INVERSES OF MEASURES ON A CLASS OF DISCRETE GROUPS

ABSTRACF. We exam[he a class of groups G, having a certain growth condition. We given an estimate for the norm of the inverse of an element in l(G) in terms of the spectral tad[us and the card[nal[ty of the support.

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 referred to in [1], are A-groups.
We shall complete this introduction with s3me deflnLt[ons and lotatfons.
We say that G is an A_-group if there is a map K: J N, where 3 is the set of all fln[te subsets of G, and a e N such that for any F E #(F n) F)+n-I)X, n e N Fn_where F F 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 tFn) a collection of n-words in the elements of F which as a subset of G consists of n al[ distinct elements of Fn.Finally we shall, denote by the convolution product We are going to show the following: THEOREM 2.1.
Let G be an A-group and let be an element in M(G), with finite support F and r() < I. Thenis invertible such that where K(F) and are constants determined from the A-group structure.Furthermore n if exp we have PROOF.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 #(Fn)l/2 r(p)n" (2.3) Now by (2.3) (F) we see that (K+n-I)X < tKX+n-l).In fact Thus n=O n and since r() < I, by the blnomlal formula, (6-) -I exists and we obtain (2.1).
In this section, we show that [FC] groups and nilpotent groups are A-groups.We also show that the class A is closed under finite extention.
First we examine the growth of [FC] groups.# f eF[f]), if] is the conjugacy class of f.
PROOF.We show that if iF] such that fgf-l=gl, and so fg gl f.
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]
2 has no more than k elements f2 and k(k-l)/2 elements fg where f, g ).
We suppose that the Theorem is true for any r < n, we show that it is also true for r no We denote by (g) the number of all appearances of a g e iF] in the words of iF] n) '.
If all elements of iF] had the same chance to appear in (iFn])' then for n n each g g IF], (g) Thus we may consider a g e IF] such that k Since for any f c iF] g, there is some fl (say) such that fg gfl" Hence without loss of generality we may assume that in any word of (iF]n) ', either there is no g or all g's keep the left place of the word.Now from each word of (iF]n) ', where g appears, cancel one g.The resulting (n-l)-words form a subset of dlstlnt we denote this set by g (iF] Hence from our hypothesis it is clear that where l(g) is the number of all appearances of g in g (iF] '.Suppose that k+n-2) In a similar way we define i(g) (l n-l) i.e. the number of appearances of -i n g In the collection g (iF] )'.
If the inequality (3.4) does not occur for any i n-i we observe that k bn_l(g) (1) In this case we write and this completes the proof.Let G be a discrete group with a normal subgroup K such that G/K is abellan, and let be the canonical map :G G/K.If FCG is such that #F # (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-l).r -I F r PROOF.Let x E G/K fixed and J () f that I and F r in this proof mean collections of r-words, which as elements of G Note r may not be distinct.
We shall denote by # Jr the cardinallty of Jr' and we shall show that # ([F]n) (#F + 2 (3.5)Let x,y F and xy 32, then yx fl2; in fact since G/K is abelian (xy) (x)(y) (yx).
Now suppose that there is a z F such that x,z (or z,x) is in J2' i.e. (x,y) (x,z) and so (y) (z) and # (F) < # F-contradictlon.
Hence it is clear that 2 < # F and (3.5) follows.
We suppose that Lemma 3.1 is true for any r > n-1 and we show this for rfn.For for some g G/K, Let -I F n.

J (g)
If all elements of F had the same chance to appear in J n' then the number of all appearances (), x F, of x in the words of Jn should be n #(x) = # (/n We consider a x F such that # j #_.F (x) 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, w2,w' I, w'2, be words in the elements of F such that w w2, '), we have (w ') and 3 w,x w Jn Since (w w 2) (w w 2 lW2 (WlW 2 x n as we claimed. is Now, the set x Jn by our inductive hypothesis has cardinallty no greater than #F+n-2 n-I and so #F+n-2 (x) where l(X) is the number of appearances of x in x Jn.
As in Proposition (3.1), (3.7) in the case where #F+n-2 1 (x)n-1 it is nothing to show.
If the inequality above is not true by (3.6) and (3.7) we obtain We complete the proof in the same arguments as in Proposition 3.1.PROPOSITION 3.2.Any nilpotent group G is an A-group with (F) #F and % 2q-l, where q is the Index of G and F Is a finite set.

PROOF.
Let G A D A ...'DA Aq {e} be the normal series of o q-1 ntlpotent group G of index q.We note that Af_l/l i is the center of G/A t (1 i q-I) and we denote by i the canonical map G G/A i.We denote by Frll the RHS of (3.8).
Let G be nilpotent of index p.Since G/A is abetlan by Lemma 3.1 if #F g # 7 I(F).
We have, iFrll k# F + r r-I)2 Let # (.I(F)) m < # (F), then F can be written as F {x ej: (t 6 m 1, j #(F)-ml} where # l(Xj) i # j, j ml, and all a.'s3 are in A By (3.8) we obtain, ml+r-I Jl -I Any element of J1 can be written as where (g) Fr' for some g xlia'31 xi2 a32 xlraJ J # F-m I} Note that if q 2, then A is the center of G, all ]'s commute with xt's and by rj # F +r-1 Corollary (3.1), F -rml ); thus by (3.10) and (3.12), (3.9) follows.
where Since A is a nilpotent of index p-l we apply our inductive hypothesis in (3.14) and for q p we obtain (3.9).Now if we replace m|,...,mq in (3.9) by #F we see that G is an A group with constants k and k =2q-l.
The class of A-groups is closed under extensions by finite group s. PROOF.
We may write G/A {dlA, d2A, ..., dsA} where dl,d2,...,d s are s representatives of all the different classes of G/A" without loss of generality let d s e the unit of G.We may also write dldj a(i,J) d(i,J) where each d(i,j) is one of dl,...,d s and each a(i,J) is in A.
Let F {d i Xl,...,d i Xm} #F=m, each d i is one of dl,...,d s and x t e A m t (1,t Let <x i> {dIxidj J 1,2,..,s} (I i m) and djlXildj2 xi2"'" djr Xir (3.15)   be a typical r-word in the elements of F. Each word in (3.15) is in the set dj x i or in <Xil> a(Jl 'j2 d(Jl 'j2)xi2 r r <xi> a(jl,J2) <x i > a(.,j3 dj x i or finally in 2 r r <x i > <x i > <x i > a.d.where a e A and d e {d l,d2,...,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 of {<Xl>, <Xm>}, which is a subset of the A-group, A. Thus G inherits the growth of A.

COROLLARY 3 .
1.If G is abelian and F is a finite subset of G then, It is obvious that for any FC G and r E N -I F r #(Fr) #(l(Fr)max {# (')3 : G/AI}.