Lehmer's Problem for arbitrary groups

TL;DR

This paper explores a generalization of Lehmer's problem to arbitrary groups, investigating whether a uniform lower bound exists for the Fuglede-Kadison determinants of certain operators over group rings.

Contribution

It introduces a group-theoretic extension of Lehmer's problem, defining a constant Lambda(G) and analyzing its existence for various groups.

Findings

For the infinite cyclic group, the problem reduces to Lehmer's original problem.

The paper establishes conditions under which Lambda(G) exists for different groups.

It connects algebraic properties of groups with spectral invariants of associated operators.

Abstract

We consider the problem whether for a group G there exists a constant Lambda(G) > 1 such that for any (r,s)-matrix A over the integral group ring ZG the Fuglede-Kadison determinant of the G-equivariant bounded operator from L^2(G)^r to L^2(G)^s given by right multiplication with A is either one or greater or equal to Lambda(G). If G is the infinite cyclic group and we consider only r = s = 1, this is precisely Lehmer's problem.