Entropy of Bernoulli convolutions and uniform exponential growth for linear groups

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