A Hyperbolic Analogue of the Rademacher Symbol

One of the most famous results of Dedekind is the transformation law of $\log \Delta(z)$. After a half-century, Rademacher modified Dedekind's result and introduced an $\mathrm{SL}_2(\mathbb{Z})$-conjugacy class invariant (integer-valued) function $\Psi(\gamma)$ called the Rademacher symbol. Inspired by Ghys' work on modular knots, Duke-Imamo\={g}lu-T\'{o}th (2017) constructed a hyperbolic analogue of the symbol. In this article, we study their hyperbolic analogue of the Rademacher symbol $\Psi_\gamma(\sigma)$ and provide its two types of explicit formulas by comparing it with the classical Rademacher symbol. In association with it, we contrastively show Kronecker limit type formulas of the parabolic, elliptic, and hyperbolic Eisenstein series. These limits give harmonic, polar harmonic, and locally harmonic Maass forms of weight 2.