Lehmer's Problem, McKay's Correspondence, and 2,3,7

Eriko Hironaka

This paper addresses a long standing open problem due to Lehmer in which the triple 2,3,7 plays a notable role. Lehmer's problem asks whether there is a gap between 1 and the next largest algebraic integer with respect to Mahler measure. The question has been studied in a wide range of contexts including number theory, ergodic theory, hyperbolic geometry, and knot theory; and relates to basic questions such as describing the distribution of heights of algebraic integers, and of lengths of geodesics on arithmetic surfaces. This paper focuses on the role of Coxeter systems in Lehmer's problem. The analysis also leads to a topological version of McKay's correspondence.