  We investigate subsets of a multifractal decomposition of the limit set of a
conformal graph directed Markov system, which is constructed from the Cayley
graph of a free group with at least two generators. The subsets we consider are
parametrised by a normal subgroup $N$ of the free group and mimic the radial
limit set of a Kleinian group. Our main results show that, regarding the
Hausdorff dimension of these sets, various results for Kleinian groups can be
generalised. Namely, under certain natural symmetry assumptions on the
multifractal decomposition, we prove that, for a subset parametrised by $N$,
the Hausdorff dimension is maximal if and only if $\F_{d}/N$ is amenable and
that the dimension is greater than half of the maximal value. We also give a
criterion for amenability via the divergence of the Poincar\'{e} series of $N$.
Our results are applied to the Lyapunov spectrum for normal subgroups of
Kleinian groups of Schottky type.
