The Markov spectra for Fuchsian groups

Abstract

Applying the Klein model <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D squared"> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">D^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the hyperbolic plane and identifying the geodesics in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D squared"> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">D^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with their poles in the projective plane, the author develops a method of determining infinite binary trees in the Markov spectrum for a Fuchsian group. The method is applied to a maximal group commensurable with the modular group and other Fuchsian groups.