On the Goncharov depth conjecture and a formula for volumes of orthoschemes

Abstract

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that correspond to decompositions of a polygon into quadrangles.

Our second result is a formula for volume of hyperbolic orthoschemes, generalizing the formula of Lobachevsky in dimension three to an arbitrary dimension. We show a surprising relation between the two results, which comes from the fact that hyperbolic orthoschemes are parametrized by configurations of points on P 1 \mathbb {P}^1 . In particular, we derive both formulas from their common generalization.