SEMINAR: Groups and Combinatorics Seminar: Triple factorisations: Geometric and group theoretic approaches

Groups and Combinatorics Seminar

Seyed Hassan Alavi (UWA)

will speak on

Triple factorisations: Geometric and group theoretic approaches

at 1pm in MLR 2 on Tuesday 14th of December

Abstract: Triple factorisations of groups G of the form G=ABA, for subgroups A and B, are fundamental in the study of Lie type groups as well as in geometry. In this talk, we first introduce and develop a general framework for studying triple factorisations, especially nondegenerate ones where G eq AB. We identify two necessary and sufficient conditions to obtain a triple factorisation in terms of the G-actions on the A-cosets and the B-cosets. We more importantly present a rationale for further study of primitive triple factorisations G=ABA in which A is maximal and both A and B are core-free.

Geometrically, triple factorisations correspond to flag-transitive collinearly connected point-line geometries in which `each pair of points lies on at least one line'. Geometries satisfying the dual condition, where each pair of lines meets in at least one point, are called concurrently connected. In this talk, we also introduce new point-line incidence geometries which arise from studying triple factorisations G=ABA and G=BAB of general linear groups G=GL(V) with A parabolic and B either parabolic, or the stabiliser of a decomposition V=V_1 plus V_2. These investigations give rise to various important examples of flag-transitive collinearly (respectively, concurrently) connected spaces. As duality is important in geometry, we also present the conditions under which these rank 2 geometries satisfy one, both or neither of connectivity properties.

This is a practice talk for an upcoming conference.