The Edge Correlation of Random Forests

The conjecture was made by Kahn that a spanning forest F chosen uniformly at random from all forests of any finite graph G has the edge-negative association property. If true, the conjecture would mean that given any two edges ε₁ and ε₂ in G, the inequality \mathbbP(e₁ ? F, e₂ ? F) ￡ \mathbbP(e₁ ? F)\mathbbP(e₂ ? F){{\mathbb{P}(\varepsilon_{1} \in \mathbf{F}, \varepsilon_{2} \in \mathbf{F}) \leq \mathbb{P}(\varepsilon_{1} \in \mathbf{F})\mathbb{P}(\varepsilon_{2} \in \mathbf{F})}} would hold. We use enumerative methods to show that this conjecture is true for n large enough when G is a complete graph on n vertices. We derive explicit related results for random trees.     

On Residual Nilpotence of Projective Crossed Modules

Abstract

Let F be a free group, (M,?, F) a non-aspherical projective F-crossed module. We prove that the action of Coker (?) on Ker (?) is faithful. Also we show that if (M,?, F) is a residually nilpotent crossed module, then Coker (?) is a residually nilpotent group. As a corollary, we get an alternative proof of Conduche's translation of Whitehead's asphericity conjecture in terms of residual nilpotence of certain crossed modules.     

The McKay conjecture for exceptional groups and odd primes

Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G ^F , such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C _G (S), the centraliser of S in G, extend to their inertia group in N _G (S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [14]. This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz fellowship.     

Approximation of analytic by Borel sets and definable countable chain conditions

LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ ₁ ¹ countable chain condition if there is aΣ ₁ ¹ equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI is definable in the codes of Borel sets, then eachΣ ₁ ¹ set is equal to a Borel set modulo a set fromI iffI fulfils theΣ ₁ ¹ countable chain condition. Further we characterize the σ-idealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ ₁ ¹ sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀F ∈ F A ∩F is meager inF} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition. Research partially supported by NSF grant DMS-9317509.     

Definability of Geometric Properties in Algebraically Closed Fields

We prove that there exists no sentence F of the language of rings with an extra binary predicat I₂ satisfying the following property: for every definable set X ? ?², X is connected if and only if (?, X) ? F, where I₂ is interpreted by X. We conjecture that the same result holds for closed subset of ?². We prove some results motivated by this conjecture.     

A Filtration on the Chow Groups of a Complex Projective Variety

Let X/ _C be a projective algebraic manifold, and further let CH ^k (X) _Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {F ^l}_{l 0} on CH ^k (X) _Q of Bloch-Beilinson type. In the event that a certain conjecture of Jannsen holds (related to the Bloch-Beilinson conjecture on the injectivity, modulo torsion, of the Abel–Jacobi map for smooth proper varieties over Q), this filtration truncates. In particular, his conjecture implies that F ^k+1 = 0.     

On the Grigorchuk–Kurchanov conjecture

 Let denote the free group of rank 2g. An automorphism φ? Aut(F _{2 g} ) is generating if N _a φ (N _b ) = F _{2 g} , where N _a is the normal closure of and N _b is defined analogously. We present a characterization of generating automorphisms in Aut(F ₂) and observe that there exists a unique (up to equivalence) epimorphism F ₂→Z×Z: this is a particular case of the Grigorchuk–Kurchanov conjecture. This leads to further investigations for splitting homomorphisms for the pairs (F _{2 g} , F _g) and (G _g, F _g) where G _g denotes the fundamental group of a closed orientable surface of genus g and a reformulation of the Poincaré and Grigorchuk–Kurchanov conjectures is derived. Received: 1 October 2001     

Approximation Presentations of Modules and Homological Conjectures

In this article we give a sufficient condition of the existence of 𝕎 ^t -approximation presentations. We also introduce property (W ^k ). As an application of the existence of 𝕎 ^t -approximation presentations we give a connection between the finitistic dimension conjecture, the Auslander–Reiten conjecture, and property (W ^k ).     

Rational curves of degree at most 9 on a general quintic threefold

ABSTRACT.

We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P ⁴. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (?1) ⊕ (?1), and in P ⁴ with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.     

Integrality of L2L^2-Betti numbers

The Atiyah conjecture predicts that the -Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it holds for G and that is a normal subgroup, for amalgamated free products . Here F is a free group and is an arbitrary semi-direct product. This includes free products G*F and semi-direct products . We also show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new (bigger) class of groups for which the Atiyah conjecture holds, which contains all free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with torsion-free elementary amenable quotient or with free quotient, and under certain direct and inverse limits. Received: 22 August 1998/ Revised: 10 Jannary 2000 / Published online: 28 June 2000     

Subgroups of Free Groups: a Contribution to the Hanna Neumann Conjecture

We prove that the strengthened Hanna Neumann conjecture, on the rank of the inter-section of finitely generated subgroups of a free group, holds for a large class of groups characterized by geometric properties. One particular case of our result implies that the conjecture holds for all positively finitely generated subgroups of the free group F(A) (over the basis A), that is, for subgroups which admit a finite set of generators taken in the free monoid over A.     

Local Shalika models and functoriality

We prove, over a p-adic local field F, that an irreducible supercuspidal representation of GL_2n (F) is a local Langlands functorial transfer from SO_2n+1(F) if and only if it has a nonzero Shalika model (Corollary 5.2, Proposition 5.4 and Theorem 5.5). Based on this, we verify (Sect. 6) in our cases a conjecture of Jacquet and Martin, a conjecture of Kim, and a conjecture of Speh in the theory of automorphic forms.     

Nilpotency of Bocksteins, Kropholler's hierarchy and a conjecture of Moore

We show that the class of pairs (Γ,H) of a group and a finite index subgroup which verify a conjecture of Moore about projectivity of modules over ZΓ satisfy certain closure properties. We use this, together with a result of Benson and Goodearl, in order to prove that Moore's conjecture is valid for groups which belongs to Kropholler's hierarchy LHF. For finite groups, Moore's conjecture is a consequence of a theorem of Serre, about the vanishing of a certain product in the cohomology ring (the Bockstein elements). Using our result, we construct examples of pairs (Γ,H) which satisfy the conjecture without satisfying the analog of Serre's theorem.     

Differential Posets and Smith Normal Forms

We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU + tI and UD + tI have Smith normal forms over . In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice Y F studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young’s lattice Y and the r-differential Cartesian products Y ^r .     

When Will Every Maximal <Emphasis Type="BoldItalic">F</Emphasis>-free Subposet Contain a Maximal Element?

Let F be a partially ordered set (poset). A poset P is called F-free if P contains no subposet isomorphic to F. A finite poset F is said to have the maximal element property if every maximal F-free subposet of any finite poset P contains a maximal element of P. It is shown that a poset F with at least two elements has the maximal element property if and only if F is an antichain or F ≅ 2 + 2.     

On the vertex group of n-manifolds

In this paper we obtain the decomposition of the vertex group of n-manifolds, extending the one given by Kauffman and Lins for dimension 3 and solving the related conjecture. The result is obtained in the more general category of gems: the vertex group of a gem , representing an n-manifold M, is the free product of n copies of the fundamental group of M and a free group F of rank N–n, where N is the number of n-residues of . In particular, for crystallizations FZ and consequently the vertex group is an invariant of M.     

Towards the Eaton-Moretó conjecture on the minimal height of characters

Abstract

In this note we prove that the Eaton-Moretó conjecture holds for all blocks of finite general linear and unitary groups for all primes. Also, we show that no block of a finite quasi-simple group of classical Lie type provides a minimal counterexample to the conjecture, and so for ? > 5 no ?-block of any quasi-simple group can be a minimal counterexample.     

Local-Global Principle for the Baum–Connes Conjecture with Coefficients

We establish the Hasse principle (local-global principle) in the context of the Baum–Connes conjecture with coefficients. We illustrate this principle with the discrete group GL(2,F) where F is any global field.     

Representation dimension of exterior algebras

We determine the representation dimension of exterior algebras. This provides the first known examples of representation dimension > 3. We deduce that the Loewy length of the group algebra over F₂ of a finite group is strictly bounded below by the 2-rank of the group (a conjecture of Benson). A key tool is the use of the concept of dimension of a triangulated category.     

Representation fields for quaternionic skew-hermitian forms

Classes of indefinite quadratic forms in a genus are in correspondence with the Galois group of an abelian extension called the spinor class field (Estes and Hsia, Japanese J. Math. 16, 341–350 (1990)). Hsia has proved (Hsia et al., J. Reine Angew. Math. 494, 129–140 (1998)) the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. This far, the corresponding result for skew-hermitian forms was known only in some special cases, e.g., when the ideal (2) is square free over the base field. In this work we prove the existence of representation fields for quaternionic skew-hermitian forms in complete generality.