Reciprocity laws for Dedekind symbols and the Euler class

Claire Burrin

Published 2016-09-12Version 1

We show that reciprocity laws for generalized Dedekind sums can be deduced from a concrete realization of the bounded Euler class for lattices in $\mathrm{SL}_2(\mathbb{R})$. Asai observed that the reciprocity law for Dedekind sums follows from a certain "splitting" of the central extension $$ 0 \to \mathbb{Z} \to \widetilde{\mathrm{SL}(2,\mathbb{R})} \to \mathrm{SL}(2,\mathbb{Z}) \to 1. $$ We develop and extend this point of view. First, to highlight that the splitting is provided by the explicit construction of a bounded representative of the Euler class. Then, to deduce the reciprocity of Dedekind symbols, which are natural generalizations of the Dedekind sums for lattices in $\mathrm{SL}_2(\mathbb{R})$. In doing so, we demonstrate that the reciprocity of Dedekind sums goes well beyond any arithmetical aspect of the associated group. As an application of our results, we prove combinatorial formulas for the Dedekind symbol for Hecke triangle groups.