NDLI: The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Content Provider	Springer Nature Link
Author	Nevo, Amos
Copyright Year	2003
Abstract	Let G denote a semisimple group, Γ a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any Γ-quasi-invariant measure η on B, and any probablity measure μ on Γ, the norm of the operator πη(μ) on L 2(B,η) is equal to ∥λΓ(μ)∥, where πη is the unitary representation in L 2(X,η), and λΓ is the regular representation of Γ.(2) In particular this estimate holds when η is Lebesgue measure on B, a Patterson–Sullivan measure, or a μ-stationary measure, and implies explicit lower bounds for the displacement and Margulis number of Γ (w.r.t. a finite generating set), the dimension of the conformal density, the μ-entropy of the measure, and Lyapunov exponents of Γ.(3) In particular, when G=PSL2(ℂ) and Γ is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ∥πη(μ)∥=∥λG(μ)∥ for any amenable action of G and μ∈L 1(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the μ-entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when Γ is any word-hyperbolic group, acting on their Poisson boundary, for example.
Starting Page	187
Ending Page	218
Page Count	32
File Format	PDF
ISSN	00465755
Journal	Geometriae Dedicata
Volume Number	100
Issue Number	1
e-ISSN	15729168
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2003-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Orbits and invariants of visible group actions

INITIAL FORMS OF STABLE INVARIANTS FOR ADDITIVE GROUP ACTIONS

Algebraic Entropy of Elementary Amenable Groups

Typical separating invariants

Elementary amenable groups are quasidiagonal

The geometric invariants $${\Omega^{n}}$$ of a product of groups

Observable actions of algebraic groups

Embedding Free Burnside Groups in Finitely Presented Groups

The algebras of semi-invariants of quivers

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Similar Documents

Orbits and invariants of visible group actions

INITIAL FORMS OF STABLE INVARIANTS FOR ADDITIVE GROUP ACTIONS

Algebraic Entropy of Elementary Amenable Groups

Typical separating invariants

Elementary amenable groups are quasidiagonal

The geometric invariants $${\Omega^{n}}$$ of a product of groups

Observable actions of algebraic groups

Embedding Free Burnside Groups in Finitely Presented Groups

The algebras of semi-invariants of quivers

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups