  We study dimension theory for the $C^*$-algebras of row-finite $k$-graphs
with no sources. We establish that strong aperiodicity - the higher-rank
analogue of condition (K) - for a $k$-graph is necessary and sufficient for the
associated $C^*$-algebra to have topological dimension zero. We prove that a
purely infinite $2$-graph algebra has real-rank zero if and only if it has
topological dimension zero and satisfies a homological condition that can be
characterised in terms of the adjacency matrices of the $2$-graph. We also show
that a $k$-graph $C^*$-algebra with topological dimension zero is purely
infinite if and only if all the vertex projections are properly infinite. We
show by example that there are strongly purely infinite $2$-graphs algebras,
both with and without topological dimension zero, that fail to have real-rank
zero.
