Local Sylow theory of totally disconnected, locally compact groups

Abstract

Totally disconnected, locally compact (t.d.l.c.) groups are a class of topological groups that occur naturally as automorphism groups of locally finite combinatorial structures, such as graphs or simplicial complexes. Compact totally disconnected groups are known as profinite groups, which can also be characterised as inverse limits of finite groups, and some familiar concepts from finite group theory generalise directly to profinite groups. The inverse limits of finite p-groups are known as pro-p groups, and for these we have a generalisation of Sylow's theorem: given a profinite group G, every pro-p subgroup of G is contained in a maximal pro-p subgroup (a 'p-Sylow subgroup'), and all p-Sylow subgroups of G are conjugate. At the same time, profinite groups play a key role in the general theory of t.d.l.c. groups, because every t.d.l.c. group has an open profinite subgroup, and all such subgroups are commensurable. Thus we can develop a 'local Sylow theory' for t.d.l.c. groups, based on the Sylow subgroups of their open compact subgroups. Starting from an arbitrary t.d.l.c. group G, we produce a new t.d.l.c. group, the 'p-localisation' of G: this is naturally determined by G up to isomorphism, embeds in G with dense image, and has an open pro-p subgroup corresponding to a local Sylow subgroup of G. I will describe the construction and some properties of the p-localisation, illustrating the concepts with the example of the automorphism group of a regular tree of finite degree.