Representation theoretic rigidity in \(PSL(2,\mathbb{R})\)
type: Item
instance of: scholarly article
title: Representation theoretic rigidity in \(PSL(2,\mathbb{R})\)
zbMATH Open document ID: 0785.22011
DOI: 10.1007/BF02392456
author: Tim Steger
author: Christopher J. Bishop
published in: Acta Mathematica
publication date: 1994-04-17T00:00:00Z
review text: Let \(G\) be a connected simple Lie group with trivial center, let \(\Gamma\) be an abstract group, and let \(\iota_ 1\) and \(\iota_ 2\) be inclusions of \(\Gamma\) in \(G\). Assume that each of the images \(\iota_ j(\Gamma)\) is a lattice subgroup, meaning that \(\iota_ j(\Gamma)\) is discrete and that the \(G\)-invariant measure on \(G/\iota_ j(\Gamma)\) has total finite mass. One says that \(\iota_ 1\) and \(\iota_ 2\) are equivalent if there is some automorphism \(\rho\) of \(G\) such that \(\iota_ 2=\rho\circ \iota_ 1\). If \(G\) is not isomorphic to \(PSL_ 2(\mathbb{R})\) then the Mostow rigidity theorem says that \(\iota_ 1\) and \(\iota_ 2\) are equivalent. This result fails for \(PSL_ 2(\mathbb{R})\). The authors prove for \(PSL_ 2(\mathbb{R})\) the following Theorem. Suppose that \(\pi_ 1\) and \(\pi_ 2\) are irreducible representations of \(PSL_ 2(\mathbb{R})\), not in the discrete series. Then \(\pi_ 1\circ \iota_ 1\) and \(\pi_ 2\circ\iota_ 2\) are equivalent representations of \(\Gamma\) if and only if \(\iota_ 1\) and \(\iota_ 2\) are equivalent inclusions and \(\pi_ 1\) and \(\pi_ 2\) are equivalent representations of \(PSL_ 2(\mathbb{R})\). The essential step in the proof of this theorem is an analytic criterion for the equivalence of \(\iota_ 1\) and \(\iota_ 2\).
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 205190
zbMATH Keywords: representation of lattices
zbMATH Keywords: connected simple Lie group
zbMATH Keywords: rigidity
reviewed by: Herbert Abels
MaRDI profile type: Publication
cites work: A geometric proof of Mostow's rigidity theorem for groups of divergence type
cites work: Inequalities for certain Fuchsian groups
cites work: Q3310101
cites work: Q5583306
cites work: Q3977203
cites work: Q2899316
cites work: Q3999944
cites work: Q4693152
cites work: Canonical polygons for finitely generated Fuchsian groups
cites work: Q3741813
cites work: Restrictions of the special representation of \(\text{Aut}(\text{tree}_ 3)\) to two cocompact subgroups
cites work: Q5179680
cites work: Q4145882
cites work: Q4075684
cites work: Discrete groups of motions of manifolds of nonpositive curvature
cites work: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: The limit set of a Fuchsian group
cites work: Q4296482
cites work: Strong rigidity of Q-rank 1 lattices
cites work: Q5599242
cites work: The infinite word problem and limit sets in Fuchsian groups
cites work: Q5607484
cites work: The density at infinity of a discrete group of hyperbolic motions
cites work: A rigidity theorem for Möbius groups
cites work: Hausdorff dimension and quasisymmetric mappings.
cites work: Strong rigidity for ergodic actions of semisimple Lie groups
cites work: Ergodic theory, group representations, and rigidity
cites work: Q3687381
full work available at URL: https://doi.org/10.1007/bf02392456
OpenAlex ID: W2018409824
description: scientific article; zbMATH DE number 205190