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).     

“具有幂零局部子群的有限群”一文的注记

称有限群$G$为一个PN-群若 $G$非幂零群,且对$G$的每一个$p$-子群$P$, 或者$P$是$G$的正规子群, 或者$P \subseteq Z_\infty(G)$, 或者$N_G(P)$是幂零群, $\forall p \in \pi(G)$. 本文证明了PN-群是亚幂零群. 特别地, PN-群是可解的 且给出了PN-群结构定理的一个初等的、直观的、简洁的证明.     

Some two-step and three-step nilpotent Lie groups with small automorphism groups

We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are ``small' in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we also get new examples of compact manifolds covered by two-step simply connected nilpotent Lie groups which do not admit Anosov automorphisms.

     

Representations of nilpotent locally compact groups

The main result of this paper is that if all the σ-representations of a separable locally compact nilpotent group are type I, then all the irreducible σ-representations are (up to equivalence) induced from one-dimensional σ-representations. This generalizes a result of Kirillov for simply connected nilpotent Lie groups. Parallels with the orbit theory for Lie groups are drawn.     

A class of symmetric and a class of Wiener group algebras

It is shown that connected groups of polynomial growth and compact extensions of nilpotent group have symmetric group algebras and that the group algebras of discrete solvable groups have the Wiener property.     

On the supports of absolutely continuous Gauss measures on connected Lie groups

The behaviour of the supports of an absolutely continuous Gauss semigroup on certain Lie groups is discussed. It is shown that on a connected nilpotent Lie group any absolutely continuous Gauss semigroup has full supports but on compact connected Lie groups which are not Abelian there exist absolutely continuous Gauss semigroups which do not have common supports.     

Compact Three-Dimensional Elation Quadrangles

We show that each compact three-dimensional EGQ is isomorphic either to the real symplectic quadrangle, or to a translation quadrangle of Tits type. In particular, the elation group is one of the two classical elation groups, and thus a simply connected nilpotent Lie group.     

AUTOMORPHISM GROUPS OF 4-VALENT CONNECTED CAYLEY GRAPHS OF p-GROUPS

61. IntroductionLet G be a trite grouP and S a subs6t of G such thst 1' S and S = S--1. The Cayleygraph X = Cay(G, S) Of G with respect to S is defined to have vertex set V(X) = G and edgeset E(X) = {(g, ag) I g E G, s E' S}. ~ the defection the following two faCts are obvious:(1) the automorphism group Ant(X) of X contains GR, the right regular representation ofG, as a subgroup, and (2) X is cormected if and only if S generates the group G.FOr a Cayley graph X = Cay(G, S) Of …     

Self homotopy groups with large nilpotency classes

We demonstrate that for any n>0 there exists a compact connected Lie group G such that the self homotopy group [G,G] has the nilpotency class greater than n, where [G,G] is a nilpotent group for a compact connected Lie group G.     

Closed geodesics in compact nilmanifolds

We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie groups N which are constructed using irreducible representations of the compact Lie group SU(2). Received: 8 November 2000 / Revised version: 9 April 2001     

Über den Typenkonvergenzsatz auf zusammenhängenden Lie Gruppen

The Convergence of Types Theorem on ^d is wellknown as an important tool for investigations on the limit behaviour of normalized sums or r.v. It is natural to look for a generalization for group-valued r.v. While for simply connected nilpotent Lie groups the Theorem is valid in general the existence of non-trivial compact subgroups causes problems. For compact extensions of nilpotent groups we prove restricted versions of the Convergence of Types Theorem.

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Herrn Prof. Dr. L. Schmetterer zum 70. Geburtstag gewidmet     

Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds

In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum. In memory of Robert Brooks     

Two-Step Nilpotent Lie Groups and Homogeneous Fiber Bundles

In this paper, we construct two-step nilpotent Lie groups from homogeneous fiber bundles over compact symmetric spaces. The structure of the constructed nilpotent groups is expressed in terms of the compact Lie groups involved in the fiber bundles. There are close relations between the geometric properties of the nilpotent groups and the total spaces of the fiber bundles. We will find new examples of nilpotent groups which are weakly symmetric and Riemannian geodesic orbit spaces.     

2-step nilpotent Lie groups of higher rank

We construct a family of simply connected 2-step nilpotent Lie groups of higher rank such that every geodesic lies in a flat. These are as Riemannian manifolds irreducible and arise from real representations of compact Lie algebras. Moreover we show that groups of Heisenberg type do not even infinitesimally have higher rank. Received: 2 July 2001 / Revised version: 19 October 2001     

Tortrat groups—Groups with a nice behavior for sequences of shifted convolution powers

For a locally compact groupG a condition in terms of probability measures and conjugation is introduced, which implies that limits of shifted convolution powers are always translates of idempotent measures. Such groups are called Tortrat groups. The connection between Tortrat groups and shifted convolution powers is established by the method of tail idempotents. Some construction principles for Tortrat groups are given and applied to show that compact groups, abelian groups, and more generally SIN-groups, as well as MAP-groups and almost connected nilpotent groups are of this type. The class of Tortrat groups is compared with another class investigated by A. Tortrat.     

Simply connected nilpotent Lie groups with quasi-standard -algebras

The problem of when the group -algebra of a locally compact group is quasi-standard is investigated for certain simply connected nilpotent Lie groups. The characterization is in terms of the coadjoint orbit structure in the dual of the Lie algebra.

     

On (co)homology locally connected spaces

We prove that there exists a cohomology locally connected compact metrizable space which is not homology locally connected. In the category of compact Hausdorff spaces a similar result was proved earlier by G.E. Bredon.     

Canonical extensions of Morita homomorphisms to the Ptolemy groupoid

Let Σ be a compact connected oriented surface with one boundary component. We extend each of Johnson’s and Morita’s homomorphisms to the Ptolemy groupoid of Σ. Our extensions are canonical and take values into finitely generated free abelian groups. The constructions are based on the 3-dimensional interpretation of the Ptolemy groupoid, and a chain map introduced by Suslin and Wodzicki to relate the homology of a nilpotent group to the homology of its Malcev Lie algebra.