Internat. J. Math. & Math. Sci.
VOL. 14 NO. 2 (1991) 261-268
261
INVERSES OF MEASURES ON A CLASS OF DISCRETE GROUPS
C. KARANIKAS
Department of Mathematics
Faculty of Science
Aristotle University oF Thessaloniki
54006 Thessalonlkl, Greece
(Received February 12, 1990)
ABSTRACF.
We
exam[he
a
class of groups G,
having a certain growth condition.
given an estimate for the norm of the inverse of an element in
l(G)
We
in terms of the
spectral tad[us and the card[nal[ty of the support.
KEY WORDS AND PHRASES. Discrete groups, spectral radius and measure algebra.
|980 AMS SUBJECT CLASSIFICATION CODES.
43A05, 43AI0, 22D05.
INTRODUCTION.
Throughout this work G is a discrete group and M(G) is the usual measure algebra
we write
for the unit of M(G), *v for the (convolution) product of two
measures
v m M(G).
on G:
,
B E
The purpose of this paper is to investigate the following problems: Let
M(G) have nite support and the (convolution) inverse of (-) exist in M(G).
Is
it
One can easily realize that this problem becomes more interesting in the "limit"
case, where the support of
is infinite. One also can realize the connection of this
with the general problem of the invertlbillty in M(G); namely the characterization of
the class of Hermltlan groups G (see [I]) or the equality of different norms and
spectrums in M(G) (see [2], [3] and [4]).
Our work here can be separated into two parts.
class of groups A and we glve an estimate of
the spectral radius
In the first part we consider the
I(-)-III
and of
r() and of the cardinallty of the support of
lexpll
.
in terms of
In the second part we examine the relation of the class A, with the class of
nilpotent groups and [FC] groups (groups having finite conjugacy class) (see [I]). We
show that nilpotent groups are A-groups and that the class A is closed under finite
extentlon. We should note that be Gromov’s well known result, any finitely generated
group G with polynomial growth is a finite extentlon of a nilpotent group, and so it
is an
A-group.
It is remarkable to note that all known Hermitlan groups, as they are
262
C. KARANIKAS
referred
to
[1], are A-groups.
in
We shall complete this introduction with
s3me
deflnLt[ons and lotatfons.
We say that G is an A_-group if there is a map
all fln[te subsets of G, and a
#(F n)
where
tFn)
Fn_-
F F
F)+n-I)X,
J
N, where 3
n e
is the set of
E
N
F (n-tlmes) and #F is the candinality of E.
An n-word in the etements of F is any (reduced) word of ength n. We denote by
a collection of n-words in the elements of F which as a subset of G consists of
al[ distinct elements of F
2.
K:
N such that for any F
e
n.
n
Finally we shall, denote by
the convolution product
NORMS OF CERTAIN INVERTIBLE MEASURES.
-
We are going to show the following:
THEOREM 2.1.
Let G be an A-group and let
<
support F and r()
where K(F) and
if exp
PROOF.
n
I.
Then
be an element in M(G), with finite
is invertible such that
are constants determined from the A-group structure.
Furthermore
we have
First we observe that for any n e
N
x EG
# (F
n)112p( )n
representation, i.e. the norm of the operator
: 12(G)
12(G):f
p*f.
Since
p(U) (r() we have
II:II
Now by (2.3)
#(Fn)l/2
r(p)n"
(2.3)
263
INVERSES OF MEASURES ON A CLASS OF DISCRETE GROUPS
+ # (F
<
n--O
Now let K
(F) we see that
2)1/2 rk) 2+
((F)+n-1)),
r)n
n
(K+n-I)X <
tKX+n-l).
In fact
Thus
n=O
and since r()
<
n
I, by the blnomlal formula, (6-)
To see (2.2) we observe that since
n
(K(F)+n-1),
n
H
J=l
<
g
>
and
-I
exists and we obtain (2.1).
J > I,
(- + I)
and
K(F)-I + I)
J
z:(F)
(2.4)
Thus by (2.3) and (2.4)
exp(K(F) Xr(u) ).
3.
THE CLASS OF GROUPS A.
In this section, we show that [FC] groups and nilpotent groups are A-groups.
also show that the class A is closed under finite extention.
First we examine the growth of
[FC] groups.
We
C. KARAN IKAS
264
Let G be an [FC] group and F be a finite subset of G.
PROPOSiTiON 3.[.
//
#
where k
PROOF.
(k+r-l)
r
F r)
f eF[f]),
if] is the conjugacy class of f.
J
We show that if iF]
F, f
fgf-l=gl,
that
g, there
if]
F
(k+l2
#k[F]2)
#kF 2)
f, g
N)
r
f
Given
Then
is
element
an
(say)
gl
iF]
[g]
in
such
gl f.
and so fg
Hence any 2-word in the elements of iF] consisting of two different letters is
equal to another 2-word in the elements of iF].
Thus iF]
F, f
f, g
#
2
has
no
k
than
more
#
k(k-l)/2 elements
fg
where
<
n, we show that it is also true for
no
We
of
and
k+
).
2
< k + k_(_k_-2
2
(F2)
We suppose that the Theorem is true for any r
r
f2
elements
It is clear that
g.
denote
by (g) the
number
of
appearances
all
of
a g e iF] in
the
words
iF] n) ’.
If
each g
elements of iF]
n
n
(iF]).
IF], (g)
the
had
all
g
same
chance
to
appear
in
(iF n])’ then for
Thus we may consider a g e IF] such that
# (iF] n)
k
(3.1)
(g)
g, there is some
Since for any f c iF]
fl
(say) such that fg
gfl" Hence
(iF]n) ’, either there is
without loss of generality we may assume that in any word of
no g or all
g’s keep the left place of the
elements of iF]
n-I
word.
Now from each word of
(iF]n) ’,
The resulting (n-l)-words form a subset of dlstlnt
where g appears, cancel one g.
we denote this set by g
-I
(iF] n).
Hence from our hypothesis it
is clear that
,k+n-2
#(g)
where
l(g)
+
n-l
(3.2)
l(g
is the number of all appearances of g in g
,l(g)
4
I_
-I (iF] n) ’.
k+n-2)
n-I
43.3)
then by (.I) and (B.2) we obtain
# (iF] n)
_k (I +
n
Suppose that
)( k+n-2
n-I
(k+n-I
n
and so in this case there is nothing to show.
If the inequality (3.3) does not occur, from (3.1) and (3.2) we ha,
INVERSES OF MEASURES ON A CLASS OF DISCRETE GROUPS
k
([Fin)
(_k_
n-1
ff
In a similar way we define
g In the collection g
As
-i
+
1))l(g)
i(g)
n
[hfl) )l (g)
(l
n-l) i.e. the number of appearances of
n (3.2)
i (g)
t=2,3,...,n-t)
n-I
and if for some
n___i
#t (g)
is
k,k+n-I
(iF] )’.
+
it
265
k+n-i-I
t3.4)
n-i
If the inequality (3.4) does not occur for any i
nothing to show.
n-i we
observe that
k
bn_l(g)
(1)
In this case we write
(k+l)
k(k+n-l)
n.(n-l)
(iF]n)
#
(k+n-l)!
(k-l)!n!
and this completes the proof.
COROLLARY 3.1.
If G is abelian and F is a finite subset of G then,
(F r)
PROOF.
(# F+r-l),
r
r e
N
Clear
LEMMA 3.1.
Let G be a discrete group with a normal subgroup K such that G/K is
be the canonical map :G
abellan, and let
# (F),
G/K. If FCG is such that #F
then the number of all r-words in the elements of F(r g
N), in a given class of G
modulo K can not be greater than
(#F +r r-l).
PROOF.
Let x
E
G/K fixed and J
-I () f F r
r
and F in this proof mean collections of r-words, which as elements of G
r
may not be distinct.
Note that I
We shall denote by #
#
Let x,y
([F]n)
F and xy
Jr
the cardinallty of
(#F
Jr’
and we shall show that
+
(3.5)
2
then yx
32,
fl2; in fact since G/K is abelian
(x)(y)
(yx).
Now suppose that there is a z
F such that x,z (or z,x)
is in
i.e. (x,y)
and
(x,z)
so
(y)
(z) and # (F) < # F-contradictlon.
J2’
Hence it is clear that 2 < # F and (3.5) follows.
(xy)
We suppose that Lemma 3.1 is true for any r
>
n-1 and we show this for
rfn.
For
C. KARAN IKAS
266
G/K, Let
for some g
-I
J
F
n.
F had the same chance to appear in
If all elements of
appearances (),
(g)
x in the words of
F, of
x
#(x)
We consider a x
=
n
#
Jn
J n’
then the number of all
should be
(/n
F such that
#j
#_.F (x)
n
(3.6)
n
From the set of all words In J n where x has at least one entry we cancel one
We denote by x Jn the collection of all the resultlng (n-l)-words. We show thatx
.
Jn
is in a class of G modulo K.
Let Wl,
be words in the elements of F such that w
w2,w’ I, w’2,
w,x w Jn
Since (w
w
2)
w2,
(WlW 2’)
w ’), we have (w
lW2
2
(w
and x 3 n
is
as we claimed.
Now, the set x Jn
#F+n-2
by our inductive hypothesis has cardinallty no greater than
and so
n-I
l(X)
where
-
#F+n-2
(x)
+
n-1
is the number of
1 (x)
Jn.
appearances of x in x
As in Proposition (3.1), (3.7) in the case where
1 (x)
it
is nothing to show.
obtain
#j
n
# F+n- 2
n-1
If the inequality above is not true by (3.6) and (3.7) we
#F #F+n-1
n-1
n
1 (x)
We complete the proof in the same arguments as in Proposition 3.1.
#F and
PROPOSITION 3.2. Any nilpotent group G is an A-group with (F)
%
2q-l, where q is the Index of G and F Is a finite set.
PROOF.
Let G
o
ntlpotent group G of index q.
(1
i
Aq
A D A ...’DA
q-I) and we denote by
q-1
We note that
i
We denote by
Frll
Af_l/l i
#(l(Fr)max
{#
E
the
normal
is the center of
the canonical map G
It is obvious that for any FC G and r
#(F r)
be
{e}
series
of
G/A t
G/A i.
N
-I (’)3 F r
:
G/AI}.
the RHS of (3.8).
We shall show that there are q positive integers ml,m2,...,mq such that
(3.8)
INVERSES OF MEASURES ON A CLASS OF DISCRETE GROUPS
#(F)
ml+m2+...+m q
and
Ir
6
2
m2+r-[
m +r-[ 2
iFrll
m
r
true for each hi[potent group of inde q
Let G be nilpotent of index p.
7
I(F).
(.I(F))
#
<
m
{x
F
where
m +r-I
3.9)
r
We suppose that (3.9)
I.
p
G/A
Since
q
is abetlan by Lemma 3.1 if
iFrll k# F +r r-I)2
We have,
Let #
+r-I 2
q-lr
Corollary k3.1) Implies (3.9).
If q=|, i.e. G is abet[an.
#F g #
267
l(Xj)
# (F), then F can be written as
(t 6 m
1,
ej:
i # j,
j
#(F)-ml}
j
and all
ml,
a.’s3
are in A
By (3.8) we obtain,
ml+r-I
Jl
where
where
-I
(g)
Fr’
Any element of
for some g
can be written as
(3.11)
xlia’31 xi2 a32 xlraJ
(XilXi_..... X,,r c each [_ (I
1,2,...,F-m
t
g’
one of
J1
r) Is one of 1,2,...,m
I.
and each
]t
is
m +r-I
By Lemma (3. t) the cardinality of all x i
in g is
x
r
r
and so
m +r-1
where
Jl
x
lXi2
g
i
xi F IXI2 F
xa
F
"r
is fixed suitably choosen from
(3.11) and belongs to g;
r
F
Corollary (3.1),
J
{aj:
Note that if q
F
rj
# F-m
I}
2, then A is the center of G, all
commute with
and by
# F
+r-1
); thus by (3.10) and (3.12), (3.9) follows.
As in (3.8), for some
]’s
-rml
xt’s
G/A 2 we have
(3.13)
Since
obtain
AI/A 2
is the center of
G/A 2 and
FIC AI,
by (3.10), (3.12) and (3.13) we
C. KARANIKAS
268
[,Flrll
where
is defined as
[Frll.
is a nilpotent of index p-l we apply our inductive hypothesis in (3.14)
Since A
p we obtain (3.9).
and for q
Now if we replace m|,...,mq in (3.9) by #F we see that G is an A group with
and k =2q-l.
constants k
PROPOSITION 3.3.
The
class of A-groups is closed under extensions by finite
group s.
We
PROOF.
may
write
{dlA, d2A, ..., dsA}
G/A
representatives of all the different classes of
where each d(i,j) is one of
(1,t
Let
<x i>
(I
a(i,J) d(i,J)
dldj
{d i
dl,d2,...,d s
are
s
We may also write
e the unit of G.
Let F
where
G/A" without loss of generality let d s
{dIxidj
J
s)
and each a(i,J) is in A.
dl,...,d s
Xl,...,d i mXm}
i,
#F=m, each
1,2,..,s} (I
d
i
is one of
t
i
dl,...,d s
and x
t
e A
m) and
(3.15)
djlXildj2 xi2"’" djr Xir
be a typical r-word in the elements of F.
<Xil> a(Jl ’j2
d(Jl ’j2)xi2
<xi> a(jl,J2)
<x i > a(.,j3
2
<x i > <x i >
2
where a e A and d e {d
dj r x i r
Each word in (3.15) is in the set
or in
dj r x i r
<x i >
or finally in
a.d.
(3.16)
r
l,d 2,...,ds }"
It is clear that the cardlnality of the r-words in the elements of F, as in
(3.15) is less than the candinality of the words in (3.16) in the elements
which is a subset of the A-group, A. Thus G inherits the growth
of
{<Xl>,
<Xm>},
of A.
REFERENCES
I.
PALMER, T. Classes of Non-Abellan, Non-Compact, Locally Compact Groups,
Moun. J. Math. 8(4) (1978), 683-741.
2.
FOUNTAIN, J.B., RUMSAY, R.W. and WILLIAMSON, J.H. "Functions of Measures on
Compact Groups, Proc. Roy.. Irish Acad. Sect. A., 76 (1976), 235-251.
3.
KARANIKAS, C. and WILLIAMSON, J.H. Norms and Spectral for Certain Subalgebras
of M(G), Math. Proc. Camb. Phil. Soc. 95 (1984), 109-122.
4.
KARANIKAS, C. Wiener Pairs of Measure Algebras, Pac. Jour. Math. 139(I),
(1989), 79-86.