Property (T), finite-dimensional representations, and generic representations

Michal Doucha, Maciej Malicki, Alain Valette

Type: Article

Publication Date: 2018-08-07

Citations: 3

DOI: https://doi.org/10.1515/jgth-2018-0030

Abstract

Abstract Let G be a discrete group with Property (T). It is a standard fact that, in a unitary representation of G on a Hilbert space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℋ</m:mi> </m:math> {\mathcal{H}} , almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation σ, then the vector is close to a sub-representation isomorphic to σ: this makes quantitative a result of P. S. Wang. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot, that a group G with Property (T) and such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {C^{*}(G)} is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>⁢</m:mo> <m:mi>e</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>ℋ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {Rep(G,\mathcal{H})} under the unitary group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>U</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>ℋ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {U(\mathcal{H})} is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>ℋ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{Rep}(G,\mathcal{H})} .

Locations

Journal of Group Theory - View - PDF
arXiv (Cornell University) - View - PDF

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