The ?2-cohomology of hyperplane complements

We compute the l^2-Betti numbers of the complement of any finite collection of affine hyperplanes in complex n-space. At most one of the l^2-Betti numbers is non-zero. <br/

Infinite groups with fixed point properties

We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\in \mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group.Comment: Version 2: 29 pages. This is the final published version of the articl

3D body scanning in the apparel industry: Do we really know where we are heading?

This paper through - the Diffusion of Innovation Model - shows that while 3D Body Scanning brings some refining and improvements of existing methods, it does not introduce new concepts that depart from traditional retail practices. 3D Body Scanning is - potentially - a powerful way of approaching size and fit in apparel and one that presents novel opportunities. Yet, despite the advantages that this technology offers, and the many initiatives that have taken place, 3D Body Scanning has not reached its full potential and has failed to produce the expected results held by many stakeholders. Stakeholders must increase collaboration to realise 3D Body Scanning’s relative advantage. Much of the potential has, however, been promoted by distinct organisations that are biases about how the diverse processes and structures will work together, whilst focussing on profit from their own incremental IP. In this paper we elicit 3D Body Scanning’s fundamental concepts, and its central goal to provide ‘glue’ needed to create an innovation. We offer further implications for researchers and policymakers about expecting and managing trends in technology. <br

Online virtual fit is not yet fit for purpose: An analysis of fashion e-commerce interfaces

To unify the methodology of Virtual Fit platforms and allowing cross platform integration of 3D Body Scanning, the current Virtual Fit platforms need to be assessed in terms of their size recommendation approach and user interaction. Digital data, interactivity, and internet technology are changing the ways we interact in online shopping, with the Virtual Fit platforms having great potential to increase retail engagement and market share. This will support online purchasing activities while minimising the perceived risk in garment returns due to the poor sizing fit information. Current research has focused on the analysis of computer modelling techniques, avatars, cloth, fabric draping simulations, and customer behaviour / aesthetic impact in the online domain. From a technical perspective, these investigations offer an interesting insight, although do not address issues of implementation or customer attitude. Therefore, to judge the current and potential impact of such technologies, it is important to understand 1) how they are being enacted online, 2) the Interaction Design elements of the user journey, 3) the application (or lack thereof) of mathematical models, and 4) how such interfaces are embedded within websites. Once these four key questions have been answered a greater understanding of how 3D Body Scanning and Technologies integrated into eCommerce and Virtual Fit platforms in the consumer market may be reached. Through analysis of nine leading Virtual Fit platforms, the persona of a single female dress form was used to work through the customer journey. Through this, screen shot data captured along each stage in relation to the four research questions listed above. Following this, the study utilised content analysis structure with NVivo as a qualitative thematic analysis tool. This study found that despite a large number of platforms using virtual fit technology, only a handful companies exist that provide such technology and interfaces; often based upon subjective ‘previous purchases’ rather than scientific prediction. This issue is made more complicated in how subjective measures such as personal perception of one’s body is required (e.g. what size are you), besides body shape; a concept shown to be ‘broken’ and not fit-for-purpose. In addition, many of the technologies use limited and often misinterpreted body measurements, the impact of which is explored in greater detail within the paper. This study contributes to the understanding of the information required from users by virtual fit platforms, and the understanding of the output as presented by virtual fit platforms. The research goal is to contribute to knowledge as a potential guideline for any future projects in virtual fit and to help direct body scanning developments to better support these platforms

Bucolic Complexes

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study various approaches to bucolic complexes: from graph-theoretic and topological perspective, as well as from the point of view of geometric group theory. In particular, we characterize bucolic complexes by some properties of their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several known results are generalized. We also show that locally-finite bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties.Comment: 45 pages, 4 figure

On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.Comment: 35 pages; abridged, revised and update