A non-compact, compactly generated, locally compact group whose proper quotients are all compact is called just-non-compact. Discrete just-non-compact groups are John Wilson’s famous just-infinite groups. In this talk, I’ll describe an ongoing project to use permutation groups to better understand the class of just-non-compact groups that are totally disconnected. An important step for this project has recently been completed: there is now a structure theorem for non-compact tdlc groups G that have a compact open subgroup that is maximal. Using this structure theorem, together with Cheryl Praeger and Csaba Schneider’s recent work on homogeneous cartesian decompositions, one can deduce a neat test for whether the monolith of such a group G is a one-ended group in the class 𝒮 of non-discrete, topologically simple, compactly generated, tdlc groups. This class 𝒮 plays a fundamental role in the structure theory of compactly generated tdlc groups, and few types of groups in 𝒮 are known.
This video was produced by Newcastle University, Australia, as part of the Symmetries in Newcastle seminar series.
