Countable Amenable Identity Excluding Groups

2004 ◽  
Vol 47 (2) ◽  
pp. 215-228 ◽  
Author(s):  
Wojciech Jaworski
Keyword(s):  
Probability Measure ◽  
Unitary Representation ◽  
Discrete Group ◽  
Polynomial Growth ◽  
Irreducible Unitary Representation ◽  
Compact Groups ◽  
Groups Of Polynomial Growth ◽  
Identity Representation ◽  
Dimensional Identity

AbstractA discrete group G is called identity excluding if the only irreducible unitary representation of G which weakly contains the 1-dimensional identity representation is the 1-dimensional identity representation itself. Given a unitary representation π of G and a probability measure μ on G, let Pμ denote the μ-average ∫π(g)μ(dg). The goal of this article is twofold: (1) to study the asymptotic behaviour of the powers , and (2) to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure μ on an identity excluding group and every unitary representation π there exists and orthogonal projection Eμ onto a π-invariant subspace such that for every a ∈ supp μ. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of FC-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

scholarly journals Mil nor's problem on the growth of groups and its consequences

2014 ◽  
Author(s):  
Rostislav Grigorchuk
Keyword(s):  
Asymptotic Behavior ◽  
Random Walks ◽  
Exponential Growth ◽  
Main Part ◽  
Polynomial Growth ◽  
Open Problems ◽  
Growth Of Groups ◽  
Significant Interest ◽  
Groups Of Polynomial Growth

This chapter presents a survey of results related to John Milnor's problem on group growth. The notion of group growth first appeared in 1955 in a paper of A. S. Schwarz, but it remained virtually unnoticed for over a decade. The situation changed after Milnor's papers from 1968, which sparked significant interest in this area. Particularly influential were two problems raised in these papers: the characterization of groups of polynomial growth and the question of the existence of groups of intermediate growth. The chapter discusses the cases of polynomial growth and exponential but not uniformly exponential growth; the main part of this chapter is devoted to the intermediate (between polynomial and exponential) growth case. A number of related topics (growth of manifolds, amenability, asymptotic behavior of random walks) are considered, and a number of open problems are suggested.

scholarly journals Besov algebras on Lie groups of polynomial growth

2012 ◽  
Vol 212 (2) ◽  
pp. 119-139 ◽  
Author(s):  
Isabelle Gallagher ◽  
Yannick Sire
Keyword(s):  
Lie Groups ◽  
Polynomial Growth ◽  
Groups Of Polynomial Growth

MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

2008 ◽  
Vol 19 (10) ◽  
pp. 1187-1201 ◽  
Author(s):  
MASAYASU MORIWAKI
Keyword(s):  
Unitary Representation ◽  
Orthogonal Group ◽  
Irreducible Unitary Representation ◽  
Unitary Representations ◽  
Minimal Representation ◽  
Infinite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Multiplicity Free ◽  
Kirillov Dimension

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on 𝕋

2009 ◽  
Vol 30 (1) ◽  
pp. 151-157 ◽  
Author(s):  
MANFRED EINSIEDLER ◽  
ALEXANDER FISH
Keyword(s):  
Probability Measure ◽  
Polynomial Growth ◽  
Borel Probability Measure ◽  
Finite Support ◽  
Multiplicative Semigroup ◽  
Logarithmic Density ◽  
Circle Group ◽  
Borel Probability

AbstractWe prove that if a Borel probability measure on the circle group is invariant under the action of a ‘large’ multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then the measure is either Lebesgue or has finite support.

Exponential Boundedness and Amenability of Open Subsemigroups of Locally Compact Groups

1994 ◽  
Vol 46 (06) ◽  
pp. 1263-1274 ◽  
Author(s):  
Wojciech Jaworski
Keyword(s):  
Probability Measure ◽  
Open Subset ◽  
Compact Support ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Complete Proof ◽  
Absolutely Continuous ◽  
Crucial Part

Abstract Let G be a connected amenable locally compact group with left Haar measure λ. In an earlier work Jenkins claimed that exponential boundedness of G is equivalent to each of the following conditions: (a) every open subsemigroup S ⊆ G is amenable; (b) given and a compact K ⊆ G with nonempty interior there exists an integer n such that (c) given a signed measure of compact support and nonnegative nonzero f ∈ L ∞(G), the condition v * f ≥ 0 implies v(G) ≥ 0. However, Jenkins‚ proof of this equivalence is not complete. We give a complete proof. The crucial part of the argument relies on the following two results: (1) an open σ-compact subsemigroup S ⊆ G is amenable if and only if there exists an absolutely continuous probability measure μ on S such that lim for every s ∈ S; (2) G is exponentially bounded if and only if for every nonempty open subset U ⊆ G.

scholarly journals A Note on Amenability of Locally Compact Quantum Groups

2014 ◽  
Vol 57 (2) ◽  
pp. 424-430 ◽  
Author(s):  
Piotr M. Sołtan ◽  
Ami Viselter
Keyword(s):  
Quantum Groups ◽  
Short Note ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups

AbstractIn this short note we introduce a notion called quantum injectivity of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. In particular, this provides a new characterization of amenability of locally compact groups.

scholarly journals Delocalized Eta Invariants, Algebraicity, and K-Theory of Group C*-Algebras

2019 ◽  
Author(s):  
Zhizhang Xie ◽  
Guoliang Yu
Keyword(s):  
Conjugacy Class ◽  
Discrete Group ◽  
Polynomial Growth ◽  
Algebraic Numbers ◽  
Eta Invariant ◽  
Trace Map ◽  
Novikov Conjecture ◽  
C Algebras ◽  
Eta Invariants ◽  
K Theory

Abstract In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $K_0$-group of its group $C^\ast$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum–Connes conjecture holds for a group, then Lott’s delocalized eta invariants take values in algebraic numbers. We also generalize Lott’s delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.

Sign in / Sign up

Export Citation Format

Share Document