A substitute for Kazhdan’s property (T) for universal nonlattices

Ozawa, Narutaka

doi:10.2140/apde.2024.17.2541

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A substitute for Kazhdan's property (T) for universal nonlattices

Narutaka Ozawa

Vol. 17 (2024), No. 7, 2541–2560

DOI: 10.2140/apde.2024.17.2541

Abstract

The well-known theorem of Shalom–Vaserstein and Ershov–Jaikin-Zapirain states that the group ${EL}_{n} (ℛ)$ , generated by elementary matrices over a finitely generated commutative ring $ℛ$ , has Kazhdan’s property (T) as soon as $n \geq 3$ . This is no longer true if the ring $ℛ$ is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients ${EL}_{n} (ℛ ∕ ℛ^{k})$ . We prove that even in such a case the group ${EL}_{n} (ℛ)$ satisfies a certain property that can substitute property (T), provided that $n$ is large enough.

Keywords

Kazhdan's property (T), real group algebras, sum of hermitian squares

Mathematical Subject Classification

Primary: 22D10

Secondary: 22D15, 46L89

Milestones

Received: 14 September 2022

Accepted: 31 March 2023

Published: 21 August 2024

Authors

Narutaka Ozawa
Research Institute for Mathematical Sciences Kyoto University Kyoto Japan

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Vol. 17, No. 7, 2024