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On Growth in Group Theory
Rostislav I. Grigorchuk
Moscow Institute of Railway Transportation Engineers
ul. Obraszowa 15, Moscow, USSR
1. General
The concept of growth appeared in group theory in the mid-fifties and now it plays
an increasing role in this theory. Among different notions concerning the idea of
growth in group theory the most important and successful is the notion of growth
of a finitely generated (f.g.) group.
We start with the main definition, then enumerate the results concerning this
definition and at the end we shall touch some other aspects of growth in group
theory.
1.1 The Main Definition
Let G be a finitely generated (f.g.) group with generator system A = {al9..., am},
and let d(g) be the length of the element g e G with respect to the system A, in other
words the minimal number k such that g can be represented in the form
g=
all...a%,
fi/ = ± 1,
j=l,...,k.
The growth function of the group G with respect to the system A is the function
y(n) = card{# e G; ô(g) < n}.
As y(n) depends on the generator system it is convenient to introduce an equivalence relatiop on the set of growth functions:
yM ~ y2(n)oiC(yM
< y2(Cn)&y2(n) <
7l(Cn))
and the preordering relation:
yi(n)<y2(n)olC(yi(n)<y2(Cn)).
The equivalence class [y/(n)] is an invariant of the group G. We shall call it the
growth degree of the group G,
Obviously the growth degrees of a group and of any of its subgroup of finite
index coincide.
The partially ordered (with respect to <) set 2B of the growth degrees of f.g.
groups will be the main object of our consideration.
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
© The Mathematical Society of Japan, 1991
326
Rostislav I. Grigorchuk
1.2 Examples
If G = Zd is the free abelian group of rank d, then y(n) ~ nd. If G = Fm is an (absolute)
free group of rank m > 2 then y(n)coen. Thus the growth of a group can be
polynomial of any degree d e Z+ and can be exponential.
1.3 Connection with Geometrical Growth
The notion of the growth function was introduced by A.S. Schwarze [1] and
independently by J. Milnor [2]. This notion is a combinatorial form of the geometrical growth notion explored by VA. Efremovich [3].
Let M be a Riemannian manifold. Its growth at infinity is characterized by the
growth when r -> oo of the function
v(r) = Vol(Bx(r)),
expressing the volume of the geodesic ball of radius r with center at the point xe M.
The examples of the Euclidean space R d and hyperbolic space H d show that this
function in the first, case grows as the polynomial rd and in the second case as the
exponential function er.
On the other hand the growth function yG(n) of the f.g. group G expresses the
volume of the ball Bx(n) of radius n with center at the unit element 1 e G if the group
G is supplied with the metric d(g, h) = d(g~1h) and Haar measure.
The connection between geometric and algebraic growths can be also illustrated
by A.S. Schwarzc's theorem [1]: if the manifold M is a universal covering manifold of
the compact Riemannian manifold M then
"MW ~ y*l(M)(r)In other words the growth of the covering manifold M is the same as the growth of
the fundamental group nx(M) of the manifold M.
2. Milnor's Problems
2.1 Some Results
Intensive investigation of the growth functions of f.g. groups began after J. Milnor's
paper [2] was published. A series of important results was obtained within a short
time interval. Here are some of them,
2.1.1
A f.g. nilpotent group G has polynomial growth (Wolf [4]) like hd, where
d=
YJkvmkQ(Gk/Gk+1)
k
and {GjJ is the lower central series of the group G (Bass [5]).
2.1.2 A f.g. solvable group has exponential growth if it is not virtually nilpotent
(Wolf [4], Milnor [6]). (Virtually nilpotent means: contains a nilpotent subgroup of
finite index).
On Growth in Group Theory
327
2.1.3 A f.g. linear group has exponential growth if it is not virtually nilpotent (Tits
[7]).
2.1.4 A free periodic Burnside group
B(m,n) = (au...,am\gn
= l,geG}
has exponential growth when m > 2 and n > 665 is odd (Adyan [8]).
In 1968 J. Milnor [9] proposed a number of problems concerning growth of
groups.
1) "Is the growth function y(n) necessarily equivalent either to a power of n or to
the exponential function TT 2) In particular, is the growth exponent
dl=lim!ogyW
»-oo
(1)
l o g 77
always either a well-defined integer or infinitelyl For which groups is d < oo? (A
possible conjecture would be that d < oo if and only if G contains a nilpotent subgroup
of finite index)".
(Division of Milnor's problem into Parts 1) and 2) has been made by me).
As a conjecture Problem 1) was formulated by Wolf [4] and Bass [5].
Conjecture: "A f.g. group of non-exponential growth is virtually nilpotent".
2.2 Two Definitions
A group whose growth function is not majorized by any function nd and is not equivalent to the function e" is called an intermediate growth group. And a group G is called a
group of subexponential growth if y(n) •< en (strong inequality), in other words, if
lim */yfr) = 1
II-+00
(the limit exists due to the semi-multiplicativity of the function y(n)).
3. Answer to Milnor's Problem 1)
Milnor's problem 1) was solved negatively in [10,11, 12].
s
3.1 On the Construction of Intermediate Growth Groups
The simplest example of such a group is the group /"from the paper [13] which
was constructed as a simple example of a f.g. infinite periodic group. T is defined as
the group of transformations of the interval [0,1] from which rational points of the
formfc/2",n — 1, 2, ..., 0 < k < 2" are removed. This group is generated by four
transformations a, b, c, d, where a is the permutation of the halves of the interval
[0, 1] and b, c, d, are defined with the help of infinite periodic words
328
Rostislav I. Grigorchuk
P
P
a\
1
0
1
0
1
d|
c |
P
T P ...
1 _ | _ |
0
Ì
U
2
3
7
4
1
g
...
.
.
1
P
1
1
I
I
J...1
b\
T
A
U
i
1
T
P P . . .
1
2
.
1
1
I
7
4
g ...
1
1
|
l
Fig. 1. (P denotes the permutation of the halves of the interval, T is the identity
transformation).
PPTPPT...
PTPPTP...
TPP TPP ...
as in Figure 1.
We constructed uncountably many intermediate growth groups on the basis
of this example (see [10-12]).
3.2 The Main Idea of the Growth Function yr(n) Upper Estimation
The group T contains a subgroup H of finite index, which allows an embedding W
in the direct product of eight copies of the group T:
H^H<rxrxrxrxrxrxrxr,
and H is a subgroup offiniteindex in 7"8.
This embedding has the following property: if g e H and
^(0) = (0i,02, •••>0 8 )ei?
then
where k is some constant less than 1. Hence
y(«)<cSr( W l )...yK)
(2)
(C is some constant), where the summation in (2) is taken over those sets of nonnegative integers nl9..., n8 for which the inequality
8
Y nt < kn
i=l
is valid.
If we assume that
lim 7y(n) = X > 1
On Growth in Group Theory
329
then we shall obtain a contradiction k < Xk from (2). So r has sub-exponential
growth.
The fact that the function y(n) grows faster than any polynomial results, for
example, from the following consideration.
The group r is commensurable with its square, in other words the groups T
and r x T contain isomorphic subgroups of finite index. Hence y(n) satisfies the
so-called ^-condition y(n) ~ y2(n), which results in the lower estimate by some
function of the type e"p, where ß > 0.
However, the lower estimate of the function yr(n) can be obtained directly. It is
proved more precisely in [11] that
e^x < yr(n) < en\
where a — log 32 31. This obviously gives a negative answer to Milnor's problem 1).
On the basis of a construction which generalizes this example we proved the
following theorem (see [11, 14]).
Theorem 1. The set of growth degrees of f.g. groups has the cardinality of the
continuum. It contains a chain of the cardinality of the continuum, and an antichain
of the cardinality of the continuum.
I would like to stress that we are speaking precisely about intermediate degrees
of groups in this theorem.
Another example of a group of intermediate growth was constructed by
J. Fabrykowski and N. Gupta [15].
3.3 Some Properties of Intermediate Growth Groups
Intermediate growth groups can be constructed both in the class of p-groups (p is
any given prime) and in the class of torsion free groups [11, 12].
The intermediate growth groups known up to this moment belong to the class
of residually finite groups. Moreover, the groups from the papers [10-12] are
residually-p-finite.
The periodic groups constructed in [11,12] have the property that all their
proper factor groups are finite. At the same time in [16] for any prime p, & f.g.
p-group of intermediate growth having continuum nonisomorphic factor groups
was constructed.
The torsion free group of intermediate growth constructed in [12] admits an
invariant linear ordering.
The intermediate growth groups from [11, 12] are not finitely presentable.
There exist recursively presentable (by generating elements and defining
relations) intermediate growth groups having solvable word problem. At the same
time there are analogous groups with unsolvable word problem [11, 14].
4. On Groups with Polynomial Growth
An exhaustive answer to Part 2) of Milnor's problem was given in the papers
[17, 18] and Milnor's conjecture suggested in this part was confirmed. Note that a
330
Rostislav I. Grigorchuk
more general result was obtained in [18], although this paper is mainly devoted to
proving Gromov's theorem by methods of nonstandard analysis.
4.1 On Gromov's Theorem
Gromov's theorem proved in [17] gives the description of f.g. groups whose growth
functions admit polynomial estimate
y(n)<nd,
(3)
(d is some constant). This theorem states that a group G is virtually nilpotent if the
estimate (3) is valid for its growth function.
Hence the estimate (3) results in an equivalence y(n) ~ nd for a suitable deZ+.
The proof of this theorem is based on geometric considerations. The left
invariant metric d(g, h) = d(g~1h), g,hsGis built on the group G and the sequence
Xn = (G,±d),
neN
of metric spaces is considered*
Gromov proved that from this sequence one can extract some subsequence
which converges in an exactly defined way to some metric space X^ if the condition
(3) is valid. A homomorphism G -> Isomf-X^) from the group G into the isometry
group of X^ arises and Isorn^^) is a Lie group with a finite number of connected
components. Some simple algebraic considerations complete the proof of this
theorem.
4.2 On L. van den Dries and A. Wilkie's Theorem
The main result of [18] is: if the inequality
y(n) < Cnd
is valid on some infinite subset N 0 ç N of the set of natural numbers then the group
G is virtually nilpotent.
From this statement if follows that the limit (1) (finite or infinite) always exists,
and also d < oo if and only if the group G is virtually nilpotent.
5. On the Growth of Cancellation Semigroups
The notion of the growth function of a f.g. semigroup is defined as in the case of
a group. The growth of a semigroup can be very strange even in the case when the
condition (3) holds true. But if we consider the class of semigroups with left and
right cancellation laws then in this case the statement similar to Gromov's theorem
is valid.
5.1 The Nilpotency of Semigroups Due to A.I. Malcev
Let x, y, Çl9 Ç2, ..., Çn, ... be symbols denoting variables running through a
semigroup S. Let us denote X0 = x, Y0 = y and after that by induction
On Growth in Group Theory
X„+i = Xnç„Yn,
331
Y)l+1 = YnçnXtr
The semigroup S whose elements satisfy the identity Xn = Yn for some n is called
nilpotent.
A.I. Malcev proved [19] that the classical nilpotency identity for groups is
equivalent to the semigroup identity X„ = Yn for corresponding n.
5.2 On Cancellation Semigroup of Polynomial Growth
Let S0 £ S be a subsemigroup. We shall say that S0 has finite index in S if there
exists a finite subset K ^ S such that for any se S there exists k e K for which
ske S0.
Theorem 2. A f.g. cancellation semigroup S has polynomial growth (Condition (3)) if
and only if S contains a nilpotent subsemigroup of finite index.
The proof of this theorem is based on the remark that for a cancellation
semigroup of subexponential growth the Ore condition is satisfied and this semigroup possesses a group Gs = S - 1 S of left quotients.
The estimate (3) on the growth function of the semigroup S makes it possible to
give a polynomial type estimate on the growth function of the group Gs after detailed
analysis. Then Gromov's theorem can be applied.
5.3 Some Remarks
In [20] it was proved that if a semigroup S with cancellation has polynomial growth,
then the group Gs of left quotients also has polynomial growth of the same degree.
It would be interesting to construct examples of semigroups with cancellation
of subexponential growth for which there is a jump of growth degree in the diagram
S^GS.
If one could construct a semigroup S with cancellations of subexponential
growth such that Gs has exponential growth, then Problem N12 from [21] will be
positively solved.
6. On Lacunae in the Set of Growth Degrees
of Residuali} Nilpotent Groups
Recall that a group G is residually-p if for any nonunit element g e G there exists a
finite p-group K and a homomorphism cp:G -> K such that cp(g) ^ 1.
Theorem 3 [22]. Let p be any prime, let the f.g. group G be residually-p and its growth
function satisfy the estimate (3). Then G is virtually nilpotent and so has polynomial
growth.
6.1 Proof of Theorem 3
Let Fp be a finite prime field, F p [G] the group algebra, A < F p [G] the augmentation
ideal, in other words the ideal generated by elements of the form g - 1, g e G, gr(G)
332
Rostislav I. Grigorchuk
Polynomial degrees
Lacunae
1 n n2 ... nd ...
Chain
e^1
.
en exp.
degree
Antichain
Fig. 2
the associative graded algebra defined by means of the powers of the augmentation
ideal
gr(G)=@A„ = ©A"/A»+\
n=0
n=0
let fGtp(t) be the Hilbert-Poincaré
:aré series of the algeb:
algebra gr(G):
where an(G) = dim F An.
It follows from Lazard's Theorem 3.11 [23] that either the sequence an(G) has
polynomial growth and then the p-completion G is p-analytic or
an(G)>e^.
Besides if y (ri) is the growth function of the group G with respect to any system of
generators then the inequality an(G) < y(n), n = 1, 2 , . . . is valid.
Using Lazard's result and Condition (3) we obtain that the growth of coefficients
an(G) when n -» oo is polynomial and so the p-completion G is analytic. It follows
from Tit's theorem [7] that either G contains a free subgroup with two generators
or G is a virtually solvable group. But the first case is impossible due to limitation
on the growth, and in the second case, we can conclude that G is virtually nilpotent
due to results of [4, 6].
Recently A. Lubotzky and A. Mann [24] have pointed out that in Theorem 3
the assumption that G is residually-p can be changed by the assumption that G is
residually nilpotent.
6.2 On the Scale of Growth Degrees of Residually-/? groups
Let p be any prime and denote by Wp the set of growth degrees of f.g. residually-p
groups. Theorem 1 formulated above for the whole class of groups is also true for
every set 2Bp. So, due to Theorem 3 the structure of the partially ordered set 2Bp
can be presented in the form as in Figure 2.
7. On the Generating Series of Growth Function
Many questions of a theoretical and applied character require more detailed investigation of the asymptotic behavior of the growth function y (ri) than up to the
On Growth in Group Theory
333
equivalence ~, defined above. For this it is sometimes useful to connect the
generating series
m = t y(r>)t"
N= 0
to the function y(n).
7.1 Some Cases When the Function T(t) is Rational for any System of Generators
i) The group is virtually abelian (Benson [25])
ii) The group is a cocompact group of isometries of the hyperbolic space WLd
(Cannon [26])
iii) The group is hyperbolic (Gromov [27]).
For many nilpotent groups the series the r(t) also represents a rational function
(see for example [28]). At the same time the conjecture that for any f.g. nilpotent
group and any of its finite generating set the function T(t) is rational proved to be
false. Namely F. Grunewald proved that for the nilpotent group
G = <fll5 a2, bu b2, z\\_au foj = \_a2,fc2]= z,ze Z(G),
f<*u a 2 ] = [Pu M = [ßu hl = \ßi> M = O
(Z(G) is the centre of the group G and the function T(t) defined using the system of
generators {al9 a2, bx, b2, z}), the generating series r(t) is not a rational function.
7.2 Information on the Growth Functions of Nilpotent Groups
Let G be a nilpotent group, y (ri) the growth function of G with respect to any system
of generators, d the power of polynomial growth of G. Then
a) there exists the limit
lim^ =C
n -»oo
n
(Pancu [29]),
b) the estimate
y(ri) = Cnd + 0(nd~112)
is valid (Grunewald, unpublished).
It is interesting to find new examples of rationality of the growth function T(i)
and also to obtain another information about the generating series of growth
functions.
8. Some Problems and Conjectures
The theory of growth degrees of groups has been developed for more than three
decades and has already accumulated a lot of unsolved problems. Let me formulate
some of them.
334
Rostislav I. Grigorchuk
8.1 (A well known problem.) Is there a f.g. group of intermediate growth with a
finite set of defining relations?
8.2 Is it true that if the growth function of a f.g. group G grows slower than the
function e^n, then G is virtually nilpotent?
8.3
Does there exist a f.g. group whose growth function is equivalent to the function
8.4
Is it true that any group of subexponential growth is residually finite?
8.5 To find the asymptotic behaviour of the growth function of the group T from
[13] (conjecture is that y (ri) ~ e*" in this case).
8.6 Is it true that for any f.g. nilpotent group the function T(t) is meromorphic or
even algebraic? (F. Grunewald).
Let
G = (A\r=
l(reÄ))
be f.g. and 3 be the Cayley graph of G with respect to the system of generators A. A
spanning tree T s ^ i s regular if it is defined by a finite automaton. A spanning tree
T is minimal if every word in T is the shortest.
8.7 Conjecture. G has a rational growth function if and only if 3 has a minimal
regular spanning tree (Machi, Schupp).
9. Other Aspects of Growth in Groups
9.1 Cogrowth
Let a f.g. group G be realized as a factor group FJH of the free group Fm of rank
m, H < Fm2L normal subgroup, the elements of which will be further considered
as reduced words over the basis of Fm. Denote by h(ri) the number of words of
length <n in H. Obviously h(ri) <, 2m(2m - l) w_1 . Let
00
aH = limsup y/h(n),
ii-»oo
J^(t) = £ h(ri)tn.
n=0
The value a H which appeared in [30] is called the. growth exponent of the normal
subgroup H and belongs to the interval (y/2m — 1, 2m — 1].
In [31] the following amenability criterion was proved.
Theorem 4. The group G = Fm/H is amenable if and only if ocH = 2m — 1.
The function J^(t) and growth exponent aH can be defined also for any subgroup
H < Fm. In [31] it was proved that if if is a f.g. group then J f (t) is a rational function
which can be effectively calculated if the system of generators of H is known.
On Growth in Group Theory
335
9.2 Subgroup Growth
Let G be a f.g. group, and let Qn(G) be the (finite) number of its subgroups of index
n. The problem of investigation of the asymptotic behavior of the sequence g(n) when
n -» oo has been drawing more and more interest. For example in [32] F. Grunewald,
D. Segal and G. Smith investigated in detail the Dirichlet series
EG(s) = £ft,(G)n-'
of the sequence gn(G) for the case when G is nilpotent.
On the other hand A. Lubotzky and A. Mann [24] showed that if G is a
residually nilpotent f.g. group then Qn(G) has polynomial growth if and only if G is
solvable offiniterank.
The general case of a residually finite group is still open. See [24] for other
references and comments.
9.3 On the Growth of Graded Algebras Associated to Groups
During investigations of groups different graded algebras associated to these groups
arise. For example the associative graded algebra gr G was defined above. Studying
the growth of algebras and in particular graded algebras associated to groups is
important from different points of view and is connected with different applied
aspects of algebra [33-37].
9.4 On the Asymptotic Behavior of Random Walks on Groups
Investigation of random walks on groups was started by H. Kesten in [38]. It is
interesting to investigate the asymptotic behavior when n -^ oo of the probability
Plf i(/,) of returning to the unit element after n steps [38, 39], to calculate the entropy
of random walks [40], to describe Martin's and Poisson's boundaries [41, 42] and
so on.
10. Applications
10.1 To the Theory of Invariant Means
The definition of an invariant means and the corresponding notion of amenable
groups belong to J. von Neumann [43].
Finite and abelian groups are amenable. The class of amenable groups is closed
under operations of extension and inductive limit.
Let us denote the class of amenable groups by AG, the class of groups without
free subgroups with two generators by NF, and the class of elementary amenable
groups constructed by J. von Neumann by EG, i.e. the minimal class of groups
containing finite groups, abelian groups and closed under the operations of extension and inductive limit. The following imbeddings EG £ AG £ NF hold true.
336
Rostislav L Grigorchuk
In 1957 M. Day [44] proposed the following two problems:
(1) Is it true that AG = NF?
(2) Is it true that AG = EG?
Some mathematicians ascribe Problem (1) to J. von Neumann but there is no
written confirmation of the fact that it was he who suggested this problem.
In 1979 to disprove the conjecture AG = NF I suggested A.Yu. Olshanskii to
apply the amenability criterion (Theorem 4) to the groups constructed by him at that
moment for the solution of some famous problems of group theory. This suggestion
was realized in the short paper [45].
A little later S.I. Adyan [46] also applied the amenability criterion to prove that
free periodic Burnside groups B(m,ri)are nonamenable when m > 2 and n > 665 is
odd (the conjecture that these groups are nonamenable was expressed by S.I. Adyan
in 1975).
The negative answer to the problem (2) was given in [11].
Theorem 5. There exist uncountably many amenable groups not belonging to the class
EG.
Hence the class AG of amenable groups is much more than the class EG of
elementary amenable groups.
Theorem 5 is an easy corollary to the existence of a continuum of nonisomorphic
intermediate growth groups.
Using intermediate growth groups we managed [48] to disprove one of
Rosenblatt's conjectures [47] (see also [21], Problem 11) about the so-called
superamenable groups.
10.2 To Riemannian Geometry
The results concerning the growth functions of groups are widely used in the
classification theory of Riemannian manifolds up to quasi-isometries, in the theory
of foliations, in the investigation of Laplace operator on Riemannian manifolds
(see [49, 50]),....
10.3 Other Applications
Information concerning the growth functions of groups is used in many, sometimes
unexpected branches of mathematics: in the theory of random walks [38, 39],
ergodic theory [50, 52], the theory of finite automata [20] and so on.
In my lecture I did not touch upon many other aspects of applications of the
notion of growth in group theory, many of which at this moment are only in their
initial stage of development.
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