Continued fractions, modular symbols, and non-commutative geometry

Yuri I. Manin, Matilde Marcolli

Published 2001-02-01, updated 2001-08-07Version 2

Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient $PGL(2,\bold{Z})\setminus\bold{P}^1(\bold{R})$ should be considered as non-commutative modular curve, and show that the modular complex can be seen as a sequence of $K_0$-groups of the related crossed-product $C^*$-algebras. This paper is an expanded version of the previous "On the distribution of continued fractions and modular symbols". The main new features are Section 4 on non-commutative geometry and the modular complex and Section 1.2.2 on the Mixmaster Universe.