POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS
OF RIGHT-ANGLED ARTIN AND COXETER GROUPS
TARAS PANOV AND YAKOV VERYOVKIN

arXiv:1603.06902v2 [] 1 Sep 2016

To the memory of Rainer Vogt
Abstract. We construct and study polyhedral product models for classifying
spaces of right-angled Artin and Coxeter groups, general graph product groups
and their commutator subgroups. By way of application, we give a criterion of
freeness for the commutator subgroup of a graph product group, and provide
an explicit minimal set of generators for the commutator subgroup of a rightangled Coxeter group.

1. Introduction
Right-angled Artin and Coxeter groups are familiar objects in geometric group
theory [Da2]. From the abstract categorical viewpoint, they are particular cases of
graph product groups, corresponding to a sequence of m groups G = (G1 , . . . , Gm )
and a graph Γ on m vertices. Informally, the graph product group G Γ consists of
words with letters from G1 , . . . , Gm in which the elements of Gi and Gj with i 6= j
commute whenever {i, j} is an edge of Γ. The graph product group G Γ interpolates
between the free product G1 ⋆ · · · ⋆ Gm (corresponding to a graph consisting of
m disjoint vertices) and the cartesian product G1 × · · · × Gm (corresponding to a
complete graph). Right-angled Artin and Coxeter groups RAΓ and RC Γ correspond
to the cases Gi = Z and Gi = Z2 , respectively.
The polyhedral product is a functorial combinatorial-topological construction
assigning a topological space (X , A)K to a sequence of m pairs of topological
spaces (X , A) = {(X1 , A1 ), . . . , (Xm , Am )} and a simplicial complex K on m vertices [BP1, BBCG, BP2]. It generalises the notion of a moment-angle complex
ZK = (D2 , S 1 )K , which is a key object of study in toric topology. Polyhedral products also provide a unifying framework for several constructions of classifying spaces
for right-angled Artin and Coxeter groups, their commutator subgroups, as well as
general graph products groups. The description of the classifying spaces of graph
product groups and their commutator subgroups was implicit in [PRV], where the
canonical homotopy fibration
m
Y
(EG, G)K −→ (BG)K −→
BGk .
k=1

of polyhedral products was introduced and studied.
To each graph Γ without loops and double edges one can assign a flag simplicial
complex K, whose simplices are the vertex sets of complete subgraphs (or cliques)
of Γ. For any flag complex K the polyhedral product (BG)K is the classifying
2010 Mathematics Subject Classification. 20F65, 20F12, 57M07.
Key words and phrases. Right-angled Artin group, right-angled Coxeter group, graph product,
commutator subgroup, polyhedral product.
The research of the first author was carried out at the IITP RAS and supported by the Russian
Science Foundation grant no. 14-50-00150. The research of the second author was supported by
the Russian Foundation for Basic Research grant no. 14-01-00537.
1

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TARAS PANOV AND YAKOV VERYOVKIN

space for the corresponding graph product group G K = G Γ , while (EG, G)K is
the classifying space for the commutator subgroup of G K . In the case of right-angled
Artin group RAΓ = RAK , each BGi = BZ is a circle, so we obtain as (BG)K the
subcomplex (S 1 )K in an m-torus introduced by Kim and Roush in [KR]. In the case
of right-angled Coxeter group RC K , each BGi = BZ2 is an infinite real projective
space RP ∞ , so the classifying space for RC K is a similarly defined subcomplex
(RP ∞ )K in the m-fold product of RP ∞ . The classifying space for the commutator
subgroup of RC K is a finite cubic subcomplex RK in an m-dimensional cube, while
the classifying space for the commutator subgroup of RAK is an infinite cubic
subcomplex LK in the m-dimensional cubic lattice. All these facts are summarised
in Theorem 3.2 and Corollaries 3.3 and 3.4.
The emphasis of [PRV] was on properties of graph products of topological (rather
than discrete) groups, as part of the homotopy-theoretical study of toric spaces and
their loop spaces. In the present work we concentrate on the study of the commutator subgroups for discrete graph product groups. Apart from a purely algebraic
interest, our motivation lies in the fact that the commutator subgroups of graph
products are the fundamental groups of very interesting aspherical spaces. From this
topological perspective, right-angled Coxeter groups RC K are the most interesting.
The commutator subgroup RC ′K is π1 (RK ) for a finite-dimensional aspherical complex RK , which turns out to be a manifold when K is a simplicial subdivision of
sphere. When K is a cycle (the boundary of a polygon) or a triangulated 2-sphere,
one obtains as RC ′K a surface group or a 3-manifold group respectively. These
groups have attracted much attention recently in geometric group theory and lowdimensional topology. The manifolds RK corresponding to (the dual complexes of)
higher-dimensional permutahedra and graph-associahedra also feature as the universal realisators in the works of Gaifullin [Ga1], [Ga2] on the problem of realisation
of homology classes by manifolds.
In Theorem 4.3 we give a simple criterion for the commutator subgroup of a graph
product group to be free. In the case of right-angled Artin groups this result was
obtained by Servatius, Droms and Servatius in [SDS]. In Theorem 4.5 we provide
an explicit minimal generator set for the finitely generated commutator subgroup
of a right-angled Coxeter group RC K . This generator set consists of nested iterated
commutators of the canonical generators of RC K which appear in a special order
determined by the combinatorics of K.
Theorems 4.3 and Theorem 4.5 parallel the corresponding results obtained
in [GPTW] for the loop homology algebras and rational homotopy Lie algebras
of moment-angle complexes. Algebraically, these results of [GPTW] can be interpreted as a description of the commutator subalgebra in a special graph product
graded Lie algebra (see Theorem 4.6). The results of Section 4 in the current paper
constitute a group-theoretic analogue of the results of [GPTW] for graded associative and Lie algebras.
We dedicate this article to the memory of Rainer Vogt, who shared his great
knowledge and ideas with T. P. during insightful collaboration in the 2000s.
The authors are grateful to Alexander Gaifullin for his invaluable comments and
suggestions, which much helped in making the text more accessible.
2. Preliminaries
We consider a finite ordered set [m] = {1, 2, . . . , m} and its subsets I =
{i1 , . . . , ik } ⊂ [m], where I can be empty of the whole of [m].
Let K be an (abstract) simplicial complex on [m], i. e. K is a collection of subsets
of [m] such that for any I ∈ K all subsets of I also belong to K. We always assume
that the empty set ∅ and all one-element subsets {i} ⊂ [m] belong to K. We

POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS

3

refer to I ∈ K as a simplex (or a face) of K. One-element faces are vertices, and
two-element faces are edges. Every abstract simplicial complex K has a geometric
realisation |K|, which is a polyhedron in a Euclidean space (a union of convex
geometric simplices). In all subsequent constructions it will be useful to keep in
mind the geometric object |K| alongside with the abstract collection K.
We recall the construction of the polyhedral product (see [BP1, BBCG, BP2]).
Construction 2.1 (polyhedral product). Let K be a simplicial complex on [m]
and let
(X , A) = {(X1 , A1 ), . . . , (Xm , Am )}
be a sequence of m pairs of pointed topological spaces, pt ∈ Ai ⊂ Xi , where pt
denotes the basepoint. For each subset I ⊂ [m] we set
(2.1)

m
Y

(X , A)I = (x1 , . . . , xm ) ∈
Xk : xk ∈ Ak

for k ∈
/I

k=1

and define the polyhedral product of (X , A) corresponding to K as
[
[ Y
Y 
(X , A)K =
(X , A)I =
Ai .
Xi ×
I∈K

I∈K i∈I

i∈I
/

In the case when all pairs (Xi , Ai ) are the same, i. e. Xi = X and Ai = A for
i = 1, . . . , m, we use the notation (X, A)K for (X , A)K . Also, if each Ai = pt , then
we use the abbreviated notation X K for (X , pt )K , and X K for (X, pt )K .
This construction of the polyhedral product has the following categorical interpretation. Consider the face category cat(K), whose objects are simplices I ∈ K
and morphisms are inclusions I ⊂ J. Let top denote the category of topological spaces. Define a cat(K)-diagram (a covariant functor from the small category
cat(K) to the “large” category top)
DK (X , A) : cat(K) −→ top,

(2.2)

I 7−→ (X , A)I ,

which maps the morphism I ⊂ J of cat(K) to the inclusion of spaces (X , A)I ⊂
(X , A)J . Then we have
(2.3)

(X , A)K = colim DK (X , A) = colim(X , A)I .
I∈K

Here colim denotes the colimit functor (also known as the direct limit functor )
from the category of cat(K)-diagrams of topological spaces to the category top.
By definition, colim is the left adjoint to the constant diagram functor. The details
of these constructions can be found, e. g., in [BP2, Appendix C].
Given a subset J ⊂ [m], consider the restriction of K to J:
KJ = {I ∈ K : I ⊂ J},
which is also known as a full subcomplex of K. Recall that a subspace Y ⊂ X is
called a retract of X if there exists a continuous map r : X → Y such that the
r
composition Y ֒→ X −→ Y is the identity. We record the following simple property
of the polyhedral product.
Proposition 2.2. (X, A)KJ is a retract of (X, A)K whenever KJ ⊂ K is a full
subcomplex.
Proof. We have

[ Y
Y
Ai ,
(X , A)K =
Xi ×
I∈K i∈I

i∈[m]\I

(X , A)KJ =

[

Y

I∈K, I⊂J i∈I

Xi ×

Y

i∈J\I


Ai .

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TARAS PANOV AND YAKOV VERYOVKIN

Since each Ai is a pointed space, there is a canonical inclusion (X , A)KJ ֒→
(X , A)K . Furthermore, for each I ∈ K there is a projection
Y
Y
Y
Y
Ai .
Ai −→
Xi ×
rI :
Xi ×
i∈I

i∈I∩J

i∈[m]\I

i∈J\I

Since KJ is a full subcomplex, the image of S
rI belongs to (X , A)KJ . The projections
rI patch together to give a retraction r = I∈K rI : (X , A)K → (X , A)KJ .

The following examples of polyhedral products feature throughout the paper.
Example 2.3.
1. Let (X, A) = (S 1 , pt ), where S 1 is a circle. The corresponding polyhedral
product (S 1 )K is a subcomplex in the m-torus (S 1 )m :
[
(2.4)
(S 1 )K =
(S 1 )I ⊂ (S 1 )m .
I∈K

In particular, when K = {∅, {1}, . . . , {m}} (which is m disjoint points geometrically), the polyhedral product (S 1 )K is the wedge (S 1 )∨m of m circles.
When K consists of all proper subsets of [m] (which geometrically corresponds
to the boundary ∂∆m−1 of an (m − 1)-dimensional simplex), (S 1 )K is known as the
fat wedge of m circles; it is obtained by removing the top-dimensional cell from the
standard cell decomposition of an m-torus (S 1 )m .
For a general K on m vertices, (S 1 )K sits between the m-fold wedge (S 1 )∨m and
the m-fold cartesian product (S 1 )m .
2. Let (X, A) = (R, Z), where Z is the set of integer points on a real line R. We
denote the corresponding polyhedral product by LK :
[
(2.5)
LK = (R, Z)K =
(R, Z)I ⊂ Rm .
I∈K

When K consists of m disjoint points, LK is a grid in m-dimensional space Rm
consisting of all lines parallel to one of the coordinate axis and passing though
integer points. When K = ∂∆m−1 , the complex LK is the union of all integer
hyperplanes parallel to coordinate hyperplanes.
3. Let (X, A) = (RP ∞ , pt ), where RP ∞ is an infinite-dimensional real projective
space, which is also the classifying space BZ2 for the 2-element cyclic group Z2 .
Consider the polyhedral product
[
(2.6)
(RP ∞ )K =
(RP ∞ )I ⊂ (RP ∞ )m .
I∈K

Similarly to the first example above, (RP ∞ )K sits between the m-fold wedge
(RP ∞ )∨m (corresponding to K consisting of m points) and the m-fold cartesian
product (RP ∞ )m (corresponding to K = ∆m−1 ).
4. Let (X, A) = (D1 , S 0 ), where D1 is a closed interval (a convenient model is the
segment [−1, 1]) and S 0 is its boundary, consisting of two points. The polyhedral
product (D1 , S 0 )K is known as the real moment-angle complex [BP1, §3.5], [BP2]
and is denoted by RK :
[
(2.7)
RK = (D1 , S 0 )K =
(D1 , S 0 )I .
I∈K

It is a cubic subcomplex in the m-cube (D1 )m = [−1, 1]m . When K consists of
m disjoint points, RK is the 1-dimensional skeleton of the cube [−1, 1]m . When
K = ∂∆m−1 , RK is the boundary of the cube [−1, 1]m . In general, if {i1 , . . . , ik }
is a face of K, then RK contains 2m−k cubic faces of dimension k which lie in the
k-dimensional planes parallel to the {i1 , . . . , ik }th coordinate plane.

POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS

5

The space RK was introduced and studied in the works of Davis [Da1] and
Davis–Januszkiewicz [DJ], although their construction was different. When |K| is
homeomorphic to a sphere, RK is a topological manifold (this follows from the
results of [Da1], see also [BP2] and [Ca, Theorem 2.3]). Furthermore, the manifold
RK has a smooth structure when |K| is the boundary of a convex polytope. In this
case RK is the universal abelian cover of the dual simple polytope P [DJ, §4.1].
The four polyhedral products above are related by the two homotopy fibrations [PRV], [BP2, §4.3]
(2.8)

LK −→ (S 1 )K −→ (S 1 )m ,

(2.9)

RK −→ (RP ∞ )K −→ (RP ∞ )m .

Construction 2.4 (right-angled Artin and Coxeter group). Let Γ be a graph on m
vertices. We write {i, j} ∈ Γ when {i, j} is an edge. Denote by F (g1 , . . . , gm ) a free
group with m generators corresponding to the vertices of Γ. The right-angled Artin
group RAΓ corresponding to Γ is defined by generators and relations as follows:

(2.10)
RAΓ = F (g1 , . . . , gm ) (gi gj = gj gi for {i, j} ∈ Γ).
When Γ is a complete graph we have RAΓ = Zm , while when Γ has no edges we
obtain the free group.
The right-angled Coxeter group RC Γ is defined as

(2.11)
RC Γ = F (g1 , . . . , gm ) (gi2 = 1, gi gj = gj gi for {i, j} ∈ Γ).

Both right-angled Artin and Coxeter groups have a categorical interpretation
similar to that of polyhedral products (see (2.3)). Namely, consider the following
cat(K)-diagrams, this time in the category grp of groups:
DK (Z) : cat(K) −→ grp,

I 7−→ ZI ,

DK (Z2 ) : cat(K) −→ grp,
I 7−→ ZI2 ,
Q
Q
where ZI = i∈I Z and ZI2 = i∈I Z2 . A morphism I ⊂ J of cat(K) is mapped to
the monomorphism of groups ZI → ZJ and ZI2 → ZJ2 respectively. Then
(2.12)

I
RAK1 = colimgrp DK (Z) = colimgrp
I∈K Z ,
I
RC K1 = colimgrp DK (Z2 ) = colimgrp
I∈K Z2 ,

where K1 denotes the 1-skeleton of K, which is a graph. Here colimgrp denotes the
colimit functor in grp.
A missing face (or a minimal non-face) of K is a subset I ⊂ [m] such that I is
not a simplex of K, but every proper subset of I is a simplex of K. A simplicial
complex K is called a flag complex if each of its missing faces consists of two vertices.
Equivalently, K is flag if any set of vertices of K which are pairwise connected by
edges spans a simplex.
A clique (or a complete subgraph) of a graph Γ is a subset I of vertices such that
every two vertices in I are connected by an edge. Each flag complex K is the clique
complex of its one-skeleton Γ = K1 , that is, the simplicial complex formed by filling
in each clique of Γ by a face.
Note that the colimits in (2.12), being the corresponding right-angled groups,
depend only on the 1-skeleton of K and do not depend on missing faces with more
than 2 vertices. For example, the colimits of the diagrams of groups D∆2 (Z) and
D∂∆2 (Z) are both Z3 . This reflects the lack of “higher” commutativity in the category of groups: when generators gi commute pairwise, they commute altogether.
This phenomenon is studied in more detail in [PRV] and [PR].

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TARAS PANOV AND YAKOV VERYOVKIN

For these reasons we denote the right-angled Artin and Coxeter groups corresponding to the 1-skeleton of K simply by RAK and RC K respectively.
By analogy with the polyhedral product of spaces X K = colimI∈K X I , we may
consider the following more general construction of a discrete group.
Construction 2.5 (graph product). Let K be a simplicial complex on [m] and
let G = (G1 , . . . , Gm ) be a sequence of m groups, which we think of as discrete
topological groups. We also assume that none of Gi is trivial, i. e. Gi 6= {1}. For
each subset I ⊂ [m] we set
m
Y

G I = (g1 , . . . , gm ) ∈
Gk : gk = 1 for k ∈
/I .
k=1

Then consider the following cat(K)-diagram of groups:
DK (G) : cat(K) −→ grp,

I 7−→ G I ,

which maps a morphism I ⊂ J to the canonical monomorphism of groups G I →
G J . Define the group
I
G K = colimgrp DK (G) = colimgrp
I∈K G .

(2.13)

The group G K depends only on the graph K1 and is called the graph product of the
groups G1 , . . . , Gm . We have canonical homomorphisms G I → G K , I ∈ K, which
can be shown to be injective.
As in the case of right-angled Artin and Coxeter groups (corresponding to Gi = Z
and Gi = Z2 respectively), one readily deduces the following more explicit description from the universal property of the colimit:
Proposition 2.6. The is an isomorphism of groups
m

Gk (gi gj = gj gi for gi ∈ Gi , gj ∈ Gj , {i, j} ∈ K),
GK ∼
=

⋆

k=1

where

⋆

m
k=1

Gk denotes the free product of the groups Gk .

Remark. We use the symbol ⋆ to denote the free product of groups, instead of the
more common ∗; the latter is reserved for the join of topological spaces.
3. Classifying spaces
Here we collect the information about the classifying spaces for graph product
groups. The results of this section are not new, but as they are spread across the
literature we find it convenient to collect everything in one place. The corresponding
references are given below.
Recall that a path-connected space X is aspherical if πi (X) = 0 for i > 2. An
aspherical space X is an Eilenberg–Mac Lane space K(π, 1) with π = π1 (X).
Given a (discrete) group G, there is a universal G-covering EG → BG whose
total space EG is contractible and the base BG, known as the classifying space
for G, has the homotopy type K(G, 1) (i. e. π1 (BG) = G and πi (BG) = 0 for
i > 2). We shall therefore switch between the notation BG and K(G, 1) freely.
Note that BZ ≃ S 1 and BZ2 ≃ RP ∞ , with the universal coverings R → S 1 and
∞
S → RP ∞ respectively.
Now we use the notation from Construction 2.5. The classifying space BG I is
the product of BGi over i ∈ I. We therefore have the polyhedral product (BG)K
corresponding to the sequence of pairs (BG, pt ) = {(BG1 , pt ), . . . , (BGm , pt )}.
Similarly, we have the polyhedral product (EG, G)K corresponding to the sequence
of pairs (EG, G) = {(EG1 , G1 ), . . . , (EGm , Gm )}. Here each Gi is included in EGi
as the fibre of the covering EGi → BGi over the basepoint.

POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS

7

The homotopy fibrations (2.8) and (2.9) can be generalised as follows.
Proposition 3.1. The sequence of canonical maps
(EG, G)K −→ (BG)K −→

m
Y

BGk

k=1

is a homotopy fibration.
When each Gk is Z, we obtain the fibration (2.8), as the pair (EZ, Z) is homotopy
equivalent to (R, Z). Similarly, when each Gk is Z2 , we obtain (2.9), as the pair
(EZ2 , Z2 ) is homotopy equivalent to (D1 , S 0 ).
Qm
Proof of Proposition 3.1. We denote k=1 BGk by BG [m] ; this is compatible with
the notation BG I . According to [PRV, Proposition 5.1], the homotopy fibre of
the inclusion (BG)K → BG [m] can be identified with the homotopy colimit
hocolimI∈K G [m] /G I of the cat(K)-diagram in top given on the objects by I 7→
G [m] /G I (where the latter is the quotient group, viewed as a discrete space) and
sending a morphism I ⊂ J to the canonical projection G [m] /G I → G [m] /G J of the
quotients. This diagram is not Reedy cofibrant, e. g. because G [m] /G I → G [m] /G J
is not a cofibration of spaces. The latter map is homotopy equivalent to the closed
cofibration (EG, G)I → (EG, G)J , which is a morphism in the cat(K)-diagram
DK (EG, G), see (2.2). The diagram DK (EG, G) is Reedy cofibrant, see [BP2,
Proposition 8.1.1]. Therefore, the homotopy fibre of the inclusion (BG)K → BG [m]
is given by
hocolimI∈K G [m] /G I ≃ colimI∈K (EG, G)I = (EG, G)K .



Now we state the following group-theoretic consequence of the homotopy fibration in Proposition 3.1.
Theorem 3.2. Let K be a simplicial complex on m vertices, and let GK be a graph
product group given by (2.13).
(a) π1 ((BG)K ) ∼
= GK .
(b) Both spaces (BG)K and (EG, G)K are aspherical if and only if K is flag.
Hence, B(GK ) = (BG)K whenever K is flag.
(c) πi ((BG)K ) ∼
= πi ((EG, G)K ) for i > 2.
(d) Q
π1 ((EG, G)K ) is isomorphic to the kernel of the canonical projection GK →
m
k=1 Gk .
Proof. To prove (a) we proceed inductively by adding simplices to K one by one
and use van Kampen’s Theorem. The base of the induction is K consisting of m
disjoint points. Then (BG)K is the wedge BG1 ∨ · · · ∨ BGm , and π1 ((BG)K ) is the
free product G1 ⋆ · · · ⋆ Gm . This is precisely G K , so (a) holds. Assume now that K′
is obtained from K by adding a single 1-dimensional simplex {i, j}. Then, by the
definition of the polyhedral product,
′

(BG)K = (BG)K ∪ (BGi × BGj ),
where the two pieces are glued along BGi ∨ BGj . By van Kampen’s Theorem,
′
π1 ((BG)K ) is the amalgamated free product π1 ((BG)K ) ⋆(Gi ⋆Gj ) (Gi × Gj ). The
latter group is obtained from π1 ((BG)K ) by adding all relations of the form gi gj =
′
gj gi for gi ∈ Gi , gj ∈ Gj . By the inductive assumption, this is precisely G K .
Adding simplices of dimension > 2 to K does not change G K and results in adding
cells of dimension > 3 to (BG)K , which does not change the fundamental group
π1 ((BG)K ). The inductive step is therefore complete, proving (a).

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TARAS PANOV AND YAKOV VERYOVKIN

Now we prove (b). The canonical homomorphisms G I → G K give rise to the
maps of classifying spaces BG I → B(G K ). These define a morphism from the
cat(K)-diagram DK (BG, pt ) to the constant diagram B(G K ), and hence a map
colimI∈K BG I = (BG)K → B(G K ).

(3.1)

According to [PRV, Proposition 5.1], the homotopy fibre of the map (3.1) can be
identified with the homotopy colimit hocolimI∈K G K /G I of the cat(K)-diagram
in top given on the objects by I 7→ G K /G I (where the latter is the right coset,
viewed as a discrete space) and sending a morphism I ⊂ J to the canonical projection G K /G I → G K /G J of cosets. By [PRV, Corollary 5.4], the homotopy colimit
hocolimI∈K G K /G I is homeomorphic to the identification space

(3.2)
Bcat(K) × G K ∼ .
Here Bcat(K) is the classifying space of cat(K), which is homeomorphic to the
cone on |K|. The equivalence relation ∼ is defined as follows: (x, gh) ∼(x, g) whenever h ∈ G I and x ∈ B(I ↓ cat(K)), where I ↓ cat(K) is the undercategory, whose
objects are J ∈ K such that I ⊂ J, and B(I ↓ cat(K)) is homeomorphic to the star
of I in K. When K is a flag complex, the identification space (3.2) is contractible
by [PRV, Proposition 6.1]. Therefore, the map (3.1) is a homotopy equivalence,
which implies that (BG)K is aspherical when K is flag.
Assume now that K is not flag. Choose a missing face J = {j1 , . . . , jk } ⊂ [m] with
k > 3 vertices and consider the corresponding full subcomplex KJ . Then (BG)KJ is
the fat wedge of the spaces {BGj , j ∈ J} (see Example 2.3.1), and it is a retract of
(BG)K by Proposition 2.2. Hence, in order to see that (BG)K is not aspherical, it
is enough to check that (BG)KJ is not aspherical. Let FW (X1 , . . . , Xk ) denote the
fat wedge of spaces X1 , . . . , Xk . According to a result of Porter [Po], the homotopy
fibre of the inclusion
k
Y
Xi
FW (X1 , . . . , Xk ) ֒→
i=1

k−1

is Σ
ΩX1 ∧ · · · ∧ ΩXk , where Σ denotes the suspension and Ω the loop
space functor.
Q In our case we obtain that the homotopy fibre of the inclusion
(BG)KJ → j∈J BGj is Σ k−1 Gj1 ∧ · · · ∧ Gjk . Since each Gj is a discrete space, the
latter suspension is a wedge
Q of (k − 1)-dimensional spheres. It has nontrivial homotopy group πk−1 . Since j∈J BGj is a K(π, 1)-space, the homotopy exact sequence
implies that πk−1 ((BG)KJ ) 6= 0 for some k > 3. Hence, (BG)KJ and (BG)K are
non-aspherical.
Asphericity of (EG, G)K and statements (c) and (d)Q
follow from the homotopy
m
exact sequence of the fibration in Proposition 3.1, as πi ( k=1 BGk ) = 0, i > 2. 
Specialising to the cases Gk = Z and Gk = Z2 respectively we obtain the following results about right-angled Artin and Coxeter
groups. Note that in these two
Q
cases the groups Gk are abelian, so G K → m
G
k=1 k is the abelianisation homo′
morphism, and its kernel is the commutator subgroup (G K ) .
Corollary 3.3. Let K be a simplicial complex on m vertices, let (S 1 )K and LK be
the polyhedral products given by (2.4) and (2.5) respectively, and let RAK be the
corresponding right-angled Artin group.
(a) π1 ((S 1 )K ) ∼
= RAK .
(b) Both (S 1 )K and LK are aspherical if and only if K is flag.
(c) πi ((S 1 )K ) ∼
= πi (LK ) for i > 2.
(d) π1 (LK ) is isomorphic to the commutator subgroup RA′K .

POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS

9

Corollary 3.4. Let K be a simplicial complex on m vertices, let (RP ∞ )K and RK
be the polyhedral products given by (2.6) and (2.7) respectively, and let RC K be the
corresponding right-angled Coxeter group.
(a) π1 ((RP ∞ )K ) ∼
= RC K .
(b) Both (RP ∞ )K and RK are aspherical if and only if K is flag.
(c) πi ((RP ∞ )K ) ∼
= πi (RK ) for i > 2.
(d) π1 (RK ) is isomorphic to the commutator subgroup RC ′K .
Remark. All ingredients in the proof of Theorem 3.2 are contained in [PRV]. The
fact that the polyhedral product (BG)K is the classifying space for the graph
product group G K whenever K is a flag complex implies that the classifying space
functor converts the colimit of groups (defining the graph product) to the colimit
of topological spaces (defining the polyhedral product). This is not the case when
K is not flag because of the presence of higher Whitehead and Samelson products
(see [PRV, PR, GT]), but the situation can be remedied by replacing colimits with
homotopy colimits. All these facts were proved in [PRV] for arbitrary well-pointed
topological groups.
Statements (a) and (b) of Corollary 3.3, implying a homotopy equivalence
(S 1 )K ≃ K(RAK , 1) for flag K, were obtained by Kim and Roush [KR, Theorem 10]. Statements (a) and (b) of Corollary 3.4, implying a homotopy equivalence
(RP ∞ )K ≃ K(RC K , 1) for flag K, are implicit in the works of Davis [Da1] and
Davis–Januszkiewicz [DJ, p. 437]. In particular, contractibility of the space (3.2)
(which is the crucial step in the proof of Theorem 3.2 (b)) in the case of right-angled
Coxeter group RC K follows from [Da1, Theorem 13.5]. The isomorphism between
π1 (RK ) and the commutator subgroup RC ′K was also obtained in the work of
Droms [Dr] (his cubic complex is the 2-dimensional skeleton of our complex RK ,
and therefore has the same fundamental group).
In the case of a general graph product G K , the result that both spaces (BG)K
and (EG, G)K are aspherical if and only if K is flag appeared in the work of
Stafa [St, Theorem 1.1].
Example 3.5. Let K be an m-cycle (the boundary of an m-gon). A simple argument with Euler characteristic shows that RK is homeomorphic to a closed orientable surface of genus (m − 4)2m−3 + 1 (this observation goes back to a 1938 work
of Coxeter, see [BP2, Proposition 4.1.8]). Therefore, the commutator subgroup of
the corresponding right-angled Coxeter group RC K is a surface group. This example
was studied in [SDS] and [Dr].
Similarly, when |K| ∼
= S 2 (which is equivalent to K being the boundary of a
3-dimensional simplicial polytope), RK is a 3-dimensional manifold. Therefore, the
commutator subgroup of the corresponding RC K is a 3-manifold group. The fact
that 3-manifold groups appear as subgroups in right-angled Artin and Coxeter
groups has attracted much attention in the recent literature.
All homology groups are considered with integer coefficients. The homology of
RK is described by the following result. For the particular case of flag K it gives a
description of the homology of the commutator subgroup RC ′K .
Theorem 3.6 ([BP1], [BP2, §4.5]). For any k > 0, there is an isomorphism
M
e k−1 (KJ ),
H
Hk (RK ) ∼
=
J⊂[m]

e k−1 (KJ ) is the reduced simplicial homology group of KJ .
where H

The cohomology ring structure of H ∗ (RK ) is described in [Ca].

10

TARAS PANOV AND YAKOV VERYOVKIN

4. The structure of the commutator subgroups
By Theorem 3.2,
m


Y
Ker G K →
Gk = π1 ((EG, G)K ).
k=1

In the case of right-angled Artin or Coxeter groups (or, more generally, when each
′
Gk is abelian), the group above is the commutator subgroup (G K ) . We want to
study the group π1 ((EG, G)K ), identify the class of simplicial complexes K for
which this group is free, and describe a minimal generator set.
We shall need the following modification of a result of Grbić and Theriault [GT]:
Proposition 4.1. Let K = K1 ∪I K2 be a simplicial complex obtained by gluing
K1 and K2 along a common face I, which may be empty. If the polyhedral products (EG, G)K1 and (EG, G)K2 are homotopy equivalent to wedges of circles, then
(EG, G)K is also homotopy equivalent to a wedge of circles.
Proof. We may assume that K has the vertex set [m] = {1, . . . , m}, K1 is the full
subcomplex of K on the first m1 vertices {1, . . . , m1 }, K2 is the full subcomplex of
K on the last m2 vertices {m − m2 + 1, . . . , m}, and the common face I is on the
k vertices {m1 − k + 1, . . . , m1 }, where m1 < m, m2 < m and m = m1 + m2 − k.
Consider the polyhedral product (CX , X )K corresponding to a sequence of pairs
(CX , X ) = {(CX1 , X1 ), . . . , (CXm , Xm )}, where CXi denotes the cone on Xi .
According to [GT, Theorem 6.12],
(CX , X )K ≃ (M1 ∗ M2 ) ∨ ((CX , X )K1 ⋊ M2 ) ∨ (M1 ⋉ (CX , X )K2 ),
Q 1
Qm
where M1 = m
i=1 Xi , M2 =
i=m−m2 +1 Xi , M1 ∗ M2 denotes the join of M1 and
M2 , X ⋊ Y denotes the right half-smash X × Y /pt × Y of two pointed spaces X, Y ,
and X ⋉ Y denotes their left half-smash X × Y /X × pt .
In our case, each Xi = Gi is a discrete space, the pair (EGi , Gi ) is homotopy
equivalent to (CGi , Gi ), and each of M1 , M2 in (4.1) is a discrete space. Hence, each
of the three wedge summands in (4.1) is a wedge of circles, and so is (EG, G)K . 
(4.1)

A graph Γ is called chordal (or triangulated ) if each of its cycles with > 4 vertices
has a chord (an edge joining two vertices that are not adjacent in the cycle).
The following result gives an alternative characterisation of chordal graphs.
Theorem 4.2 (Fulkerson–Gross [FG]). A graph is chordal if and only if its vertices
can be ordered in such a way that, for each vertex i, the lesser neighbours of i form
a clique.
Such an ordering of vertices is called a perfect elimination ordering.
Theorem 4.3. Let K be a flag simplicial complex on m vertices, let G =
(G1 , . . . , Gm ) be a sequence of m nontrivial groups, and let GK be the graph product
group given by (2.13). The following conditions are equivalent:
Qm
(a) Ker(GK → k=1 Gk ) is a free group;
(b) (EG, G)K is homotopy equivalent to a wedge of circles;
(c) Γ = K1 is a chordal graph.
Proof. (b)⇒(a) This follows from Theorem 3.2 (d) and the fact that the fundamental group of a wedge of circles is free.
(c)⇒(b) Here we use the argument from [GPTW, Theorem 4.6]. However, that
argument contained an inaccuracy, which was pointed out by A. Gaifullin and
corrected in the argument below.

POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS

11

Assume that the vertices of K are in perfect elimination order. We assign to each
vertex i the clique Ii consisting of i and the lesser neighbours of i. Since K is a
flag complex,
Sm each clique Ii is a face. All maximal faces are among I1 , . . . , Im , so
we have i=1 Ii = K. Furthermore, for each k = 1, . . . , m the perfect elimination
ordering on K induces such an ordering on the full subcomplex K{1,...,k−1} , so we
Sk−1
Sk−1
have i=1 Ii = K{1,...,k−1} . In particular, the simplicial complex i=1 Ii is flag as
Sk−1
a full subcomplex in a flag complex. The intersection Ik ∩ i=1 Ii is a clique, so it
Sk−1
is a face of i=1 Ii . An inductive argument using Proposition 4.1 then shows that
(EG, G)K is a wedge of circles.
Qm
(a)⇒(c) Let Ker(G K → k=1 Gk ) be a free group. Suppose that the graph
Γ = K1 is not chordal, and choose a chordless cycle J with |J| > 4. Then the full
subcomplex KJ is the same cycle (the boundary of a |J|-gon).
We first consider the case when each Gk is Z2 , so that (EG, G)K is RK . Then
RKJ is homeomorphic to a closed orientable surface of genus (|J| − 4)2|J|−3 + 1
by [BP2, Proposition 4.1.8]. In particular, the fundamental group π1 (RKJ ) is not
free. On the other hand, RKJ is a retract of RK by Proposition 2.2, so π1 (RKJ ) is
a subgroup of the free group π1 (RK ) = Ker(RC K → (Z2 )m ). A contradiction.
Now consider the general case. Note that the pair (EGk , Gk ) is homotopy equivalent to (CGk , Gk ), so we can consider (CG, G)K instead of (EG, G)K . Since each
Gk is discrete and nontrivial, we may fix an inclusion of a pair of points S 0 ֒→ Gk ;
then there is a retraction Gk → S 0 (it does not have to be a homomorphism
of groups). It extends to a retraction of cones, so we have a retraction of pairs
(CGk , Gk ) → (D1 , S 0 ). These retractions give rise to a retraction of polyhedral
products (CG, G)K → (D1 , S 0 )K = RK . Hence, we have a composite retraction
(CG, G)K → RK → RKJ , so π1 (RKJ ) includes as a subgroup in the free group
Q
1
π1 (EG, G)K = Ker(G K → m
k=1 Gk ). On the other hand, if K contains a chordless cycle J with |J| > 4, then π1 (RKJ ) is the fundamental group of a surface of
positive genus, so it is not free. A contradiction.

Corollary 4.4. Let RAK and RC K be the right-angled Artin and Coxeter groups
corresponding to a simplicial complex K.
(a) The commutator subgroup RA′K is free if and only if K1 is a chordal graph.
(b) The commutator subgroup RC ′K is free if and only if K1 is a chordal graph.
Part (a) of Corollary 4.4 is the result of Servatius, Droms and Servatius [SDS].
The difference between parts (a) and (b) is that the commutator subgroup RA′K
is infinitely generated, unless RAK = Zm , while the commutator subgroup RC ′K is
finitely generated. We elaborate on this in the next theorem.
Let (g, h) = g −1 h−1 gh denote the group commutator of two elements g, h.
Theorem 4.5. Let RC K be the right-angled Coxeter group corresponding to a simplicial complex K on m vertices. The commutator subgroup RC ′K has a finite minP
e 0 (KJ ) iterated commutators
imal generator set consisting of J⊂[m] rank H

(4.2)

(gj , gi ),

(gk1 , (gj , gi )),

...,

(gk1 , (gk2 , · · · (gkm−2 , (gj , gi )) · · · )),

where k1 < k2 < · · · < kℓ−2 < j > i, ks 6= i for any s, and i is the smallest vertex
in a connected component not containing j of the subcomplex K{k1 ,...,kℓ−2 ,j,i} .
Theorem 4.5 is similar to a result of [GPTW] describing the commutator subalgebra of the graded Lie algebra given by


(4.3)
LK = FLhu1 , . . . , um i [ui , ui ] = 0, [ui , uj ] = 0 for {i, j} ∈ K ,

where FLhu1 , . . . , um i is the free graded Lie algebra on generators ui of degree
one, and [a, b] = −(−1)|a||b|[b, a] denotes the graded Lie bracket. The commutator

12

TARAS PANOV AND YAKOV VERYOVKIN

subalgebra is the kernel of the Lie algebra homomorphism LK → CLhu1 , . . . , um i
to the commutative (trivial) Lie algebra.
The graded Lie algebra (4.3) is a graph product similar to the right-angled
Coxeter group RC K . It has a colimit decomposition similar to (2.13), with each
Gi replaced by the trivial Lie algebra CLhui = FLhui/([u, u] = 0) and the colimit
taken in the category of graded Lie algebras.
Theorem 4.6 ([GPTW, Theorem 4.3]). The commutator subalgebra of the graded
P
e 0 (KJ )
Lie algebra LK has a finite minimal generator set consisting of J⊂[m] rank H
iterated commutators
[uj , ui ],

[uk1 , [uj , ui ]],

...,

[uk1 , [uk2 , · · · [ukm−2 , [uj , ui ]] · · · ]],

where k1 < k2 < · · · < kℓ−2 < j > i, ks 6= i for any s, and i is the smallest vertex
in a connected component not containing j of the subcomplex K{k1 ,...,kℓ−2 ,j,i} .
Although the scheme of the proof of Theorem 4.5 is similar to that for Theorem 4.6, more specific techniques are required to work with group commutators, as
opposed to Lie algebra brackets. Nevertheless, most of these techniques are quite
standard, and can be extracted from the classical texts like [MKS].
Proof of Theorem 4.5. The first part is a standard argument applicable to the commutator subgroup of an arbitrary group. An element of RC ′K is a product of commutators (a, b) with a, b ∈ RC K . Writing each of a, b as a word in the generators
g1 , . . . , gm and using the Hall identities
(a, bc) = (a, c)(a, b)((a, b), c),
(4.4)
(ab, c) = (a, c)((a, c), b)(b, c),
n

ni

we express each element of RC ′K in terms of iterated commutators (gi1i1 , . . . , giℓ ℓ )
with nik ∈ Z and arbitrary bracketing. Since we have relations gi2 = 1 in RC K ,
we may assume that each nik is 1. We refer to ℓ > 2 as the length of an iterated
commutator. If an iterated commutator (gi1 , . . . , giℓ ) contains a commutator (a, b)
where each of a, b is itself a commutator, then we can remove such (gi1 , . . . , giℓ )
from the list of generators by writing (a, b) as a word in shorter commutators a, b
and using (4.4) iteratively. We therefore obtain a generators set for RC ′K consisting
only of nested iterated commutators, i. e. those not containing (a, b) where both a, b
are commutators. The next step is to use the identity
((a, b), c) = (b, a)(c, (b, a))(a, b)
and the identities (4.4) to express each nested commutator in terms of canonical
nested commutators (gi1 , (gi2 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · )).
The most important part of the proof is to express each canonical nested commutator in terms of canonical nested commutators in which the generators gi appear
in a specific order. This will be done by a combination of algebraic and topological
arguments and use the specifics of the group RC K .
We first prove a particular case of the statement, corresponding to K consisting
of m disjoint points. The group RC K is then a free product of m copies of Z2 .
Lemma 4.7. Let G be a free product of m copies of Z2 , given by the presentation

G = F (g1 , . . . , gm ) (gi2 = 1, i = 1, . . . , m).

Then the commutator subgroup G′ is a free group freely generated by the iterated
commutators of the form
(gj , gi ),

(gk1 , (gj , gi )),

...,

(gk1 , (gk2 , · · · (gkm−2 , (gj , gi )) · · · )),

where k1 < k2 < · · · < kℓ−2 < j > i and ks 6= i for any s. Here, the number of
commutators of length ℓ is equal to (ℓ − 1) m
ℓ .

POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS

13

Proof. The statement is clear for m = 1 (then G = Z2 ) and for m = 2 (then
G = Z2 ⋆ Z2 and G′ ∼
= Z with generator (g2 , g1 )). For m = 3, the lemma says that
the commutator subgroup of G = Z2 ⋆ Z2 ⋆ Z2 is freely generated by
(g2 , g1 ), (g3 , g1 ), (g3 , g2 ), (g1 , (g3 , g2 )), (g2 , (g3 , g1 )).
This is easy to see geometrically, by identifying RC ′K with π1 (RK ). In our case
RK is the 1-skeleton of the 3-cube (see Example 2.3.4). We have (g1 , (g3 , g2 )) =
g1 (g2 , g3 )g1 (g3 , g2 ), (g2 , (g3 , g1 )) = g2 (g1 , g3 )g2 (g3 , g1 ), and the elements (g2 , g1 ),
(g3 , g1 ), (g3 , g2 ), g1 (g2 , g3 )g1 , g2 (g1 , g3 )g2 correspond to the loops around five different faces of RK , which freely generate its fundamental group.
The general statement for arbitrary m can be proved by a similar topological
argument, by identifying G′ with the fundamental group of the 1-skeleton of the
m-dimensional cube (see [St, Proposition 3.6]). However, we record an algebraic
argument for subsequent use. We have the commutator identity
(4.5) (gq , (gp , x)) = (gq , x)(x, (gp , gq ))(gq , gp )(x, gp )(gp , (gq , x))(x, gq )(gp , gq )(gp , x),
which can be deduced from the Hall–Witt identity, or checked directly. Note that
if x is a canonical nested commutator, then the factor (x, (gp , gq )) can be expressed
via nested commutators as in the beginning of the proof of Theorem 4.5. In this case
we can use (4.5) to swap gp and gq in the commutator (gq , (gp , x)), by expressing
it through (gp , (gq , x)) and canonical nested commutators of lesser length.
In the subsequent arguments, we shall swap elements in an iterated commutator.
Such a swap will change the element of the group represented by the commutator,
but the two elements will always differ by a product of commutators of lesser
length, as in the case of (gp , (gq , x)) and (gq , (gp , x)) in the argument above. If we
want to swap elements gp and gq inside a canonical nested commutator of the form
(· · · , (gp , (gq , x)) · · · ), where x is a smaller canonical nested commutator, then we
need to use the first identity of (4.4) alongside with (4.5). Note that if both b and
c in the identity (a, bc) = (a, c)(a, b)((a, b), c) are canonical nested commutators,
then (a, c) and (a, b) are also canonical nested commutators, while ((a, b), c) =
(b, a)c−1 (a, b)c is a product of nested commutators of lesser length.
Therefore, using (4.5) together with the identities (4.4) we can change the order of any two generators in a commutator (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) within
the positions i1 to iℓ−2 . We first use this observation to eliminate commutators (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) which contain a pair of repeating generators gi (i. e. have ip = iq for some p 6= q). Namely, if the repeating pair occurs within the positions i1 to iℓ−2 , then we use (4.5) to reduce the commutator to the form (· · · (gi , (gi , x)) · · · ), where x = (giℓ−1 , giℓ ), and use the relation
(gi , (gi , x)) = (gi , x)(gi , x) to reduce the commutator to a product of commutators of lesser length. (Note that here we use the relation gi2 = 1 in G.) If
one of the repeating generators is on the position iℓ−1 or iℓ , then we use (4.5)
to reduce the commutator to the form (· · · (gi , (gi , gj )) · · · ) and use the relation
(gi , (gi , gj )) = (gi , gj )(gi , gj ). As a result, we obtain a generator set for G′ consisting of commutators (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) with all different gi . This already
shows that G′ is finitely generated.
Now we use (4.5) to put the generators gi in (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · ) in an
order. Choose the generator gik with the largest index ik . If it is not within the last
three positions, then we use (4.5) to move it to the third-to-last position. The case
m = 3 considered above shows that the commutator (gj , (gi , gk )) can be expressed
through (gi , (gj , gk ), (gk , (gj , gi )) and commutators of lesser length. This allows us
to move the generator gik with the largest index ik in (gi1 , · · · (giℓ−2 , (giℓ−1 , giℓ )) · · · )
to the second-to-last position, and set j = ik . Then we use (4.5) and (4.4) to put
the first ℓ − 2 generators in the commutator in an increasing order, and redefine

14

TARAS PANOV AND YAKOV VERYOVKIN

their indices as k1 < · · · < kℓ−2 . As a result, we obtain a generator set for G′
consisting of commutators of the required form (gk1 , (gk2 , · · · (gkℓ−2 , (gj , gi )) · · · ))
where k1 < k2 < · · · < kℓ−2 < j > i and ks 6= i for any s.
It remains to show that the
of G′ is free. This gen generating set
Pmconstructed
m
m−1
erating set consists of N = ℓ=2 (ℓ − 1) ℓ = (m − 2)2
+ 1 commutators. On
the other hand, G′ ∼
= π1 (RK ), where RK the 1-skeleton of the m-cube. Then RK is
homotopy equivalent to a wedge of N circles (as easy to see inductively or by computing the Euler characteristic), and π1 (RK ) is a free group of rank N . We therefore
have a system of N generators in a free group of rank N . This system must be free,
by the classical result that a free group of finite rank cannot be isomorphic to its
proper quotient group, see [MKS, Theorem 2.13].


Now we resume the proof of Theorem 4.5. We shall exclude some commutators
(gk1 , (gk2 , · · · (gkℓ−2 , (gj , gi )) · · · )) from the generating set using the new commutativity relations.
First assume that j and i are in the same connected component of the
complex K{k1 ,...,kℓ−2 ,j,i} . We shall show that the corresponding commutator
(gk1 , (gk2 , · · · (gkℓ−2 , (gj , gi )) · · · )) can be excluded from the generating set. We
choose a path from i to j, i. e. choose vertices i1 , . . . , in from k1 , . . . , kℓ−2 with
the property that the edges {i, i1 }, {i1, i2 }, . . . , {in−1 , in }, {in , j} are all in K. We
proceed by induction on the length of the path. The induction starts from the
commutator (gj , gi ) = 1 corresponding to a one-edge path {i, j} ∈ K. Now assume that the path consists of n + 1 edges. Using the relation (4.5) we can move
the elements gi1 , gi2 , . . . , gin in (gk1 , (gk2 , · · · (gkℓ−2 , (gj , gi )) · · · )) to the right and
restrict ourselves to the commutator (gi1 , (gi2 , · · · (gin , (gj , gi )) · · · )). Observe that
in the presence of the commutation relation (gp , gq ) = 1 the identity (4.5) does
not contain the factor (x, (gp , gq )) and therefore it allows us to change the order
of gp and gq without assuming x to be a commutator. We therefore can convert the commutator (gi1 , (gi2 , · · · (gin , (gj , gi )) · · · )) (with {in , j} ∈ K) to the
commutator (gj , (gi1 , · · · (gin−1 , (gin , gi )) · · · )). The latter contains a commutator
(gi1 , · · · (gin−1 , (gin , gi )) · · · ) corresponding to a shorter path {i, i1 , . . . , in }. By inductive hypothesis, it can be expressed through commutators of lesser length, and
therefore excluded from the set of generators.
We therefore obtain a generator set for RC ′K consisting of nested commutators
(gk1 , · · · (gkℓ−2 , (gj , gi )) · · · ) with j and i in different connected components of the
complex K{k1 ,...,kℓ−2 ,j,i} . Consider commutators (gk1 , · · · (gkℓ−2 , (gj , gi1 )) · · · ) and
′
′
, j, i2 }
(gk1′ , · · · (gkℓ−2
, (gj , gi2 )) · · · ) such that {k1 , . . . , kℓ−2 , j, i1 } = {k1′ , . . . , kℓ−2
and i1 , i2 lie in the same connected component of K{k1 ,...,kℓ−2 ,j,i1 } which is different
from the connected component containing j. We claim that one of these commutators can be expressed through the other and commutators of lesser length. To see
this, we argue as in the previous paragraph, by considering a path in K{k1 ,...,kℓ−2 ,j,i1 }
between i1 and i2 , and then reducing it inductively to a one-edge path. This leaves
us with the pair of commutators (gi2 , (gj , gi1 )) and (gi1 , (gj , gi2 )) where {i1 , i2 } ∈ K,
{i1 , j} ∈
/ K, {i2 , j} ∈
/ K. The claim then follows easily from the relation (gi1 , gi2 ) = 1
(compare the case m = 3 of Lemma 4.7).
Thus, to enumerate independent commutators, we use the convention of writing
(gk1 , · · · (gkℓ−2 , (gj , gi )) · · · ) where i is the smallest vertex in its connected component within K{k1 ,...,kℓ−2 ,j,i} . This leaves us with precisely the set of commutators (4.2). It remains to show that this generating set is minimal. For this we once
again recall that RC ′K = π1 (RK ). The first homology group H1 (RK ) is isomorphic
to RC ′K /RC ′′K , where RC ′′K is the commutator subgroup of RC ′K . On the other

POLYHEDRAL PRODUCTS AND COMMUTATOR SUBGROUPS

hand, we have
H1 (RK ) ∼
=

M

J⊂[m]

15

e 0 (KJ )
H

by Theorem 3.6. Hence, the number of generators in the abelian group H1 (RK ) ∼
=
P
e 0 (KJ ). The latter number agrees with the number of
RC ′K /RC ′′K is J⊂[m] rank H
e 0 (K) is one less the number of
iterated commutators in the set (4.2), as rank H
connected components of K.

r3
Example 4.8.
1. Let K be the simplicial complex 1 r r 2 r 4 on four vertices. Then the commutator subgroup RC ′K is free, and Theorem 4.5 gives the following free generators:
(g3 , g1 ), (g4 , g1 ), (g4 , g2 ), (g4 , g3 ),
(g2 , (g4 , g1 )), (g3 , (g4 , g1 )), (g1 , (g4 , g3 )), (g3 , (g4 , g2 )),
(g2 , (g3 , (g4 , g1 ))).
2. Let K be an m-cycle with m > 4 vertices. Then K1 is not a chordal graph, so
the group RC ′K is not free. One can see that RK is an orientable surface of genus
(m − 4)2m−3 + 1 (see Example 3.5), so RC ′K ∼
= π1 (RK ) is a one-relator group.
When m = 4, we get a 2-torus as RK , and Theorem 4.5 gives the generators
a1 = (g3 , g1 ) and b1 = (g4 , g2 ). The single relation is obviously (a1 , b1 ) = 1. For
m > 5 we do not know the explicit form of the single relation in the surface group
RC ′K ∼
= π1 (RK ) in terms of the generators provided by Theorem 4.5. Compare [Ve],
where the corresponding problem is studied for the commutator subalgebra of the
graded Lie algebra from Theorem 4.6.
References
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[BP1]
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[KR]
[MKS]

[PR]

[PRV]

[Po]
[SDS]
[St]
[Ve]

TARAS PANOV AND YAKOV VERYOVKIN

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Mat. Zh. (2016), no. 1, 9–23 (in Russian); arXiv:1512.00283.

Department of Mathematics and Mechanics, Moscow State University, Leninskie
Gory, 119991 Moscow, Russia,
Institute for Theoretical and Experimental Physics, Moscow, Russia and
Institute for Information Transmission Problems, Russian Academy of Sciences
E-mail address: tpanov@mech.math.msu.su
URL: http://higeom.math.msu.su/people/taras/
Department of Mathematics and Mechanics, Moscow State University, Leninskie
Gory, 119991 Moscow, Russia,
Steklov Mathematical Institute, Russian Academy of Sciences
E-mail address: verevkin j.a@mail.ru

