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The Andersen-Masbaum-Ueno conjecture for the derived subgroup of the Johnson kernel

Renaud Detcherry

Abstract

A conjecture of Andersen, Masbaum and Ueno states that for any compact oriented surface $Σ_{g,n}$ and any pseudo-Anosov $f\in \mathrm{Mod}(Σ_{g,n}),$ the matrix $ρ_r(f)$ has infinite order for any large $r,$ where $ρ_r$ is the $\mathrm{SO}(3)$-WRT quantum representation of the mapping class group $\mathrm{Mod}(Σ_{g,n})$ at a primitive $r$-th root of unity. We prove this conjecture for prime $r$ and any $f\in [J_2(Σ_{g,n}),J_2(Σ_{g,n})],$ where $J_2(Σ_{g,n})$ is the Johnson kernel.

The Andersen-Masbaum-Ueno conjecture for the derived subgroup of the Johnson kernel

Abstract

A conjecture of Andersen, Masbaum and Ueno states that for any compact oriented surface and any pseudo-Anosov the matrix has infinite order for any large where is the -WRT quantum representation of the mapping class group at a primitive -th root of unity. We prove this conjecture for prime and any where is the Johnson kernel.

Paper Structure

This paper contains 3 sections, 11 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. A number-theoretic criterion for elements of $GL_d(\mathbb{Z}[\zeta_p])$ to be of infinite order
  3. The $h$-adic expansion and the AMU conjecture

Key Result

Theorem 1.1

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ and $n$ boundary components. Let $f\in [J_2(\Sigma_{g,n}),J_2(\Sigma_{g,n}))]$ be non-trivial. Then for any prime $p$ large enough, $\rho_p(f)$ has infinite order.

Theorems & Definitions (26)

  • Conjecture 1
  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • ...and 16 more