Projective Rigidity of Once-Punctured Torus Bundles via the Twisted Alexander Polynomial

TL;DR

This paper introduces a method to certify infinitesimal projective rigidity of hyperbolic once-punctured torus bundles using twisted Alexander polynomials related to the holonomy, connecting algebraic and geometric properties.

Contribution

It establishes a link between twisted Alexander polynomials and the infinitesimal projective rigidity of hyperbolic once-punctured torus bundles, providing a new algebraic criterion.

Findings

Certifies infinitesimal projective rigidity via twisted Alexander polynomials.

Relates the polynomial to an induced action on the tangent space of the character variety.

Shows the induced action matches the group theoretic action in the spectral sequence.

Abstract

In this paper we provide a means of certifying infinitesimal projective rigidity relative to the cusp for hyperbolic once punctured torus bundles in terms of twisted Alexander polynomials of representations associated to the holonomy. We also relate this polynomial to an induced action on the tangent space of the character variety of the free group of rank 2 into PGL(4,R) that arises from the holonomy of a hyperbolic once-punctured torus bundle. We prove the induced action on the tangent space of the character variety is the same as the group theoretic action that arises in the Lyndon Hochschild Serre spectral sequence on cohomology.