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On the positivity of twisted $L^2$-torsion for 3-manifolds

Jianru Duan

Abstract

For any compact orientable irreducible 3-manifold $N$ with empty or incompressible toral boundary, the twisted $L^2$-torsion is a non-negative function defined on the representation variety $\operatorname{Hom}(π_1(N),\operatorname{SL}(n,\mathbb C))$. The paper shows that if $N$ has infinite fundamental group, then the $L^2$-torsion function is strictly positive. Moreover, this torsion function is continuous when restricted to the subvariety of upper triangular representations.

On the positivity of twisted $L^2$-torsion for 3-manifolds

Abstract

For any compact orientable irreducible 3-manifold with empty or incompressible toral boundary, the twisted -torsion is a non-negative function defined on the representation variety . The paper shows that if has infinite fundamental group, then the -torsion function is strictly positive. Moreover, this torsion function is continuous when restricted to the subvariety of upper triangular representations.
Paper Structure (12 sections, 21 theorems, 124 equations)

This paper contains 12 sections, 21 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Notations and some algebraic facts
  3. Twisting $\mathbb{C} G$-modules via $\operatorname{SL}(n,\mathbb{C})$ representations
  4. $L^2$-torsion theory
  5. Twisted $L^2$-torsion for CW complexes
  6. Twisted $L^2$-torsion for 3-manifolds
  7. Twisted $L^2$-torsion for graph manifolds
  8. Twisted $L^2$-torsion for hyperbolic or mixed manifolds
  9. Continuity of twisted $L^2$-torsion on representation varieties
  10. $L^2$-Alexander torsions
  11. Alexander multi-twists of matrices
  12. Applications to 3-manifolds

Key Result

Theorem 1.1

Let $N$ be a compact orientable irreducible 3-manifold with empty or incompressible toral boundary. Suppose $N$ has infinite fundamental group, then the twisted $L^2$-torsion $\tau^{(2)}(N,\rho)$ is positive for any group homomorphism $\rho:\pi_1(N)\rightarrow\operatorname{SL}(n,\mathbb{C})$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Definition 3.1: Admissible triple
  • ...and 33 more