Growth of connected locally compact groups

The asymptotic behavior of the Haar measure of Uⁿ = U · U ··· U, where U is a compact neighborhood of the identity in a separable, connected locally compact group, is considered. It is shown that for a given group G, the measure of Uⁿ grows at most polynomially for all U ? G, or at least exponentially for all U ? G. The groups having polynomial growth are characterized in terms of uniformly discrete free subsemigroups and in terms of the eigenvalues of elements occurring in the adjoint group of an approximating Lie group.     

Nonstandard representations of locally compact groups

In the note, it is proved that, under natural conditions, any infinite-dimensional unitary representation T of a direct product of groups G = K × N, where K is a compact group and N is a locally compact Abelian group, is imaged by a representation of the nonstandard analog \(\tilde G\) of the group G in the group of nonstandard matrices of a fixed nonstandard size.     

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.     

Saturation on locally compact abelian groups

Let G be an arbitrary locally compact abelian group. It is the purpose of the present paper to establish saturation theorems for approximation processes generated by families (_t)_{t > 0} of complex bounded Radon measures on G and operating on a submodule of the Banach module L^p(G), L^p(G), over the convolution algebra. A basic tool is the Fourier transform method and, in the case p>1 for noncompact G, its interpretation in the context of the theory of quasimeasures on G.     

Midconvex functions in locally compact groups

The theorems of Bernstein-Doetsch, and Ostrowski, concerning the continuity of midconvex functions are extended to open subsets of locally compact and root-approximable topological groups.

     

Compactness properties of locally compact groups

Compactness propertiesC _n andC P _n for locally compact groupsG are introduced generalizing the finiteness propertiesF _n andF P _n for discrete groups. The propertyC ₁ resp.C ₂ is equivalent withG having a compact set of generators, resp. having a compact presentation. Some basic properties of the compactness propertiesC _n are shown. A local-global principle is proved by the second named author in the adjacent paper of this volume.