Dedekind sums, reciprocity, and non-arithmetic groups

Abstract

Dedekind sums, arithmetic correlation sums that arose in Dedekind's study of the modular transformation of the logarithm of the eta-function, are surprisingly ubiquitous. Their arithmetic properties attracted the attention of number theorists, combinatorists, and theoretical computer scientists alike, and they appear more broadly in geometry, topology, and physics. Accordingly, there is a similarly vast literature on variations and generalizations of Dedekind sums. This note surveys recent developments of some aspects of Dedekind sums for (non-uniform) lattices in SL_2(R), otherwise referred to as Dedekind symbols.