Commutators in $SL_2$ and Markoff surfaces I

Ghosh, Amit; Meiri, Chen; Sarnak, Peter

doi:10.53733/198

Cited by 3 publications

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References 21 publications

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“…(b). In the recent paper [25], the Hasse failure (i) is exploited to give failures of profinite local to global principles for commutator equations in SL 2 (O) for O a ring of S-integers. (c).…”

Section: Theorem 12 (I)mentioning

confidence: 99%

Integral points on Markoff type cubic surfaces

Ghosh

Sarnak

2022

Invent. math.

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For integers k, we consider the affine cubic surface V k given byWe show that for almost all k the Hasse Principle holds, namely that V k (Z) is non-empty if V k (Z p ) is non-empty for all primes p, and that there are infinitely many k's for which it fails. The Markoff morphisms act on V k (Z) with finitely many orbits and a numerical study points to some basic conjectures about these "class numbers" and Hasse failures. Some of the analysis may be extended to less special affine cubic surfaces.

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Section: Theorem 12 (I)mentioning

confidence: 99%

Integral points on Markoff type cubic surfaces

Ghosh

Sarnak

2022

Invent. math.

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Tautological characteristic classes II: the Witt class

Dymara,

Januszkiewicz

2024

Math. Ann.

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Let K be an arbitrary infinite field. The cohomology group $$H^{2}(SL(2,K);H_2 SL(2,K))$$ H 2 ( S L ( 2 , K ) ; H 2 S L ( 2 , K ) ) contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in SL(2, K) it is useful to have classes stable under deformations (Fenchel-Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovàř. The Milnor-Wood inequality asserts that an $$SL(2,\textbf{R})$$ S L ( 2 , R ) -bundle over a surface of genus g admits a flat structure if and only if its Euler number is $$\le (g - 1)$$ ≤ ( g - 1 ) . We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general.

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