A cubic graph G is cyclically 5-connected if G is simple, 3–connected, has at least 10 vertices and for every set F of edges of size at most four, at most one component of G\F contains circuits. We prove that if G and H are cyclically 5-connected cubic graphs and H topologically contains G, then either G and H are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-con...

Let G be an abelian group. A collection of (G, k, λ) disjoint difference families, {F0,F1, · · · ,Fs−1}, is a complete set of disjoint difference families if ∪0≤i≤s−1∪B∈FiB form a partition of G− {0}. In this paper, several construction methods are provided for complete sets of disjoint difference families. Applications to one-factorizations of complete graphs and to cyclically resolvable cycli...

In this article we will describe a finitely presented subgroup of the group of piecewise projective homeomorphisms of the real projective line. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. It is in fact the first such example which is torsion free. We will also develop a means for representing t...

This article shows that the Cremona group of the plane is compactly presented. To do this we prove that it is a generalised amalgamated product of three of its algebraic subgroups (automorphisms of the plane and Hirzebruch surfaces) divided by one relation.