Quasi-randomness and the distribution of copies of a fixed graph

We show that if a graph G has the property that all subsets of vertices of size n/4 contain the “correct” number of triangles one would expect to find in a random graph G(n, 1/2), then G behaves like a random graph, that is, it is quasi-random in the sense of Chung, Graham, and Wilson [6]. This answers positively an open problem of Simonovits and Sós [10], who showed that in order to deduce that G is quasi-random one needs to assume that all sets of vertices have the correct number of triangles. A similar improvement of [10] is also obtained for any fixed graph other than the triangle, and for any edge density other than 1/2. The proof relies on a theorem of Gottlieb [7] in algebraic combinatorics, concerning the rank of set inclusion matrices.     

Compactifications and algebraic completions of limit groups

This paper considers the existence of nondiscrete embeddings Γ ↦ G, where Γ is an abstract limit group and G is topological group. Namely, it is shown that a locally compact group G that admits a nondiscrete nonabelian free subgroup F admits a nondiscrete copy of every nonabelian limit group L. In some cases, for instance if the F is of rank 2 and its closure in G is compact or semisimple algebraic, or if L is a surface group (as considered in [6]), L can be chosen with the same closure as F.     

Distal actions and shifted convolution property

A locally compact group G is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure μ on G, either sup _x∈G μ ⁿ (Cx) → 0 for all compact subsets C of G, or there exist x ∈ G and a compact subgroup K normalised by x such that μ ⁿ x ⁻ⁿ → ωK, the normalised Haar measure on K. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors by compact groups invariant under the action and for factors by the connected component of the identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (pointwise distality). This gives some interesting corollaries to distality of certain actions and Choquet-Deny measures which actually motivated SCP and pointwise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.     

Volterra-like Banach Algebras and their Second Duals

 This paper presents and studies a class of algebras which includes the usual Volterra algebra. Roughly speaking, they relate to the Volterra algebra in the way a general locally compact group relates to ℝ. We show that they can be viewed as quotients of some semigroup algebras introduced by Baker and Baker [1]. Their sets of nilpotent elements are dense. We investigate the second duals of these algebras and find that most of the properties found in [7] for the biduals of the group algebras L ¹(G) for compact G are retained here.     

Courants invariants par une action propre

The purpose of this paper is to characterise the invariant sections-distributions by a proper action. More precisely, we show that if G is a connected Lie group acting on a differentiable vector bundle E→V such that the induced action on V is proper, then the topological vector space of the G-invariant linear functionals (on the space of C ^∞ sections with compact support) equipped with the induced weak-topology (resp. the strong-topology), is isomorphic to the weak (resp. strong) topological dual of the space (of all G-invariant sections σ with compact quotient supp(σ)/G) equipped with a suitable topology; this coincides with the usual C ^∞-topology if the orbit space is compact, and with the Schwartz-topology if the group G is compact. Received: 8 June 1998 / Revised version: 22 September 1998     

Volterra-like Banach Algebras and their Second Duals

 This paper presents and studies a class of algebras which includes the usual Volterra algebra. Roughly speaking, they relate to the Volterra algebra in the way a general locally compact group relates to ℝ. We show that they can be viewed as quotients of some semigroup algebras introduced by Baker and Baker [1]. Their sets of nilpotent elements are dense. We investigate the second duals of these algebras and find that most of the properties found in [7] for the biduals of the group algebras L ¹(G) for compact G are retained here. Received 8 July 1997; in revised form 17 November 1997     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

Finite solvable groups in which the Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups are either bicyclic or of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript>

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p ³ for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A₄-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p ² + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.     

Real loci of based loop groups

Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K is the fixed point set of an involutive automorphism σ. This induces an involutive automorphism τ of the based loop space Ω(G). There exists a maximal torus T ⊂ G such that the canonical action of T × S ¹ on Ω(G) is compatible with τ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat’s convexity theorem. Namely, the images of Ω(G) and Ω(G) ^τ (fixed point set of τ) under the T × S ¹ moment map on Ω(G) are equal. The space Ω(G) ^τ is homotopy equivalent to the loop space Ω(G/K) of the Riemannian symmetric space G/K. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in \mathbbZ₂ {\mathbb{Z}_2} of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [BS] had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe [HHP].     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

Rigid geometric structures,isometric actions,and algebraic quotients

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group G on a smooth or analytic manifold M with a rigid A-structure σ. It generalizes Gromov’s centralizer and representation theorems to the case where R(G) is split solvable and G/R(G) has no compact factors, strengthens a special case of Gromov’s open dense orbit theorem, and implies that for smooth M and simple G, if Gromov’s representation theorem does not hold, then the local Killing fields on [(M)\tilde]{\widetilde{M}} are highly non-extendable. As applications of the generalized centralizer and representation theorems, we prove (1) a structural property of Iso(M) for simply connected compact analytic M with unimodular σ, (2) three results illustrating the phenomenon that if G is split solvable and large then π ₁(M) is also large, and (3) two fixed point theorems for split solvable G and compact analytic M with non-unimodular σ.     

Classical theorems of probability on Gelfand pairs—Khinchin’s theorems and Cramér’s theorem

We prove Khinchin’s Theorems for Gelfand pairs (G, K) satisfying a condition (*): (a)G is connected; (b)G is almost connected and Ad (G/M) is almost algebraic for some compact normal subgroupM; (c)G admits a compact open normal subgroup; (d) (G,K) is symmetric andG is 2-root compact; (e)G is a Zariski-connectedp-adic algebraic group; (f) compact extension of unipotent algebraic groups; (g) compact extension of connected nilpotent groups. In fact, condition (*) turns out to be necessary and sufficient forK-biinvariant measures on aforementioned Gelfand pairs to be Hungarian. We also prove that Cramér’s theorem does not hold for a class of Gaussians on compact Gelfand pairs. This author was supported by the European Commission (TMR 1998–2001 Network Harmonic Analysis).     

Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen

In this paper we study the class of all locally compact groupsG with the property that for each closed subgroupH ofG there exists a pair of homomorphisms into a compact group withH as coincidence set, and the class of all locally compact groupG with the property that finite dimensional unitary representations of subgroups ofG can be extended to finite dimensional representations ofG. It is shown that [MOORE]-groups (every irreducible unitary representation is finite dimensional) have these two properties. A solvable group in is a [MOORE]-group. Moreover, we prove a structure theorem for Lie groups in the class [MOORE], and show that compactly generated Lie groups in [MOORE] have faithful finite dimensional unitary representations.     

Symplectic convexity for orbifolds

Let a connected compact Lie group G act on a connected symplectic orbifold of orbifold fundamental group Г. If the action preserves the symplectic structure and there is a G-equivariant and mod-Г proper momentum map for the lifted action on the universal branch covering orbifold, and if the lifted G-action commutes with that of Г, then the symplectic convexity theorem is still true for this kind of lifted Hamiltonian action.     

Duality of Fourier type with respect to locally compact Abelian groups

It is shown that a Banach space X has Fourier type p with respect to a locally compact abelian group G if and only if the dual space X′ has Fourier type p with respect to G if and only if X has Fourier type p with respect to the dual group of G. This extends previously known results for the classical groups and the Cantor group to the setting of general locally compact abelian groups. Supported by DFG grant Hi 584/2-2. Partially supported by a DAAD-grant A/02/46571.     

On existence of invariant measures

Let G be a Lie group, H ≤ G a closed subgroup and M ≈ G/H. In [14] André Weil gave a necessary and sufficient condition for the existence of invariant measures on homogeneous spaces of arbitrary locally compact groups. For Lie groups using the structure theory we give a neater necessary and sufficient condition for the existence of a G-invariant measure on M, cf. Theorems (2.1) and (3.2) in the introduction.     

Strong Arens irregularity of Beurling algebras with a locally convex topology

Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L₀^∞ (G,1/ω) can be identified with the strong dual of L¹(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L¹(G, ω), τ)**, and prove that the topological center of (L¹(G, ω), τ)** is (L¹(G, ω); this enables us to conclude that (L¹(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L₀^∞ (G, 1/ω)¹. Received: 8 March 2005     

The Socle and finite dimensionality of some Banach algebras

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact groupG, G is compact if there exists a measure μ in Soc(L ¹(G)) such that μ(G) ≠ 0. We also prove thatG is finite if Soc(M(G)) is closed and every nonzero left ideal inM(G) contains a minimal left ideal.     

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT ₁ primitive ideal space if the groupC ^*-algebra,C ^*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC ^*(G), denoted by PrimG. This means of course that every primitive ideal inC ^*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT ₁ primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT ₁ primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC ^*(G) are always maximal which implies of course that PrimG isT ₁.