  Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a
non-degenerate finitely supported symmetric random walk such that its harmonic
measure on the flag space is singular. The main ingredients of the proof are:
(1) a new upper estimate for the Hausdorff dimension of the projections of the
harmonic measure onto Grassmannians in $R^d$ in terms of the associated
differential entropies and differences between the Lyapunov exponents; (2) an
explicit construction of random walks with uniformly bounded entropy and
Lyapunov exponents going to infinity.
