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<p>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 the Wada Invariant 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\text {PGL}(4,\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>PGL</mtext> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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 from the monodromy action of the once-punctured torus bundle on cohomology.</p>

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Projective Rigidity of Once-Punctured Torus Bundles via Twisted Invariants

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  • 发表日期：

    2025年11月4日

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  • Charles Daly

摘要

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 the Wada Invariant 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 \(\text {PGL}(4,\mathbb {R})\) 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 from the monodromy action of the once-punctured torus bundle on cohomology.