  The exponential growth rate of non polynomially growing subgroups of $GL_d$
is conjectured to admit a uniform lower bound. This is known for non-amenable
subgroups, while for amenable subgroups it is known to imply the Lehmer
conjecture from number theory. In this note, we show that it is equivalent to
the Lehmer conjecture. This is done by establishing a lower bound for the
entropy of the random walk on the semigroup generated by the maps $x\mapsto
\lambda\cdot x\pm 1$, where $\lambda$ is an algebraic number. We give a bound
in terms of the Mahler measure of $\lambda$. We also derive a bound on the
dimension of Bernoulli convolutions.
