Mahler measure and volumes in hyperbolic space

Abstract

The Mahler measure of the polynomials $t(x^m-1) y - (x^n-1) \in \dC[x,y]$ is essentially the sum of volumes of a certain collection of ideal hyperbolic polyhedra in $\HH^3$, which can be determined a priori as a function on the parameter $t$. We obtain a formula that generalizes some previous formulas given by Cassaigne and Maillot \cite{M} and Vandervelde \cite{V}. These examples seem to be related to the ones studied by Boyd \cite{B1}, \cite{B2} and Boyd and Rodriguez Villegas \cite{BRV2} for some cases of the $A$-polynomial of one-cusped manifolds.