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EMBEDDABILITY OF HIGHER-RANK GRAPHS IN GROUPOIDS AND THE STRUCTURE OF THEIR
$\mathbf {C^*}$-ALGEBRAS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 31 October 2025, pp. 1-33
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- Article
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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.
