  We show that the C*-algebra of a row-finite source-free k-graph is
Rieffel-Morita equivalent to a crossed product of an AF algebra by the
fundamental group of the k-graph. When the k-graph embeds in its fundamental
groupoid, this AF algebra is a Fell algebra; and simple-connectedness of a
certain sub-1-graph characterises when this Fell algebra is Rieffel--Morita
equivalent to a commutative C*-algebra. We provide a substantial suite of
results for determining if a given k-graph embeds in its fundamental groupoid,
and provide a large class of examples, arising via work of Cartwright,
Robertson, Steger et al. from the theory of $\tilde{A_2}$-groups, that do
embed.
