EMBEDDABILITY OF HIGHER-RANK GRAPHS IN GROUPOIDS AND THE STRUCTURE OF THEIR $\mathbf {C^*}$ -ALGEBRAS

Abstract We show that the $C^*$ -algebra of a row-finite source-free k-graph is Rieffel–Morita equivalent to a crossed product of an approximately finite-dimensional (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 et al. [‘Groups acting simply transitively on the vertices of a building of type $\tilde{\rm A}_2$ I’, Geom. Dedicata 47 (1993), 143–166], Cartwright et al. ‘Groups acting simply transitively on the vertices of a building of type $\tilde{\rm A}_2$ II’, Geom. Dedicata 47 (1993), 167–226] and Robertson and Steger [‘Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras’, J. reine angew. Math. 513 (1999), 115–144] from the theory of $\tilde {A_2}$ -groups, which do embed.