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Let $\Gamma$ be a group and $(\Gamma_n)_{n=1} ^{\infty}$ be a descendingsequence of finite-index normal subgroups. We establish explicit upper boundson the diameters of the directed Cayley graphs of the $\Gamma/\Gamma_n$ , undersome natural hypotheses on the behaviour of power and commutator words in$\Gamma$. The bounds we obtain do not depend on a choice of generating set.Moreover under reasonable conditions our method provides a fast algorithm forconstructing directed Cayley graphs of diameter satisfying our bounds. Theproof is closely analogous to the the Solovay-Kitaev procedure, which only usescommutator words, but also only constructs small-diameter undirected Cayleygraphs. As an application we give directed diameter bounds on finite quotientsof two very different groups: $SL_2 (\mathbb{F}_q [[t]])$ (for $q$ even) and agroup of automorphisms of the ternary rooted tree introduced by Fabrykowski andGupta.

Constructing Directed Cayley Graphs of Small Diameter: A Potent Solovay-Kitaev Procedure