Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the spectral gap conjecture for pairs in SU(2)

Oleg Pikhurko, Kohki Sakamoto

Abstract

For $n \ge 2$, Gamburd, Jakobson, and Sarnak [J. Eur. Math. Soc. 1, 51-85 (1999)] conjectured that almost every $n$-tuple in $\mathrm{SU}(2)$ has a spectral gap. Toward this conjecture, Fisher [Int. Math. Res. Not. (2006)] established a zero-one law for $n \ge 3$, but obtained only a partial result for $n=2$. In this paper, we prove that the zero-one law also holds for $n=2$. We also remark that a Baire categorical analogue of this result holds.

On the spectral gap conjecture for pairs in SU(2)

Abstract

For , Gamburd, Jakobson, and Sarnak [J. Eur. Math. Soc. 1, 51-85 (1999)] conjectured that almost every -tuple in has a spectral gap. Toward this conjecture, Fisher [Int. Math. Res. Not. (2006)] established a zero-one law for , but obtained only a partial result for . In this paper, we prove that the zero-one law also holds for . We also remark that a Baire categorical analogue of this result holds.
Paper Structure (4 sections, 7 theorems, 23 equations)

This paper contains 4 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the zero-one law for $n=2$
  4. Baire categorical version

Key Result

Theorem 1.1

The set of pairs with spectral gap $\mathbf{SG}_{2} \subset \mathrm{SU}(2)^{2}$ has either zero measure or full measure with respect to the Haar measure.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: Fisher, Theorem 2.3 in MR2250018
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 5 more