Local Sylow theory of totally disconnected, locally compact groups (Q2869863)

In the important work [Math. Ann. 300, No. 2, 341--363 (1994; Zbl 0811.22004)], \textit{G. Willis} addressed the notoriously difficult problem of developing a structure theory of general totally disconnected, locally compact groups. More recently, the author has been studying various local aspects of profinite groups.NEWLINENEWLINE\noindent In the paper under review, the author comes up with an interesting local Sylow theory of totally disconnected, locally compact groups; he defines a local \(p\)-Sylow subgroup of such a group \(G\) to be a maximal pro-\(p\) subgroup of an open compact subgroup of \(G\). Using a local \(p\)-Sylow subgroup \(S\), he defines the \(p\)-localization \(G_{(p)}\) of \(G\) to be the topological group formed by giving the commensurator subgroup of \(G\) in \(S\), the coarsest topology so that \(S\) is open. The author discovers useful properties of \(G_{(p)}\) like its embeddability as a dense subgroup of \(G\). These properties are closely connected with the question of how far a \(p\)-Sylow subgroup of an open compact subgroup \(U\) of \(G\) is from being normal in \(U\). The author also proves results relating the modular function as well as the scale function of \(G\) with the corresponding functions for its \(p\)-localizations. In addition, he analyzes the scale functions for groups acting on trees. In the last section he obtains results on pro-solvable groups in terms of local Sylow bases.