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A human property (T) proof for high-rank $Aut(F_n)$

Martin Nitsche

Abstract

Existing property (T) proofs for $Aut(F_n)$, $n\geq 4$, rely crucially on extensive computer calculations. We give a new proof that $Aut(F_n)$ has property (T) for all but finitely many $n$ that is inspired by the semidefinite programming approach but does not use the computer in any step. More specifically, we prove property (T) for a certain extension $Γ_n$ of $SAut(F_n)$ as $n\to\infty$.

A human property (T) proof for high-rank $Aut(F_n)$

Abstract

Existing property (T) proofs for , , rely crucially on extensive computer calculations. We give a new proof that has property (T) for all but finitely many that is inspired by the semidefinite programming approach but does not use the computer in any step. More specifically, we prove property (T) for a certain extension of as .
Paper Structure (4 sections, 6 theorems, 28 equations, 1 figure)

This paper contains 4 sections, 6 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on harmonic cocycles
  3. The setting for SAut(Fn), n->inf
  4. Proof of the main result

Key Result

Theorem 1

All but finitely many of the groups $\Gamma_n$ have property (T). Consequently, the same is true for the groups $\mathrm{Aut}(F_n)$.

Figures (1)

  • Figure 1: The distances between the elements $\mathbf{1},{E_{{ac}}},{E_{{bc}}},{E_{{ac}}}{E_{{bc}}},{E_{{ab}}}$

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2
  • proof
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 4 more