Subgroups of Word Hyperbolic Groups in Dimension 2

If G is a word hyperbolic group of cohomological dimension 2,then every subgroup of G of type FP₂ is also word hyperbolic.Isoperimetric inequalities are defined for groups of type FP₂and it is shown that the linear isoperimetric inequality inthis generalized context is equivalent to word hyperbolicity.A sufficient condition for hyperbolicity of a general graphis given along with an application to ‘relative hyperbolicity’.Finitely presented subgroups of Lyndon's small cancellationgroups of hyperbolic type are word hyperbolic. Finitely presentedsubgroups of hyperbolic 1-relator groups are hyperbolic. Finitelypresented subgroups of free Burnside groups are finite in thestable range.     

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, N_G(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of N_G(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.     

Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

Our first result is a ‘sum-product’ theorem forsubsets A of the finite field F_p, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

Necessary and Sufficient Tauberian Conditions, Under Which Convergence Follows From Summability (C, 1)

Let (s_k: k = 0, 1, ...) be a sequence of real numbers whichis summable (C, 1) to a finite limit. We prove that (s_k) isconvergent if and only if the following two conditions are satisfied: where _n denotes the integer part of the productn. Both conditions are clearly satisfied if (s_k) is slowly decreasingin the sense of R. Schmidt and G. H. Hardy. The symmetric counterparts of the conditions above are thosewhen ‘limsup’ and ‘liminf’ are interchangedon the left-hand sides, while the inequality sign ‘ ’is changed for the opposite ‘ ’ in them. Next, let (s_k) be a sequence of complex numbers which is summable(C, 1) to a finite limit. We prove that (s_k) is convergent ifand only if one of the following conditions is satisfied: We also prove a general Tauberian theorem forsequences in ordered linear spaces.     

Totaro's Question on Zero-Cycles on G2, F4 and E6 Torsors

In a 2004 paper, Totaro asked whether a G-torsor X that hasa zero-cycle of degree d > 0 will necessarily have a closedétale point of degree dividing d, where G is a connectedalgebraic group. This question is closely related to severalconjectures regarding exceptional algebraic groups. Totaro gavea positive answer to his question in the following cases: Gsimple, split, and of type G₂, type F₄, or simply connectedof type E₆. We extend the list of cases where the answer is‘yes’ to all groups of type G₂ and some nonsplitgroups of type F₄ and E₆. No assumption on the characteristicof the base field is made. The key tool is a lemma regardinglinkage of Pfister forms.     

Symmetries of Surface Singularities

The study of reductive group actions on a normal surface singularityX is facilitated by the fact that the group Aut X of automorphismsof X has a maximal reductive algebraic subgroup G which containsevery reductive algebraic subgroup of Aut X up to conjugation.If X is not weighted homogeneous then this maximal group G isfinite (Scheja, Wiebe). It has been determined for cusp singularitiesby Wall. On the other hand, if X is weighted homogeneous butnot a cyclic quotient singularity then the connected componentG₁ of the unit coincides with the C* defining the weighted homogeneousstructure (Scheja, Wiebe, Wahl). Thus the main interest liesin the finite group G/G₁. Not much is known about G/G₁. Ganterhas given a bound on its order valid for Gorenstein singularitieswhich are not log-canonical. Aumann-Körber has determinedG/G₁ for all quotient singularities. We propose to study G/G₁ through the action of G on the minimalgood resolution of X. If X is weightedhomogeneous but not a cyclic quotient singularity, let E₀ bethe central curve of the exceptional divisor of . We show that the natural homomorphism GAut E₀ haskernel C* and finite image. In particular, this re-proves therest of Scheja, Wiebe and Wahl mentioned above. Moreover, itallows us to view G/G₁ as a subgroup of Aut E₀. For simple ellipticsingularities it equals (Z_bxZ_b)Aut₀ E₀ where –b is theself-intersection number of E₀, Z_bxZ_b is the group of b-torsionpoints of the elliptic curve E₀ acting by translations, andAut₀ E₀ is the group of automorphisms fixing the zero elementof E₀. If E₀ is rational then G/G₁ is the group of automorphismsof E₀ which permute the intersection points with the branchesof the exceptional divisor while preserving the Seifert invariantsof these branches. When there are exactly three branches weconclude that G/G₁ is isomorphic to the group of automorphismsof the weighted resolution graph. This applies to all non-cyclicquotient singularities as well as to triangle singularities.We also investigate whether the maximal reductive automorphismgroup is a direct product GG₁xG/G₁. This is the case, for instance,if the central curve E₀ is rational of even self-intersectionnumber or if X is Gorenstein such that its nowhere-zero 2-form has degree ±1. In the latter case there is a ‘natural’section G/G₁G of GG/G₁ given by the group of automorphisms inG which fix . For a simple elliptic singularity one has GG₁xG/G₁if and only if –E₀ · E₀ = 1.     

A Bound on the Presentation Rank of a Finite Group

Let G be a finite group, and let I_G be the augmentation idealof ZG. We denote by d(G) the minimum number of generators forthe group G, and by d(I_G) the minimum number of elements ofI_G needed to generate I_G as a G-module. The connection betweend(G) and d(I_G) is given by the following result due to Roggenkamp]14]: where pr(G) is a non-negative integer, called the presentationrank of G, whose definition comes from the study of relationmodules (see [4] for more details). 1991 Mathematics SubjectClassification 20D20.     

Latin Squares and the Hall-Paige Conjecture

The Hall–Paige conjecture deals with conditions underwhich a finite group G will possess a complete mapping, or equivalentlya Latin square based on the Cayley table of G will possess atransversal. Two necessary conditions are known to be: (i) thatthe Sylow 2-subgroups of G are trivial or non-cyclic, and (ii)that there is some ordering of the elements of G which yieldsa trivial product. These two conditions are known to be equivalent,but the first direct, elementary proof that (i) implies (ii)is given here. It is also shown that the Hall–Paige conjecture impliesthe existence of a duplex in every group table, thereby provinga special case of Rodney's conjecture that every Latin squarecontains a duplex. A duplex is a ‘double transversal’,that is, a set of 2n entries in a Latin square of order n suchthat each row, column and symbol is represented exactly twice.2000 Mathematics Subject Classification 05B15, 20D60.     

Homology and derived p-series of groups

We give homological conditions that ensure that a group homomorphisminduces an isomorphism modulo any term of the derived p –series, in analogy to Stallings's 1963 result for the p-lowercentral series. In fact, we prove a stronger theorem that isanalogous to Dwyer's extensions of Stallings’ results.It follows that spaces that are _p-homology equivalent have isomorphicfundamental groups modulo any term of their p-derived series.Various authors have related the ranks of the successive quotientsof the p-lower central series and of the derived p-series ofthe fundamental group of a 3-manifold M to the volume of M,to whether certain subgroups of ₁(M) are free, to whether finiteindex subgroups of ₁(M) map onto non-abelian free groups, andto whether finite covers of M are ‘large’ in variousother senses.     

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L₂ norm error estimator is proposed for thePoisson equation. The error estimator can be applied to anisotropictetrahedral or triangular finite element meshes. The estimatoris rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropicbubble functions and the corresponding inverse inequalities.The upper error bound utilizes non-standard anisotropic interpolationestimates. Its proof requires H² regularity of the Poisson problem,and its quality depends on how good the anisotropic mesh resolvesthe anisotropy of the problem. This is measured by a so-called‘matching function’. A numerical example supports the anisotropic error analysis.     

Locally Finite Groups All of Whose Subgroups are Boundedly Finite over Their Cores

For n a positive integer, a group G is called core-n if H/H_Ghas order at most n for every subgroup H of G (where H_G is thenormal core of H, the largest normal subgroup of G containedin H). It is proved that a locally finite core-n group G hasan abelian subgroup whose index in G is bounded in terms ofn. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.     

Uniqueness cases in odd-type groups of finite Morley rank

There is a longstanding conjecture, due to Gregory Cherlin andBoris Zilber, that all simple groups of finite Morley rank aresimple algebraic groups. One of the major theorems in the areais Borovik's trichotomy theorem. The ‘trichotomy’here is a case division of the generic minimal counterexampleswithin odd type, that is, groups with a large and divisibleSylow° 2-subgroup. The so-called ‘uniqueness case’in the trichotomy theorem is the existence of a proper 2-generatedcore. It is our aim to drive the presence of a proper 2-generatedcore to a contradiction, and hence bind the complexity of theSylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilberconjecture. This paper shows that the group in question is aminimal connected simple group and has a strongly embedded subgroup,a far stronger uniqueness case. As a corollary, a tame counterexampleto the Cherlin–Zilber conjecture has Prüfer rankat most two.     

Shape-preserving interpolation by G1 and G2 PH quintic splines

The interpolation of a planar sequence of points p₀, ..., p_Nby shape-preserving G¹ or G² PH quintic splines with specifiedend conditions is considered. The shape-preservation propertyis secured by adjusting ‘tension’ parameters thatarise upon relaxing parametric continuity to geometric continuity.In the G² case, the PH spline construction is based on applyingNewton–Raphson iterations to a global system of equations,commencing with a suitable initialization strategy—thisgeneralizes the construction described previously in NumericalAlgorithms 27, 35–60 (2001). As a simpler and cheaperalternative, a shape-preserving G¹ PH quintic spline schemeis also introduced. Although the order of continuity is lower,this has the advantage of allowing construction through purelylocal equations.     

The Tits Alternative for Cat(0) Cubical Complexes

A Tits alternative theorem is proved in this paper for groupsacting on CAT(0) cubical complexes. That is, a proof is givento show that if G is assumed to be a group for which there isa bound on the orders of its finite subgroups, and if G actsproperly on a finite-dimensional CAT(0) cubical complex, theneither G contains a free subgroup of rank 2, or G is finitelygenerated and virtually abelian. In particular, the above conclusionholds for any group G with a free action on a finite-dimensionalCAT(0) cubical complex. 2000 Mathematics Subject Classification20F67, 20E08.     

On Manifolds Whose Tangent Bundle is Big and 1-Ample

A line bundle over a complex projective variety is called bigand 1-ample if a large multiple of it is generated by globalsections and a morphism induced by the evaluation of the spanningsections is generically finite and has at most 1-dimensionalfibers. A vector bundle is called big and 1-ample if the relativehyperplane line bundle over its projectivisation is big and1-ample. The main theorem of the present paper asserts that any complexprojective manifold of dimension 4 or more, whose tangent bundleis big and 1-ample, is equal either to a projective space orto a smooth quadric. Since big and 1-ample bundles are ‘almost’ample, the present result is yet another extension of the celebratedMori paper ‘Projective manifolds with ample tangent bundles’(Ann. of Math. 110 (1979) 593–606). The proof of the theorem applies results about contractionsof complex symplectic manifolds and of manifolds whose tangentbundles are numerically effective. In the appendix we re-provethese results. 2000 Mathematics Subject Classification 14E30,14J40, 14J45, 14J50.     

A Generic Identification Theorem for Groups of Finite Morley Rank

The paper contains a final identification theorem for the ‘generic’K^*-groups of finite Morley rank.     

Modular Subgroup Arithmetic and a Theorem of Philip Hall

A surprising relationship is established in this paper, betweenthe behaviour modulo a prime p of the number S_n G of index nsubgroups in a group G, and that of the corresponding subgroupnumbers for a normal subgroup in G normal subgroup in p-powerorder. The proof relies, among other things, on a twisted versiondue to Philip Hall of Frobenius' theorem concerning the equationx^m=1 in finite groups. One of the applications of this result,presented here, concerns the explicit determination modulo pof S_n G in the case when G is the fundamental group of a treeof groups all of whose vertex groups are cyclic of p-power order.Furthermore, a criterion is established (by a different technique)for the function S_n G to be periodic modulo p. 2000 MathematicsSubject Classification 20E06, 20F99 (primary); 05A15, 05E99(secondary).     

For Rewriting Systems the Topological Finiteness Conditions FDT and FHT are Not Equivalent

A finite rewriting system is presented that does not satisfythe homotopical finiteness condition FDT, although it satisfiesthe homological finiteness condition FHT. This system is obtainedfrom a group G and a finitely generated subgroup H of G througha monoid extension that is almost an HNN extension. The FHTproperty of the extension is closely related to the FP₂ propertyfor the subgroup H, while the FDT property of the extensionis related to the finite presentability of H. The example systemseparating the FDT property from the FHT property is then obtainedby applying this construction to an example group.     

Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formulafor the equivariant Poincaré polynomial of a linear transformationg which normalises a unitary reflection group G, acting on thecohomology of the corresponding hyperplane complement. Thisformula involves a certain function (called a Z-function below)on the centraliser CG(g), which was proved to exist only incertain cases, for example, when g is a reflection, or is G-regular,or when the centraliser is cyclic. In this work we prove theexistence of Z-functions in full generality. Applications includereduction and product formulae for the equivariant Poincarépolynomials. The method is to study the poset L(C_G(g)) of subspaceswhich are fixed points of elements of C_G(g). We show that thisposet has Euler characteristic 1, which is the key propertyrequired for the definition of a Z-function. The fact aboutthe Euler characteristic in turn follows from the ‘join-atom’property of L(C_G(g)), which asserts that if [X₁,..., X_k} isany set of elements of L(C_G(g)) which are maximal (set theoretically)then their setwise intersection lies in L(C_G(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40.