Reflections in abstract Coxeter groups

Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take $r$ to a reflection in $W'$. The aim of this paper is to show how to determine, by inspection of the Coxeter graph, the intersection of the maximal finite sugroups containing $r$. In particular we show that the condition above is satisfied whenever $W$ is infinite and irreducible, and has the property that all rank two parabolic subgroups are finite. So in this case all isomorphisms map reflections to reflections.Comment: 25 pages, 0 figure

On the direct indecomposability of infinite irreducible Coxeter groups and the Isomorphism Problem of Coxeter groups

In this paper we prove, without the finite rank assumption, that any irreducible Coxeter group of infinite order is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group, the centralizers of the normal subgroups that are generated by involutions. As a consequence, we show that the problem of deciding whether two general Coxeter groups are isomorphic, as abstract groups, is reduced to the case of irreducible Coxeter groups, without assuming the finiteness of the number of the irreducible components or their ranks. We also give a description of the automorphism group of a general Coxeter group in terms of those of its irreducible components.Comment: 30 page

Local Government Revenue Mobilisation in Anglophone Africa

This paper examines opportunities and constraints facing local revenue mobilisation in anglophone Africa, with an emphasis on urban settings. It discusses specific revenue instruments and their effects on economic efficiency, income distribution and accountability. In particular, it addresses political and administrative constraints facing various revenue instruments and factors affecting citizens’ compliance. The analysis is illustrated with examples from across anglophone Africa. A general conclusion emerging from the study is that local revenues mobilised in most local government authorities in Africa are necessary but not sufficient to develop and supply adequate services for the fast-growing population. On this basis, areas for further research on local government revenue mobilisation in Africa are identifiedDfI