J. Group Theory 18 (2015), 793 – 804
DOI 10.1515/ jgth-2015-0007
© de Gruyter 2015
On the length of the shortest non-trivial element in
the derived and the lower central series
Abdelrhman Elkasapy and Andreas Thom
Communicated by Miklós Abért
Abstract. We provide upper and lower bounds on the length of the shortest non-trivial
element in the derived series and lower central series in the free group on two generators.
The techniques are used to provide new estimates on the nilpotent residual finiteness
growth and on almost laws for compact groups.
1 Introduction
It is a well-known and remarkable theorem of Friedrich Levi [6, 7] that any nested
series of subgroups which are characteristic in each other in a free group either
stabilizes or has trivial intersection. This is non-trivial to prove directly even for
the derived series (see Section 2 for definitions). Using his non-commutative differential calculus, Ralph Fox [4] has extended this result to the lower central series
and given a conceptual explanation – for the lower central series he proved that
the length of the shortest non-trivial element in the n-th step of this series has
length at least n=2. It is an interesting question to determine the precise asymptotics of this quantity. Equivalently, one could ask for some information on the
smallest integer m such that every element of length n in the free group survives
in some quotient which is m-step solvable resp. m-step nilpotent. Hence, we are
trying to make the fact that the free group is residually solvable and residually
nilpotent quantitative. Similar questions have been asked in the context of residual
finiteness, see [2, 3, 5, 10] for some recent work on this problem.
In this note, we want to provide upper bounds for the growth rate of the length
of the shortest non-trivial element in the derived series and the lower central series.
An upper bound of n2 for the derived series was proved by Malestein–Putman [9]
and conjectured to be asymptotically sharp. We disprove this conjecture with
a concrete construction.
For a group € and a; b 2 €, we write Œa; b D aba 1 b 1 . We note the basic
identities Œa; b 1 D Œb; a and Œa; a D Œa; a 1  D Œa; e D e for all a; b 2 €. If
ƒ1 ; ƒ2 € are subgroups, we write Œƒ1 ; ƒ2  for the subgroup generated by
¹Œ1 ; 2  j 1 2 ƒ1 ; 2 2 ƒ2 º.
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For functions f; gW N ! R, we write f .n/ D O.g.n// if
lim sup
n!1
jf .n/j
< 1:
jg.n/j
We write f .n/ D o.g.n// if
lim
n!1
jf .n/j
D0
jg.n/j
and f .n/ g.n/ if there is a constant C such that f .n/ Cg.C n/ for all n 2 N.
2 Growth of girth in the lower central and derived series
Let F2 be the free group on two generators a and b. We denote the word length
function with respect to the generating set ¹a; a 1 ; b; b 1 º by `W F2 ! N. Recall
that the lower central series is a nested family of normal subgroups of a group €
which is defined recursively by
1 .€/ WD €
nC1 .€/ WD Œn .€/; €;
and
n 1:
We also consider the derived series, which is defined by the recursion
€ .0/ WD €
and € .nC1/ WD Œ€ .n/ ; € .n/ ;
n 0:
It is a well-known fact that Œn .€/; m .€/ nCm .€/, and hence induction can
be used to show the inclusions
n .m .€// nm .€/
and
€ .n/ 2n .€/
for all n; m 2 N:
(2.1)
Moreover, it is clear from the definition that
.€ .n/ /.m/ D € .nCm/ :
(2.2)
In this section, we want to study the growth of the functions
˛.n/ WD min¹`.w/ j w 2 n .F2 / n ¹eºº;
.n/
ˇ.n/ WD min¹`.w/ j w 2 F2
n ¹eºº:
It is clear from (2.1) that
˛.2n / ˇ.n/:
(2.3)
We can think of ˛.n/ resp. ˇ.n/ as the girth of the Cayley graph of the group
F2 =n .F2 / resp. F2 = € .n/ with respect to the image of the natural generating set
of F2 . It is clear that ˛.1/ D ˇ.0/ D 1 and that ˛ and ˇ are monotone increasing.
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Fox [4, Lemma 4.2] showed ˛.n/ n=2 and this was improved by Malestein
and Putman to ˛.n/ n ([9, Theorem 1.2]). Since Œn .F2 /; m .F2 // nCm .F2 /,
we get
˛.n C m/ 2.˛.n/ C ˛.m//:
Since in particular ˛.2n/ 4˛.n/, this suggests an asymptotic behaviour of the
form ˛.n/ D O.n2 /. This indeed was shown by Malestein and Putman [9] (on
an infinite subset of N) and conjectured to be sharp. However, already the simple
computation
`.ŒŒa; b; Œb; a
1
/ D `.aba
1
b
1
ba
1
b
1
abab
1
a
1
a
1
bab
1
/
8`.a/ C 6`.b/
(2.4)
and the observation ŒŒn .F2 /; n .F2 /; Œn .F2 /; n .F2 / 4n .€/ suggests that it
is enough to multiply the length by 14 in order to increase the depth in the central
series by a factor of 4. So, this then suggests ˛.n/ D O.n / for D log4 .14/ < 2.
In what follows we want to make these considerations precise and try to minimize . It remains an open question if D 1 C " for all " > 0 is possible to
achieve.
Lemma 2.1. We have
²
log2 .˛.n//
inf
log2 .n/
and
²
log2 .ˇ.n//
inf
n
ˇ
³
ˇ
ˇ n 2 N D lim log2 .˛.n//
ˇ
n!1 log .n/
2
ˇ
³
ˇ
ˇ n 2 N D lim log2 .ˇ.n// :
ˇ
n!1
n
Proof. From the first inclusion in (2.1), we see that ˛.nm/ ˛.n/˛.m/. Indeed,
let w 2 F2 be the shortest non-trivial word in m .F2 /. Then, it is easy to see that w
and some cyclic rotation w 0 of w are free and of length ˛.m/. Applying the shortest
non-trivial word in n .F2 / to w and w 0 yields some non-trivial element in nm .F2 /
of length less than or equal ˛.n/˛.m/. Now, the first part of the lemma is implied
by Fekete’s sub-additivity lemma. The second part follows in a similar way from
equation (2.2).
In view of the preceding lemma, we set
˛ WD lim
n!1
log2 .˛.n//
log2 .n/
and
ˇ WD lim
n!1
log2 .ˇ.n//
:
n
By Fox’s result [4, Lemma 4.2] and inequality (2.3), we get 1 ˛ ˇ. Our main
result is the following.
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Theorem 2.2. Let F2 be the free group on two generators and .˛.n//n2N , ˛,
.ˇ.n//n2N and ˇ be defined as above.
(i) We have
˛
log2 .3 C
p
17/ 1
D 1; 4411 : : :
p
log2 .1 C 2/
or equivalently
˛.n/ n
p
log2 .3C 17/ 1
p
C"
log2 .1C 2/
(ii) We have
log2 .3/ ˇ log2 .3 C
p
for all " > 0.
1 D 1:8325 : : :
17/
or equivalently
log2 .3/ n log2 .ˇ.n// .log2 .3 C
p
1/ n C o.n/:
17/
It is currently unclear to us how one could improve the upper bounds. Unfortunately, it seems even more unclear how to provide lower bounds for ˛. The proof
of the upper bounds follows from an explicit construction of short elements in
the next section. The lower bound for ˇ is a consequence of Theorem 4.1, see
Corollary 4.2.
3 The construction
Recall that we consider F2 to be generated by letters a and b. We set a0 WD a,
b0 WD b and define recursively
anC1 WD Œbn 1 ; an ;
bnC1 WD Œan ; bn  for all n 2 N:
Lemma 3.1. For all n 2 N, the products an an , bn bn , an 1 bn , bn 1 an , an bn 1 ,
bn an 1 ; an 1 bn 1 , and bn an involve no cancellation.
Proof. We prove the claim by induction, where the case n D 0 is obvious. We
check
an 1 bn D Œbn 11 ; an
The claim follows since bn
1
1
Œan
1 an 1
1 ; bn 1 
D Œan
1
1 ; bn 1 Œan 1 ; bn 1 :
involves no cancellation. Similarly,
an bn 1 D Œbn 11 ; an
1 Œbn 1 ; an 1 
(and hence bn an 1 ) involves no cancellation since an 11 bn
an 1 bn 1 D Œan
1
has no cancellation;
1
1 ; bn 1 Œbn 1 ; an 1 
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Length of shortest non-trivial elements
(and hence bn an ) has no cancellation since bn 1 bn 1 has no cancellation. Now,
similarly
an an D Œbn 11 ; an 1 Œbn 11 ; an 1 
has no cancellation since an 11 bn 11 has no cancellation, and finally
bn bn D Œan
has no cancellation since bn 11 an 1
1 ; bn 1 Œan 1 ; bn 1 
has no cancellation. This proves the claim.
Lemma 3.2. We have `.an / D `.bn / 2n for all n 2 N.
Proof. It follows from Lemma 3.1 that
`.bn / D `.an
D `.an
1
1
1 bn 1 an 1 bn 1 /
1 bn 1 /
D `.bn 11 an
C `.bn
1/
C `.an
1/
1
1 bn 1 an 1 /
D `.an /:
Now, it is obvious from this computation that `.bn / 2`.bn
hence `.bn / 2n for all n 2 N. This proves the claim.
1 / for all n
2 N, and
Lemma 3.3. For all n 2 N, we have
`.bn / 3 `.bn
In particular, there exists a constant
all n 2 N.
C0
1/
C 2 `.bn
2 /:
p
> 0 such that `.bn / C 0 . 3C2 17 /n for
Proof. We estimate the length of bn in a straightforward way:
`.bn / D `.Œan
1 ; bn 1 /
D `.ŒŒbn 12 ; an
`..bn 12 an
2 ; Œan 2 ; bn 2 /
1
1
1
2 bn 2 an 2 an 2 bn 2 an 2 bn 2 //
C `.Œan
`.bn 12 an
2 bn 2 /
C `.Œan
3 `.bn
1/
1
2 ; bn 2 /
C `.Œbn
C `.bn
1
2 ; bn 2 /
C 2 `.bn
2/
2 ; an 2 /
C `.an 12 / C `.bn 12 /
C `.Œbn
2 ; an 2 /
2 /;
where we used the equation `.bn 12 an 2 bn 2 / D `.an 1 / `.an 2 / (a consequence of Lemma
3.1) in the last equality. The estimate follows from the fact
p
1
that 2 .3 C 17/ is the largest root of the polynomial p./ D 2 3 2. This
proves the claim.
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Our first result concerns the growth of the girth in the derived series.
Proposition 3.4. Let WD
p
3C 17
2
D 3:56155 : : : . We have
ˇ.n/ C 0 n
for some constant C 0 > 0 and infinitely many n 2 N. In particular, we get
ˇ log2 ./ D 1:8325 : : : :
.n/
Proof. We set ı.w/ WD max¹n 2 N j w 2 F2 º. It is clear from the construction
that ı.bn / n. Moreover, we clearly have ˇ.ı.w// `.w/. Thus,
ˇ.ı.bn // D `.bn / C 0 n C 0 ı.bn / :
This finishes the proof.
Since ˛.2n / ˇ.n/, the previous resultpsuggests ˛.n/ C 0 nlog2 ./ . We can
improve the exponent by a factor log2 .1 C 2/. Let
WD
log2 .3 C
p
17/ 1
D 1:44115577304 : : : :
p
log2 .1 C 2/
Proposition 3.5. We have ˛.n/ C 0 n for infinitely many n 2 N and thus ˛ .
Proof. Note that we have the identities
ŒŒa
1
; b; Œa; b D ŒŒŒa
1
; b; a; Œa; b;
ŒŒa
1
; b; Œb; a D ŒŒŒa
1
; b; a; Œb; a:
(3.1)
Indeed, we just check ŒŒa 1 ; b; a D a 1 bab 1 aba 1 b 1 aa 1 D Œa 1 ; bŒa; b
and use that Œa; b commutes with both Œa; b and Œb; a D Œa; b 1 . This proves
equation (3.1). We set
.w/ WD max¹n j w 2 n .F2 /º for all w 2 F2 :
(3.2)
Clearly, .w1 w2 / min¹.w1 /; .w2 /º and .Œw1 ; w2 / .w1 / C .w2 /. In
order to proceed we need the following lemma.
Lemma 3.6. We have .bn / 2.bn 1 / C .bn 2 / for all
p n 2 N. In particular,
there exists a constant C > 0 such that .bn / C .1 C 2/n .
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799
Proof. We compute
bn D Œan
1 ; bn 1 
D ŒŒbn 12 ; an
(3.1)
2 ; Œan 2 ; bn 2 
D ŒŒŒbn 12 ; an
D ŒŒan
2 ; bn 2 ; Œan 2 ; bn 2 
1 ; bn 2 ; bn 1 :
This proves the claim since .an 1 / D .bn 1 / as bn 2 an 1 bn 12 D bn 1 . The
estimate on .bn / follows as before by a study of the growth of the recursively
defined sequence n WD 2n 1 C n 2 .
We are ready to prove the upper bounds on ˛.n/. Note that ˛..bn // `.bn /
for all n 2 N. Thus, as a consequence of Lemma 3.6 and Lemma 3.3, we get
n
log2 ..bn //
log .C /
p 2
log2 .1 C 2/
and hence
˛..bn // `.bn /
C 0 n
log./ .log2 ..bn // log2 .C //
0
C exp
p
log2 .1 C 2/
log./ log2 .C /
0
..bn // :
D C exp
p
log2 .1 C 2/
This proves the claim.
Question 3.7. Can we prove better bounds of the form .w/ `.w/ı for some
ı < 1?
4
Lower bounds for the derived series
Again, we consider F2 – the free group with generators a; b. For a subgroup
ƒ F2 , we define
girth.ƒ/ WD min¹`.w/ j w 2 ƒ n ¹eºº:
Theorem 4.1. Let ƒ F2 be a normal subgroup. Then the following holds:
girth.Œƒ; ƒ/ 3 girth.ƒ/:
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Proof. Recall, a subset S F2 is called Nielsen reduced if
(i) u ¤ e for all u 2 S ,
(ii) `.uv/ max¹`.u/; `.v/º for all u; v 2 S ˙1 with uv ¤ e,
(iii) `.uvw/ > `.u/ `.v/C`.w/ for all u; v; w 2 S ˙1 with uv ¤ e and vw ¤ e.
It is well known that ƒ has a Nielsen reduced basis [8, Proposition 2.9] – let us
denote it by S. We use the notation
1
jwj WD min¹`.vwv
/ j v 2 F2 º:
We will show that jwj 3 girth.ƒ/ for all non-trivial w 2 Œƒ; ƒ. Every element
w D Œƒ; ƒ is a product of elements in S ˙1 , so that the exponent sum of each individual s 2 S is equal to zero. Hence, we may assume that w D sw1 tw2 s 1 w3 t 1
or w D sw1 s 1 w2 t w3 t 1 for some elements w1 ; w2 ; w3 2 ƒ and s; t 2 S ˙ such
that st ¤ e and st 1 ¤ e. Since we are assuming that our basis for ƒ is Nielsen
reduced, the cancellations from the left and right inside some element of S cannot
overlap and each will never touch more that one half of the word. Without loss of
generality, we may assume that the cancellation that occurs in the product t 1 s
is the longest among the cancellations between all other letters that appear in w.
Let us write t D at1 and s D as1 so that t1 1 s1 is reduced. Let us discuss the first
case, i.e., w D sw1 tw2 s 1 w3 t 1 . Now, the cancellation in the product of sw1
and tw2 s 1 must be an initial segment b of a, and similarly the cancellation in
the product of tw2 s 1 and w3 t 1 must be an initial segment c of a. Since ƒ is
a normal subgroup, we get that
1
girth.ƒ/ `.t w2 s
/
girth.ƒ/ `.sw1 /
girth.ƒ/ `.w3 t
2`.a/;
2`.b/;
1
/
2`.c/:
Hence,
3 girth.ƒ/ `.sw1 / C `.t w2 s
1
1
/ C `.w3 t
/
2`.a/
2`.b/
2`.c/ D jwj:
1 w t w t 1 , we consider the words sws 1 ; w
2
3
2
In the second case, i.e., w D sw1 s
and tw3 t 1 and argue in a similar way. Indeed, the word b cancelled in the product
of sws 1 and w2 must be an initial segment of a. Similarly for the word c cancelled in the product of w2 and t w3 t 1 . Without loss of generality, c is an initial
segment of b. Now, we get
girth.ƒ/ `.t w2 t
1
girth.ƒ/ `.sw1 s
1
girth.ƒ/ `.w2 /
2`.c/:
/
/
2`.a/;
2`.a/;
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Hence, also in this case we get
3 girth.ƒ/ `.sw1 s
1
/ C `.w2 / C `.t w3 t
1
/
4`.a/
2`.c/ jwj:
This proves the claim.
Corollary 4.2. We have
.n/
girth.F2 / 3n :
In particular, we get ˇ log2 .3/ D 1:5849 : : : .
5
5.1
Some applications
Nilpotent residually finiteness growth
Following Khalid Bou-Rabee [2] we define FFnil2 .n/ to be the smallest integer k so
that for every element w 2 F2 of length less than or equal n, there exists a homomorphism to a finite nilpotent group of cardinality at most k which does not map
w to the neutral element. Following [2], the growth behaviour determined by FFnil2
is called the nilpotent residual finiteness growth of the free group. Claim 1 in the
proof of [2, Theorem 3] stated
exp.n1=2 / FFnil2 .n/:
Using the upper bound on ˛ in Theorem 2.2, we can improve a little bit on this.
The proof is identical to [2, proof of Claim 1, p. 705].
Theorem 5.1. We have exp.nı / FFnil2 .n/ with with
p
log2 .1 C 2/
ıD
D 0:69391 : : : :
p
log2 .3 C 17/ 1
p
p
Proof. Indeed, an 2 .F2k / for some k C.1 C 2/n and `.an / C 0 . 3C2 17 /n .
Since any finite nilpotent group of size less than 2k has nilpotency class k, we
conclude that any finite nilpotent
quotient of F2 in which an survives must be of
p
size at least exp.C 00 .1 C 2/n /. This implies the claim.
5.2
Almost laws for compact groups
For every group G, an element w 2 F2 gives rise to a word map wW G G ! G,
which is just given by evaluation. In [11], the second author proved that there
exists a sequence of non-trivial elements .wn /n in the free group on two generators
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such that for every compact group G and every neighbourhood V G of the
neutral element, there exists an m 2 N such that wn .G G/ V for all n m.
This statement is already non-trivial for a fixed compact group such as SU.2/.
Following [1, Section 5.4], we call such a sequence an almost law for the class of
compact groups.
For a specific group like SU.k/ with a natural metric, say d.u; v/ WD ku vk
where k k denotes the operator norm, it is natural to ask how long a word w 2 F2
necessarily has to be, if we demand that d.1k ; w.u; v// < " for all u; v 2 SU.k/.
We set
Lk .w/ WD max¹d.1k ; w.u; v// j u; v 2 SU.k/º:
In [11, Remark 3.6], it was claimed that there is a construction of an almost law
.wn /n as above such that for every " > 0, there exists a constant C > 0 (which
depends also on k) such that
Lk .wn / exp. C `.wn /log14 4 " /
with log14 4 D 0:5252 : : : . This construction relies on the basic idea that was
already mentioned in connection with equation (2.4). The more refined study in
this paper yields:
Theorem 5.2. Let k 2 N. There exists an almost law .wn /n for SU.k/ such that
the following holds. There exists a constant C > 0 such that
ı
Lk .wn / exp C `.wn /
with
p
log2 .1 C 2/
ıD
D 0:69391 : : : :
p
log2 .3 C 17/ 1
Proof. Our basic method is a well-known contraction property of the commutator
map in a Banach algebra. Let k be fixed. In terms of the function Lk , Lemma 2.1
in [11] says
Lk .Œw; v/ 2 Lk .w/Lk .v/:
(5.1)
We conclude from [11, Corollary 3.3] that there exist words w; v 2 F2 which generate a free subgroup and satisfy Lk .w/; Lk .v/ 31 . Let us set wn WD an .w; v/.
It is clear that
p 3 C 17 n
00
`.wn / C (5.2)
2
for some constant C 00 > 0. On the other side, equation (5.1) and the equation
bn D ŒŒan
1 ; bn 2 ; bn 1 
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from the proof of Lemma 3.6 show that
Lk .wn / 4 Lk .wn
2
1 / Lk .wn 2 /
or equivalently
log.2Lk .wn // 2 log.2Lk .wn
1 //
log.2Lk .wn
2 //:
Thus – precisely as in the proof of Lemma 3.6 – there exists a constant D > 0
such that
p
(5.3)
log.2Lk .wn // D .1 C 2/n
for some constant D > 0. Hence,
(5.3)
Lk .wn / 1
exp
2
D .1 C
p
2/n
(5.2)
exp
C `.wn /ı
for some constant C . This implies the claim.
It would be interesting to find a more direct relationship between the growth
of the girth of the lower central series and the asymptotics encountered in Theorem 5.2. It is presently unclear if 1 C " for any " > 0 (or even for " D 0) is enough
in Theorem 5.2, see also [1, Section 5.4] for a discussion of this question.
Acknowledgments. We want to thank Jan-Christoph Schlage-Puchta and Dan
Titus Salajan for interesting comments. We thank the unknown referee for helpful
comments.
Bibliography
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[7] F. Levi, Über die Untergruppen der freien Gruppen II, Math. Z. 37 (1933), no. 1,
90–97.
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Received January 7, 2014; revised January 5, 2015.
Author information
Abdelrhman Elkasapy, MPI-MIS, Inselstraße 22, 04103 Leipzig, Germany;
and Mathematics Department, South Valley University, Qena, Egypt.
E-mail: [email protected]
Andreas Thom, Universität Leipzig, PF 100920, 04009 Leipzig, Germany.
E-mail: [email protected]
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