NONCOMMUTATIVE REAL ALGEBRAIC GEOMETRY OF KAZHDAN’S PROPERTY (T)

2013/12/31 by Narutaka Ozawa · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic number #Combinatorics #Geometry #Mathematical analysis #Mathematics #Noncommutative geometry #Pure mathematics #math.GR #math.OA #msc:16A27 #msc:22D10 #msc:46L89

paper · pdf · doi:10.1017/s1474748014000309

published as Journal of the Institute of Mathematics of Jussieu 15 (2014) 85-90 · 6 pages; a few improvement (v2); update (v3)

openalex publication_date 2014/07/30 · arxiv created 2015/01/26 · arxiv updated 2015/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

It is well known that a finitely generated group \rmΓ has Kazhdan’s property (T) if and only if the Laplacian element \rmΔ in ℝ[\rmΓ] has a spectral gap. In this paper, we prove that this phenomenon is witnessed in ℝ[\rmΓ] . Namely, \rmΓ has property (T) if and only if there exist a constant \itκ>0 and a finite sequence \itξ₁,… ,\itξ_n in ℝ[\rmΓ] such that \rmΔ²-\itκ\rmΔ=∑ _i\itξ_i^∗ \itξ_i . This result suggests the possibility of finding new examples of property (T) groups by solving equations in ℝ[\rmΓ] , possibly with the assistance of computers.

Cited by

• \text Aut(\mathbb F₅) has property (T)