Markov Operators on the Solvable Baumslag Solitar Groups

Transcription

1 Markov Operators on the Solvable Baumslag Solitar Groups Florian Martin and Alain Valette CONTENTS. Introduction. Connectedness of Spectra. The Nonsymmetric Case. The Symmetric Case 5. An Application to Wavelet Theory Acknowledgements References We consider the solvable Baumslag Solitar group BS n = ha, b j aba = b n i, for n, and try to compute the spectrum of the associated Markov operators M S, either for the oriented Cayley graph (S = fa, bg), or for the usual Cayley graph (S = fa, b g). We show in both cases that Sp M S is connected. For S = fa, bg (nonsymmetric case), we show that the intersection of Sp M S with the unit circle is the set C n of (n )-st roots of, and that Sp M S contains the n circles fz C : jz!j = g, for! C n, together with the n + curves given by wk w k exp i = 0, where w C n+, 0, ]. Conditional on the Generalized Riemann Hypothesis (actually on Artin s conjecture), we show that Sp M S also contains the circle fz C : jzj = g. This is confirmed by numerical computations for n =,, 5. For S = fa, b g (symmetric case), we show that Sp M S =, ] for n odd, and Sp M S =, ] for n =. For n even, at least, we only get Sp M S = r n, ], with < r n sin n (n + ). We also give a potential application of our computations to the theory of wavelets.. INTRODUCTION Let be a nitely generated group. Fix a nite, generating, not necessarily symmetric subset S in. To these data we associate the oriented Cayley graph (or Cayley digraph) G( ; S), whose vertex set is, and whose set of oriented edges is f(x; xs) : x ; s Sg. If S is symmetric (S = S ), it is customary to replace a pair of opposite edges by a single, nonoriented edge. c A K Peters, Ltd /000 $0.50 per page Experimental Mathematics 9:, page 9

2 9 Experimental Mathematics, Vol. 9 (000), No. On G( ; S), consider the simple random walk, in which a particle jumps from x to xs with a probability of jsj. Denote by p(n) (x; y) the probability of a transition in n steps from x to y. It can be expressed by means of the Markov operator M S on `( ), dened by (M S )(x) = jsj X ss (xs); for `( ); x : Indeed p (n) (x; y) = hms n x j y i, where ( x ) x is the canonical basis of `( ). The operator M S is a contraction (km S k ) on `( ); its spectrum Sp M S is therefore a closed nonempty subset of the unit disk in C. A number of results, going back to Kesten 959] in the symmetric case, and to Day 96] in the general case, exhibit a close relationship between the properties of Sp M S and those of the pair ( ; S). Here is a sample; we denote by T the multiplicative group of complex numbers of modulus, and by C n the group of n-th roots of in C. Theorem... is amenable if and only if Sp M S.. If is amenable, then (Sp M S ) \ T is a closed subgroup of T ; moreover, for z T, the following statements are equivalent: a. z Sp M S ; b. Sp M S is invariant under multiplication by z; c. there exists a homomorphism :! T such that (S) = fzg.. Sp M S = C n if and only if ' Z =nz and jsj = ; and Sp M S = T if and only if ' Z and jsj =. For proofs, see respectively Day 96; de la Harpe et al. 99, Proposition III; de la Harpe et al. 99, Proposition ]. Gromov has asked which properties of Sp(M S ) are invariant under quasi-isometries, pointing out that the Kesten{Day result mentioned above provides an example (since amenability is a quasi-isometry invariant). To attack Gromov's question, one dif- culty lies with the lack of examples of explicitly computed spectra of Markov operators. The aim of this paper is to present a class of solvable, non virtually nilpotent groups, for which some explicit calculations are possible. In this paper we deal with the solvable Baumslag{ Solitar groups, a family of one-relator groups dened by the presentations BS n = ha; b j aba = b n i; for n. These groups belong to a two-parameter family of one-relator groups introduced by Baumslag and Solitar 96]. There has been recent activity around the groups BS n ; for example, here is a remarkable result by B. Farb and L. Mosher 998]: Theorem.. The following statements are equivalent:. The groups BS m and BS n are quasi-isometric.. The groups BS m and BS n are commensurable.. There exists an integer r such that m and n are powers of r. C. Pittet and L. Salo-Coste 999] have determined the isoperimetric prole and the rate of decay of the heat kernel on BS n. Our goal is to study the spectrum of the Markov operator M S, where we take either S = fa; bg or S = fa ; b g as generating subset of BS n. Our results are as follows: We show in Section that Sp M S is always connected. For S = fa; bg (nonsymmetric case), we show in Section that (Sp M S ) \ T = C n, and that Sp M S contains the n circles fz C : jz!j = g; for! C n ; together with the n + curves given by wk w k exp i = 0; where w C n+, 0; ] (and also their images under the action of the symmetry group of Sp M S, which turns out to be the dihedral group D n of order n ). Assuming the Generalized Riemann Hypothesis or just Artin's conjecture Murty 988] we show that Sp M S also contains the circle fz C : jzj = g. This is conrmed by numerical computations for n = ; ; 5. For S = fa ; b g (symmetric case), we show in Section that ; ] if n is odd, Sp M S = r n ; ] if n is even,

3 Martin and Valette: Markov Operators on the Solvable Baumslag Solitar Groups 9 where < r n sin n (n + ) (notice that lim n! r n = ). For n =, we get the exact value r =. In Section 5, we give a potential application of our study to the theory of wavelets (see, for instance, Daubechies 99]); to explain the link, notice that BS is isomorphic to the subgroup of the ane group of the real line, generated by translation by and dilation by : these are exactly the two transformations used in multiresolution analysis. All of our computations rest on the following lemma, useful in constructing points of Sp M S. Lemma.. Let be a nitely generated amenable group. Dene h S = jsj X ss s C : For every unitary representation of, one has Sp (h S ) Sp M S. Proof. Denoting by the right regular representation of on `( ), one has M S = (h S ). For a group, we shall denote by C r the reduced C*-algebra of, i.e. the C*-algebra generated by ( ) (notice that M S C r ). When is amenable, any unitary representation of extends to C r, hence denes a quotient of C r. But passing to a quotient only decreases the spectrum. For this reason we are interested in nding families of representations of C r, and especially separating families. Our interest in separating families lies in the fact that, at least for self-adjoint elements in C r, they \approximate" well the spectrum (see formula ({) and the proof of Theorem.). We construct several of them in this paper.. CONNECTEDNESS OF SPECTRA We begin by realizing BS n in a more concrete way. For a commutative, unital ring A, we denote by A (A) the ane group of A (or \ax + b" group): this is the semidirect product of the additive group of A by the multiplicative group. It is easy to check that BS n can be identied with the subgroup of A (Q ) generated by the dilation a : x 7! nx and the translation b : x 7! x +. Lemma.. Every element of C r BS n has a connected spectrum. Proof. The proof is in the same spirit as that of Beguin et al. 997, Proposition ]; the statement to be proved is equivalent to the conjecture of idempotents for C r BS n. Recall that, for a torsion-free group (notice that BS n is a torsion-free group), the conjecture of idempotents for says that the only idempotents in C r ( ) are 0 and. This in turn is a consequence of the Baum{Connes conjecture for, which says that the analytical assembly map (or index map) 0 : RK 0 (B )! K 0 (C r ( )) is an isomorphism; here RK 0 (B ) denotes the K- homology with compact support of the classifying space B, and K 0 (C r ( )) denotes the Grothendieck group of nite type projective modules over C r ( ) (see Baum et al. 99; Valette 989]). There are at least dierent proofs of the Baum{Connes conjecture for BS n. Kasparov et Skandalis 99] have shown that the Baum{Connes conjecture is true for every torsion-free discrete subgroup of A (K ) A (K m ), where the K i 's are local elds. Then one may appeal to an arithmetic realization of BS n : if p ; : : : ; p k is the list of prime divisors of n, the diagonal embedding BS n,! A (Q p ) A (Q pk ) A (R ) has discrete image. The Baum{Connes conjecture has been proved for torsion-free one-relator groups Beguin et al. 999]. Higson and Kasparov 997] proved the Baum{ Connes conjecture for all torsion-free amenable groups, in particular for all torsion-free solvable groups.. THE NONSYMMETRIC CASE In this section, we set S = fa; bg. We may already deduce some qualitative informations about the spectrum of M S. Theorem.. Sp M S is a connected subset of the closed unit disk of C, such that:

4 9 Experimental Mathematics, Vol. 9 (000), No.. (Sp M S ) \ T = C n ;. the symmetry group of Sp M S is D n, the dihedral group of order n ;. Sp M S contains the n circles fz C : jz!j = g; for! C n :. Sp M S contains the n + curves given by Proof. wk w k exp i = 0; where w C n+, 0; ].. From the presentation BS n = ha; b j aba = b n i, it is clear that any homomorphism from BS n to an abelian group satises (b) n = ; on the other hand, for any! C n one may dene a homomorphism! : BS n! C n by! (a) =! (b) =!. The result then follows from Theorem.... For every group and every nite generating subset S, the spectrum of M S is symmetric with respect to the real axis in C de la Harpe et al. 99, Proposition (ii)]. So the result is a consequence of part and Theorem.... Dene an epimorphism ab : BS n! Z Z =(n )Z by ab (a) = (; 0) and ab (b) = (0; ). The Pontryagin dual of Z Z =(n )Z is T C n, and C r (Z Z =(n )Z ) is identied via the Fourier transform with the algebra of continuous functions on T C n. The Fourier transform of ab (h S ) is the function T C n! C : (z;!) 7! z +! : The spectrum of ab (h S ) is the range of this function, i.e. the union of n circles appearing in the theorem. By Lemma., these circles are contained in Sp M S.. Consider the epimorphism of BS n onto ha; b j aba = b ; b n+ = i. This group can be identi- ed with the semidirect product Z =(n+)z o Z (where the action is given by m (k) = ( ) m k). It contains the normal subgroup Z =(n+)z Z. So by Mackey theory, the irreducible representations of the semidirect product, which are obtained by inducing the characters of the normal abelian subgroup, are given by and 0 k; (a) = k; (b) = e i e i 0 w k 0 0 w k ; where w = e n+ i ; k = ; : : : ; n + ; 0; ]. Now, by computing the spectra of k; (h S ), we obtain the curves appearing in the theorem, and again we conclude by Lemma.. In fact, part follows from and. Indeed, = 0;] Sp( n+; (h S )) = z C : z ; and by using part of the theorem, we get part. It is easy to check that ab is just the abelianization homomorphism of BS n (that is, ker ab is the commutator subgroup of BS n ). In other words, Theorem.. describes the contribution to Sp M S of the abelianized group of BS n. To proceed, we clearly need other representations which do not factor through the abelianization of BS n and through the group ha; b j aba = b ; b n+ = i: We now construct a family of such representations, viewing BS n as an \arithmetic" group. For a prime p, we denote by F p the eld with p elements, and by ` 0(F p ) the orthogonal complement of the constants in `(F p ). We begin by recalling the representation theory of the nite group A (F p ). Lemma.. The group A (F p ) has p irreducible representations, namely: the p characters 0 ; : : : ; p coming from the epimorphism A (F p )! F p ; one representation p of degree p, on the space ` 0(F p ), associated with the action of A (F p ) on F p. Proof. A standard exercise in the representation theory of nite groups; see, for example, Robert 98, pp. 59{60].

5 Martin and Valette: Markov Operators on the Solvable Baumslag Solitar Groups 95 Let us come back to the group BS n. From the description as a subgroup of A (Q ), it follows that BS n is in fact a subgroup of A (Z n ), where Z n is the subring of Q generated by. n If p is a prime not dividing n, reduction modulo p from Z n onto F p, induces a homomorphism p : BS n! A (F p ): Denote by p the regular representation of A (F p ). Proposition.. For every innite set S of primes not dividing n, the family of representations ( p p ) ps is separating for C r BS n. Proof. We show that the regular representation BSn of BS n is weakly equivalent (in the sense of Dixmier 977,..5]) to L ps p p. The latter is weakly contained in BSn, because of amenability of BS n. For the converse, for p S let us denote by p the characteristic function of the identity of A (F p ), viewed as a vector in `(A (F p )); consider the positive denite function ' p on BS n, dened by ' p (g) = h p ( p (g)) p j p i; with g BS n. Clearly if g ker p, ' p (g) = 0 otherwise. Since any element g BS n feg belongs to a nite number of subgroups ker p, we have lim ' p (g) = g;e = h(g) e j e i; p!; pp n by Dixmier 977, 8..], this shows that BSn weakly contained in L ps p p, and concludes the proof. For a xed prime p, the homomorphism p is onto if and only if n is a primitive root modulo p, i.e. is a generator of the multiplicative group of F p. Set a p = p (a) and b p = p (b). Proposition.. Let p be an odd prime. If n is a primitive root modulo p, then Sp p (a p + b p ) consists of 0 and the (p )-st roots of, distinct from. Proof. In fact we shall determine Sp p (a p + b p ), which is the image of the desired spectrum under complex conjugation, and which will turn out to be invariant under complex conjugation. Set! = e i We work in the basis of characters of ` 0(F p ): e i (j) =! ij (i = ; : : : ; p ; j F p ): is p. Clearly e i is an eigenvector of p (b p ), with eigenvalue! i. On the other hand p (a p )(e i ) = e ni. The assumption allows us to re-arrange the basis of e i 's according to powers of n; thus we work in the basis e ; e n ; e n ; : : : ; e n p. Then: p (a p +b p ) = ! n 0 0 0! n ! np 0 0 0! np C A To compute the characteristic polynomial of this matrix, we develop the determinant according to the rst row, and get (since p is odd): det( p (a p Y p + b p ) ) = (! ni ) = i=0 p Y j= (! j ) = p = ( p ) ; where the second equality follows from the assumption on n. The result is now clear. The preceding proposition is false when n is not a primitive root modulo p; this is clearly visible on Figure, which shows, for n = ; ; 5, the union of the sets Sp p (a p +b p ) for p running over the rst 00 primes. Corollary.5. Let p be an odd prime. If n is a primitive root modulo p, the spectrum of p ( p (h S )) consists of : 0 with multiplicity p; +exp ij p with multiplicity, for j = 0,..., p distinct from (p ); ik exp with multiplicity p, for k =,..., p p. Proof. Notice that p (h S ) = (a p + b p ). Let 0,..., p be the characters of A (F p ) (see Lemma.). In view of the assumption, j is determined by its value on a p, and we may assume j (a p ) = exp( ij ), p so that (a j p + b p ) = + exp ij p : :

6 96 Experimental Mathematics, Vol. 9 (000), No. FIGURE. p running For n = (left), n = (middle) and n = 5 (right), the graphs show the union of Sp p (ap + bp ) for over the rst 00 primes. The regular representation p decomposes into p (with multiplicity p ) and the j 's (each with multiplicity ). The result follows from this and the previous proposition. Remark. For p = 7, there are some coincidences between eigenvalues of the second and third kind in Corollary.5. The reader will have no di culty to compute the correct multiplicities. The previous proposition and corollary raise a natural question: how many primes p are there, such that n is a primitive root modulo p? It turns out that this is an open problem in number theory! For n, denote by Pn the set of primes p such that n is a primitive root modulo p. Artin's conjecture for the integer n is the following statement: Conjecture.6. The set Pn is in nite. For an excellent introduction to Artin's conjecture, see Murty 988]. We collect in the theorem below some striking results on Artin's conjecture. Part is due to Hooley 967], and parts and to HeathBrown 986]. Theorem.7.. Artin's conjecture for any n follows from the Generalized Riemann Hypothesis (the statement that the Dedekind -function of any number eld K satis es the Riemann Hypothesis ).. Artin's conjecture holds for prime n, with at most two possible exceptions.. Artin's conjecture holds for square-free n, with at most three possible exceptions. Since spectra are closed subsets of C, it follows immediately from Corollary.5 (together with Lemma.): Theorem.8. If Artin's conjecture holds for n, then Sp MS fz C : jzj = g: We will apply Proposition. and Theorem.7 to a classical problem in operator theory, namely the description of the spectrum of a direct sum of operators. Indeed, if A is a C*-algebra and ( i)ii is a separating family of representations of A, the equality Sp x = ii Sp i(x) ( ) holds provided that x is a normal element in A (e.g., a self-adjoint element). The classical example, showing that this equality fails in general, is given in Halmos 967, solution to Problem 8]. Here we get new examples of the same situation. Corollary.9. For at least one n f; ; 5g, the family ( p p )ppn is separating for Cr BSn, but the inclusion ppn Sp( p p )(hs ) Sp MS is strict. Proof. By Theorem.7., the set Pn is in nite for at least one n f; ; 5g; for this n, the family of

7 representations ( p p ) ppn is separating for C r BS n (Proposition.). By Corollary.5, we have pp n Sp( p p )(h S ) = fz C : jzj = g z C : jz j = : But a glance at Figure shows that, in every case, Sp M S contains points outside of the union of these two circles. For n = ; 5, we may also appeal to the fact that Sp M S contains (by Theorem..). Martin and Valette: Markov Operators on the Solvable Baumslag Solitar Groups 97 The description of BS n as a subgroup of A Z n makes it clear that BS n is actually a semidirect product: BS n = Z ] o n a Z : We are going to consider representations of BS n induced from characters of the normal subgroup Z n. For R, we denote by the character of the real line dened by (x) = e ix, for x R. Lemma.. The family of representations. THE SYMMETRIC CASE In this section we set S = fa ; b g. The following result is analogous to Theorem.. Theorem.. ; ] if n is odd, Sp M S = r n ; ] if n is even, where < r n sin n (n + ) : Proof. Amenability guarantees that the spectrum is r n ; ] for some r n (Theorem.. and Lemma.). The case of n odd is trivial. Indeed Sp(M S ) = ; ] if and only if there is a homomorphism : BS n! C = f; g mapping a and b to - (Theorem..); and such a homomorphism exists if and only if the relation in the group is of even length (i.e. n is odd). Now assume that n is even. The preceeding remark shows immediatly that r n >. To get the upper bound on r n, we use the representations k; dened in the proof of Theorem... The spectrum of k; (h S ) is k (cos cos ) and is contained n+ in Sp(M S ) (Lemma.). For k = n and = 0, we get the minimal value cos n n + n = sin (n + ) : Note that the contribution to Sp M S of the abelianized group of BS n (by considering ab (h S )) does not improve the upper bound for r n. The subtlety here is that formula ({) does not hold for every element in a C -algebra. Pretending that it does leads to a quick disproof of the Generalized Riemann Hypothesis, just by glancing at Figure ; the second author used this as the basis of an April fool's joke (a la Bombieri). is separating on C r BS n. Ind BS n Z=n] RestZ=n] R R Proof. Since the dual group of R is dense in the dual group of Z n, the family of characters Rest Z=n] R R is weakly equivalent to the regular representation Z=n] of Z n. By continuity of weak containment with respect to induction, the family Ind BS n Z=n] RestZ=n] R R is weakly equivalent to Ind BS n Z=n] Z=n] ' BSn. Using the semidirect product decomposition of BS n, we see that the representation =: Ind BS n Z=n] RestZ=n] is canonically realized on `(Z ); in that picture, the generator a acts by the bilateral shift on `(Z ), while the generator b acts by R ( (b))(k) = e in k (k); for `(Z ), k Z. Therefore (h S ) is a tridiagonal operator: ( (h S ))(k) = (k ) + (k + ) + cos(n k )(k) : To estimate the spectrum of a tridiagonal operator, one may appeal to the following remarkable result by R. Szwarc 998]: Proposition.. Let J be the operator on `(Z ) dened by J(k) = k+ (k + ) + k (k) + k (k ); where ( k ) kz ; ( k ) kz are real, bounded sequences, with k > 0 for all k. Let m be such that m <

8 98 Experimental Mathematics, Vol. 9 (000), No. inf kz k. Assume there exists a sequence (h k ) kz in ]0; such that k (m k )(m k ) h k( h k ) for every k Z. Then Sp J m; +. From this we deduce: Theorem.. For n = and S = fa ; b g, the spectrum of the Markov operator M S on BS is Sp M S = ; ]. Proof. We begin by showing that belongs to the spectrum of M S. For this, we consider the prime p = and the representation, of degree, appearing in Lemma.. From the formulae in the proof of Proposition., it is clear that ( (h S )) = : The spectrum of this matrix is ;, and it is contained in Sp M S by Lemma.. To show the converse inclusion, we nd a sequence (h n ) nz ]0; satisfying ( + cos (n ) ')( + cos n ') h n( h n ); where ' =. By Proposition., this will imply that Sp( (h S )) ;, for all R. If we dene h n = + n + cos n ' for all n Z, we have to search for a sequence ( n ) nz such that + cos (n ) ' n + cos n ' + n and cos n ' < n < + cos n ': A candidate is n = f( n '), where f is dened on 0; ] by with f(x) = 8 >< >: 0 if x 0; ] 5 ; ]; u (x) if x ; ; 5 ; u (x) if x ; ], u (x) := ( +cos x)+ +cos x ; u (x) := ( + +cos x)+ cos x +cos x : Next extend f to be periodic of period. To show that + cos x f(x) + cos x + f(x) we must verify several conditions: + cos x + cos x 0 for x 0; 6 6 ;. This follows from simple trigonometry estimates and is clear from the graph of the function on the left-hand side of the inequality: cos x + cos x + u (x) 0 for x ; 6 5 ; 6. The denition of u was cooked up exactly so that this is satised. + cos x u (x) + cos x + u (x) 0 for x ; ; 5. Again, this comes from the denition of u. +cos x u (x) +cos x+u (x) 0 for x ; ; 6. A slightly tedious computation shows that this is equivalent to u (x) 0 for the same range of x, and again this is clear from the graph of u : + cos x + cos x) cos x u (x) + cos x 0 for x 5 ; 7 ]. This is equivalent to u 6 6 (x) + cos x + cos x for the same range of x (see pre- ceding graph). 7 6 u

9 Martin and Valette: Markov Operators on the Solvable Baumslag Solitar Groups 99 Finally, each u i, for i = ;, satises cos x u i (x) + cos x for x 0; ]: + cos x ( + cos x) u u This implies that cos n ' < n < + cos n ', and the result follows. The value in Theorem 7 was discovered experimentally, by computing numerically the spectrum of p (a p + a p + b p + b p ) for small primes p. For n larger than and less than 8, we also approximated the smallest value of Sp p (a p + a p + b p + b p ) by numerical computations for p running over the rst 00 primes, but that does not improve the upper bound in Theorem AN APPLICATION TO WAVELET THEORY As observed, the connection with wavelet theory comes from the fact that BS is isomorphic to the subgroup of A (R ) generated by translation by and dilation by. These are exactly the two transformations used in multiresolution analysis Daubechies 99; Bultheel 995]; for this reason, we think that BS deserves to be called the wavelet group. We recall some notations from wavelet theory. On L (R ), dene the unitary operators for r R, L (R ), and (T r )(x) = (x r); (D s )(x) = p s x s ; for s > 0, L (R ). Setting (a) = D n and (b) = T then denes a unitary representation of BS n on L (R ). Theorem 5.. The map extends to a faithful representation of C r BS n. Proof. We have to show that is weakly equivalent to BSn. Once more, weak containment of in BSn follows from amenability of BS n. To prove the converse, dene a function L (R ) by 8p n on 0;, n (x) = < : p n on n ; n, 0 otherwise. Clearly k k = ; note that, for n =, the function is just the Haar wavelet. For k; m Z, set k;m(x) = (D n kt m )(x) = n k (n k x m): The k;m 's are orthonormal (but not a basis for n > ): indeed, considerations of supports show that two k;m 's of the same scale (same value of k) never overlap; on the other hand, if k < k 0, then the support of k;m lies totally in a region where k 0 ;m is constant, so that 0 h k;mj k 0 ;m0i = 0. For g BS n, the operator (g) can be written uniquely (g) = D n jt r, with j Z and r Z n. For k N, h(g) k;0 j k;0 i = hd n jd n kt r D n k j i = hd n jt rn k j i: But, for k big enough, rn k is an integer N, so that h(g) k;0 j k;0 i = h j;n j i = e;g ; by orthonormality of the k;m 's. This shows that BSn is weakly contained in, so the proof is complete. Remark. In the case n =, we used in the above proof the Haar wavelet, but any wavelet basis would do as well. From this result and the connectedness of spectra in C r BS n (see Lemma.), we immediately deduce: Corollary 5.. On L (R ), operators of the form X k;mz c k;m D n kt m (with c k;m C, only nitely many nonzero c k;m 's), have connected spectra. In particular, this applies to the operators X mz c m D T m appearing in the two-scale relation (or dilation equation) in multiresolution analysis Bultheel 995, x 5].

10 00 Experimental Mathematics, Vol. 9 (000), No. ACKNOWLEDGEMENTS We thanks K. Taylor and A. Zuk for many useful conversations around the contents of this paper. REFERENCES Baum et al. 99] P. Baum, A. Connes, and N. Higson, \Classifying spaces for proper actions and K-theory of group C*-algebras", pp. {9 in C -algebras 9{ 99 : a fty year celebration, edited by R. S. Doran, Contemp. Math. 67, 99. Baumslag and Solitar 96] G. Baumslag and D. Solitar, \Some two-generator one-relator non-hopan groups", Bull. Amer. Math. Soc. 68 (96), 99{0. Beguin et al. 997] C. Beguin, A. Valette, and A. Zuk, \On the spectrum of a random walk on the discrete Heisenberg group, and the norm of Harper's operator", J. Geometry and Physics (997), 7{56. Beguin et al. 999] C. Beguin, H. Bettaieb, and A. Valette, \K-theory for C -algebras of one-relator groups", K-Theory 6: (999), 77{98. Bultheel 995] A. Bultheel, \Learning to swim in a sea of wavelets", Bull. Belg. Math. Soc. (995), {. Daubechies 99] I. Daubechies, Ten lectures on wavelets, CBMS/NSF regional conference series in applied mathematics 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 99. Day 96] M. M. Day, \Convolutions, means and spectra", Illinois J. Math. 8 (96), 00{. Dixmier 977] J. Dixmier, C -algebras, North Holland, Amsterdam, 977. Farb and Mosher 998] B. Farb and L. Mosher, \A rigidity theorem for the solvable Baumslag-Solitar groups", Invent. Math. (998), 9{5. Halmos 967] P. R. Halmos, A Hilbert space problem book, Van Nostrand, New York, 967. Reprinted by Springer, 98. de la Harpe et al. 99] P. de la Harpe, A. G. Robertson, and A. Valette, \On the spectrum of the sum of generators of a nitely generated group", Israel J. Math. 8 (99), 65{96. de la Harpe et al. 99] P. de la Harpe, A. G. Robertson, and A. Valette, \On the spectrum of the sum of generators of a nitely generated group, II", Colloquium Math. 65 (99), 87{0. Heath-Brown 986] D. R. Heath-Brown, \Artin's conjecture for primitive roots", Quart. J. Math. 7 (986), 7{8. Higson and Kasparov 997] N. Higson and G. Kasparov, \Operator K-theory for groups which act properly and isometrically on Hilbert space", Electron. Res. Announc. Amer. Math. Soc. (997), {. Hooley 967] C. Hooley, \On Artin's conjecture", J. reine angew. Math. 6 (967), 09{0. Kasparov and Skandalis 99] G. G. Kasparov and G. Skandalis, \Groups acting on buildings, operator K- theory, and Novikov's conjecture", K-theory (99), 0{7. Kesten 959] H. Kesten, \Symmetric random walks on groups", Trans. Amer. Math. Soc. 9 (959), 6{5. Murty 988] M. R. Murty, \Artin's conjecture for primitive roots", Math. Intelligencer 0 (988), 59{ 67. Pittet and Salo-Coste 999] C. Pittet and L. Salo- Coste, \Amenable groups, isoperimetric proles and random walks", pp. 9{6 in Geometric group theory down under (Canberra, 996), edited by J. Cossey et al., de Gruyter, Berlin, 999. Robert 98] A. Robert, Introduction to the representation theory of compact and locally compact groups, London Math. Soc. Lect. Note Ser. 80, Cambridge Univ. Press, Cambridge, 98. Szwarc 998] R. Szwarc, \Norm estimates of discrete Schrodinger operators", Coll. Math. 76 (998), 5{ 60. Valette 989] A. Valette, \The conjecture of idempotents: a survey of the C*-algebraic approach", Bull. Soc. Math. Belg. (989), 85{5. Florian Martin, Institut de Mathematiques, Rue Emile Argand, CH-007 Neuch^atel, Switzerland (orian.martin@maths.unine.ch) Alain Valette, Institut de Mathematiques, Rue Emile Argand, CH-007 Neuch^atel, Switzerland (alain.valette@maths.unine.ch) Received February, 999; accepted in revised form September, 999