Dedekind sums, reciprocity, and non-arithmetic groups

Dedekind sums, arithmetic correlation sums that arose in Dede-kind’s study of the modular transformation of the logarithm of the η \eta -function [De1892], are surprisingly ubiquitous. Their arithmetic properties attracted the attention of number theorists, combinatorists, and theoretical computer scientists alike [RG1972,Me1957,Po1993,Kn1981,BR2004], and they appear more broadly in geometry, topology, and physics [At1987,KM1994,BG1992]. Accordingly, there is a similarly vast literature on variations and generalizations of Dedekind sums. It is the goal of this note to survey some of the aspects of Dedekind sums for (non-uniform) lattices in S L 2 ( R ) \mathrm {SL}_2(\mathbb {R}) , otherwise referred to as Dedekind symbols. Intrinsically, this gives us a framework in which to investigate the rôle ‘arithmeticity’ plays in defining properties of Dedekind sums. In this note, we discuss the reciprocity law for Dedekind symbols associated to lattices that are not necessarily arithmetic. We will see that the reciprocity law holds given an algebraic structure similar to that of S L 2 ( Z ) \mathrm {SL}_2(\mathbb {Z}) , e.g. Hecke triangle groups.