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On quasi-homomorphism rigidity for lattices in simple algebraic groups

Guillaume Dumas

Abstract

Property $(TTT)$ was introduced by Ozawa as a strengthening of Kazhdan's property $(T)$ and Burger and Monod's property $(TT)$. In this paper, we improve Ozawa's result by showing that any simple algebraic group of rank $\geq 2$ over a local field has property $(TTT)$. We also show that lattices in a second countable locally compact group inherits property $(TTT)$. Finally, we study to what extent Lie groups with infinite center fail to have properties $(TT)$ and $(TTT)$.

On quasi-homomorphism rigidity for lattices in simple algebraic groups

Abstract

Property was introduced by Ozawa as a strengthening of Kazhdan's property and Burger and Monod's property . In this paper, we improve Ozawa's result by showing that any simple algebraic group of rank over a local field has property . We also show that lattices in a second countable locally compact group inherits property . Finally, we study to what extent Lie groups with infinite center fail to have properties and .
Paper Structure (9 sections, 19 theorems, 94 equations)

This paper contains 9 sections, 19 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Related properties
  3. Positive definite kernels and completely bounded norm
  4. Property (Tp) and (Tq)
  5. Measurable factorisation
  6. Relation between properties
  7. The symplectic group Sp4(K)
  8. Algebraic groups over local fields
  9. Simple Lie groups with infinite center

Key Result

Theorem 1

Let $G$ be a locally compact second countable group and $\Gamma$ a lattice in $G$. Then $G$ has property $(TTT)$ if and only if $\Gamma$ has property $(TTT)$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1
  • Theorem 2
  • Definition 2.1: Ozawa+2011+89+104
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • ...and 28 more