1,721,018 research outputs found
THE UNIVERSITY OF SOUTHAMPTON NATIONAL CIPHER CHALLENGE
No full textThe National Cipher Challenge is a web based competition aimed at UK high schools. In its first two years it has attracted entries from over 3,000 pupils. Here we examine how and why it was set up, look at some of the problems involved in the operation and ask the question: what next
National Cipher Challenge at twenty
No full textA report on the first twenty years of the schools outreach programme, the National Cipher Challenge
A geometric proof of Stallings' Theorem on groups with more than one end
No full textStallings showed that a finitely generated group which has more than one end splits as an amalgamated free product or an HNN extension over a finite subgroup. Dunwoody gave a new geometric proof of the theorem for the class of almost finitely presented groups, and separately, using somewhat different methods, generalised it to a larger class of splittings. Here we adapt the geometric method to the class of finitely generated groups using Sageev's generalisation of Bass Serre theory concerning group pairs with more than one end, and show that this new proof simultaneously establishes Dunwoody's generalisation
Double coset decompositions of groups
No full textWe show that residually finite or word hyperbolic groups which can be decomposed as a finite union of double cosets of a cyclic subgroup are necessarily virtually cyclic, and apply this result to the study of Frobenius permutation groups. We show that in general finite double coset decompositions of a group can be interpreted as an obstruction to splitting a group as a free product with amalgamation or an HNN extension
A topological splitting theorem for Poincaré duality groups and high dimensional manifolds
No full textWaldhausen's celebrated torus theorem plays a central role in the classification of topological 3-manifolds. It also led to a number of algebraic splitting theorems for discrete groups including Kropholler's algebraic torus theorem for Poincaré duality groups and to the algebraic annulus theorems of Dunwoody/Sageev and Scott/Swarup. Here, in the same spirit, we offer topological and algebraic decomposition theorems in the context of high dimensional aspherical manifolds, providing an algebraic splitting theorem for Poincar\'e duality groups and exploiting Cappell's splitting theory to extract the required topological splittings. As a result we show that for a wide class of manifold pairs with , every, -injective map f factorises up to homotopy as a finite cover of an embedding. As an application of this we show that under certain circumstances the vanishing of the first Betti number for is an obstruction to the existence of such maps
Groups acting on CAT(0) cube complexes
No full textWe show that groups satisfying Kazhdan's 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
Separability properties of free groups and surface groups
No full textAbstractA subset X of a group G is said to be separable if it is closed in the profinite topology. Separable subgroups are very useful in low-dimensional topology and there has been some interest in separable double cosets. A new method for showing that a double coset is separable is introduced and it is used to obtain a short proof of the result of Gitik and Rips, that in a free group every double coset of finitely generated subgroups is separable. In addition it is shown that this property is shared by Fuchsian groups and the fundamental groups of Seifert fibred 3-manifolds
Extended Affine Weyl groups, the Baum-Connes correspondence and Langlands duality
Full text linkIn this paper we consider the Baum-Connes correspondence for the affine and extended affine Weyl groups of a compact connected semisimple Lie group. We show that the Baum-Connes correspondence in this context arises from Langlands duality for the Lie group
Hilbert space compression and exactness of discrete groups
No full textWe show that the Hilbert space compression of any (unbounded) finite-dimensional CAT(0) cube complex is 1 and deduce that any finitely generated group acting properly, co-compactly on a CAT(0) cube complex is exact, and hence has Yu's Property A. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)–T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams
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