New Directions in Locally Compact Groups

Pierre-Emmanuel Caprace, Nicolas Monod

• Publisher: Cambridge University Press
• Year: 2018
• Print ISBN: 9781108413121
• eBook ISBN: 9781108351942
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This collection of expository articles by a range of established experts and newer researchers provides an overview of the recent developments in the theory of locally compact groups. It includes introductory articles on totally disconnected locally compact groups, profinite groups, p-adic Lie groups and the metric geometry of locally compact groups. Concrete examples, including groups acting on trees and Neretin groups, are discussed in detail. An outline of the emerging structure theory of locally compact groups beyond the connected case is presented through three complementary approaches: Willis' theory of the scale function, global decompositions by means of subnormal series, and the local approach relying on the structure lattice. An introduction to lattices, invariant random subgroups and L2-invariants, and a brief account of the Burger–Mozes construction of simple lattices are also included. A final chapter collects various problems suggesting future research directions.

Table of contents

• Cover
• Series information
• Title page
• Copyright information
• Table of contents
• Foreword by George Willis
• 1 On the role of totally disconnected groups in the structure of locally compact groups
• 2 Locally compact groups as metric spaces
• 3 A short primer on profinite groups
• 4 Lectures on Lie groups over local fields
• 5 Abstract quotients of profinite groups, after Nikolov and Segal
• 6 Automorphism groups of trees: generalities and prescribed local actions
• 7 Simon Smith’s construction of an uncountable family of simple, totally disconnected, locally compact groups
• 8 The Neretin groups
• 9 The scale function and tidy subgroups
• 10 Contraction groups and the scale
• 11 The Bader–Shalom normal subgroup theorem
• 12 Burger–Mozes’ simple lattices
• 13 A lecture on invariant random subgroups
• 14 L[sup(2)]-Betti number of discrete and non-discrete groups
• 15 Minimal normal closed subgroups in compactly generated tdlc groups
• 16 Elementary totally disconnected locally compact groups, after Wesolek
• 17 The structure lattice of a totally disconnected locally compact group
• 18 The centraliser lattice
• 19 On the quasi-isometric classification of locally compact groups
• 20 Future directions in locally compact groups: a tentative problem list
• Index