Abstract
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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.