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<p>We give a new geometric proof of a theorem of Heuer showing that, in the presence of letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups), and in particular in RAAGs, there is a sharp lower bound of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1\over 2\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation> for stable commutator length. Our approach is to show that letter-quasimorphisms give rise to negatively curved angle structures on admissible surfaces. This generalises Duncan and Howie’s proof of the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(1\over 2\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation>-lower bound in free groups, and can also be seen as a version of Bavard duality for letter-quasimorphisms.</p>

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From letter-quasimorphisms to angle structures and spectral gaps for scl

  • 期刊论文

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  • 发表日期：

    2026年3月26日

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  • Alexis Marchand

摘要

We give a new geometric proof of a theorem of Heuer showing that, in the presence of letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups), and in particular in RAAGs, there is a sharp lower bound of \(1\over 2\) 1 2 for stable commutator length. Our approach is to show that letter-quasimorphisms give rise to negatively curved angle structures on admissible surfaces. This generalises Duncan and Howie’s proof of the \(1\over 2\) 1 2 -lower bound in free groups, and can also be seen as a version of Bavard duality for letter-quasimorphisms.