The group of automorphisms of a connected locally finite graph is naturally a totally disconnected locally compact topological group, when equipped with the permutation topology. It therefore makes sense to ask for which graphs is the topology not discrete. We show that in case of Cayley graphs of Coxeter groups, one can fully characterize the discrete ones in terms of the symmetries of the corresponding Coxeter system.
Actions on trees are ubiquitous in group theory. The standard approach to describing them is known as Bass–Serre theory, which presents the group acting on the tree as assembled from its vertex and edge stabilizers. However, a different approach emerges if instead of considering vertex and edge stabilizers as a whole, we focus on local actions, that is, the action of a vertex stabilizer only on the immediate neighbours of that vertex. Groups acting on trees defined by their local actions are especially important as a source of examples of simple totally disconnected locally compact groups, with a history going back to a 1970 paper of Tits. I will go through some highlights of this theory and then present some recent joint work with Simon Smith: we develop a counterpart to Bass–Serre theory for local actions, which describes all possible local action structures of group actions on trees.
We provide a new axiomatic framework, inspired by the work of Ol'shanskii, to describe explicitly certain irreducible unitary representations of second-countable non-discrete unimodular totally disconnected locally compact groups. We show that this setup applies to various families of automorphism groups of locally finite semiregular trees and right-angled buildings.
It is well-known that the Galois group of an (infinite) algebraic field extension is a profinite group. When the extension is transcendental, the automorphism group is no longer compact, but has a totally disconnected locally compact structure (TDLC for short). The study of TDLC groups was initiated by van Dantzig in 1936 and then restarted by Willis in 1994. In this talk some of Willis' concepts, such as tidy subgroups, the scale function, flat subgroups and directions are introduced and applied to examples of automorphism groups of transcendental field extensions. It remains unknown whether there exist conditions that a TDLC group must satisfy to be a Galois group. A suggestion of such a condition is made.
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.
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