The cone topology on masures

Masures are generalizations of Bruhat–Tits buildings and the main examples are associated with almost split Kac–Moody groups G over non-Archimedean local fields. In this case, G acts strongly transitively on its corresponding masure Δ as well as on the building at infinity of Δ, which is the twin building associated with G. The aim of this article is twofold: firstly, to introduce and study the cone topology on the twin building at infinity of a masure. It turns out that this topology has various favorable properties that are required in the literature as axioms for a topological twin building. Secondly, by making use of the cone topology, we study strongly transitive actions of a group G on a masure Δ. Under some hypotheses, with respect to the masure and the group action of G, we prove that G acts strongly transitively on Δ if and only if it acts strongly transitively on the twin building at infinity ∂ Δ. Along the way a criterion for strong transitivity is given and the existence and good dynamical properties of strongly regular hyperbolic automorphisms of the masure are proven.

OPTIMAL LINE PACKINGS FROM FINITE GROUP ACTIONS

We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.

Actions of Semitopological Groups

We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups.