arXiv subjects

Samuel B. G. Webster

Publications and source records attributed to Samuel B. G. Webster.

7 recordsLinked to original sources

arXiv2020

Computing the fundamental group of a higher-rank graph

We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about $k$-graphs when we show that the abelianisation of fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

math.DS↗

arXiv2015

von Neuman algebras of strongly connected higher-rank graphs

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$_\infty$. 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.

math.OA↗

arXiv2014

Fractal dual substitution tilings

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 the Čech cohomology of a (not necessarily border-forcing) tiling using a graph iterated function system of a fractal tiling.

math.DS↗

arXiv2012

The path space of a higher-rank graph

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $Λ$. We identify the boundary-path space $\partialΛ$ as the spectrum of a commutative $C^*$-subalgebra $D_Λ$ of $C^*(Λ)$. Then, using a construction similar to that of Farthing, we construct a finitely aligned $k$-graph $\wtΛ$ with no sources in which $Λ$ is embedded, and show that $\partialΛ$ is homeomorphic to a subset of $\partial\wtΛ$ . We show that when $Λ$ is row-finite, we can identify $C^*(Λ)$ with a full corner of $C^*(\wtΛ)$, and deduce that $D_Λ$ is isomorphic to a corner of $D_{\wtΛ}$. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.

math.OA↗

arXiv2011

On some fundamental results about higher-rank graphs and their C*-algebras

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.

math.CO↗

arXiv2011

The Path Space of a Directed Graph

We construct a locally compact Hausdorff topology on the path space of a directed graph $E$, and identify its boundary-path space $\partial E$ as the spectrum of a commutative $C^*$-subalgebra $D_E$ of $C^*(E)$. We then show that $\partial E$ 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.

math.OA↗

arXiv2010

A direct approach to co-universal algebras associated to directed graphs

We prove directly that if E is a directed graph in which every cycle has an entrance, then there exists a C*-algebra which is co-universal for Toeplitz-Cuntz-Krieger E-families. In particular, our proof does not invoke ideal-structure theory for graph algebras, nor does it involve use of the gauge action or its fixed point algebra.

math.OA↗