Even Artin Groups: Structure and Rigidity

• Even Artin groups are Artin groups whose every finite defining label is even, characterized by relations of the form (st)^n=(ts)^n.
• They exhibit distinctive Lie-theoretic invariants and explicit homology computations, including notable torsion phenomena and nontrivial cohomology generators.
• Their rich subgroup structure features canonical retractions, poly-freeness, and strong parabolic closure properties that underpin strong rigidity results.

Even Artin groups are Artin groups whose every finite defining label is even. Equivalently, if the Coxeter matrix has finite off-diagonal entries $m_{s,t}=2n$, then all defining relations are of the form $(st)^n=(ts)^n$. Right-angled Artin groups arise as the special case in which every finite label is $2$. Across recent work, even Artin groups appear as a class where low-degree Lie-theoretic invariants, homology, subgroup structure, and several $p$-local constructions remain strongly graph-controlled, while also exhibiting phenomena absent in the right-angled case, such as torsion in degree-$2$ Lie data and degree-$2$ generators in cohomology (Paris et al., 2019, Ferrer et al., 29 Jun 2026).

1. Definitions and combinatorial encoding

Let $S$ be a finite set and $M=(m_{s,t})_{s,t\in S}$ a Coxeter matrix with $m_{s,s}=1$, $m_{s,t}=m_{t,s}\ge 2$ for $(st)^n=(ts)^n$0, and possibly $(st)^n=(ts)^n$1. The associated Artin group is

$(st)^n=(ts)^n$2

where $(st)^n=(ts)^n$3 is the alternating word of length $(st)^n=(ts)^n$4. In the even case, every finite off-diagonal entry is even, $(st)^n=(ts)^n$5, and the relation becomes

$(st)^n=(ts)^n$6

Thus an even Artin group is exactly an Artin group all of whose defining relations are of that form (Paris et al., 2019).

The Coxeter graph encodes the presentation: vertices correspond to generators, an edge labeled $(st)^n=(ts)^n$7 is recorded when $(st)^n=(ts)^n$8, the label $(st)^n=(ts)^n$9 means commutation, and the label $2$0 means no relation. This graph-theoretic encoding interacts especially well with product decompositions. If $2$1 is the $2$2-join of two labeled graphs, then

$2$3

so direct products are built by adjoining only commuting edges between the factors (Blasco-Garcia et al., 2018).

A structural feature specific to even Artin groups is the existence of canonical retractions onto standard parabolic subgroups. For $2$4, the subgroup $2$5 generated by $2$6 is again an Artin group on the induced subgraph, and the map sending generators outside $2$7 to $2$8 defines a retraction $2$9. This retraction property recurs in the subgroup, homological, and rigidity theories of the class (Blasco-Garcia et al., 2017).

2. FC-type, parabolics, and reduction mechanisms

A particularly tractable subclass is formed by even Artin groups of FC-type. In the even setting, FC-type admits a simple graph criterion: every triangle in the defining graph must have at least two edges labeled $p$0. This class contains right-angled Artin groups but is strictly larger, and it supports a strong parabolic theory (Antolín et al., 2023).

For even FC-type groups, standard parabolic subgroups are stable under intersection in the strongest possible sense: the intersection of two parabolic subgroups is again parabolic, and in fact arbitrary intersections of parabolic subgroups are parabolic as well (Antolín et al., 2022). This permits the definition of parabolic closure as an actual minimal parabolic subgroup containing a given subset, rather than merely an abstract intersection.

The same FC-type setting also supports a root-closure theorem for parabolics. If $p$1 is an even FC-type Artin group, $p$2, and $p$3, then

$p$4

Equivalently, parabolic subgroups are closed under taking roots: if $p$5 for a parabolic subgroup $p$6, then $p$7. This statement is a central ingredient in later Bass–Serre arguments on subgroup structure (Antolín et al., 2023).

Cone points provide a different reduction mechanism. If $p$8 is the set of cone points of the defining graph, then

$p$9

Hence the center problem for a cone-defined Artin group reduces to the Artin subgroup generated by the cone points. In particular, if the graph has exactly one cone point $2$0, then the Center Conjecture holds, and

$2$1

These statements apply without parity restrictions and therefore specialize directly to even Artin groups (Jankiewicz et al., 2024).

3. Rigidity and the isomorphism problem

One of the most detailed rigidity results for even Artin groups uses the truncated lower-central-series Lie algebra

$2$2

a $2$3-step nilpotent graded Lie algebra obtained by truncating the graded Lie algebra of the lower central series above degree $2$4. For an even Artin group $2$5, this invariant admits an explicit presentation: degree $2$6 is free on the vertices, while degree $2$7 is generated by classes $2$8, with torsion of order $2$9 when the edge label is $2$0, and free generators for $2$1-labels. The basic relation is that a defining relation $2$2 yields

$2$3

in degree $2$4 (Paris et al., 2019).

This invariant solves the isomorphism problem for a broad arithmetic family. If $2$5, $2$6, $2$7, and the Artin exponents belong to

$2$8

then isomorphism of the truncated Lie algebras forces isomorphism of the Coxeter matrices. Equivalently, within this family, the even Artin group is determined by its presentation up to permutation of generators. On the other hand, the invariant is not complete for all even Artin groups: the paper exhibits non-isomorphic even Artin groups with isomorphic truncated Lie algebras (Paris et al., 2019).

A distinct $2$9-local rigidity statement appears for FC-type groups. For every prime $S$0, one can form the $S$1-part $S$2 of the defining graph by collapsing odd edges and retaining only the $S$3-power part of each even label. If two even Artin groups of FC type are isomorphic, then for every prime $S$4, their $S$5-parts are isomorphic: $S$6 This does not solve the full isomorphism problem, but it imposes a strong family of necessary conditions on any isomorphism (Ferrer et al., 29 Jun 2026).

These results also delimit common misconceptions. Low-degree invariants can be decisive on carefully controlled families, but neither the truncated Lie algebra nor the full family of $S$7-parts is a complete invariant for all even Artin groups (Paris et al., 2019, Ferrer et al., 29 Jun 2026).

4. Homology, cohomology, and $S$8-local structure

The second integral homology of an even Artin group admits a direct graph-theoretic description. If $S$9 is an even Coxeter matrix and $M=(m_{s,t})_{s,t\in S}$0 is the set of finite edges $M=(m_{s,t})_{s,t\in S}$1, then

$M=(m_{s,t})_{s,t\in S}$2

with basis represented by

$M=(m_{s,t})_{s,t\in S}$3

where $M=(m_{s,t})_{s,t\in S}$4. Thus every finite edge contributes one $M=(m_{s,t})_{s,t\in S}$5-summand, and the rank of $M=(m_{s,t})_{s,t\in S}$6 is exactly the number of finite-labeled edges. The associated cup products satisfy

$M=(m_{s,t})_{s,t\in S}$7

and every class in $M=(m_{s,t})_{s,t\in S}$8 is a linear combination of Pontryagin products (Akita, 6 Jul 2025).

The full cohomology ring also has an explicit presentation. For an even Artin group $M=(m_{s,t})_{s,t\in S}$9 and a coefficient ring $m_{s,s}=1$0, $m_{s,s}=1$1 is generated by degree-$m_{s,s}=1$2 classes $m_{s,s}=1$3 for vertices and degree-$m_{s,s}=1$4 classes $m_{s,s}=1$5 for edges of label $m_{s,s}=1$6, with relations

$m_{s,s}=1$7

when $m_{s,s}=1$8 has label $m_{s,s}=1$9, together with vanishing products whenever the corresponding spherical-subgraph conditions fail. In contrast with the RAAG case, degree $m_{s,t}=m_{t,s}\ge 2$0 may contribute genuinely new generators; the cohomology ring need not be generated by $m_{s,t}=m_{t,s}\ge 2$1 (Ferrer et al., 29 Jun 2026).

The same paper identifies the pro-$m_{s,t}=m_{t,s}\ge 2$2 completion of an Artin group purely from $m_{s,t}=m_{t,s}\ge 2$3, and in the FC case gives a presentation of the $m_{s,t}=m_{t,s}\ge 2$4-Zassenhaus restricted Lie algebra. For a pro-$m_{s,t}=m_{t,s}\ge 2$5 FC Artin group, the restricted Lie algebra is generated by the vertices with relations

$m_{s,t}=m_{t,s}\ge 2$6

for an edge of label $m_{s,t}=m_{t,s}\ge 2$7 with $m_{s,t}=m_{t,s}\ge 2$8. The same framework yields residual $m_{s,t}=m_{t,s}\ge 2$9, $(st)^n=(ts)^n$00-Magnus, and cohomological $(st)^n=(ts)^n$01-completeness statements for $(st)^n=(ts)^n$02 (Ferrer et al., 29 Jun 2026).

Artin kernels provide a further homological direction. For even Artin groups of FC-type, the homology of cocyclic subgroups $(st)^n=(ts)^n$03 is computed explicitly over fields of characteristic $(st)^n=(ts)^n$04, with partial results in prime characteristic (Blasco-García et al., 2021).

5. Subgroup structure, poly-freeness, and acylindrical hyperbolicity

Even Artin groups of FC-type admit strong recursive decompositions. One fundamental theorem states that every even Artin group of FC type is poly-free, and hence residually finite as well (Blasco-Garcia et al., 2017). A complementary result proves that every large even Artin group is poly-free and that every even Artin group based on a triangle graph is also poly-free (Blasco-Garcia, 2020). Both arguments use the even-case retractions onto parabolic subgroups and decompositions with free kernel.

For finitely generated even Artin groups of FC-type, the subgroup theory is markedly rigid. There exists a finite-index normal subgroup $(st)^n=(ts)^n$05 such that for every subgroup $(st)^n=(ts)^n$06, either $(st)^n=(ts)^n$07 lies in a group of the form

$(st)^n=(ts)^n$08

with $(st)^n=(ts)^n$09 free abelian, or $(st)^n=(ts)^n$10 maps onto a non-abelian free group. Here $(st)^n=(ts)^n$11 is the Klein bottle group. This is stronger than the classical Tits alternative, and the finite-index passage is necessary: already the even dihedral Artin group

$(st)^n=(ts)^n$12

contains a Klein bottle subgroup $(st)^n=(ts)^n$13 (Antolín et al., 2023).

The same class also supports a subgroup dichotomy in the sense of acylindrical hyperbolicity. For a non-virtually-cyclic subgroup $(st)^n=(ts)^n$14 of an even Artin group of FC-type, either $(st)^n=(ts)^n$15 is contained in a product $(st)^n=(ts)^n$16 of proper Artin subgroups, or $(st)^n=(ts)^n$17 is acylindrically hyperbolic. Equivalently, either $(st)^n=(ts)^n$18 contains two nontrivial normal subgroups with trivial intersection, or it is acylindrically hyperbolic (Zearra et al., 2024).

A broader algebraic statement applies to all Artin groups: the derived subgroup is free if and only if the Artin group is coherent. In particular, coherent even Artin groups have free commutator subgroup. In the even setting this interacts well with parabolic subgroups, since for a standard parabolic $(st)^n=(ts)^n$19,

$(st)^n=(ts)^n$20

holds in every even Artin group (Zearra et al., 2024).

6. Algorithmic, geometric, algebro-geometric, and logical aspects

Even Artin groups satisfy a uniform isoperimetric bound coming from a more general theorem for Artin groups with no edge labeled $(st)^n=(ts)^n$21. For such groups, $(st)^n=(ts)^n$22 is an isoperimetric function, and therefore the word problem is solvable. Since even Artin groups have only even finite labels, they automatically satisfy the hypothesis $(st)^n=(ts)^n$23, so every even Artin group has solvable word problem. The same work states that $(st)^n=(ts)^n$24 is a proved upper bound and explicitly suggests that a quadratic bound may be true, but does not prove it (Juhasz, 22 Jul 2025).

Quasi-projectivity is extremely restrictive in even type. An even Artin group $(st)^n=(ts)^n$25 is quasi-projective if and only if its defining graph is a $(st)^n=(ts)^n$26-join of finitely many copies of the discrete graphs $(st)^n=(ts)^n$27, the two-vertex graphs $(st)^n=(ts)^n$28, and the triangle $(st)^n=(ts)^n$29. Moreover, every quasi-projective even Artin group is realized by a quasi-projective $(st)^n=(ts)^n$30, in fact by a plane curve complement (Blasco-Garcia et al., 2018). This sharply separates the exceptional quasi-projective cases from the generic even Artin group.

Plane-curve constructions provide concrete examples. The complement of a parabola together with two parallel tangent lines has fundamental group

$(st)^n=(ts)^n$31

identified in the paper as the Euclidean Artin group $(st)^n=(ts)^n$32, hence an even Artin group. By contrast, the main hypocycloid family in that work yields polygonal Artin groups with adjacent label $(st)^n=(ts)^n$33, so it lies outside the even class (Bartolo et al., 2017).

In first-order terms, non-abelian Artin groups of even FC type are not superstable. More generally, the paper reduces non-superstability to an $(st)^n=(ts)^n$34-purity property for dihedral parabolic subgroups, a condition especially natural for rank-two even Artin subgroups $(st)^n=(ts)^n$35 (Cassella et al., 29 Jul 2025).

Even Artin groups therefore occupy a distinctive position in Artin-group theory. They retain enough combinatorial structure to support explicit homological and $(st)^n=(ts)^n$36-local calculations, strong subgroup theorems, and several rigidity results, while also displaying genuine departures from the right-angled model: quasi-projectivity is rare, low-degree invariants are not universally complete, and the geometry of subgroup actions is mediated by noncommuting even dihedral pieces rather than by pure commutation alone.