Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated to a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterisation, originally due to Robertson and Sims, of simplicity of the C * -algebra of a row-finite k-graph with no sources.

We construct a locally compact Hausdorff topology on the path space of a directed graph E, and identify its boundary-path space ∂E as the spectrum of a commutative C * -subalgebra D E of C * (E). We then show that ∂E is homeomorphic to a subset of the infinite-path space of any desingularisation F of E. Drinen and Tomforde showed that we can realise C * (E) as a full corner of C * (F ), and we deduce that D E is isomorphic to a corner of D F . Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $\Lambda$. We identify the boundary-path space $\partial\Lambda$ as the spectrum of a commutative $C^*$-subalgebra $D_\Lambda$ of $C^*(\Lambda)$. Then, using a construction similar to that of Farthing, we construct a finitely aligned $k$-graph $\wt\Lambda$ with no sources in which $\Lambda$ is embedded, and show that $\partial\Lambda$ is homeomorphic to a subset of $\partial\wt\Lambda$ . We show that when $\Lambda$ is row-finite, we can identify $C^*(\Lambda)$ with a full corner of $C^*(\wt\Lambda)$, and deduce that $D_\Lambda$ is isomorphic to a corner of $D_{\wt\Lambda}$. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.Comment: 30 pages, all figures drawn with TikZ/PGF. Updated numbering and minor corrections to coincide with published version. Updated 29-Feb-2012 to fix a compiling error which resulted in the arXiv PDF output containing two copies of the articl

Abstract. We investigate the factor types of the extremal KMS states for the preferred dynamics on the Toeplitz algebra and the Cuntz-Krieger algebra of a strongly connected finite k-graph. For inverse temperatures above 1, all of the extremal KMS states are of type I∞. At inverse temperature 1, there is a dichotomy: if the k-graph is a simple k-dimensional cycle, we obtain a finite type I factor; otherwise we obtain a type III factor, whose Connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph.

Abstract. Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing geometric and combinatorial, rather than topological, features of the original. Our method is easy to apply to particular substitution tilings, permits experimentation, and can be used to construct border-forcing substitution rules. For a large class of examples we show that the combinatorial dual tiling has a realization as a substitution tiling. Since the boundaries of our new tilings are fractal we are led to compute their fractal dimension. As an application of our techniques we show how to compute theČech cohomology of a (not necessarily border-forcing) tiling using a graph iterated function system of a fractal tiling.