Reciprocity of Dedekind sums and the Euler class

Dedekind sums are arithmetic sums that were first introduced by Dedekind in the context of elliptic functions and modular forms, and later recognized to be surprisingly ubiquitous. Among the variations and generalizations introduced since, there is a construction of Dedekind sums for lattices in $\mathrm {SL}_2(\mathrm {R})$. Building upon work of Asai, we prove the reciprocity law for these Dedekind sums, based on a concrete realization of the Euler class. As an application, we obtain an explicit formula for Dedekind sums on Hecke triangle groups in terms of continued fractions.