New models of higher-rank graph algebras arising from buildings, and computation of their K-theory

Abstract

Formally, a k-rank graph is defined as a small category Λ together with a functor d which associates to each morphism λ ∈ Hom(Λ) an element of Nk , called its degree. The functor d must satisfy a special factorisation property which at first glance might not seem to restrict much, but actually heralds intricate and complex properties, even in low dimensions. In practise, it is often more instructive to regard k-rank graphs as generalisations of graphs. These manifest themselves as directed graphs with each edge painted one of k different colours, which can be decomposed into a set of squares: subgraphs with four edges which resemble a geometric square. Hazlewood, Raeburn, Sims and Webster demonstrated that, if G is a k-coloured graph with a decomposition into squares such that the squares can be assembled into cubes, then G induces a k-rank graph. In this thesis, we generate two main new infinite families of higher-rank graphs, exploiting this cubical structure in order to do so. In two dimensions, we use a theorem of Vdovina to construct for each complete connected bipartite graph a so-called tile complex which induces two different 2-rank graphs. In higher dimensions, we define a class of groups called domino groups which act freely and transitively on a k-dimensional affine building. The quotient of this action on the building defines a k-dimensional cube complex, which in turn induces a k-rank graph. Perhaps most importantly, to each k-rank graph can be associated a C ?-algebra. These higher-rank graph algebras are extremely versatile, and have arisen in connection with quantum spheres, Yang–Baxter equations, Thompson’s groups, and braid groups. We show that the C ?-algebras corresponding to all of our models of k-rank graphs are separable, nuclear, unital, purely infinite, and simple, and hence that they are determined by their K-groups. Some of our main theorems are dedicated to computations of the K-theory of these infinite families of algebras, and the Kirchberg–Phillips Classification Theorem tells us that our examples are indeed new. Until now, there have been very few explicit computations of the K-theory of such algebras, so these results furnish a large part of this thesis. We also identify some of the relationships between the structure of k-rank graph C ?-algebras and the algebras of lower-rank subgraphs, which we hope will simplify hitherto difficult computations in high dimensions. Each of the tile complexes and domino groups we employ in this thesis can be viewed geometrically as a cube complex, and we examine their topological properties.We compute the cellular homology groups and again show how these can be retrieved from the geometry of lower-rank subgraphs. Thus we introduce a concrete link between the C ?-algebra–based theory of higher-rank graphs and the geometrical theory of affine buildings. We used the computer algebra package MAGMA for many of the computations, and wrote most of the algorithms for constructing the k-rank graphs in Python.