PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 106, Number 3, July 1989
A SHORT PROOF OF THE GRIGORCHUK-COHEN
COGROWTH THEOREM
RYSZARDSZWARC
(Communicatedby J. MarshallAsh)
There are exactly
Part of them, say
Yn representsidentity element of G. Let y = limsupyn . We give a short
proof of the theoremof Grigorchukand Cohenwhich states that G is amenable
if and only if y = 2r - 12. Moreoverwe derive some new propertiesof the
generatingfunction
Ynzn
ABSTRACT. Let G be a group generated by gl,...,gr.
of length n.
2r(2r - l)n-1 reduced words in gl,..,gr
E
Let G be a finitely generated discrete group. Consider G as an epimorphic
image it: Fr -- G of the free group Fr on r generators. Thus G is isomorphic
to the quotient group Fr/N where N = ker X is a normal subgroup of Fr . Once
we fix a set of free generatorsin Fr we introduce IxIthe length of the word x in
Fr with respectto the generatorsand their inverses. Let Nn = {x E N: IxI = n},
= cardNn and y = lim sup 1In which is called the growth exponent of N
is Fr with respect to the fixed set of free generatorsin Fr. Because there are
I
exactly 2r(2r - 1)n elements of length n in Fr y2 < 2r - 1 . Grigorchuk [2]
and Cohen [1] have shown that a group G is amenable if and only if y attains
maximal possible value, i.e. y = 2r - 1 . We propose a new rathersimple proof
without any estimates which allows us to draw out new information on the
behaviour of the generatingfunction N(z) = E z
As in [1] and [2] we will base our proof on a characterizationof discrete
amenable groups given by Kesten [3]. Any absolutelysummable function f on
a group G defines a translationinvariantoperatorof 12(G) as g ~-*f * g. The
norm of correspondingmap we denote as usual by IIfII'c(G) .
Theorem1 (Kesten [3]). Let f be any selfadjointsummablepositivefunction on
G whosesupportgenerates G. Then G is amenable if and only if IfIIC (G) =
ExEG f(x).
Received by the editors May 24, 1988.
1980 MathematicsSubject Classification(1985 Revision). Primary 43A07, 20F05; Secondary
22D25.
This work was written while the author held C.N.R. fellowship at the University of Rome
"LaSapienza".
? 1989 American Mathematical Society
0002-9939/89
663
$1.00 + $.25 per page
RYSZARDSZWARC
664
As in [1] let Xn, n = 0, 1 , 2, . .. denote the characteristicfunction of the
set of works of length n in Fr. Observethat the support of the function 7r(X1)
generates G by assumption. We are going to compute the norm Jj7r(x1) IC (G).
Theorem 2 (Grigorchuk[2], Cohen [1]). Let q = 2r - 1. then 117r(Xl)IlC(G) =
y + q/y only if y > 1 (or equivalently ker Xris nontrivial).
Together with Kesten's theorem it gives immediately the following
Theorem3 (Grigorchuk[2], Cohen [1]). G is amenable if and only if y = 2r- 1.
Proofof Theorem2. The linear functional Cled(G) : f 4 f(e) is the faithful
trace Tr on C*d(G) thus (see [6])
IIfIIc*(G) = lim(Trf
2n)l/2n
= lim (Trf *2n
/
Let t be a positive number such that 0 < t < q 2. If we let a(t) = qt + t
then by [4, proof of Theorem 3.1, p. 128] or by [5, Theorem 1] we have
(a(t)5e
Thus using (a -x)
-Xf)'(x)
=
>aa
l 1
1
=
)xn we obtain
00
Za(t)
in C*d(Fr)*
00
=
-(n
tXnt
n=O
n=O
Applying successively Xr and then Tr to both sides and observingthat Tr7r(Xn)
=
Yn gives
00
(*)
t
-(t)-(n+lT7=
STrU(Xl)
tn
it
n=O
(all this makes sense at least for values of t small enough).
Now the point is that (*) relates the radii of convergence: r, of the series EYn tn and r2 of E Trz7(XI)nyn+l.
re(X)Id*
r2 = 11r(Xl)IIId(G). Furthermore
r2 <
q1/2
Clearly we have r1 = y2 1 and
=XIc(F)
-
2q1/2 implies
(see [5, Corollary 2], [1, Theorem 4]). Next using the fact that the
function y(t) a(t<-: [0,q"/2]
is increasing and the coeffi[?2q-"1/2]
cients of the both senriesare nonnegative yields y(rl) = r2. But this gives the
= y + q/ly.
desired 117r(Xl)llc*d(G)
= Y + q/y. Then the function z i'4 Tr[a(z)65 Remarks. Let IIir(X1)IIC*d(G)
-
is analytic inthedomain {z e C: a(z) 0 [-(y+qy2
particular it is analytic in D = {z E C: Izl < q 12}\([-q
7r(X1)]
2,
), y+qy2' 1]} In
).
Y]U [_,
Hence by (*) the function z/(1 - z 2)EYnzn as well as EYnZn can be continued analytically to D. It means that -y 1 and y2 are the only possible
singularpoints on the circle of convergenceof the power series N(z) = E ynZn.
THE GRIGORCHUK-COHENTHEOREM
665
Addedin proof. After submission of the manuscriptI was informed of the paper
by W. Woess, Cogrowthof groups and simple random walks, Archiv der mathematic 41 (1983), 363-370, where the same argumentsare used. In particulara
formula similar to (*) is proved there.
REFERENCES
1. J. M. Cohen, Cogrowth and amenability of discrete groups, J. Functional Analysis 48 (3) (1982),
301-309.
2. R. I. Grigorchuk, Symmetrical random walks on discrete groups, in "Multicomponent Random
Systems" ed. R. L. Dobrushin, Ya. G. Sinai, pp. 132-152, Nauka, Moscow 1978 (English
translation: Advances in Probability and Related Topics, Vol. 6, pp. 285-325, Marcel Dekker
1980.
3. H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 149-156.
4. T. Pytlik, Radial functions on free groups and a decompositionof the regularrepresentation
into irreducible components, J. Reine Angew. Math. 326 (1981), 124-135.
5. R. Szwarc,An analyticseries of irreduciblerepresentationsof thefree group,Ann. Inst. Fourier
(Grenoble) 38 (1) (1988), 87-1 10.
6. S. Wagon, Elementary problems E 3226, Amer. Math. Monthly 94 (8) (1987), 786-787.
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