Lehmer’s famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at 1. In 2019, Lück extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer’s constants of the group to measure such a gap.In this paper, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups, which yields the new upper bound 2 3 for Lehmer’s constants of all torsion-free groups which have non-cyclic free subgroups.Our proofs use relations between Fuglede-Kadison determinants and random walks on Cayley graphs, as well as works of Bartholdi and Dasbach-Lalin.Furthermore, via the gluing formula for L 2 -torsions, we show that the Lehmer’s constants of an infinite number of fundamental groups of hyperbolic 3-manifolds are bounded above by even smaller values than 2 3.