Zeta Functions of Loop Groups

Abstract

We will make a preparation for defining the Selberg zeta function of $PSL(2, \mathbf{Z}[T])$, which is a discrete subgroup of the loop group $G$ of $PSL(2, \mathbf{C})$. Conjugacy classes of $PSL(2, \mathbf{Z}[T])$ will be classified and the definition of the norm of hyperbolic classes will be proposed.