arXiv:1611.07453v1 [] 22 Nov 2016

ON DISTORTION OF NORMAL SUBGROUPS
HUNG CONG TRAN
Abstract. We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group
divergence. We suggest a method computing the distortion of normal
subgroups by decomposing the whole group into smaller subgroups. We
apply our work to compute the distortion of normal subgroups of graph
of groups and normal subgroups of right-angled Artin groups that induce infinite cyclic quotient groups. We construct normal subgroups
of CAT(0) groups introduced by Macura and introduce a collection of
normal subgroups of right-angled Artin groups. These groups provide
a rich source to study the connection between subgroup distortion and
group divergence on CAT(0) groups.

1. Introduction
Subgroup distortion is a famous tool to study the geometric connection
between a finitely generated group and its finitely generated subgroups. A
metric on a subgroup can be inherited from the metric of the whole group.
Also a finitely generated subgroup can be a metric space itself. Therefore,
the subgroup distortion notion was introduced to measures the difference
between the two metric structures of a finitely generated subgroup. In this
article, we only focus on distortion of finitely generated normal subgroups.
The divergence is a quasi-isometry invariant which arose in the study of
non-positively curved manifolds and metric spaces. Roughly speaking, the
divergence of a connected metric space measures the complement distance
of a pair of points in a sphere as a function of the radius. For example,
the divergence of the Euclidean plane is linear and the divergence of the
hyperbolic plane is exponential. The concept of finitely generated groups
are defined via their Cayley graphs.
Connection between group divergence and distortion of normal subgroups
was first shown by Gersten [Ger94]. He proved the distortion of a finitely
generated normal subgroup H in a finitely generated G is bounded below
by the divergence of G when the quotient group G/H is an infinite cyclic
group. We generalize the result of Gersten by the following theorem:
Theorem 1.1. Let G be a finitely generated group and H a finitely generated
infinite index infinite normal subgroup of G. Then the divergence of G is
dominated by the subgroup distortion of H in G.
Date: January 31, 2018.
2000 Mathematics Subject Classification. 20F67, 20F65.
1

ON DISTORTION OF NORMAL SUBGROUPS

2

By the above theorem, we can use the divergence of the whole group as a
lower bound for distortion of any finitely generated normal subgroup. Also,
we can use subgroup distortion as a tool to compute the upper bound for
group divergence. In this paper, we revisit Macura’s examples on CAT(0)
groups with polynomial divergence (see [Mac13]) and show an alternative
way using subgroup distortion to compute the upper bound of the divergence
of these groups (see Remark 3.6).
In [Sis], Sisto proved that the divergence of a one-ended nontrivial relatively hyperbolic group is at least exponential. We are not sure if the study
of the distortion of finitely generated normal subgroups in finitely generated relatively hyperbolic groups is recorded in the literature. However, the
following corollary is a direct application of Theorem 1.1.
Corollary 1.2. Let G be a one-ended relatively hyperbolic group with respect
to some collection of proper subgroups and H a finitely generated infinite
index infinite normal subgroup of G. Then the distortion of H in G is at
least exponential.
We study the upper bound for the distortion of a normal subgroup by
decomposing the whole group into smaller subgroups and computing the
distortion of the normal subgroup on each of these subgroups.
Theorem 1.3. Let G be a finitely generated group and H normal subgroup
of G with the canonical projection p : G → G/H. Suppose there is a finite
collection A of finitely generated subgroups of G that satisfies the following
conditions:
(1) The union of all subgroups in A generates G and the subgroup A ∩ H
is finitely generated for each subgroup A in A.
(2) For each pair A and A′ in A, there is a sequence of subgroups in A
A = A1 , A2 , · · · , An = A′ such that p(Ai ∩ Ai+1 ) is a finite index
subgroup of G/H for each i in {1, 2, · · · , n − 1}.
Then, the subgroup H is finitely generated and the distortion DistH
G is domA∩H
inated by the function f (n) = n max{ DistA (n) | A ∈ A }.
The above theorem can be used to compute the distortion of normal
subgroups of a simply connected finite graph of groups (see the following
corollary).
Corollary 1.4. Let G be the fundamental group of a simply connected finite graph of finitely generated groups and H a finitely generated normal
subgroup of G not contained in any edge subgroup. We assume that the
intersection of H with any vertex subgroup is finitely generated. Then, the
A∩H (n) |
distortion DistH
G is dominated by the function f (n) = n max{ DistA
A is a vertex subgroup }.
We also use Theorem 1.3 to compute the distortion of normal subgroups
of right-angled Artin groups that induce infinite cyclic quotient groups.

ON DISTORTION OF NORMAL SUBGROUPS

3

Corollary 1.5. Let Γ be a simplicial connected graph with at least two
vertices. Let p be an arbitrary group homomorphism from the right-angled
Artin group AΓ to Z and N the kernel of the map p. Let Γ′ be the induced
subgraph of Γ with the vertices which are mapped to nonzero numbers in Z
by p. If the graph Γ′ is connected and the star of Γ′ in Γ contains all vertices
of Γ, then the subgroup N is finitely generated and the distortion of N in AΓ
is at most quadratic. Moreover, if each vertex of Γ is mapped to a nonzero
number in Z, then the distortion of the N in AΓ is linear when Γ is a join
and the distortion is quadratic otherwise.

In the above corollary, the subgroup N is the Bestvina-Brady subgroup
of AΓ if p maps each vertex of Γ to 1. We remark that the distortion
of Bestvina-Brady subgroups were also computed by the author in [Tra].
The kernel of the group homomorphism p : AΓ → Z was also studied by
Papadima-Suciu in [PS09]. The divergence of right-angled Artin groups was
studied by Behrstock–Charney in [BC12] and we used the divergence of the
right-angled Artin group AΓ to prove for the lower bound of the distortion of
N . However, the upper bound of the distortion of N is calculated by using
Theorem 1.3. Moreover, the upper bound of the distortion of N provides
an alternative method to compute the upper bound of the divergence of the
right-angled Artin group AΓ .
The notion of algebraic thickness of groups was introduced in [BDM09]
to study the geometry of non-relatively hyperbolic groups. The collection of
algebraically thick groups includes many types of groups such as mapping
class groups of surfaces S with 3 × genus(S) + #punctures ≥ 5, Aut(Fn )
and Out(Fn ) for n ≥ 3, non-relatively hyperbolic Coxeter groups, various
Artin groups, and others. Behrstock-Druţu [BD14] used algebraic thickness
of a group to compute the upper bound of its divergence. Since right-angled
Artin groups with connected defining graphs are examples algebraically thick
groups, Corollary 1.5 gives us a hint to use Theorem 1.3 to study distortion
of normal subgroups of algebraically thick groups.
More precisely, the algebraic thick order of group G is defined inductively
through a decomposition of a finite index subgroup G1 into a finite collection H of subgroups with lower algebraic thick order that satisfies some
certain conditions (see [BDM09] for the precise definition). Therefore, we
can use Theorem 1.3 to compute the upper bound of distortion of a normal subgroup H in G if the subgroup H ∩ G1 and the collection H satisfy
all hypothesis in the theorem and the distortion of H on each subgroup in
H is well-understood. Also divergence functions of many thick groups were
known. Therefore, we can apply Theorem 1.1 to study the distortion normal
subgroups of these groups.
Macura [Mac13] introduced a class of CAT(0) groups (i.e. groups which
act properly and cocompactly on some CAT(0) spaces) to study group divergence. She showed that for each positive integer d there is a CAT(0) groups

ON DISTORTION OF NORMAL SUBGROUPS

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with polynomial of degree d divergence function. We revisit Macura’s examples and construct normal subgroups for each example with distortion
function as the same as the divergence function of the whole group.
Theorem 1.6. For each integer d ≥ 2 let Gd = ha0 , a1 , · · · , ad |a0 a1 =
a1 a0 , a−1
i a0 ai = ai−1 , for 2 ≤ i ≤ di. Then Gd acts properly and cocompactly
on some CAT (0) space with polynomial divergence function of degree d and
we can find a normal finitely generated free subgroup Fd inside such that the
quotient Gd /Fd is an infinite cyclic subgroup and the distortion of Fd is also
a polynomial function of degree d.
We remark that each group Gd in the above theorem was constructed
by Macura [Mac13] and the property that each group Gd acts properly and
cocompactly on some CAT(0) space with polynomial divergence function of
degree d was also proved by her. We only construct the free normal subgroup
Fd in each Gd with polynomial distortion function of degree d. We also
remark that the existence of subgroups of CAT(0) groups with polynomial
distortion was known before (see [BB06] for example). However, the purpose
of the above theorem is to emphasis on the connection between distortion of
normal free subgroups and group divergence functions on CAT(0) groups.
We also hope that the theorem can shed a light for the positive answer to
the following conjecture raised by Gersten [Ger94]:
Conjecture 1.7. [Ger94] Let F be a finitely generated normal free subgroup
in a group G such that the quotient G/F is an infinite cyclic group. Then
the distortion of F in G is equivalent to the divergence of G.
We introduce a collection of normal subgroups of right-angled Artin groups,
called generalized Bestvina-Brady subgroups (see Section 4 for the precise
definition). By using the work of Meier-VanWyk [MV95], we give a necessary and sufficient condition for these groups to be finitely generated.
Theorem 1.8. Let Γ be a finite simplicial graph. Let HΦ be a generalized
Bestvina-Brady subgroup of AΓ . Then HΦ is finitely generated iff each basis
subgraph of Γ with respect to Φ is connected and dominating.
We compute the distortion of groups of this collection with some additional hypothesis by mainly using Theorem 1.1 and Theorem 1.3.
Theorem 1.9. Let Γ be a simplicial connected graph. Let HΦ be a strong
generalised Bestvina-Brady subgroup of AΓ with basis subgraphs {Γi }1≤i≤d .
Then, the distortion of HΦ in AΓ is quadratic if there is a non-empty subset
I of {1, 2, · · · , d} such that the subgraph Γ′ generated by {Γi }i∈I is not a
join. Moreover, the distortion of HΦ in AΓ is linear if one of the following
conditions holds:
(1) The generalised Bestvina-Brady subgroup HΦ is a special generalised
Bestvina-Brady subgroup.
(2) Each basis subgraphs in {Γi }1≤i≤d is a join and each pair of basis
subgraphs in {Γi }1≤i≤d commutes.

ON DISTORTION OF NORMAL SUBGROUPS

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The above theorem provides a rich source of examples to study the connection between group divergence and subgroup distortion. We also hope
that the collection of generalized Bestvina-Brady subgroups also provides a
good source of examples to study normal subgroups of right-angled Artin
groups.
Acknowledgments. I would like to thank Prof. Christopher Hruska and
Hoang Nguyen for their helpful comments and suggestions. I also want to
thank Prof. Jason Behrstock for his very useful conversation. Especially,
he gives me an idea that leads to Theorem 1.3. I also would like thank
Prof. Ruth Charney and Prof. Thomas Koberda for their fruitful conversations on subgroups of right-angled Artin groups. I also thank Kevin Schreve
for advice that improved the exposition of the paper.
2. Distortion of normal subgroups
In this section, we review the concepts of group divergence and subgroup
distortion. We investigate the connection between these concepts for a pair
of finitely generated groups (G, H), where H is a normal subgroup of G.
More precisely, we prove that the divergence of G is a lower bound for the
distortion of H in G. We compute an upper bound for the distortion of
H in G by dividing the group G into a collection of smaller subgroups and
computing the distortion of H on these subgroups.
Before we define the concepts of group divergence and subgroup distortion, we need to define the notions of domination and equivalence. These
notions are the key tool to measure group divergence and subgroup distortion.
Definition 2.1. Let M be the collection of all functions from [0, ∞) to
[0, ∞]. Let f and g be arbitrary elements of M. The function f is dominated
by the function g, denoted f  g, if there are positive constants A, B, C
and D such that f (x) ≤ Ag(Bx) + Cx for all x > D. Two function f and g
are equivalent, denoted f ∼ g, if f  g and g  f .
Remark 2.2. A function f in M is linear, quadratic or exponential... if
f is respectively equivalent to any polynomial with degree one, two or any
function of the form abx+c , where a > 1, b > 0.
Definition 2.3. Let {δρ } and {δρ′ } be two families of functions of M, indexed over ρ ∈ (0, 1]. The family {δρ } is dominated by the family {δρ′ }, denoted {δρ }  {δρ′ }, if there exists constant L ∈ (0, 1] such that δLρ  δρ′ . Two
families {δρ } and {δρ′ } are equivalent, denoted {δρ } ∼ {δρ′ }, if {δρ }  {δρ′ }
and {δρ′ }  {δρ }.
Remark 2.4. A family {δρ } is dominated by (or dominates) a function f in
M if {δρ } is dominated by (or dominates) the family {δρ′ } where δρ′ = f for
all ρ. The equivalence between a family {δρ } and a function f in M can be
defined similarly. Thus, a family {δρ } is linear, quadratic, exponential, etc if

ON DISTORTION OF NORMAL SUBGROUPS

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{δρ } is equivalent to the function f where f is linear, quadratic, exponential,
etc.
Definition 2.5. Let G be a group with a finite generating set S and H a
subgroup of G with a finite generating set T . The subgroup distortion of H
in G is the function DistH
G : (0, ∞) → (0, ∞) defined as follows:

H
DistG (r) = max |h|T h ∈ H, |h|S ≤ r .
Remark 2.6. It is well-known that distortion does not depend on the choice
of finite generating sets.
We now recall Gersten’s definition of divergence from [Ger94]. Let X be a
geodesic space and x0 one point in X. Let Br (x0 ) be the open ball centered
at x0 and radius r and Sr (x0 ) the sphere with the same center and radius.
Let dr,x0 be the induced length metric on the complement of the open ball
with radius r about x0 . If the point x0 is clear from context, we will use the
notation dr instead of using dr,x0 .
Definition 2.7. Let X be a geodesic space and x0 one point in X. We define
the divergence of X, denoted Div(X), as a family {δρ }ρ∈(0,1] of functions,
where each function δρ : [0, ∞) → [0, ∞) is given as follows:
For each r, let δρ (r) = sup dρr (x1 , x2 ) where the supremum is taken over
all x1 , x2 ∈ Sr (x0 ) such that dρr (x1 , x2 ) < ∞.
Remark 2.8. The divergence of a geodesic space X does not depend on
the choice of x0 . Moreover, the concept of divergence is a quasi-isometry
invariant. The divergence of a finitely generated group G, denoted Div(G),
is the divergence of a Cayley graph Γ(G, S) for some finite generating set S.
Before we examine the connection between the group divergence and the
distortion of a normal subgroup, we need to examine the relationship between the distance between cosets and the distance on the quotient group
as follows:
Proposition 2.9. Let G be a finitely generated group with a finite generating
set S = {si }i∈I and H a normal subgroup of G. For each element g in G let g
be the image of g under the canonical projection p : G → G/H. Then the set
S = {si }i∈I generates the quotient group G/H and dS (aH, bH) = dS (a, b)
for every a and b in G.
Proof. It is obvious that S is a generating set of G/H and we now prove
that dS (aH, bH) = dS (a, b) every a and b in G. Let ah1 and bh2 be elements
in aH and bH respectively such that the distance between ah1 and bh2 is
the same as the distance between aH and bH. Let w = s1 s2 · · · sn be a
word in S such that n = dS (aH, bH) and ah1 = (bh2 )(s1 s2 · · · sn ). This
implies that a = b(s1 s2 · · · sn ). Therefore, dS (a, b) ≤ n = dS (aH, bH).
Conversely, let m = dS (a, b) and w′ = s′1 s′2 · · · s′m be a word in S such
that a = b(s′ 1 s′ 2 · · · s′ m ). Therefore, c = b(s1 s2 · · · sm ) is an element in

ON DISTORTION OF NORMAL SUBGROUPS

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aH and dS (b, c) ≤ m. Therefore, dS (aH, bH) ≤ m = dS (a, b). Thus,

dS (aH, bH) = dS (a, b) for every a and b in G.
Theorem 2.10. Let G be a finitely generated group and H a finitely generated infinite index infinite normal subgroup of G. Then the divergence
Div(G) of G is dominated by the subgroup distortion DistH
G of H in G.
Proof. Let S = {si }i∈I be a finite generating set of G. We assume that S
also contains a finite generating set T of H. Therefore, we can consider the
Cayley graph Γ(H, T ) as a subgraph of Γ(G, S). For each element g in G
let g be the image of g under the canonical projection p : G → G/H. By
Proposition 2.9, the set S = {si }i∈I generates the quotient group G/H and
dS (aH, bH) = dS (a, b) for every a and b in G.
Let Div(G) = {δρ }ρ∈(0,1] and we will prove that δρ  DistG
H for each ρ in
(0, 1/2]. Let r be an arbitrary positive number. We can assume that r is an
integer. Let g be an element in G such that dS (H, gH) = |g|S = r. For each
x in Sr (e) we are going to construct a path ℓ outside the open ball Br/2 (e)
connecting x and g with the length at most 2DistG
H (32r) + 6r.
Let α be an arbitrary geodesic containing x. Let α1 and α2 be two rays
obtained from α with the initial point x. Since the distance between x and
e is exactly r, either of α1 or α2 must lie outside the open ball Br/2 (e). We
assume that α1 lies outside the open ball Br/2 (e). Let x1 be the point on
α1 such that the distance between x and x1 is exactly 4r and let ℓ1 the
subsegment of α1 connecting x and x1 . Therefore, ℓ1 must lie outside the
open ball Br/2 (e).
We now find a point x2 in G such that dS (x2 H, H) ≥ r and a path ℓ2
outside the open ball Br/2 (e) connecting x1 and x2 with length at most 2r.
If dS (x1 H, H) ≥ r, then we let x2 = x1 and ℓ2 the degenerate segment of
length 0 at x2 . We now assume that dS (x1 H, H) < r. Let u be an element in
G such that dS (x1 , u) = 2r. Since the distance between e and x1 is at most r
with respect to the metric dS , the distance between e and u is at least r. Let
w = s1 s2 · · · s2r be a word in S such that u = x1 w. Let x2 = x1 w and ℓ2 the
path connecting x1 and x2 with label by w. Then the length of ℓ2 is at most
2r. Also, dS (e, x1 ) ≥ dS (x, x1 ) − dS (e, x) ≥ 4r − r ≥ 3r. Therefore, ℓ2 must
lie outside the open ball Br/2 (e). Obviously, dS (x2 H, H) = dS (x2 , e) ≥ r.
We also see that the distance between x2 and e is at most 7r and the distance
between x2 and g is at most 8r.
Let h be an element in H of length 16r with respect to the metric dS and
let η is a path in Γ(H, T ) connecting e and h with length at most DistG
H (16r).
Let x3 = x2 h and ℓ3 = x2 η. Then ℓ3 is a path connecting x2 and x3 with
length at most DistG
H (16r). Since all vertices of ℓ3 lie in x2 H, ℓ3 must lie
outside the open ball Br/2 (e). We observe that the distance between x2 and
x3 is exactly 16r and the distance between x3 and e lies between 9r and 23r.
Since dS (x2 H, gH) ≤ dS (x2 , g) ≤ 8r and x3 lies in x2 H, there is a point
x4 in gH such that the distance between x3 and x4 is at most 8r. Let ℓ3 be
a geodesic connecting x3 and x4 . Then the length of ℓ3 is at most 8r. Since

ON DISTORTION OF NORMAL SUBGROUPS

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the distance between x3 and e is at least 9r, the path ℓ3 must lie outside
the open ball Br/2 (e). We also see that the distance between x4 and e is
at most 31r and the distance between x4 and g is at least 32r. Therefore,
there is a path ℓ4 with all vertices in gH connecting x4 and g such that the
length of ℓ4 is bounded above by DistG
H (32r). Since the distance between
H and gH is exactly r, ℓ4 must lie outside the open ball Br/2 (e).
Finally, let ℓ = ℓ1 ∪ ℓ2 ∪ ℓ3 ∪ ℓ4 then ℓ is a path outside Br/2 (e) connecting
G
x and g of length at most DistG
H (32r) + DistH (16r) + 6r. Also, the distortion function DistG
H is increasing. Then the length of ℓ is bounded above
by 2DistG
(32r)
+
6r.
Thus, any two points in the sphere Sr (e) can be conH
nected by a path outside Br/2 (e) with length at most 4DistG
H (32r) + 12r.
G
This implies that δρ (r) is bounded above by 4DistH (32r) + 12r for each
ρ in (0, 1/2]. Therefore, the divergence Div(G) of G is dominated by the
subgroup distortion DistH
G of H in G.

The above theorem provides us with a lower bound for the distortion of a
normal subgroup. The following theorem helps us compute an upper bound.
Theorem 2.11. Let G be a finitely generated group and H normal subgroup
of G with the canonical projection p : G → G/H. Suppose there is a finite
collection A of finitely generated subgroups of G that satisfies the following
conditions:
(1) The union of all subgroups in A generates G and the subgroup A ∩ H
is finitely generated for each subgroup A in A.
(2) For each pair A and A′ in A, there is a sequence of subgroups in A
A = A1 , A2 , · · · , An = A′ such that p(Ai ∩ Ai+1 ) is a finite index
subgroup of G/H for each i in {1, 2, · · · , n − 1}.
Then, the subgroup H is finitely generated and the distortion DistH
G is domA∩H
inated by the function f (n) = n max{ DistA (n) | A ∈ A }.
Proof. We define function g = max{ DistA∩H
| A ∈ A }. Let K be the
A
number of elements of A.
For
each
subgroup
A
in
A we fix a finite generating
S
set SA for A. Let S = A∈A SA and S = p(S). Then S is a finite generating
set for G and S is a finite generating set for G/H. Moreover, |p(g)|S ≤ |g|S
for each g in G. Let F be the intersection of all p(A ∩ A′ ), where A and A′
are elements in A such that p(A ∩ A′ ) is a finite index subgroup of G/H.
Therefore, F is a finite index subgroup of G/H. Thus, there is a finite
generating set J of F and a positive constant M such that for each v in F
(1/M )|v|S − 1 ≤ |v|J ≤ M |v|S + M.
Also, there is a finite set I of group elements in G that contains the identity
element e of G such that
[
G/H =
p(a)F.
a∈I

ON DISTORTION OF NORMAL SUBGROUPS

9

Let C be the maximum on the lengths of all elements in I with respect to
the finite generating set S.
For each b, b′ in I and A in A such that bAb′−1 ∩ H 6= ∅, we fix a group
element a(b,b′ ,A) in A such that ba(b,b′ ,A) b′−1 belongs to H. Let
L1 = max{ |a(b,b′ ,A) |SA | bAb′−1 ∩ H 6= ∅, b, b′ ∈ I, A ∈ A }
and
T1 = { a(b,b′ ,A) | bAb′−1 ∩ H 6= ∅, b, b′ ∈ I, A ∈ A }.
For each subgroup A in A we fix a finite generating set TA for A ∩ H.
Let T2 be the union of all sets bTA b−1 , where b is an element in I and A
is an element in A. Let T = T1 ∪ T2 and we will prove that T is a finite
generating set of H. Moreover, we will use the pair of generating sets (S, T )
to compute the distortion of H in G.
For each element v in J and A, A′ in A such that p(A ∩ A′ ) is a finite
index subgroup of G/H we choose an element uv,A,A′ in A ∩ A′ such that
p(uv,A,A′ ) = v. We let
ℓ(v, A, A′ ) = max{|uv,A,A′ |SA , |uv,A,A′ |SA′ }
and
L2 = max{ ℓ(v, A, A′ ) | v ∈ J and A, A′ ∈ A such that [G/H : p(A ∩ A′ )] < ∞ }.
It is not hard to see that for each element q in F and A, A′ in A such
that p(A ∩ A′ ) is a finite index subgroup of G/H we can choose an element
g in A ∩ A′ such that p(g) = q and |g|SA , |g|SA′ are both bounded above by

L2 |q|J . Therefore, both |g|SA , |g|SA′ are bounded above by L2 M |q|S + M .
Let n be an arbitrary integer and h element in H such that |h|S ≤ n.
Then there is a word w = w1 w2 · · · wm in S that represents element h with
the following properties:
(1) The number m is at most Kn and each wi is a word in the finite
generating set SAi of some group Ai in A. (We accept that some of
subword wi may be empty.)
(2) For each i in {1, 2, 3, · · · , m − 1}, p(Ai ∩ Ai+1 ) is a finite index subgroup of G/H.
(3) The sum ℓ(w1 )+ℓ(w2 )+· · ·+ℓ(wm ) is bounded above by the number
n.
For each i in {1, 2, · · · , m − 1} let bi in I and such that p(w1 w2 · · · wi ) is
an element in p(bi )F . Let ui in Ai ∩Ai+1 such that p(ui ) = p(b−1
i w1 w2 · · · wi )

−1
and both |ui |SAi , |ui |SAi+1 are bounded above by L2 M |p(bi w1 w2 · · · wi )|S + M .
−1
Also, |p(b−1
i w1 w2 · · · wi )|S ≤ |bi w1 w2 · · · wi |S ≤ n + C. Then both |ui |SAi ,
|ui |SAi+1 are bounded above by L2 M (n + C + 1). It is obvious that

p(w1 w2 · · · wi ) = p(bi ui ).

ON DISTORTION OF NORMAL SUBGROUPS

10

−1
−1 −1
We let h1 = w1 u−1
1 b1 , hm = bm−1 um−1 wm , and hi = bi−1 (ui−1 wi ui )bi
for each i in {2, 3, · · · , m − 1}. Obviously,

h = h1 h2 · · · hm and p(hi ) = eG/H for each i in {1, 2, · · · , m − 1}.
Therefore, hi is an element of H for each i in {1, 2, 3, · · · , m}. We now prove
that the length of each element hi with
 respect to T is bounded above by
g (2L2 M + 1)n + 2L2 M (C + 1) + L1 + 1.
−1
Since h1 is an element A1 b−1
1 ∩ H, the set A1 b1 ∩ H is not empty. Let
a(e,b1 ,A1 ) be the element in A1 defined as above such that x1 = a(e,b1 ,A1 ) b−1
1
−1 −1
is
an
element
in
is an element in H. Therefore, h1 x−1
=
w
u
a
1 1
1
(e,b1 ,A1 )
A1 ∩ H and
−1
|w1 u−1
1 a(e,b1 ,A1 ) |SA1 ≤ n+2L2 M (n+C+1)+L1 ≤ (2L2 M +1)n+2L2 M (C+1)+L1 .

Therefore,

A1 ∩H
−1 −1
|h1 x−1
(2L2 M +1)n+2L2 M (C+1)+L1 .
1 |T ≤ |w1 u1 a(e,b1 ,A1 ) |TA1 ≤ DistA1
This implies that

|h1 x−1
1 |T ≤ g (2L2 M + 1)n + 2L2 M (C + 1) + L1 .
Also, x1 is an element in T . Then

|h1 |T ≤ g (2L2 M + 1)n + 2L2 M (C + 1) + L1 + 1.
Since hi is an element bi−1 Ai b−1
i ∩ H for each i in {2, 3, · · · , m − 1}, the
−1
set bi−1 Ai bi ∩ H is not empty. Let a(bi−1 ,bi ,Ai ) be the element in Ai defined
as above such that xi = bi−1 a(bi−1 ,bi ,Ai ) b−1
is an element in H. Therefore,
i
−1
−1 −1
)b
is
an
element in bi−1 Ai b−1
hi x−1
=
b
(u
w
u
a
i−1 i−1 i i
i−1 ∩ H and
i
(bi−1 ,bi ,Ai ) i−1
−1
|ui−1 wi u−1
i a(bi−1 ,bi ,Ai ) |SAi ≤ n+2L2 M (n+C+1)+L1 ≤ (2L2 M +1)n+2L2 M (C+1)+L1 .

Therefore,

Ai ∩H
−1 −1
|hi x−1
(2L2 M +1)n+2L2 M (C+1)+L1 .
i |T ≤ |ui−1 wi ui a(bi−1 ,bi ,Ai ) |TAi ≤ DistAi
This implies that

|hi x−1
i |T ≤ g (2L2 M + 1)n + 2L2 M (C + 1) + L1 .
Also, xi is an element in T . Then

|hi |T ≤ g (2L2 M + 1)n + 2L2 M (C + 1) + L1 + 1.
Since hm is an element bm−1 Am ∩ H, the set bm−1 Am ∩ H is not empty.
Let a(bm−1 ,e,Am ) be the element in Am defined as above such that xm =
−1
−1
bm−1 a(bm−1 ,e,Am ) is an element in H. Therefore, hm x−1
m = bm−1 (um−1 wm a(bm−1 ,e,Am ) )bm−1
is an element in bm−1 Am b−1
m−1 ∩ H and

|um−1 wm a−1
(bm−1 ,e,Am ) |SAm ≤ n+2L2 M (n+C+1)+L1 ≤ (2L2 M +1)n+2L2 M (C+1)+L1 .
Therefore,

Am ∩H
−1
|hm x−1
(2L2 M +1)n+2L2 M (C+1)+L1 .
m |T ≤ |um−1 wm a(bm−1 ,e,Am ) |TAm ≤ DistAm

ON DISTORTION OF NORMAL SUBGROUPS

11

This implies that

|hm x−1
m |T ≤ g (2L2 M + 1)n + 2L2 M (C + 1) + L1 .
Also, xm is an element in T . Then

|hm |T ≤ g (2L2 M + 1)n + 2L2 M (C + 1) + L1 + 1.
Therefore, the group element h can be represented by a word in T and
|h|T is bounded above by Kng (2L2 M + 1)n + 2L2 M (C + 1) + L1 + Kn.
This implies that the subgroup H is finitely generated and the distortion
DistH

G is dominated by the function f (n).
Corollary 2.12. Let G be the fundamental group of a simply connected
finite graph of finitely generated groups and H a finitely generated normal
subgroup of G not contained in any edge subgroup. We assume that the
intersection of H with any vertex subgroup is finitely generated. Then, the
A∩H (n) |
distortion DistH
G is dominated by the function f (n) = n max{ DistA
A is a vertex subgroup }.
Proof. Let p : G → G/H be the canonical projection. Let A be the collection
of all vertex subgroups of G. Since G is the fundamental group of a simply
connected finite graph of groups, the group G is generated by the union
of all vertex subgroups in A. Also, the intersection of H with any vertex
subgroup is finitely generated. Therefore, A and H satisfy condition (1) of
Theorem 2.11. By Theorem 10 in [KS70], the image of each edge subgroup
of G under p has finite index in G/H. Thus, A and H satisfy condition
(2) of Theorem 2.11. Therefore, the distortion DistH
G is dominated by the
A∩H
function f (n) = n max{ DistA (n) | A is a vertex subgroup of G }.

One application of Theorem 2.10 and Theorem 2.11 is to compute the
distortion of collection of normal subgroups of right-angled Artin groups
which all induce infinite cyclic quotient group. We first recall the concept of
right-angled Artin groups and then state the result on divergence of rightangled Artin groups proved by Behrstock-Charney [BC12]. We will use the
divergence of right-angled Artin groups for the lower bound of the distortion
we are working.
Definition 2.13 (Right-angled Artin groups). For each Γ a finite simplicial
graph the associated right-angled Artin group AΓ has generating set S the
vertices of Γ, and relations st = ts whenever s and t are adjacent vertices.
Theorem 2.14 (Behrstock-Charney [BC12]). Let Γ be a connected graph
with at least 2 vertices. AΓ has linear divergence if and only if Γ is a join;
otherwise its divergence is quadratic.
Corollary 2.15. Let Γ be a simplicial connected graph with at least two
vertices. Let p be an arbitrary group homomorphism from AΓ to Z and
N the kernel of the map p. Let Γ′ be the induced subgraph of Γ with the
vertices which are mapped to nonzero numbers in Z by p. If the graph Γ′ is
connected and the star of Γ′ in Γ contains all vertices of Γ, then the subgroup

ON DISTORTION OF NORMAL SUBGROUPS

12

N is finitely generated and the distortion of N in AΓ is at most quadratic.
Moreover, if each vertex of Γ is mapped to a nonzero number in Z, then the
distortion of the N in AΓ is linear when Γ is a join and the distortion is
quadratic otherwise.
Proof. We can assume that the map p is surjective. We let A is the collection of all subgroups Ae , where each Ae is the right-angled Artin subgroup
induced by some edge e with at least one endpoint in Γ′ . We can verify
easily that the collection A and the subgroup N satisfy all hypothesis of
Theorem 2.11. Also each subgroup Ae is isomorphic to Z2 . Therefore, it
is not hard to see that the subgroup N ∩ Ae is finitely generated and the
distortion of N ∩ Ae in Ae is linear. Therefore, the subgroup N is finitely
generated and the distortion of the N in AΓ is at most quadratic.
We now assume that each vertex of Γ is mapped to a nonzero number in
Γ. If Γ is not a join, then the divergence of Γ is quadratic (see Theorem
2.14). Also, the subgroup N is a finitely generated normal subgroup of AΓ .
Therefore, the distortion of HΓ in AΓ is exactly quadratic.
We now assume that Γ is a join of Γ1 and Γ2 . We will prove that the
distortion of the N in AΓ is linear. It is a well-known result that AΓ is
the direct product of AΓ1 and AΓ2 . Since p(AΓ1 ) and p(AΓ2 ) are nontrivial subgroups of Z, there are finite index subgroup A1 of AΓ1 and finite
index subgroup A2 of AΓ2 such that p(A1 ) = p(A2 ) = p(A1 × A2 ). Let
p1 = p|A1 ×A2 : A1 × A2 → p(A1 × A2 ) and N1 is the kernel of p1 . It
is obvious that A1 × A2 is a finite index subgroup of AΓ and N1 is a finite
index subgroup of N . Therefore, it is sufficient to prove that the distortion
of N1 in A1 × A2 is linear.
In fact, let S1 be a finite generating set of A1 and S2 a finite generating
set of A2 . Then, S = S1 ∪ S2 is a finite generating set of A1 × A2 . We
can assume that p(A1 × A2 ) = Z and there are elements a ∈ S1 , b ∈ S2
such that p1 (a) = p1 (b) = 1. Let T1 be the set of all elements of the form
sb−p1(s) where each s is element in S1 . Similarly, let T2 be the set of all
elements of the form sa−p1 (s) where each s is element in S2 . We will prove
that T = T1 ∪ T2 is a finite generating set of N1 . Moreover, we will use finite
generating sets (S, T ) to prove the distortion of N1 in A1 × A2 is linear.
Let K = maxs∈S |p1 (s)|. Let n be an arbitrary positive integer and h
be an arbitrary element in N1 such that |h|S ≤ n. Then we can write
mk
nℓ
n1 n2
1 m2
h = (am
1 a2 · · · ak )(b1 b2 · · · bℓ ) such that:
(1) Each ai is element in S1 and
 each bj is element in S2 .
(2) |m1 | + |m2 | + · · · + |mk | + |n1 | + |n2 | + · · · + |nℓ | ≤ n.
(3) m1 p1 (a1 ) + m
 2 p1 (a2 ) + · · · + mk p1 (ak ) + n1 p1 (b1 ) + n2 p1 (b2 ) +
· · · + nℓ p1 (bℓ ) = 0
Let m = m1 p1 (a1 ) + m2 p1 (a2 ) + · · · + mk p1 (ak ). Then, n1 p1 (b1 ) + n2 p1 (b2 ) +
· · · + nℓ p1 (bℓ ) = −m and |m| ≤ Kn. Since a commutes with each bj , b

ON DISTORTION OF NORMAL SUBGROUPS

13

b

a1

a2

a3

a4

Figure 1
commutes with each ai and a, b commute, we can rewrite h as follows:
mk −m
1 m2
h = (am
)(ba−1 )m (am bn1 1 bn2 2 · · · bnℓ ℓ ).
1 a2 · · · ak b

Also,
mk −m
1 m2
am
= (a1 b−p1 (a1 ) )m1 (a2 b−p1 (a2 ) )m2 · · · (ak b−p1 (ak ) )mk
1 a2 · · · ak b

am bn1 1 bn2 2 · · · bnℓ ℓ = (b1 a−p1 (b1 ) )n1 (b2 a−p1 (b2 ) )n2 · · · (bℓ a−p1 (bℓ ) )nℓ
and ai b−p1 (ai ) , bj a−p1 (bj ) and ba−1 all belong to T . Therefore,


|h|T ≤ |m1 | + |m2 | + · · · + |mk | + |n1 | + |n2 | + · · · + |nℓ | + |m| ≤ (K + 1)n.
1
Therefore, the distortion function DistN
A1 ×A2 is bounded above by (K + 1)n.


Remark 2.16. If the map p in the above corollary maps each vertex of
Γ to 1, then the subgroup N is the Bestvina-Brady subgroup of AΓ (see
[BB97]). We remark that the distortion of Bestvina-Brady subgroups were
also computed by the author in [Tra].
In Corollary 2.15, the hypothesis that the graph Γ′ is connected and its
star in Γ contains all vertices of Γ is also a necessary condition for the
subgroup N to be finitely generated (see Theorem 6.1 in [MV95]).
In the above corollary, we observe that if each vertex of Γ is mapped
to a nonzero number in Z, then the distortion of the subgroup N and the
divergence of the whole group AΓ are equivalent (they are both linear or
both quadratic). However, this fact is no longer true in general when the
image of some vertex of Γ under p is zero (see Proposition 2.17).
The normal subgroups of right-angled Artin groups in Corollary 2.15 all
induce infinite cyclic quotient groups. Therefore, we can use Theorem 4.1
in [Ger94] instead of Theorem 2.10 in this paper to prove the lower bound
of the distortion of N . However, we will examine distortion of normal subgroups of right-angled Artin groups that induce non cyclic quotient groups
in Section 4. In this case, we can only use Theorem 2.10 to compute the
lower bound of the distortion.
Proposition 2.17. Let Γ be the graph in Figure 1. Let p be a group homomorphism from AΓ to Z that maps each ai to 1 and b to zero. Let N be the
kernel of the map p. Then the distortion of N in AΓ is quadratic.

ON DISTORTION OF NORMAL SUBGROUPS

14

Proof. Let Γ1 be the subgraph of Γ induced by a1 , a2 , a3 , and a4 . Therefore,
the group AΓ is the direct product of the subgroup AΓ1 and the cyclic
subgroup generated by b. We also observe that the group N is the direct
product of the Bestvina-Brady subgroup N1 of AΓ1 and the cyclic subgroup
generated by b. Since the distortion of N1 in AΓ1 is quadratic (see Corollary
2.15 or [Tra]), the distortion of N in AΓ is also quadratic.

Remark 2.18. In the above proposition, the divergence of the whole group
AΓ is linear but the distortion of N in AΓ is quadratic. Therefore, Proposition 2.17 provides an example of the non equivalent relation between the
group divergence and normal subgroup distortion.
3. Macura’s examples
Macura [Mac13] introduced a class of CAT(0) groups (i.e. groups which
act properly and cocompactly on some CAT(0) spaces) to study group divergence. We revisit these groups and investigate the distortion of their
normal subgroups to study a connection between group divergence and subgroup distortion. First, we will review the class of groups introduced by
Macura [Mac13].
For each integer d ≥ 2, we define
Gd = ha0 , a1 , · · · , ad |a0 a1 = a1 a0 , a−1
i a0 ai = ai−1 , for 2 ≤ i ≤ di.
The divergence of each group Gd is given by the following theorem.
Theorem 3.1. [Mac13] For each integer d ≥ 2 the divergence of each Gd is
a polynomial of degree d.
We now construct a normal subgroup in each Gd with the distortion the
same as the divergence of Gd . We note that there is another presentation
for each Gd which is given by
Gd = hx1 , x2 , · · · , xd , t|tx1 t−1 = x1 , txi t−1 = xi x−1
i−1 , for 2 ≤ i ≤ di.
We can see that the above two presentations of Gd are equivalent by
the map on generators a0 7→ t−1 , ai 7→ t−1 xi and its inverse t 7→ a−1
0 ,
xi 7→ a−1
a
.
From
the
second
presentation,
we
observe
that
the
subgroup
i
0
Fd of Gd generated by x1 , x2 , · · · , xd is a free group of rank d. Moreover, the
group endomorphism Φd that maps x1 to x1 , xi to xi x−1
i−1 for each 2 ≤ i ≤ d
is a group automorphism with the inverse Ψd that maps xi to xi xi−1 · · · x1
for each 1 ≤ i ≤ d. Obviously, each Gd is a free by cyclic group Fd ⋊Φd Z.
The following proposition shows an upper bound on the distortion of each
subgroups Fd in Gd .
Proposition 3.2. For each integer d ≥ 2 let Gd and Fd be groups defined
as above. Then the subgroup distortion of each Fd in Gd is bounded above
by a polynomial of degree d.
The above proposition can be proved by the following three lemmas.

ON DISTORTION OF NORMAL SUBGROUPS

15

Lemma 3.3 (Gersten [Ger94]). Let G = F ⋊Φ Z be a free by cyclic group,
where F is a free group generated by a set S of d elements x1 , x2 , · · · , xd .
For each positive number n we define
φ(n) = max{ |Φj (xi )|S | 1 ≤ i ≤ d, |j| ≤ n }.
Then the distortion of F in G is dominated by the function nφ(n).
We now use the above proposition and the following two lemmas to show
that the subgroup distortion of each Fd in Gd is bounded above by a polynomial of degree d.
Lemma 3.4. Let Fd be a free group generated by a set S of d elements
x1 , x2 , · · · , xd . Let Φd be a free group automorphism of Fd that maps x1 to
n
x1 and xi to xi x−1
i−1 for each 2 ≤ i ≤ d. Then |Φd (xi )|S is bounded above by
i−1
2n
for each 1 ≤ i ≤ d and each positive integer n.
Proof. We will prove the above lemma by induction on i. For i = 1 we
have Φnd (x1 ) = x1 . Therefore, |Φnd (x1 )|S is exactly 1 and it is obviously
bounded above by 2ni−1 for i = 1. Assume that |Φnd (xi−1 )|S is bounded
above by 2ni−2 for each positive integer n. We need to prove that |Φnd (xi )|S
is bounded above by 2ni−1 for each positive integer n. We now prove this
by induction on n.
For n = 1 we have Φd (xi ) = xi x−1
i−1 . Therefore, |Φd (xi )|S is exactly 2 and
it is obviously bounded above by 2ni−1 for n = 1. Assume that |Φdn−1 (xi )|S
is bounded above by 2(n − 1)i−1 . We have Φnd (xi ) = Φdn−1 (xi x−1
i−1 ) =
 n−1
−1
n−1
n
Φd (xi ) Φd (xi−1 ) . Therefore, |Φd (xi )|S is bounded above by 2(n −
1)i−2 + 2(n − 1)i−1 . Also 2(n − 1)i−2 + 2(n − 1)i−1 = 2(n − 1)i−2 n ≤ 2ni−1 .

Thus, |Φnd (xi )|S is bounded above by 2ni−1 .
Lemma 3.5. Let Fd be a free group generated by a set S of d elements
x1 , x2 , · · · , xd . Let Ψd be a free group automorphism of Fd that maps xi to
xi xi−1 · · · x1 for each 1 ≤ i ≤ d. Then |Ψnd (xi )|S is bounded above by ini−1
for each 1 ≤ i ≤ d and each positive integer n.
Proof. We will prove the above lemma by induction on i. For i = 1 we have
Ψnd (x1 ) = x1 . Therefore, |Ψnd (x1 )|S is exactly 1 and it is obviously bounded
above by ini−1 for i = 1. Assume that each |Ψnd (xk )|S is bounded above by
knk−1 for each k ≤ i − 1 and each positive integer n. We need to prove that
|Ψnd (xi )|S is bounded above by ini−1 for each positive integer n. We now
prove this by induction on n.
For n = 1 we have Ψd (xi ) = xi xi−1 · · · x1 . Therefore, |Ψd (xi )|S is exactly i and it is obviously bounded above by ini−1 for n = 1. Assume
that |Ψdn−1 (xi )|S is bounded above by i(n − 1)i−1 . We have Ψnd (xi ) =
Ψdn−1 (xi xi−1 · · · x1 ) = Ψdn−1 (xi )Ψdn−1 (xi−1 ) · · · Ψdn−1 (x1 ). Also, each |Ψdn−1 (xk )|S
is bounded above by k(n − 1)k−1 for each k ≤ i. Therefore, |Ψnd (xi )|S is
bounded above by 1(n − 1)0 + 2(n − 1)1 + 3(n − 1)2 + · · · + i(n − 1)i−1 . Also

ON DISTORTION OF NORMAL SUBGROUPS

16

1(n − 1)0 + 2(n − 1)1 + 3(n − 1)2 + · · · + i(n − 1)i−1 ≤ ini−1 .
Thus, |Ψnd (xi )|S is bounded above by ini−1 .



Remark 3.6. Theorem 2.10 and Proposition 3.2 give an alternative proof
of the upper bound for divergence of Gd in Theorem 3.1.
Theorem 3.7. For each integer d ≥ 2 let Gd and Fd be groups defined as
above. Then the subgroup distortion of each Fd in Gd is a polynomial of
degree d.
The proof of the above theorem is given by Theorem 2.10 (or Theorem
4.1 in [Ger94]), Theorem 3.1, and Proposition 3.2.
4. On distortion of normal subgroups in right-angled Artin
groups
In this section, we study the distortion of normal subgroups in rightangled Artin groups. Most of our computations are based on Theorem 2.10
and Theorem 2.11.
Definition 4.1. Let Γ1 and Γ2 be two graphs, the join of Γ1 and Γ2 is a
graph obtained by connecting every vertex of Γ1 to every vertex of Γ2 by
an edge. A graph Γ is join if it can be decomposed as the join of its two
subgraphs.
Definition 4.2. Let Γ be a graph with the vertex set S.
(1) A subgraph Γ1 of Γ is dominating if for every vertex v in Γ − Γ1 ,
d(v, Γ1 ) = 1.
(2) A subgraph Γ1 is strongly dominating if there is a vertex u in Γ1
that is adjacent to all vertices in Γ − Γ1 .
(3) A subgraph Γ1 is specially dominating if there is a vertex u in Γ1
that is adjacent to all vertices in Γ − {u}.
(4) Let S1 be a subset of the vertex set S of Γ. The induced subgraph Γ1
with vertex set S1 is the subgraph with vertex set S1 and all edges
in Γ with both endpoints in S1 .
(5) Let A be a collection of subgraphs of Γ. A subgraph Γ1 is generated
by A if it is the induced subgraph with the vertex set S1 which is
the union of vertex sets of all subgraphs in A.
(6) Two subgraphs Γ1 and Γ2 commutes if each vertex in Γ1 is connected
to all vertices in Γ2 and vice versa.
Definition 4.3. Let Γ be a simplicial graph and AΓ the associated rightangled Artin group. Let p be a group epimorphism from AΓ to Z. The living
subgraph L(p) of Γ with respect to p is the induced subgraph of Γ with the
vertices corresponding to generators which are not mapped to 0 by p.

ON DISTORTION OF NORMAL SUBGROUPS

17

Figure 2. Illustration of some right-angles Artin groups
with defining graphs not joins and their generalised BestvinaBrady subgroups.

Figure 3. Illustration of some right-angled Artin groups
with join defining graphs and their generalised BestvinaBrady subgroups.
The following definition is a generalization of Bestvina-Brady subgroups
in [BB97].
Definition 4.4 (Generalized Bestvina-Brady subgroups). Let Γ be a finite
simplicial graph with the vertex set S. Let d be a positive integer that is
less than or equal the number of elements of S. Label each vertex of Γ with
an integer in {1, 2, . . . d}. We construct an epimorphism Φ from AΓ to Zd by
sending each vertex with label i to ei = (0, 0, · · · , 1ith , · · · , 0). The subgroup
HΦ = Ker(Φ) is a generalized Bestvina-Brady subgroup of AΓ with respect
to Φ.
Remark 4.5. The subgroup HΦ is the commutator subgroup if d = |S| and
HΦ is the Bestvina-Brady subgroup (see [BB97]) if d = 1.
For each 1 ≤ i ≤ d let Si be the set of vertices of Γ mapped to ei by Φ.
Let Γi be the induced subgraph of Γ with vertex set Si (i.e. Γi is the union
of all edges of Γ with both endpoints in Si ). We call Γi basis subgraphs of Γ
with respect to Φ.
Theorem 4.6. Let Γ be a finite simplicial graph. Let HΦ be a generalized
Bestvina-Brady subgroup of AΓ . Then HΦ is finitely generated iff each basis
subgraph of Γ with respect to Φ is connected and dominating.
Proof. Assume that the group HΦ is finitely generated. For each i in
{1, 2, · · · d} let pi be a group homomorphism from Zd to Z that maps ei
to 1 and each ej to 0 for j 6= i. It is obvious that the map pi ◦ Φ is a a
group epimorphism from AΓ to Z and each basis subgraph Γi is the living
graph L(pi ◦ Φ) of the group epimorphism pi ◦ Φ. Therefore, each subgraph
is connected and dominating by Proposition 6.2 in [MV95]. The proof of
the opposite direction can be deduced from the proof of Corollary 6.6 in the
same article [MV95].


ON DISTORTION OF NORMAL SUBGROUPS

18

Remark 4.7. The above theorem gives us a necessary and sufficient condition for a generalized Bestvina-Brady subgroup to be finitely generated.
However, finding a finite generating set of a generalized Bestvina-Brady
subgroup and computing its distortion in the right-angled Artin group is
quite difficult. Therefore, we only focus on a subcollection of generalised
Bestvina-Brady subgroups.
Definition 4.8. Let Γ be a simplicial graph and AΓ the associated rightangled Artin group. A generalized Bestvina-Brady subgroup HΦ is strong
if each basis subgraph of Γ with respect to Φ is connected and strongly
dominating. A generalized Bestvina-Brady subgroup HΦ special if each basis
subgraph of Γ with respect to Φ is connected and specially dominating.
Remark 4.9. A special generalized Bestvina-Brady subgroup is obviously
strong. The collection of strong generalized Bestvina-Brady subgroups of
a right-angled Artin group includes its real Bestvina-Brady subgroup. If a
right-angled Artin group AΓ contains a special generalized Bestvina-Brady
subgroup, then Γ must be a join.
Example 4.10. Let Γ be one of the graphs in Figure 2 or Figure 3. Let
Φ be a group homomorphism from AΓ to Z that maps each red vertex to
e1 = (1, 0) and each blue vertex to e2 = (0, 1). Let HΦ be the generalised
Bestvina-Brady subgroup with respect to Φ. Then all HΦ are finitely generated except for the case Γ is the graph on the right of Figure 2. If Γ is
one of the graphs on Figure 3 or the graph on the left of Figure 2, then
the generalised Bestvina-Brady subgroup HΦ is strong. If Γ is the graph in
the middle of Figure 2, then the generalised Bestvina-Brady subgroup HΦ
is not strong. If Γ is the graph on the right of Figure 3, then the generalised
Bestvina-Brady subgroup HΦ is special.
We recall that each Si is the set of vertices of Γ that are mapped to ei
via Φ and each basis subgraph Γi is the induced subgraph of Γ with vertex
set Si . Let Ti be the set of group elements of the form uv −1 , where u and v
are two adjacent vertices in Γi and T the union of all sets Ti . Let HT be a
subgroup of AΓ generated by T . For each i in {1, 2, · · · , d} let pi be a group
homomorphism from Zd to Z that maps ei to 1 and each ej to 0 for j 6= i.
Let K = diam Γ + 2. Before we compute a upper bound of the distortion of
a strong generalized Bestvina-Brady subgroup HΦ in AΓ , we need to prove
several lemmas. For the following two lemmas, we assume that each basis
subgraph Γi is connected and strongly dominating.
Lemma 4.11. Let n be an arbitrary integer and s1 , s2 two elements in some
−n+1
Si . Then the group elements sn1 s2 s1−n−1 and sn1 s−1
both belong to HT
2 s1
and their length with respect to T are both bounded above by 2K |n| + 1 .
n+1 −n−1
). Since Γi is cons1
Proof. We can write sn1 s2 s1−n−1 = (sn1 s−n
2 )(s2
nected, there is a sequence s1 = u1 , u2 , · · · , uℓ = s2 of elements in Si , where

ON DISTORTION OF NORMAL SUBGROUPS

19

ℓ ≤ diam Γ + 1 and uj , uj+1 commutes for each i in {1, 2, · · · , ℓ − 1}. Therefore,
−n
−1 n
−1 n
−1 n
n −n
n −n
n
sn1 s−n
2 = (u1 u2 )(u2 u3 ) · · · (uℓ−1 uℓ ) = (u1 u2 ) (u2 u3 ) · · · (uℓ−1 uℓ ) .

This implies that the element sn1 s−n
belongs to HT and |sn1 s−n
2
2 |T ≤ ℓ|n| ≤
n+1 −n−1
K|n|. Similarly,
the
element
s
s
also
belongs
to
H
and
|sn+1
s1−n−1 |T ≤
T
2
1
2

K |n| + 1 . Therefore, the group element sn1 s2 s1−n−1 belongs to HT and

−n+1
|sn1 s2 s1−n−1 |T ≤ 2K |n| + 1 . Similarly, we can prove sn1 s−1
belongs to
2 s1

−1 −n+1
n
|T ≤ 2K |n| + 1

HT and |s1 s2 s1
Lemma 4.12. Let w be a word in Si and sa generator in Si for some i. Let
m be an arbitrary integer and n = pi ◦ Φ (w). Then the word sm ws−m−n

represents an element in HT and |sm ws−m−n |T ≤ 2K |m|ℓ(w) + ℓ2 (w) . If
s commutes with all generators in the word w, then |sm ws−m−n |T ≤ ℓ(w).
Proof. Assume w = sn1 1 sn2 2 · · · snℓ ℓ , where ℓ is the length of w, each sj is
element in Si , nj = 1 or − 1, and n1 + n2 + · · · + nℓ = n. We can write
sm ws−m−n = (sm sn1 1 s−m−n1 )(sm+n1 sn2 2 s−m−n1 −n2 ) · · · (sm+n1 +n2 +···+nℓ−1 snℓ ℓ s−m−n1 −n2 −···−nℓ ).
By Lemma 4.11, the word sm ws−m−n represents an element in HT and



|sm ws−m−n |T ≤ 2K |m| + 1 + 2K |m + n1 | + 1 + · · · + 2K |m + n1 + n2 + · · · + nℓ−1 | + 1



≤ 2Kℓ |m| + ℓ ≤ 2Kℓ(w) |m| + ℓ(w) ≤ 2K |m|ℓ(w) + ℓ2 (w) .
If s commutes with all generators in the word w, then
sm ws−m−n ≡AΓ sn1 1 sn2 2 · · · snℓ ℓ s−n
≡AΓ (s1 s−1 )n1 (s2 s−1 )n2 · · · (sℓ s−1 )nℓ .
Therefore, |sm ws−m−n |T ≤ ℓ(w).



Proposition 4.13. Let Γ be a finite simplicial graph. Let HΦ be a generalized Bestvina-Brady subgroup of AΓ . If the generalized Bestvina-Brady
subgroup HΦ is strong, then the distortion of HΦ in AΓ is at most quadratic.
Moreover, if the generalized Bestvina-Brady subgroup HΦ is special, then the
distortion of HΦ in AΓ is linear.
Proof. For each positive integer n let h be an arbitrary element in HΦ such
that |h|S ≤ n. There is a word w = w1 w2 · · · wk in S that represent h with
the following properties:
(1) The length of w is less than or equal n.
(2) For each j in {1, 2, · · · , k} the subword wj can be decomposed as
(1) (2)
(d)
(i)
wj = uj uj · · · uj , where each uj is an empty word or a word in
Si .
We will modify w to obtain a new word w = w 1 w 2 · · · wk that also represents h with the following properties:

ON DISTORTION OF NORMAL SUBGROUPS

20

For each j in {1, 2, · · · , k} the subword wj can be decomposed as wj =
(1) (2)
(d)
(i)
uj uj · · · uj , where each uj is an empty word or a word in Si such that
(i)
(i)
(i)
(i) 
uj represent an element in HT and |uj |T ≤ 2K nℓ(uj ) + ℓ2 (uj ) .
Since Γ1 is strongly dominating, then there is a vertex s1 in S1 that
(1)
commutes with all elements in S − S1 . For each j in {1, 2, · · · , k} each uj
(1)
(1)
is an empty word or a word in S1 . Let nj = p1 ◦ Φ(uj ). Then |nj | ≤ ℓ(uj )
and n1 + n2 + · · · + nk = 0. For each j in {0, 1, 2, · · · , k} we let m0 = 0 and
mj = n1 + n2 + · · · + nj . We observe that mj = mj−1 + nj and mk = 0.
m
(1)
(1) −m
(1)
Let uj = s1 j−1 uj s1 j . By Lemma 4.12, uj represent an element in HT
(1)
(1)
(1) 
(1)
(1) 
and |uj |T ≤ 2K |mj−1 |ℓ(uj ) + ℓ2 (uj ) ≤ 2K nℓ(uj ) + ℓ2 (uj ) .
We note that the vertex s1 in S1 that commutes with all elements in
S − S1 . Therefore,
−mj−1 mj−1 (1) −mj mj (2)
(d)
s1
)uj (s1 s1 )uj · · · uj
−m
m
m (2)
(1) −m
(d)
s1 j−1 (s1 j−1 uj s1 j )(s1 j uj · · · uj )
−m
(1) (2)
(d) m
s1 j−1 (uj uj · · · uj )s1 j .

wj ≡AΓ (s1
≡ AΓ
≡ AΓ

This implies that h can be represented by the word
(1) (2)

(d)

(1) (2)

(d)

(1) (2)

(d)

(u1 u1 · · · u1 )(u2 u2 · · · u2 ) · · · (uk uk · · · uk ).
By repeating the same process, the element h can be represented by a
word w = w 1 w 2 · · · wk with the following property:
For each j in {1, 2, · · · , k} the subword wj can be decomposed as wj =
(1) (2)
(d)
(i)
uj uj · · · uj , where each uj is an empty word or a word in Si such that
(i)
(i)
(i)
(i) 
uj represent an element in HT and |uj |T ≤ 2K nℓ(uj )+ℓ2 (uj ) . Therefore, the word w j represents elements in HT and |w j |T is bounded above by
P
P
(i)
(i) 
2K n di=1 ℓ(uj )+ di=1 ℓ2 (uj ) . This implies that |w j |T is bounded above

by 2K nℓ(wj ) + ℓ2 (wj ) . Thus, h is an element in HT and |h|T is bounded

P
P
P
above by 2K n kj=1 ℓ(wj ) + kj=1 ℓ2 (wj ) . Since kj=1 ℓ(wj ) ≤ n, |h|T is
bounded above by 4Kn2 . Therefore, the distortion of HΦ in AΓ is at most
quadratic.
We now assume that the generalized Bestvina-Brady subgroup HΦ is special. Then for each basis subgraph Γi of Γ with respect to Φ there is a vertex
si that commutes with all vertices of Γ. Therefore, the length of the element
(i)
(i)
in HT represented by the word uj must be less than or equal to ℓ(uj ) by
Lemma 4.12. Therefore, the length of the element in HT represented by
the word w j is bounded above by ℓ(wj ). This implies that |h|T is bounded
above by ℓ(w). Therefore, the distortion of HΦ in AΓ is linear.


ON DISTORTION OF NORMAL SUBGROUPS

x1

x2

x3

x4

x5

x6

21

Figure 4. Illustration of some right-angled Artin group
with defining graphs not joins and their generalised BestvinaBrady subgroups which are not strong.
We now compute the lower bound on the distortion of a strong generalized
Bestvina-Brady subgroup HΦ . The following proposition is deduced from
Theorem 2.10, Theorem 2.14, and Proposition 4.13.
Proposition 4.14. Let Γ be a simplicial connected graph with at least 2
vertices. Let HΦ be a strong generalized Bestvina-Brady subgroup. If the
graph Γ is not a join, then the distortion of HΦ in AΓ is quadratic.
Remark 4.15. In the above proposition, we see the equivalence between
subgroup distortion of a strong generalized Bestvina-Brady subgroup HΦ
in AΓ and the divergence of group AΓ . However, this equivalence does not
always occur if Γ is a join graph. In the rest of this section, we will investigate
the distortion of strong generalized Bestvina-Brady subgroups.
Theorem 4.16. Let Γ be a simplicial connected graph. Let HΦ be a strong
generalised Bestvina-Brady subgroup of AΓ with basis subgraphs {Γi }1≤i≤d .
Then, the distortion of HΦ in AΓ is quadratic if there is a non-empty subset
I of {1, 2, · · · , d} such that the subgraph Γ′ generated by {Γi }i∈I is not a
join. Moreover, the distortion of HΦ in AΓ is linear if one of the following
conditions holds:
(1) The generalised Bestvina-Brady subgroup HΦ is special.
(2) Each basis subgraphs in {Γi }1≤i≤d is a join and each pair of basis
subgraphs in {Γi }1≤i≤d commutes.
Proof. We assume that there is a non-empty subset I of {1, 2, · · · , d} such
that the subgraph Γ′ generated by {Γi }i∈I is not a join. We will prove that
the distortion of HΦ in AΓ is quadratic. We recall that each Si is the set
of vertices of Γ that is mapped to ei = (0, 0, · · · , 1ith , · · · , 0) in Zd via Φ
and each basis subgraph Γi is the induced subgraph of Γ with vertex set Si .
Each Ti is the set of group elements of the form uv −1 , where u and v are
two adjacent vertices in Γi and T is the union of all sets Ti . Let S ′ be the
union of all sets in {Si }i∈I . Then Γ′ is the induced subgraph with vertex set
S ′.
′
Let d′ be the number of elements in I. We can consider Zd as the subgroup
of Z d generated by {ei }i∈I . Then there is a group homomorphism Φ′ from

ON DISTORTION OF NORMAL SUBGROUPS

22

′

Γ′ to Zd such that Φ′ = Φ|AΓ′ . Therefore, the generalised Bestvina-Brady
subgroup HΦ′ of AΓ′ is also the subgroup of HΦ and {Γi }i∈I is the collection
of basis subgraphs of Γ with respect to Φ′ . Since each graph Γi is connected
and strongly dominating in Γ, each graph in {Γi }i∈I is also connected and
strongly dominating in Γ′ . Therefore, the generalised Bestvina-Brady subgroup HΦ′ of AΓ′ is strong. Also, the subgraph Γ′ is not a join. Therefore,
the distortion of HΦ′ in AΓ′ is quadratic by Proposition 4.14. Also, the
distortion of AΓ′ in AΓ is linear. Therefore, the distortion of HΦ′ in AΓ is
quadratic.
We now prove that the distortion of HΦ′ in HΦ is linear. Let T ′ be the
union of all sets in {Ti }i∈I . Then T ′ is a finite generating set for HΦ′ . There
is a group homomorphism f from AΓ to AΓ′ that maps each vertex of Γ − Γ′
to the identity and each vertex of Γ′ to itself. Therefore, the restriction of
f on AΓ′ is the identity map. Moreover, the maps f maps each element in
T ′ to itself and element in T − T ′ to the identity. This fact implies that the
distortion of HΦ′ in HΦ is linear. Therefore, the distortion of HΦ in AΓ is
least quadratic. By Proposition 4.13, the distortion of HΦ in AΓ is exactly
quadratic.
If HΦ is a special generalised Bestvina-Brady subgroup, then the distortion of HΦ in AΓ is linear by Proposition 4.13. We now assume that each
basis subgraphs in {Γi }1≤i≤d is a join and each pair of basis subgraphs in
{Γi }1≤i≤d commutes. In this case, the group AΓ can be written as the direct
product AΓ1 × AΓ2 × · · · × AΓd and the subgroup HΦ can be written as the
direct product HΓ1 × HΓ2 × · · · × HΓd , where each HΓi is the Bestvina-Brady
subgroup of AΓi . Since each graph Γi is a join, the distortion of HΓi in AΓi
is linear (see [Tra]). Therefore, the distortion of HΦ in AΓ is linear.

Remark 4.17. By the above theorem, we observe that the group divergence and the subgroup distortion can be equivalent for some cases (see the
graph on the left of Figure 2 or the graph on the right of Figure 3 for example). However, the group divergence and the subgroup distortion are not
necessarily equivalent (see the graph on the left of Figure 3 for example).
We now compute the distortion of one generalised Bestvina-Brady subgroup which is not strong. More precisely, we compute the distortion of the
generalised Bestvina subgroup illustrated by the middle graph of Figure 2
(we call graph Γ). We label each vertex of this graph as in Figure 4. Obviously, the graph Γ is not a join. Therefore, the distortion of the subgroup
HΦ is at least quadratic by Theorem 2.10 and Theorem 2.14. We now use
Theorem 2.11 to prove the upper bound for the distortion of HΦ . Let G1
be the subgroup generated by {x1 , x4 , x5 }, G2 the subgroup generated by
{x1 , x2 , x5 }, G3 the subgroup generated by {x2 , x5 , x6 }, and G4 the subgroup
generated by {x2 , x3 , x6 }. Then the collection A = {G1 , G2 , G3 , G4 } and the
normal subgroup HΦ satisfy the hypothesis of Theorem 2.11. It is not hard
to see that the distortion of each subgroup Gi ∩ HΦ is undistorted in Gi .

ON DISTORTION OF NORMAL SUBGROUPS

23

Therefore, the distortion of the subgroup HΦ in AΓ is at most quadratic.
Thus, the distortion of the subgroup HΦ in AΓ is exactly quadratic.
Conjecture 4.18. Let Γ be a simplicial graph and HΦ some finitely generated generalised Bestvina-Brady subgroup of AΓ . Then the distortion of HΦ
in AΓ is linear or quadratic.
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Hung Cong Tran. Geometric embedding properties of Bestvina-Brady subgroups. Submitted. arXiv:1606.00539.
Department of Mathematics, The University of Georgia, 1023 D. W. Brooks
Drive, Athens, GA 30605, USA
E-mail address: hung.tran@uga.edu

