeroth-dimensional cohomology of a minimal complex -> the complex
DECOMPOSES as a direct sum. But every family member E_L(x) is
indecomposable. Contradiction.
Hence:
PROPOSITION 15.3 (Spherical containment). For spherical Y and
any x in A: either all tests on Y at x are acyclic and H(Y, x) =
A_Y, or H(Y, x) is contained in p^{-1} A_J p for some positive
word p and some PROPER subset J of Y.
That closes SUBCASE A of the stabilizer theorem.
-- Sections 16-19 in the next chunk --
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16. PROOF OF THE MAIN THEOREM
================================================================
We are now ready to prove the main theorem. Recall:
THEOREM. For finite Coxeter matrix M on S, subsets X, Y of S,
and elements g, h of A, there are Z ⊆ S and k in A with
g A_X g^{-1} ∩ h A_Y h^{-1} = k A_Z k^{-1}.
Additional input needed: the "initial case".
PROPOSITION 16.1 (Initial stabilizer). Fix X ⊆ S, and let f be
the special coloring vertex of the construction. Using the family
{E_L(1)}_L produced by the algebra Z, we have
H(S, 1) = A_X.
PROOF. Direction (⊇). Every generator s of X commutes with the
auxiliary vertex f, so B_s preserves each P_{f, L}; therefore every
s in X lies in A_X ⊆ H(S, 1).
Direction (⊆). Consider the vertex v_0 = (empty, {1}) in P(S, 1).
Its family has only one nonzero layer, the level-0 unit vector at f.
This is a MINIMUM-HEIGHT vertex (any competitor with a nonzero higher
block is strictly later, and at level 0 the coordinate sum is
minimal). Apply Proposition 12.5 with this minimum. Its visible
set is T_+ = empty. Its invisible set K is X (colours not adjacent
to f). So D = X and z = 1, giving
H(S, 1) ⊆ A_X. QED
PROOF OF THE MAIN THEOREM.
STEP 1. Set x = h^{-1} g in A (so g = h x^{-1}). Then
g A_X g^{-1} ∩ h A_Y h^{-1}
= h ( x A_X x^{-1} ∩ A_Y ) h^{-1}.
(Both sides are equal after multiplying the middle expression by
h on left and h^{-1} on right.)
STEP 2. CLAIM: A_Y ∩ x A_X x^{-1} = H(Y, x).
Proof of claim. For a in A_Y:
a ∈ x A_X x^{-1}
iff x^{-1} a x ∈ A_X
iff (by Prop 16.1) B_{x^{-1} a x} preserves E_L(1) for every L
iff B_a preserves E_L(x) = B_x P_{f, L} for every L
iff a ∈ H(Y, x).
So the intersection of x A_X x^{-1} with A_Y is exactly H(Y, x).
STEP 3. By the stabilizer theorem (14.1), H(Y, x) = c^{-1} A_Z c
for some Z ⊆ Y and c in A_Y. Combining with Step 2:
x A_X x^{-1} ∩ A_Y = c^{-1} A_Z c.
STEP 4. Plug into Step 1:
g A_X g^{-1} ∩ h A_Y h^{-1}
= h (c^{-1} A_Z c) h^{-1}
= (h c^{-1}) A_Z (h c^{-1})^{-1}
= k A_Z k^{-1} with k = h c^{-1}.
This is what the theorem claims. QED.
================================================================
17. CONSEQUENCES
================================================================
COROLLARY A (arbitrary intersections, parabolic closure).
For every family F of parabolics of A, there is a subfamily
F_0 ⊆ F with |F_0| ≤ |S| and
(intersection over all P in F) = (intersection over all P in F_0).
In particular, arbitrary intersections of parabolics are themselves
parabolics. And every subset E ⊆ A lies in a unique smallest
parabolic subgroup PC(E), called the PARABOLIC CLOSURE of E.
Under inclusion, the parabolics form a complete lattice (meet =
intersection, join = parabolic closure of union).
PROOF. Start with Q = A_S (the full group), with an empty list of
selected parabolics. While Q is not contained in some P ∈ F,
replace Q by Q ∩ P and add P to the list.
- By the main theorem, Q ∩ P is a parabolic. Its defining subset
is a proper subset of Q's current one, since Q was not contained
in P.
- Proper here means: rank strictly decreases.
- Rank decreases by at least 1 each step, and rank ≤ |S|.
- So the loop terminates in at most |S| steps.
The resulting Q is contained in every member of F, and equals the
intersection of the selected (≤ |S|) parabolics. Hence intersection
over F = intersection over F_0. The lattice structure follows at
once. QED
COROLLARY B (word and parabolic membership problems).
There is an algorithm which, given (M, X, u, v) where M is a
finite Coxeter matrix, X ⊆ S, and u, v are words in the generators
and their inverses, decides whether the element represented by v
lies in u A_X u^{-1}.
In particular, the WORD PROBLEM is decidable for every finitely
generated Artin group.
PROOF SKETCH. Let w = u^{-1} v u. Build the algebra Z of Section 8
explicitly over the number field K = Q(exp(2 pi i / N)), where N
is the least common multiple of 2m over all finite m ≥ 4 (over Q if
no such m). Then:
- Deciding "w ∈ A_X" ⇔ "B_w P_{f, L_0} is homotopy-equivalent to
P_{f, L_0} for one fixed label L_0" (Prop 16.1 plus the fact
that the family is preserved in a compatible way).
- Compute B_w P_{f, L_0} explicitly as a bounded complex of graded
projective Z-modules. This uses finite tensor products and
Gaussian elimination over the finite-dimensional algebra Z.
- Cancel invertible scalar differential entries (standard
"minimal model" reduction). Terminates because it removes two
summands per step.
- The result is minimal and hence unique up to isomorphism. Test
whether it has exactly one generator: P_{f, L_0} in placements
(0, 0). Yes: w ∈ A_X. No: w ∉ A_X.
For X = empty, this solves the word problem: w = 1 iff w ∈ A_empty
iff P_{f, 1} is preserved. QED
No complexity bound is asserted. The algorithm is finite-time but
its running time can be huge; that is not the point.
COROLLARY C (non-clique dynamics).
Suppose M is irreducible (not a disjoint union of two Coxeter
matrices with all cross labels = 2), and m(s, t) = infinity for
some s ≠ t. Then:
(i) A is acylindrically hyperbolic.
(ii) Every proper parabolic subgroup P of A admits a conjugate
g P g^{-1} with P ∩ g P g^{-1} = {1}.
Part (i) is due to a separate 2025-2026 line of work (Kato-Oguni
/ Charney-Martin-Morris-Wright), not the main theorem. Part (ii)
uses the main theorem: the intersection P ∩ g P g^{-1} is a
parabolic (main theorem), it is finite (weak malnormality of proper
parabolics, from the same second source), and any nontrivial
parabolic contains a conjugate of a generator s, which has infinite
order (exponent-sum homomorphism A → Z sends s to 1). So a finite
parabolic must be {1}.
COROLLARY D (complete parabolics, FA', lcH-slender).
A parabolic is COMPLETE if it is a conjugate of A_T for a subset T
with no INFINITE label (labels 2, 3, 4, ... allowed, sphericity NOT
required). Then:
(i) Every subgroup of A with property FA' (fixed-point property
for actions on trees without inversions) lies in a complete
parabolic.
(ii) Every abstract homomorphism from an almost-connected locally
compact Hausdorff group into A has image in a complete
parabolic.
(iii) If every complete standard parabolic of A is lcH-slender,
then A is lcH-slender.
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>?
Suppose x ∈ <a, b> ∩ <b, c> is not in <b>. Then x is a
nonempty reduced word in {a, b, a^{-1}, b^{-1}} (in the first
subgroup) that is also a reduced word in {b, c, b^{-1}, c^{-1}}
(in the second). In the free product structure underlying A, an
element is uniquely determined by its normal form. An element
in the subgroup generated by a, b has normal form using only
a-appearances and b-appearances; an element in the subgroup
generated by b, c has normal form using only b-appearances and
c-appearances. An element in BOTH must use only b-appearances
(else it would have an a or c appearance, contradicting
membership in the other subgroup; the subgroup <a, b> really
contains elements with an "a" in the normal form, and similarly
for c). So x ∈ <b>. Therefore:
P ∩ Q = <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
================================================================