On the homology classes defined by closed geodesics on the modular curve

Closed geodesics on the modular curve define certain homology classes of SL(2,Z). These homology classes are very interesting objects that are related to the arithmetic of real quadratic fields, half-integral weight modular forms, etc. . For example, it is known that the pairing between such homology classes and the Eisenstein class (a cohomology class of SL(2,Z) defined by Eisenstein series) gives special values of zeta functions of real quadratic fields, leading to many applications.

In this talk, I will discuss the size of the subgroup (in the homology of SL(2,Z)) generated by such homology classes defined by closed geodesics and its consequences. This talk is based on joint work in progress with Ryotaro Sakamoto.