Pro-Lie groups approximable by discrete subgroups

2016 ◽  
Vol 28 (1) ◽  
Author(s):  
Hatem Hamrouni ◽  
Bilel Kadri
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Discrete Subgroups

AbstractA locally compact group

A BIJECTION OF INVARIANT MEANS ON AN AMENABLE GROUP WITH THOSE ON A LATTICE SUBGROUP

2021 ◽  
pp. 1-6
Author(s):  
JOHN HOPFENSPERGER
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Discrete Subgroups ◽  
Invariant Means ◽  
Lattice Subgroup

Abstract Suppose G is an amenable locally compact group with lattice subgroup $\Gamma $ . Grosvenor [‘A relation between invariant means on Lie groups and invariant means on their discrete subgroups’, Trans. Amer. Math. Soc.288(2) (1985), 813–825] showed that there is a natural affine injection $\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$ and that $\iota $ is a surjection essentially in the case $G={\mathbb R}^d$ , $\Gamma ={\mathbb Z}^d$ . In the present paper it is shown that $\iota $ is a surjection if and only if $G/\Gamma $ is compact.

scholarly journals Some properties of locally compact groups

1967 ◽  
Vol 7 (4) ◽  
pp. 433-454 ◽  
Author(s):  
Neil W. Rickert
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Locally Compact Group ◽  
Factor Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Arcwise Connected ◽  
New Formulation

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.

scholarly journals Non-minimal tree actions and the existence of non-uniform tree lattices

2004 ◽  
Vol 70 (2) ◽  
pp. 257-266
Author(s):  
Lisa Carbone
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Connected Graph ◽  
Locally Compact Group ◽  
Minimal Tree ◽  
Locally Compact ◽  
Discrete Subgroups ◽  
Finite Tree ◽  
Locally Finite ◽  
Uniform Tree

A uniform tree is a tree that covers a finite connected graph. Let X be any locally finite tree. Then G = Aut(X) is a locally compact group. We show that if X is uniform, and if the restriction of G to the unique minimal G-invariant subtree X0 ⊆ X is not discrete then G contains non-uniform lattices; that is, discrete subgroups Γ for which Γ/G is not compact, yet carries a finite G-invariant measure. This proves a conjecture of Bass and Lubotzky for the existence of non-uniform lattices on uniform trees.

Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups

2005 ◽  
Vol 57 (6) ◽  
pp. 1193-1214
Author(s):  
Nick Dungey
Keyword(s):  
Compact Group ◽  
Probability Density ◽  
Lie Groups ◽  
Continuous Time ◽  
Locally Compact Group ◽  
Heat Kernels ◽  
Locally Compact ◽  
Convolution Power ◽  
Convolution Powers

AbstractLet K be a function on a unimodular locally compact group G, and denote by Kn = K * K * · · · * K the n-th convolution power of K. Assuming that K satisfies certain operator estimates in L2(G), we give estimates of the norms and for large n. In contrast to previous methods for estimating , we do not need to assume that the function K is a probability density or nonnegative. Our results also adapt for continuous time semigroups on G. Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.

Contraction subgroups and semistable measures on p-adic Lie groups

1991 ◽  
Vol 110 (2) ◽  
pp. 299-306 ◽  
Author(s):  
S. G. Dani ◽  
Riddhi Shah
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Probability Measures ◽  
Compact Groups ◽  
Locally Compact ◽  
Significant Class

Continuous one-parameter semigroups {μt}t≥0 of probability measures on a locally compact group which are semistable with respect to some automorphism τ of the group, namely such that τ(μt) = μct for all t ≥ 0, for a fixed c ∈ (0, 1), have attracted considerable attention of various researchers in recent years (cf. [3], [5] and other references cited therein). A detailed study of semistable measures on (real) Lie groups is carried out in [5]. In this context it is of interest to study semistable measures on the class of p-adic Lie groups, which is another significant class of locally compact groups.

Collapsible probability measures and concentration functions on Lie groups

1997 ◽  
Vol 122 (1) ◽  
pp. 105-113 ◽  
Author(s):  
S. G. DANI ◽  
RIDDHI SHAH
Keyword(s):  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Compact Group ◽  
Lie Groups ◽  
Locally Compact Group ◽  
Probability Measures ◽  
Locally Compact ◽  
Concentration Functions ◽  
Prove Theorem ◽  
Divergent Sequences

Given a locally compact group G and a probability measure μ on G it is of interest to know, in various situations, whether there exist divergent sequences {gn} such that {gn μg−1n is relatively compact (see for example [DM3] and [DS]); this phenomenon may be viewed as ‘collapsing’ of the measure. It is the purpose of this note to prove Theorem 1 below and give certain applications to the asymptotic behaviour of concentration functions.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals Interpolation in wavelet spaces and the HRT-conjecture

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Eirik Berge
Keyword(s):  
Compact Group ◽  
Wavelet Transforms ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Space Structure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Positive Type ◽  
Time Frequency ◽  
Hrt Conjecture

AbstractWe investigate the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })\subset L^{2}(G)$$ W g ( H π ) ⊂ L 2 ( G ) arising from square integrable representations $$\pi :G \rightarrow \mathcal {U}(\mathcal {H}_{\pi })$$ π : G → U ( H π ) of a locally compact group G. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong restrictions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time–frequency analysis, this problem turns out to be equivalent to the HRT-conjecture. Finally, we consider the problem of whether all the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })$$ W g ( H π ) of a locally compact group G collectively exhaust the ambient space $$L^{2}(G)$$ L 2 ( G ) . We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.

The Superstability of the Generalized D'alembert Functional Equation

2003 ◽  
Vol 10 (3) ◽  
pp. 503-508 ◽  
Author(s):  
Elhoucien Elqorachi ◽  
Mohamed Akkouchi
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Compact Group ◽  
Locally Compact Group ◽  
Complex Numbers ◽  
Locally Compact

Abstract We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

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