Mark A. Ronan

Topological groups of Kac--Moody type, right-angled twinnings and their lattices

We construct new groups satisfying the combinatorial axioms of twin root data, by amalgamating spherical parabolic subgroups. The corresponding buildings have right-angled Coxeter groups as Weyl groups. The possibility of mixing several ground fields, not available for Kac--Moody groups, leads to strong non-linearity properties for the groups and some of their subgroups. We also discuss a completion procedure for groups with twin root data which is the starting point for defining a large family of topologically simple groups.

Universal presheaves on group geometries, and modular representations

Abstract We extend the ideas of sheaf homology on buildings (M. A. Ronan and S. D. Smith, J. Algebra 96 (1985), 319–346) to more general geometries with transitive automorphism groups. The main technical result is the construction of a universal extension of certain partial sheaves. This guarantees a rich supply of sheaves on such geometries. Computation of zero-homology then affords representations of the group. We consider various applications of the technique, for example to modular irreducibles for sporadic simple groups.

Twin trees II: Local structure and a universal construction

We investigate the local structure of twin tress, and produce for any semihomogeneous treeT a “universal twin” that contains, in a natural way, all trees twinned withT. Our methods allow us to show that there are uncountably many isomorphism classes of twinnings involving any semihomogeneous tree.

A note on the 3D4(q) generalized hexagons

Abstract The 3 D 4 ( q ) generalized hexagons are characterized by means of the regulus condition.

Semi-regular graph automorphisms and generalized quadrangles☆

Abstract We use topological techniques to verify the existence of non-trivial semi-regular automorphisms of certain graphs, and use this to obtain a characterization of certain families of Moufang quadrangles.