Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Invariant measures for the horocycle flow on periodic hyperbolic surfaces

We classify the ergodic invariant Radon measures for the horocycle flow on geometrically infinite regular covers of compact hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the surface. Two consequences arise: if the group of deck transformations G is of polynomial growth, then these measures are classified by the homomorphisms from G ₀ to ℝ where G ₀ ≤ G is a nilpotent subgroup of finite index; if the group is of exponential growth, then there may be more than one Radon measure which is invariant under the geodesic flow and the horocycle flow. We also treat regular covers of finite volume surfaces. The first author was supported by NSF grant 0500630. The second author was supported by NSF grant 0400687.     

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.Research supported by NSERC under grant no. 90749-00.Research supported by NSERC under grant no. 227043-00.     

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringo _K[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗o _K[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL₂(ℤ _p ).     

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.     

Geometric cycles with local coefficients and the cohomology of arithmetic subgroups of the exceptional group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

We study the cohomology of a compact locally symmetric space attached to an arithmetic subgroup of a rational form of a group of type G ₂ with values in a finite dimensional irreducible representation E of G ₂. By constructing suitable geometric cycles and parallel sections of the bundle [(E)\tilde]{\tilde{E}} we prove non-vanishing results for this cohomology. We prove all possible non-vanishing results compatible with the known vanishing theorems regarding unitary representations with non-zero cohomology in the case of the short fundamental weight of G ₂. A decisive tool in our approach is a formula for the intersection numbers with local coefficients of two geometric cycles.     

Low dimensional actions of semisimple groups

For a simple non-compact Lie groupG with finite center we determine the smallest integern(G) such thatG has an almost effective action on a compact manifold of dimensionn(G) and characterize the compact manifolds of dimensionn(G) on whichG acts. We study actions of a semisimple groupG on compact manifolds of dimensionn(G)+1 and determine the orbit structure of the action ofG and its maximal compact subgroup. We give several examples to illustrate the results. This work was supported by an NSF postdoctoral Research Fellowship. An erratum to this article is available at .     

On the commutativity degree of compact groups

In any finite group G, the commutativity degree of G (denoted by d(G)) is the probability that two randomly chosen elements of G commute. More generally, for every n ≥ 2 the nth commutativity degree (denoted by d _n (G)) is the probability that a randomly chosen ordered (n + 1)-tuple of the group elements is mutually commuting. The aim of this paper is to generalize the definition of d(G) and d _n (G) to every compact group G (infinite and even uncountable). We shall state some results concerning compact groups and we will extend some results in Erfanian et al. (Comm. Algebra 35 (2007), 4183–4197) and Lescot (J. Algebra 177 (1995), 847–869).     

Subfactors associated to compact Kac algebras

We construct inclusions of the form (B ₀P) ^G (B ₁P) ^G , whereG is a compact quantum group of Kac type acting on an inclusion of finite dimensional C^*-algebrasB ₀B ₁ and on aII ₁ factorP. Under suitable assumptions on the actions ofG, this is a subfactor, whose Jones tower and standard invariant can be computed by using techniques of A. Wassermann. The subfactors associated to subgroups of compact groups, to projective representations of compact groups, to finite quantum groups, to finitely generated discrete groups, to vertex models and to spin models are of this form.     

Quasi‐convexly dense and suitable sets in the arc component of a compact group

Let G be an abelian topological group. The symbol $\widehat{G}Let G be an abelian topological group. The symbol $\widehat{G}$ denotes the group of all continuous characters $\chi :G\rightarrow {\mathbb T}$ endowed with the compact open topology. A subset E of G is said to be qc‐dense in G provided that χ(E)?φ([? 1/4, 1/4]) holds only for the trivial character $\chi \in \widehat{G}$, where $\varphi : {\mathbb R}\rightarrow {\mathbb T}={\mathbb R}/{\mathbb Z}$ is the canonical homomorphism. A super‐sequence is a non‐empty compact Hausdorff space S with at most one non‐isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super‐sequence converging to 0 that is qc‐dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism $r:\widehat{G}\rightarrow \widehat{G}_a$ defined by $r(\chi )=\chi \upharpoonright _{G_a}$ for $\chi \in \widehat{G}$, is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component G_a contains a super‐sequence converging to the identity that is qc‐dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group.     

Noncyclic Graph of a Group

We associate a graph Γ _G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ? G| 〈x, y〉 is cyclic for all y ? G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ _G is finite if and only if Γ _G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ _G  ? Γ _H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ _G  ? Γ _H for some group H, then G ? H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ _G  ? Γ _H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.     

On Finite Groups with Few Automorphism Orbits

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) ≤5, then G is isomorphic to one of the groups A₅, A₆, PSL(2, 7), or PSL(2, 8). We also consider the case when ω(G) = 6 and show that, if G is a nonsolvable finite group with ω(G) = 6, then either G ≈ PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≈ A₅.     

Some finiteness results for groups with bounded algebraic entropy

We prove that the number of elements of any generating set S of a discrete group G is bounded from above if we assume that the algebraic entropy of G with respect to S is smaller than some universal constant and the existence of a finite index subgroup of G with hyperbolicity properties. We deduce some finiteness results for the pairs (G, S) when G is a word-hyperbolic group or the fundamental group of a compact Riemannian manifold.     

Amenability of the Second Dual of Hypergroup Algebras

The amenability of the Banach algebra L ¹(G), the measure algebra M(G) and their second duals of a locally compact group have been considered by a number of authors. During these investigations it has been shown that L ¹(G)^** is amenable if and only if G is finite. If LUC (G)^*, the dual of the space of left uniformly continuous functions on G, is amenable, then G is compact and M(G) is amenable. Finally, if M(G)^** is amenable, then G is finite. The aim of this paper is to generalize all of the above results to the locally compact hypergroups.     

On geodesic structures of weakly median graphs—II: Compactness, the role of isometric rays

We prove that the vertex set of a K_ℵ0-free weakly median graph G endowed with the weak topology associated with the geodesic convexity on V(G) is compact if and only if G has one of the following equivalent properties: (1) G contains no isometric rays; (2) any chain of interval of G ordered by inclusion is finite; (3) every self-contraction of G fixes a non-empty finite regular weakly median subgraph of G. We study the self-contractions of K_ℵ0-free weakly median graphs which fix no finite set of vertices. We also follow a suggestion of Imrich and Klavzar [Product Graphs, Wiley, New York, 2000] by defining different centers of such a graph G, each of them giving rise to a non-empty finite regular weakly median subgraph of G which is fixed by all automorphisms of G.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

On s-semipermutable subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. We say that H is ssemipermutable in G if HGv = GpH for any Sylow p-subgroup Gp of G with （p, ｜H｜） = 1. We investigate the influence of s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized and unified.     

A characterization of a class of [Z] groups via Korovkin theory

We characterize all the central topological groupsG for which the centreZ(L ¹(G)) of the group algebra admits a finite universal Korovkin set. It is proved thatZ(L ¹(G)) has a finite universal Korovkin set iffĜ is a finite dimensional, separable metric space. This is equivalent to the fact thatG is separable, metrizable andG/K has finite torsion free rank, whereK is a compact open normal subgroup of certain direct summand ofG.     

On existence of finite universal Korovkin sets in the centre of group algebras

In this paper, we characterize compact groupsG as well as connected central topological groupsG for which the centreZ(L ¹(G)) admits a finite universal Korovkin set. Also we prove that ifG is a non-connected central topological group which has a compact open normal subgroupK such thatG=KZ, thenZ(L ¹(G)) admits a finite universal Korovkin set if is a finite-dimensional separable metric space or equivalentlyG is separable metrizable andG/K has finite torsion-free rank.