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Higher property T and below-rank phenomena of lattices

Uri Bader, Roman Sauer

Abstract

The purpose of this paper is twofold. We explore higher property T as an abstract group-theoretic property. In particular, we provide new operator-algebraic characterizations of higher property T. Then we turn to lattices in semisimple Lie groups. We relate higher property T to other cohomological, rigidity and geometric phenomena below the real rank. The second part outlines a conjectural framework that unifies these aspects and reviews recent advances.

Higher property T and below-rank phenomena of lattices

Abstract

The purpose of this paper is twofold. We explore higher property T as an abstract group-theoretic property. In particular, we provide new operator-algebraic characterizations of higher property T. Then we turn to lattices in semisimple Lie groups. We relate higher property T to other cohomological, rigidity and geometric phenomena below the real rank. The second part outlines a conjectural framework that unifies these aspects and reviews recent advances.

Paper Structure

This paper contains 30 sections, 59 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Higher property T as an abstract property of groups
  3. Finiteness properties
  4. The role of the Laplace operators
  5. Banach $\Gamma$-modules
  6. Ultrapowers
  7. Hilbert $C^\ast$-modules
  8. Unitary $\Gamma$-modules
  9. Universal envelopes
  10. The augmentation ideals and the Kazhdan projection
  11. C*-algebraic reformulation of higher T
  12. von Neumann algebras
  13. Group algebra reformulation of higher T
  14. Permanence properties
  15. Products
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\mathbf{G}$ be a simple algebraic group of rank $r$ over a characteristic 0 local field $F$ and let $\Gamma<G=\mathbf{G}(F)$ be a lattice. Then $\Gamma$ has property $(T_{r-1})$. If the field $F$ is non-archimedean then $\Gamma$ has property $[T_{r-1}]$.

Figures (2)

  • Figure 1: Snake arrows are conjectural. Dotted arrows are conditional.
  • Figure 2: Relations between conjectures for $\Gamma<G$ irreducible higher rank lattice. Extra assumptions are in blue.

Theorems & Definitions (143)

  • Theorem 1
  • Definition 2: badsau
  • Remark 3
  • Remark 4
  • Conjecture 5
  • Conjecture 6
  • Theorem 7: BBBS
  • Corollary 8
  • Remark 9
  • Theorem 10
  • ...and 133 more