  Lehmer's famous problem asks whether the set of Mahler measures of
polynomials with integer coefficients admits a gap at 1. In 2019, L\"uck
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 $\frac{2}{\sqrt{3}}$
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 $\frac{2}{\sqrt{3}}$.
