Graham A. Niblo

The geometry of cube complexes and the complexity of their fundamental groups

We investigate the geometry of geodesics in CAT(0) cube complexes. A group which acts cocompactly and properly discontinuously on such a complex is shown to have a biautomatic structure. There is a family of natural subgroups each of which is shown to be rational.

Groups acting on CAT(0) cube complexes

We show that groups satisfying Kazhdans property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.

FROM WALL SPACES TO CAT(0) CUBE COMPLEXES

We explain how to adapt a construction due to M. Sageev in order to construct a proper action of a group on a CAT(0) cube complex starting from a proper action of the group on a wall space.

Hierarchical spectral clustering of power grids

A power transmission system can be represented by a network with nodes and links representing buses and electrical transmission lines, respectively. Each line can be given a weight, representing some electrical property of the line, such as line admittance or average power flow at a given time. We use a hierarchical spectral clustering methodology to reveal the internal connectivity structure of such a network. Spectral clustering uses the eigenvalues and eigenvectors of a matrix associated to the network, it is computationally very efficient, and it works for any choice of weights. When using line admittances, it reveals the static internal connectivity structure of the underlying network, while using power flows highlights islands with minimal power flow disruption, and thus it naturally relates to controlled islanding. Our methodology goes beyond the standard k-means algorithm by instead representing the complete network substructure as a dendrogram. We provide a thorough theoretical justification of the use of spectral clustering in power systems, and we include the results of our methodology for several test systems of small, medium and large size, including a model of the Great Britain transmission network.

Groups acting on cubes and Kazhdan’s property (T)

We show that a group G contains a subgroup K with e(G,K) > 1 if and only if it admits an action on a connected cube that is transitive on the hyperplanes and has no fixed point. As a corollary we deduce that a countable group Gwith such a subgroup does not satisfy Kazhdans property (T).

Benevolent Characteristics Promote Cooperative Behaviour among Humans

Cooperation is fundamental to the evolution of human society. We regularly observe cooperative behaviour in everyday life and in controlled experiments with anonymous people, even though standard economic models predict that they should deviate from the collective interest and act so as to maximise their own individual payoff. However, there is typically heterogeneity across subjects: some may cooperate, while others may not. Since individual factors promoting cooperation could be used by institutions to indirectly prime cooperation, this heterogeneity raises the important question of who these cooperators are. We have conducted a series of experiments to study whether benevolence, defined as a unilateral act of paying a cost to increase the welfare of someone else beyond ones own, is related to cooperation in a subsequent one-shot anonymous Prisoners dilemma. Contrary to the predictions of the widely used inequity aversion models, we find that benevolence does exist and a large majority of people behave this way. We also find benevolence to be correlated with cooperative behaviour. Finally, we show a causal link between benevolence and cooperation: priming people to think positively about benevolent behaviour makes them significantly more cooperative than priming them to think malevolently. Thus benevolent people exist and cooperate more.

Subgroup separability, knot groups and graph manifolds

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. We characterise the Graph Manifolds with subgroup separable fundamental group as precisely the geometric ones, i.e. the Seifert Fibered 3-manifolds and the Sol manifolds, and show that there is a specific non-subgroup separable group which is a subgroup in all other cases.

The singularity obstruction for group splittings

Abstract We study an obstruction to splitting a finitely generated group G as an amalgamated free product or HNN extension over a given subgroup H and show that when the obstruction is “small” G splits over a related subgroup. Applications are given which generalise decomposition theorems from low dimensional topology.

AMENABLE ACTIONS, INVARIANT MEANS AND BOUNDED COHOMOLOGY

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a point our result reduces to a classic theorem of B.E. Johnson characterising amenability of groups. In the case when the compact space is the Stone-\v{C}ech compactification of the group we obtain a cohomological characterisation of exactness for the group, answering a question of Higson.

A cohomological characterisation of Yu's property A for metric spaces

We introduce the notion of an asymptotically invariant mean as a coarse averaging operator for a metric space and show that the existence of such an operator is equivalent to Yu’s property A. As an application we obtain a positive answer to Higson’s question concerning the existence of a cohomological characterisation of property A. Specifically we provide coarse analogues of group cohomology and bounded cohomology (controlled cohomology and asymptotically invari- ant cohomology, respectively) for a metric space X, and provide a cohomological characterisation of property A which generalises the results of Johnson and Ringrose describing amenability in terms of bounded cohomology. These results amplify Guentner’s observation that property A should be viewed as coarse amenability for a metric space. We further provide a generalisation of Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This is used to derive Nowak’s theorem that the union of finite cubes of all dimensions does not have property A.