On Palindromic Widths of Nilpotent and Wreathe Products

0.15 MB·English

by  Valeriy G. Bardakov

ON PALINDROMIC WIDTHS OF NILPOTENT AND WREATHE
PRODUCTS
VALERIYG. BARDAKOV,OLEGV. BRYUKHANOV,AND KRISHNENDUGONGOPADHYAY
5
1
0 Abstract. We prove that nilpotent product of a set groups A1,...,As has finite palin-
2 dromic width if and only if palindromic widths of Ai,i = 1,...,s, are finite. We give a
newproofthatthecommutatorwidthofFn≀K isinfinite,whereFn isafreegroupofrank
r
p n≥2andK a finitegroup. Thisresult, combiningwith aresult ofFink [12]gives exam-
A ples of groups with infinite commutator width but finite palindromic width with respect
to some generating set.

2

]
R
G 1. Introduction
h. Let G be a group with a set of generators X. A reduced word in the alphabet X±1 is a
t palindrome if it reads the same forwards and backwards. The palindromic length l (g) of
a P
m an element g in G is the minimum number k such that g can be expressed as a product of
k palindromes. The palindromic width of G with respect to X is defined to be pw(G,X) =
[
sup lP(g). When there is no confusion about the underlying generating set X, we simply
2
g∈G
v
denotethepalindromicwidthwithrespecttoX bypw(G). Palindromicwidthoffreegroups
0
was investigated by Bardakov, Shpilrain and Tolstykh [7] who proved that the palindromic
7
1 width of a non-abelian free group is infinite. This result was generalized by Bardakov and
5 Tolstykh [8] who proved that almost all free products have infinite palindromic width; the
0
only exception is given by the free product of two cyclic groups of order two, when the
.
1
palindromic width is two. Piggot [19] studied the relationship between primitive words and
0
palindromes in free groups of rank two. Recently, there have been a series of work that aim
5
1 to understand palindromic widths in several other classes of groups. In a series of papers
: [4, 5, 6] the authors have proved finiteness of palindromic widths of finitely generated free
v
i nilpotentgroupsandcertainsolvablegroups. Someboundsofthewidthswerealsoobtained
X
in many cases. Palindromic widths of wreath products and Grigorchuk groups have been
r
a investigated byFink[12,14]. RileyandSalehaveinvestigated palindromicwidthsincertain
wreath products and solvable groups [21] using finitely supported functions from Zr to the
given group. Fink and Thom [13] have studied palindromic widths in simple groups.
For g, h in G, the commutator of g and h is defined as [g,h] = g−1h−1gh. If C is the
set of commutators in some group G then the commutator subgroup G′ is generated by C.
The commutator length l (g) of an element g ∈ G′ is the minimal number k such that g
C
Date: April 3, 2015.
2000 Mathematics Subject Classification. Primary 20F16; Secondary 20F65, 20F19, 20E22.
Key words and phrases. palindromic width;commutator width; wreath products; nilpotent product.
The authors gratefully acknowledge the support of the Indo-Russian DST-RFBR project grant
DST/INT/RFBR/P-137.
Bardakov is partially supported by Laboratory of Quantum Topology of Chelyabinsk State University
(Russian Federation government grant 14.Z50.31.0020) .
1
2 VALERIYG.BARDAKOV,OLEGV.BRYUKHANOV,ANDKRISHNENDUGONGOPADHYAY
can be expressed as a product of k commutators. The commutator width of G is defined
by sup l (g) and is denoted by cw(G). Commutator width in groups have been studied
C
g∈G′
by several authors, for eg. see [1, 2, 20]. It is well known [20] that the commutator width
of a free non-abelian group is infinite, but the commutator width of a finitely generated
nilpotent group is finite. An algorithm of the computation of the commutator length in
free non-abelian groups can be found in [2].
In this paper, we address two problems related to palindromic widths in groups. First,
we investigate the palindromic widths of nilpotent products of groups. Bardakov and
Gongopadhyay [4, 5] proved that the palindromic width of a free nilpotent group is finite.
We extend this result for nilpotent products of groups. Recall that the concept of nilpotent
productsarises fromthework of Golovin [9], also see [10,11]. Theconstruction of nilpotent
productsalso follow fromtheconstruction of socalled verbalproductsof groups,see Moran
[15,16,17]. Thisconstruction appearedtoansweraquestionbyKuroshwhoasked whether
there are any other products, other than the free and the direct products of groups, which
also have the following properties:
(a) the products are commutative;
(b) the products are associative;
(c) there are generating subgroups of the products, that is, given a product G, it has a
subgroup S such that G= hSi.
(d) the intersection of a given one of these subgroups with the normal subgroups gen-
erated by the rest of these subgroups is the identity.
Theconstructionofthenilpotentproductsgeneralizethefreeanddirectproductsofgroups.
Every nilpotent group is a quotient of a nilpotent product. In this paper we prove that any
nilpotent product of set of groups A = hX i,...,A = hX i has finite palindromic width
1 1 s s
s
with respectto thegenerating subset X if and only if pw(A ,X ),i = 1,...,s, arefinite.
i i i
iS=1
We prove this result in Section 3.
The later part of the paper addresses relationship between commutator and palindromic
widths. In [4, Problem 2] we ask for the relationship between commutator and palindromic
widths of groups and provided some relationship between them in [6]. We know that the
palindromic and commutator widths of the free non-abelian group F = hXi is infinite. On
n
the other side, Akhavan-Malayeri [1] proved that the commutator width of wreath product
F ≀ Zm is finite. Analogous result for the palindromic width has been proved by Fink
n
[12]. Recently, Fink and Thom [13] have shown that there exists finitely generated simple
groups havinginfinitecommutator width butfinite palindromicwidthwith respect to some
generating set.
Ontheotherhand,forK afinitegroup,Fink[12]hasprovedthatthereexistsagenerating
set S such that pw(F ≀K,S) is finite. We show that for K a finite group, the commutator
n
width of F ≀K, n ≥ 2, is infinite. This result is already known from the work of Nikolov
n
[18]. However, we give a different proof of this result using standard ideas of constructing a
quasi-homomorphism,for eg. see [20,7]. ThusF ≀K provides firstexampleof anon-simple
n
groupthathasinfinitecommutator widthbutfinitepalindromicwidthwithrespecttosome
generating set.
For completeness of this paper, we demonstrate Fink’s ideas by considering the simple
example of F ≀S where S is the symmetric group of three letters. Using Fink’s ideas we
2 3 3
ON PALINDROMIC WIDTHS OF NILPOTENT AND WREATHE PRODUCTS 3
show that the palindromic width of this group is at most 20 with respect to the canonical
setofgenerators. Thisisactually animprovementoftheboundofFinkthatturnsouttobe
40 in this case. In [6], we proved that the palindromic width of finite extension of a group
with finite palindromic width is finite. But here we see that F ≀S is a finite extension of
2 3
the group F6 that has infinite palindromic width. So, this gives example of a group that
2
has infinite palindromic width but finite extension has finite palindromic width.
2. Two approaches to Palindromes
There are two notions of palindromic words in a group that have been implicit in recent
literature. In the following we compare these notions.
First notion. Let A bean alphabet. A word in these alphabets is a sequence of letters
u= a a ...a , a ∈ A. An empty word is denoted by 1. A word u= a a ...a is equal to
1 2 k i 1 2 k
a word v = b b ...b if k = l and a = b , a = b ,...,a = b . Let u¯ denote the reverse
1 2 l 1 1 2 2 k k
word u¯ = a a ...a . We say that u is a palindrome with respect to A if u = u¯, i.e.
k k−1 1
a = a ,a = a ,...,a = a ,....
1 k 2 k−1 i k−i+1
LetGbeagroupwithageneratingsetA. WeassumethatAissymmetric, i.e. A= A−1.
Let A∗ bethe free monoid over the alphabet A. There is a homomorphismρ :A∗ → G that
sends every word a a ...a to some element in G. Evidently, for an element g ∈ G, there
1 2 k
are a lot of words u in A∗ such that ρ(u) = g; we denote this set of elements by ρ−1(g).
An element g in G is a palindrome (or, word palindrome) with respect to A if there is a
palindrome in the set ρ−1(g) ⊂ A∗. In this paper we shall follow this notion.
Second notion. This notion has been used by Fink [12]. We call an element g in G a
group palindrome ifitis representedbyawordg = a a ...a ,a ∈ A, suchthatg¯represent
1 2 k i
the same element in G. Evidently if g is a word palindrome then it is a group palindrome.
But the converse is not true. In [12, Lemma 3.1] Fink has implicitly used this notion to
assert that g = g¯ in any Abelian group. Using the first notion, it is not true that g = g¯ in
an Abelian group, see the following example.
Example 2.1. Let G= Z⊕n be the free abelian group of rank n. Let {a ,a ,...,a } be a
1 2 n
basisofG. Usingthefirstdefinitionweprovedthatpw(G,A) = n,see[4]. Usingthesecond
definition we see that every element of g is a palindrome, since aα1...aαn = aαn...aα1
1 n n 1
and hence for any g = aα1...aαn in G we have g = g¯. Hence by the second definition
1 n
palindromic width of G is 1.
If we denote by P the set of group palindromes and P the set of word palindromes
G A
in A, then P ⊂ P . Also, the palindromic width with respect to the group palindromes
A G
does not exceed pw(G,A). If F is a free group with basis X, then P = P .
X F
It would be interesting to compare the results of Fink [12, 13] from the above point of
views.
3. Palindromic width of nilpotent product of groups
Bardakov and Gongopadhyay [4, 5] investigated the palindromic width of the free nilpo-
tent groups. In this section the palindromic width of nilpotent product of groups is inves-
tigated. Recall the construction of the nilpotent product of groups. This construction was
defined in the paper of Golovin [9].
4 VALERIYG.BARDAKOV,OLEGV.BRYUKHANOV,ANDKRISHNENDUGONGOPADHYAY
Let A ∗ B be the free product of some groups A and B. Cartesian subgroup and
lower central series of A ∗ B are denoted by [A,B] and γ (A ∗ B),n = 1,2,..., respec-
n
tively. The n-th nilpotent product G = A(n)B, n ≥ 2, is defined as the quotient
n
A∗B/[A,B]∩γ (A∗B). It is clear that A(1)B = A×B is the direct sum.
n+1
Let us list some common properties of the nilpotent product G from [9].
n
(1) A,B ≤ G , AGn ∩B = 1, A∩BGn = 1 and G = hA,Bi.
n n
(2) Any element g ∈G can be uniquely written as a product a·b·w(g), where a ∈ A,
n
b ∈ B, w(g) ∈ [A,B].
(3) Given normal subgroups A E A, B E B so that A ,B E G , let A = A/A ,
0 0 0 0 n 0
B = B/B . Then the group homomorphisms A → A/A and B → B/B are
0 0 0
extended to the homomorphism of the nilpotent products Φ :G → A(n)B with
n
Ker Φ = A B where A B ∩[A,B] = 1.
0 0 0 0
(4) [g ,g ,...,g ] = 1 where g ∈ A∪B.
1 2 n+1 i
(5) (A(n)B)(n)C = A(n)(B(n)C) = (A(n)C)(n)B.
For subsets X, Y in G , let C (Y) denote the centralizer of Y in X.
n X
Lemma 3.1. Given nilpotent product G , the following holds.
n
(i) γ (A) ≤ C (B), γ (B) ≤ C (A).
n A n B
(ii) C (B),C (A)EG .
A B n
(iii) C (B)C (A)∩[A,B] = 1.
A B
Proof. (i) It follows from the inclusions [γ (A),B],[γ (B),A] ⊆ [A,B]∩γ (A∗B)= 1.
n n n+1
(ii) Let us prove that the subgroupsC (B) and C (A) are characteristic subgroups in A
A B
and B accordingly. Note that any pair of automorphisms ϕ ∈ AutA and ψ ∈AutB induce
the automorphism ζ ∈ Aut(A∗ B). Given commutator [a,b] ∈ [A,B] ∩γ (A∗ B) we
n+1
have, [aϕ,bψ] = [a,b]ζ ∈([A,B]∩γ (A∗B))ζ = [A,B]∩γ (A∗B). It follows that the
n+1 n+1
subgroups C (B) and C (A) are characteristic subgroups in A and B accordingly and so
A B
C (B)EA and C (A)EB. This implies that C (B),C (A)EG .
A B A B n
(iii) This statement is clear following the proved statement (ii) and property (3) of the
list above. (cid:3)
It is clear that groups A = A/C (B), B = B/C (A) are nilpotent of step ≤ n−1 and
A B
nilpotent product G = A(n)B is nilpotent of step ≤ n. Let X and Y be generating sets
n
of groups A and B with |X|= m , |Y| = m .
A B
Theorem 3.2. Let A and B are finitely generated groups with generating set X and Y
respectively. Given the nilpotent product G = A(n)B, the following holds.
n
(i) max{pw(A,X),pw(B,Y)} ≤ pw(G ,X∪Y) ≤ pw(A,X)+pw(B,Y)+3(m +m ).
n A B
(ii) If A = C (B) or B = C (A) then
A B
max{pw(A,X),pw(B,Y)} ≤ pw(G ,X ∪Y) ≤ pw(A,X)+pw(B,Y).
n
Proof. (i) Lemma 3.1 and property (3) imply that there is ahomomorphism G → G with
n n
kernel C (B)C (A)EG and C (B)C (A)∩[A,B] = 1. Let X and Y be generating sets
A B n A B
of groups A and B, and let X and Y be the corresponding generating sets of groups A and
B.
ON PALINDROMIC WIDTHS OF NILPOTENT AND WREATHE PRODUCTS 5
Next, let X ⊆ X and Y ⊆ Y be sets of representatives of elements of X and Y
0 0
respectively. Let g ∈G be homomorphic image of g ∈ G , then g = p (X,Y)···p (X,Y)
n n 1 s
and
g = p (X ,Y )···p (X ,Y )·ab
1 0 0 s 0 0
for some palindromes p (X,Y), a ∈ C (B), b ∈ C (A). Hence,
i A B
pw(G ,X ∪Y)≤ pw(A,X)+pw(B,Y)+pw(G ,X ∪Y).
n n
The nilpotent group G is the homomorphic image of some free nilpotent group N with
n m,s
rank m = m +m and step s ≥ n. Therefore, pw(G ,X∪Y)≤ pw(N ) ≤ 3(m +m ).
A B n m,s A B
The last inequality follows from [4, Theorem 1.1]. Further, A×B is homomorphic image
of G and groups A and B are homomorphic images of A×B, so
n
max{pw(A,X),pw(B,Y)} ≤ pw(A×B,X ∪Y) ≤ pw(G ,X ∪Y).
n
Finally,
max{pw(A,X),pw(B,Y)} ≤ pw(G ,X∪Y) ≤ pw(A,X)+pw(B,Y)+3(m +m ).
n A B
(ii) If A= C (B) or B = C (A) then G =A×B and so
A B n
max{pw(A,X),pw(B,Y)} ≤ pw(G ,X ∪Y) ≤ pw(A,X)+pw(B,Y).
n
This proves the lemma. (cid:3)
Using properties (1) – (5), one can define nilpotent products
(n){A ,...,A } =(...(A (n)A )(n)...)(n)A
1 s 1 2 s
of a set of groups A ,...,A inductively that also preserve properties (1) – (5), see [9].
1 s
Let A = hX i, C = C (A ∪...∪A ∪A ∪A ∪...∪A ), A = A /C , A = hX i,
i i i Ai 1 i−1 i i+1 s i i i i i
where Xi is the homomorphic image of Xci and, |Xi| = mi for all i = 1,...,s. It is
clear that C = C (A ). Then, it is followed from Lemma 3.1 that γ (A ) ≤ C
i Ai k n i i
kT6=i
and C E (n){A ,...,A }. Hence, (n){A ,...,A }/C ···C = (n){A /C ,...,A /C }
i 1 s 1 s 1 s 1 1 s s
s s
is a nilpotent group with generating set X that consists m elements. Finally, if
i i
iS=1 iP=1
A = C , then (n){A ,...,A } = A ×(n){A ,...,A ,...,A }. Further, arguments same
k k 1 s k 1 k s
as in theorem 3.2(i, ii) prove the followed statement.c
Theorem 3.3. Given the nilpotent product G = (n){A ,...,A }, the following holds.
n 1 s
s
s s
(i) max {pw(A ,X )} ≤pw(G , X ) ≤ pw(A ,X )+3 m .
i i n i i=1 i i i=1 i
i=1,...,s iS=1 P P
(ii) If A = C , G (A ) = (n){A ,...,A ,...,A } then
k k n k 1 k s
c c s
max{pw(A ,X ),pw(G (A ), X )≤ pw(G , X ) ≤ pw(A ,X )+pw(G (A ), X ).
k k n k [ i n [ i k k n k [ i
c i6=k i=1 c i6=k
The last theorem means that nilpotent product (n){A ,...,A } has finite palindromic
1 s
s
width with respect to the generating set X if and only if all palindromic widths
i
iS=1
pw(A ,X ), i = 1,...,s, are finite.
i i
6 VALERIYG.BARDAKOV,OLEGV.BRYUKHANOV,ANDKRISHNENDUGONGOPADHYAY
4. On Commutator and Palindromic Widths
Definition 4.1. LetGbeagroup. Wesaythatamapf :G → Zisaquasi-homomorphism
if there is some constant c such that for every g,h ∈ G, we have
|f(gh)−f(g)−f(h)| ≤ c,
Define a function tr :Z → {−1,0,1} by
−1 if m ≡ −1 mod 3,

tr(m)= 0 if m ≡ 0 mod 3,

1 if m ≡ 1 mod 3.

The following is easy to prove.
Lemma 4.2. If m,n ∈ Z, then
(1) tr(m)+tr(n)−3 ≤ tr(m+n)≤ tr(m)+tr(n)+3.
(2) tr(−m)=−tr(m).
Let F = hx ,...,x i, n ≥ 2 be the free group of rank n. Let w = xα1xα2...xαt be a
reducednword in1 F . Dnefine ql : F → Z by i1 i2 it
n n
t
ql(w) = tr(α ).
i
X
i=1
The following properties of ql follows from the work of Bardakov [2] who uses ideas of
Rhemtulla [20] to prove these properties.
Lemma 4.3. For f,g ∈ F ,
n
(i) ql(f)+ql(g)−3≤ ql(fg) ≤ ql(f)+ql(g)+3.
(ii) ql(g−1)= −ql(g).
(iii) −9≤ ql([f,g]) ≤9.
Proof. (i) Follows from (1) of Lemma 4.2. We prove (ii).
Let g = xα1xα2...xαt. Then
i1 i2 it
ql(g) = tr(α )+tr(α )+···+tr(α ).
1 2 t
Take g−1 = x−αtxαt−1...xα1. Then
it it−1 i1
ql(g−1) = tr(−α )+···+tr(−α ) = −ql(g).
t 1
(iii) follows from (i) and (ii). (cid:3)
Theorem 4.4. Let K be a finite group and F be a free group of rank n ≥ 2. The
n
commutator width of F ≀K is infinite.
n
Let G = F ≀K. Let K = {k ,...,k }. To prove the above theorem, we shall use ql to
n 1 l
define a quasi-homomorphism ∆ : G → Z on G. Note that G = P ⋊K, where P is the
l
direct product P = F , each F is an isomorphic copy of F . Thus, every element
i=1 ki ki n
Q
g ∈ G has a form g = (f ,f ,...,f )k, where k ∈ K, f ∈ F . The group K acts on
k1 k2 kl ki n
P by the natural action: (fk1,...,fkl)k = (fk−1k1,...,fk−1kl), and this further induces an
action of the symmetric group Sl on P: (fk−1k1,...,fk−1kl) = (fkσ(1),...,fkσ(l)), for some
σ ∈ S .
l
ON PALINDROMIC WIDTHS OF NILPOTENT AND WREATHE PRODUCTS 7
Define a quasi-homomorphism ∆ : G → Z by
l
For g = (f ,f ,...,f )k ∈ G let, ∆(g) = ql(f ).
k1 k2 kl X ki
i=1
It follows from the above lemmas using standard methods that:
Lemma 4.5. Let g,h ∈ G. Then the following holds.
(1) |∆(gh)−∆(g)−∆(h)| ≤ 3l.
(2) |∆(g)+∆(g−1)| ≤ 3l.
(3) |∆([g,h])| ≤ 15l.
Proof. Let g = (f ,...,f )k, h= (f′ ,...,f′ )k′. Then
k1 kl k1 kl
gh = (f f′ ,...,f f′ )kk′,
k1 kk1 kl kkl
and we have,
l l l
|∆(gh)−∆(g)−∆(h)| = | ql(f f′ )− ql(f )− ql(f′ )|
X ki kki X ki X ki
i=1 i=1 i=1
l
≤ |ql(f f′ )−ql(f )−ql(f′ )|
X ki kki ki ki
i=1
≤ 3l (by Lemma 4.3).
For (2), take h = g−1 in (1). Noting that ∆(1) = 0 the inequality follows.
(3) follows from (1) and (2):
|∆([g,h])−∆(g−1)−∆(h−1)−∆(g)−∆(h)| ≤ 9l.
This implies, |∆([g,h])| ≤ 15l. (cid:3)
Corollary 4.6. If g in G is a product of m commutators, then |∆(g)| ≤ 3l(6m−1).
Proof. By repeated application of the above lemma,
|∆(g)| ≤ 15lm+3l(m−1) = 3l(6m−1). (cid:3)
4.1. Proof of Theorem 4.4. We consider the sequence q = (a ,1,...,1) in G, where
j j
a = x−3jx−3j(x x )3j ∈F ⊂ G. Then
j 2 1 2 1 k1
∆(q ) = ql(x−3jx−3j(x x )3j) = 6j.
j 2 1 2 1
So,
∆(q ) < ∆(q ) < ... < ∆(q ) < ...
1 2 j
Thus the sequence of elements q is unbounded on the ∆-values. On the other hand, by
j
Corollary 4.6, every element of g that is a product of bounded number of commutators,
must have a finite ∆-value. This shows that G can not have a bounded commutator width.
8 VALERIYG.BARDAKOV,OLEGV.BRYUKHANOV,ANDKRISHNENDUGONGOPADHYAY
4.2. Palindromic width of F ≀S . Let S be the symmetric group of three symbols. It
2 3 3
is isomorphic to the Dihedral group
S = hs ,s | s2 = s2 = (s s )3 = 1i.
3 1 2 1 2 1 2
Following Fink [12], we add a new generator c= s s to the generating set of S :
1 2 3
S = hs ,s ,c | s2 =s2 = (s s )3 = 1, c = s s i.
3 1 2 1 2 1 2 1 2
Let G =F ≀S . Express an element f ∈ G in the form:
2 3
f = (f ,f ,f ,f ,f ,f )s,
1 2 3 4 5 6
where f = xαiyβig , α ,β ∈ Z, g ∈ F′, s ∈ S . Let F6 denote the direct product of six
i i i i i 2 3 2
copies of F .
2
Lemma 4.7. Every element of F6 can be written as a product of 18 palindromes with
2
respect to the generating set {x,y,s ,s ,c}.
1 2
Proof. Note that the element
(xα1yβ1,xα2yβ2,...,xα6yβ6)= (xα1,xα2,...,xα6)(yβ1,yβ2,...,yβ6),
is a product of twelve palindromes.
First, we consider the case g is a commutator to demonstrate the general method. Let
g = ([x,y],1,1,1,1,1). Let r = c(s s )2. Then r = 1 in S . But r¯= (s s )2c 6= 1. We can
1 2 3 2 1
write [x,y] = rx−1r−1y−1rxr−1y. Note that [x,y] = yr¯−1xr¯y−1r¯−1x−1r¯. Then
g¯ = (yr¯−1xr¯y−1r¯−1x−1r¯,1,1,1,1,1)
= (yy−1,1,1,1,1,1)(1,1,xx−1,1,1,1) = 1.
Then g = gg¯ is a palindrome.
Consider w ∈ F2′. Then w = xa1yb1...xalybl, ai = bi = 0. We re-write w as
w = rxa1r−1yb1rxa2r−1yb2...rxalr−1ybl. Then w¯ =P1. P
Consider g = (g ,...,g ) ∈ (F′)6. Using the above expression, arbitrary element g ∈
1 6 2
(F′)6 can be written such that g¯= 1. Hence we can write
2
g = (g ,g ,g ,g ,g ,g )(g¯ ,g¯ ,g¯ ,g¯ ,g¯ ,g¯ ),
1 2 3 4 5 6 1 1 3 4 5 6
where each g¯ = 1. Consequently, g is a product of six palindromes.
i
Thus an element (f ,··· ,f ) ∈ F6 can be written as a product of 18 palindromes with
1 6 2
respect to the generating set {x,y,s ,s ,c}. (cid:3)
1 2
Further,
S = {1,s ,s ,s s ,s s ,s s s } = {1,s ,s ,c,c−1,cs }
3 1 2 1 2 2 1 1 2 1 1 2 1
and so every element of S can be written as a product of not more then two palindromes.
3
So, every element of Gcan bewritten as aproductof ≤ 20 palindromes. From Fink’s result
[12, Theorem 4.7], it follows that pw(F ≀S ,S) ≤ 40. So, our construction above improves
2 3
the bound of the palindromic width of F ≀S .
2 3
ON PALINDROMIC WIDTHS OF NILPOTENT AND WREATHE PRODUCTS 9
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Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk 630090, Rus-
siaandLaboratoryofQuantumTopology,ChelyabinskStateUniversity,Brat’evKashirinykh
street 129, Chelyabinsk 454001, Russia
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Siberian University of Consumer Cooperatives, Novosibirsk 630087, Russia
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