On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

Singular Cardinal Problem: Shelah's Theorem on 2N{omega}

This is an expository paper giving a complete proof of a theoremof Saharon Shelah: if 2 < for all n < , then 2 < ₄.     

Spaces Universal under Closed Embeddings for Finite-Dimensional Complete Metric Spaces

We shall prove that for every natural number n and every cardinalnumber there exists an n-dimensional complete metric spaceX_n, of weight such that every n-dimensional complete metricspace of weight is embeddable in X_n, as a closed subset.     

Finite groups in Stone-Cech compactifications

Let be an infinite cardinal and let G = ₂. Now let β Gbe the Stone–ech compactification of G as a discrete semigroup,and let =_<c_{β G} {xG\{0}:minsupp (x)}. We show that thesemigroup contains no nontrivial finite group.     

Correction: on the Intersection Matrices of Curves lying on Irregular Algebraic Surfaces

Professor W. F. Hammond has kindly drawn my attention to a blunderin 4 of the above paper. He referred to the ( – 2r) xß submatrix D of the skew-symmetric matrix displayednear the top of page 181, of which it is asserted that it issquare and non-singular, and pointed out that, from the factthat the matrix of which D forms part is regular, it may onlybe deduced that the columns of D are linearly independent; thatis, it only follows that – 2r ß. The validity of the equation – 2r = ß is essentialto the succeeding argument and, fortunately, may be establishedby alternative means. Using the nomenclature of the paper, wehave on F the set ₁^*, ..., _2r^*, ₁^*, ..., _ß^* of independent3-cycles (independent because they cut independent 1-cycleson the curve C), which may be completed, to form a basis forsuch cycles on F, by a further set ₁', ..., _2q–2r–pof independent 3-cycles, each of which meets C in a cycle homologousto zero on C. The cycles ₁^*, ..., ^* are invariant cycles andare independent on F so that, if > 2r + ß, thereis a non-trivial linear combination ^* of these having zero intersectionon C with each of the cycles ₁^*, ..., _2r^*, ₁^*, ..., _ß^*.Thus we have. (^* ._k^*)c = 0 = (^* ._i^*)c i.e. (^* ._k^*) = 0 = (^* ._i^* on F (1 k 2r; 1 i ß). Furthermore, (_j . C) 0 on C and we have (^* ._j .C)_C = 0 i.e. (^* ._j) = 0 on F (1 j 2q – 2r – ß). It now follows that ^* 0 on F (for it has zero intersectionwith every member of a basic set of 3-cycles on F). But thiscondradicts the assumption that ^* is a non-trivial linear combinationof the independent cycles ₁^*, ...,^*; and hence < 2r + ß.     

Cardinal Invariants and Eventually Different Functions

In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants a_e(), b() and a certain cardinal invariant m_d(⁺)on ⁺. As a corollary we get for example the following result.For a successor cardinal , even assuming that ^< = and 2= ⁺, the following is not provable in Zermelo–Fraenkelset theory. There is a ⁺-cc poset which does not collapse andwhich forces a() = ⁺ < a_e() = ⁺⁺ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary).     

A Variation on a Theorem of Galvin and Hajnal

Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =₍₎₊₍₎, < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product_</I admits a cofinal -scale of ordinal functions. Then ₊, where =||||_I is the I-norm of .     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.     

Boundaries of reduced free group C*-algebras

We prove that the crossed product C*-algebra C*_r(, ) of a freegroup with its boundary sits naturally between the reducedgroup C*-algebra C*_r and its injective envelope I(C*_r). In otherwords, we have natural inclusion C*_r C*_r(, ) I(C*_r) of C*-algebras.     

The Set of {omega}-Limit Points of A Map of the Circle is Closed

Let f be a continuous self-map of the unit circle, S¹. The -limitpoints (x) of a point x are the set of all limit points of thesequence of iterates of f acting on x. We shall show that theset of all -limit points _xS¹(x) a closed set in S¹.     

Packing, Tiling, Orthogonality and Completeness

Let R^d be an open set of measure 1. An open set DR^d is calleda ‘tight orthogonal packing region’ for if D–Ddoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of R^d. The main contribution of this paperis a new way of proving the following result: D tiles R^d whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L²(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in R^d, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ R^d withtranslates of a certain non-negative function and, additionally,we have completeness of E in L²() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.     

The A-Polynomial has Ones in the Corners

1. Definition of the A-polynomial The A-polynomial was introduced in [3] (see also [5]), and wepresent an alternative definition here. Let M be a compact 3-manifoldwith boundary a torus T. Pick a basis , µ of ₁T, whichwe shall refer to as the longitude and meridian. Consider thesubset R_U of the affine algebraic variety R = Hom (₁M, SL₂C)having the property that () and (µ) are upper triangular.This is an algebraic subset of R, since one just adds equationsstating that the bottom-left entries in certain matrices arezero. There is a well-defined eigenvalue map given by taking the top-left entries of () and (µ).1991 Mathematics Subject Classification 57M25, 57M50.     

The Series of Norms in a Soluble p-Group

The norm of a group G is the subgroup of elements of G whichnormalise every subgroup of G. We shall denote it (G). An ascendingseries of subgroups _i(G) in G may be defined recursively by:₀(G) = 1 and, for i 0, _i+1(G)/_i(G) = (G/_i(G)). For each i,the section _i+1(G)/i(G) clearly contains the centre of the groupG/_i(G). A result of Schenkman [8] gives a very close connectionbetween this norm series and the upper central series: _i(G) _i(G) _2i(G). 1991 Mathematics Subject Classification 20E15.     

The structure of the normalisers of the congruence subgroups of the Hecke group G5

Let = 2cos (/5) and let []. Denote the normaliser ofG₀() of the Hecke group G₅ in PSL₂() by N(G₀()). Then N(G₀())= G₀(/h), where h is the largest divisor of 4 such that h² divides. Further, N(G₀())/G₀() is either 1 (if h = 1), ₂ x ₂ (if h= 2) or ₄ x ₄ (if h = 4).     

On the Behaviour of the First Eigenfunction of the p-Laplacian Near its Critical Points

In this paper, the behaviour of the positive eigenfunction of in u_| = 0, p > 1, isstudied near its critical points. Under some convexity and symmetryassumptions on , is seen to have a unique critical point atx = 0; also, the behaviour of both and is determined nearby.Positive solutions u to some general problems –_pu = f(u)in , u_| = 0, are also considered, with some convexity restrictionson u. 2000 Mathematics Subject Classification 35B05 (primary),35J65, 35J70 (secondary).     

Correction: Abelian Varieties defined over their Fields of Moduli, I

Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat v^m–1 ... = 1, where v is the canonical isomorphism(A^m, I^m, ^m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order p^m for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that p^m _m = . Proof. If (b) is false for a given m, then there exists an element G, of order p^r for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (p^m) G ^pm G). There exists an opensubgroup G_o of G such that (G₀) = 0 and has order p^r in G/G₀.Choose H to be the subgroup of G generated by , and then aneasy application to G/G₀ of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k₁ F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'_ab/k₂) whose image $$\stackrel{\&macr;}{\sigma}$$ in Gal (k₁/k₂) generates this last group. The error occursin the statement that the canonical map v : A^P A acts on pointsby sending a^p a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that ^p = 1 (in Gal (F'/k₂)). By applying Lemma 2 to G = Gal (F'_ab/k₂) and the map G Gal(k₁/k₂) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that ^pm = 1, for some m, andG = G' x H where G' acts trivially on k₁ and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k₁ k₂is a cyclic Galois extension of degree p^m. In the first case, we let K F'_ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k₂. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\&macr;}{\sigma}$$, I$$\stackrel{\&macr;}{\sigma }$$, $$\stackrel{\&macr;}{\sigma}$$) be an isomorphism defined over k₁ and let v ... ^p–1 = µ(R). If is replaced by for some Aut_k1((A, I, )) then is replacedby ^P. Thus, as µ(R) is finite, we may assume that ^pm–1= 1 for some m. Choose K, as in (b), to be of degree p^m overk₂. Let _m be a generator of Gal (K/k₂) whose restriction tok₁ is $$\stackrel{\&macr;}{\sigma }$$. Then : (A, I, ) (A^{$$\stackrel{\&macr;}{\sigma }$$}, I^{$$\stackrel{\&macr;}{\sigma}$$}, ^{$$\stackrel{\&macr;}{\sigma }$$} = (A^{$$\stackrel{\&macr;}{\sigma}$$m}, I^{$$\stackrel{\&macr;}{\sigma }$$m}, ^{$$\stackrel{\&macr;}{\sigma}$$m} is an isomorphism defined over K and v ^mpm–1, ... ^m =^pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk₂ which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of _R^*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).     

Beurling and Lipschitz Algebras

It is well known that there exist infinite closed subsets Eof T such that A(E) = C(E) (see, for example, [3]). Such setsare called Helson sets. Let E be a closed subset of T, let 0< < 1, and let A(E) be the restriction of the Beurlingalgebra A(T). Then A(E) lipE. We shall show that A(E) = lipEif and only if E is finite. This answers a question raised byPedersen [5], where partial results were obtained. 1991 MathematicsSubject Classification 46J10.     

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

An Upper Bound for the Hyperbolic Metric of a Convex Domain

In [1], Beardon introduced the Apollonian metric defined forany domain D in Rⁿ by This metric is Möbius invariant, and for simply connectedplane domains it satisfies the inequality _D2_D, where _D denotesthe hyperbolic distance in D, and so gives a lower bound onthe hyperbolic distance. Furthermore, it is shown in [1, Theorem6.1] that for convex plane domains, the Apollonian metric satisfies, and, by considering the example of the infinite strip {x + iy:|y|<1}, that the best possibleconstant in this inequality is at least . In this paper we makethe following improvements.     

Asymptotic Cones of Finitely Generated Groups

Answering a question of Gromov [7], we shall present an exampleof a finitely generated group and two non-principal ultrafiltersA, B such that the asymptotic cones Con_A and Con_B are nothomeomorphic. 1991 Mathematics Subject Classification 20F06,20F32.