Metric geometry of locally compact groups

Abstract

This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study "Infinite groups as geometric objects", as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can favorably be played for locally compact groups. The development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. Word metrics for compactly generated groups play a major role. In the particular case of finitely generated groups, they were introduced by Dehn around 1910 in connection with the Word Problem. Some of the results exposed concern general locally compact groups, such as criteria for the existence of compatible metrics on locally compact groups. Other results concern special classes of groups, for example those mapping onto the group of integers (the Bieri-Strebel splitting theorem for locally compact groups). Prior to their applications to groups, the basic notions of coarse and large-scale geometry are developed in the general framework of metric spaces. Coarse geometry is that part of geometry concerning properties of metric spaces that can be formulated in terms of large distances only. In particular coarse connectedness, coarse simple connectedness, metric coarse equivalences, and quasi-isometries of metric spaces are given special attention. The final chapters are devoted to the more restricted class of compactly presented groups, generalizing finitely presented groups to the locally compact setting. They can indeed be characterized as those compactly generated locally compact groups that are coarsely simply connected.

Book in preparation (233p). Comments and corrections welcome! an index was added in v3

References in corpus (7)

• Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups
• Asymptotic dimension in Bedlewo
• Graphs, permutations and topological groups
• Large scale geometry of metrisable groups
• Coarse classification of abelian groups and amenable shift-homogeneous metric spaces
• Zooming in on the large-scale geometry of locally compact groups
• Large scale geometry of automorphism groups

Cited by in corpus (45)

• Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces
• Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups
• Amenability and uniform Roe algebras
• Low dimensional properties of uniform Roe algebras
• Approximate lattices
• Locally compact wreath products
• Characterizing a vertex-transitive graph by a large ball
• Rational discrete cohomology for totally disconnected locally compact groups
• Vanishing of ${\ell }^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence
• Finiteness properties of totally disconnected locally compact groups
• Amenability of coarse spaces and K-algebras
• Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings
• Measure equivalence classification of transvection-free right-angled Artin groups
• Arens-Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type, and homological epimorphisms
• From homogeneous metric spaces to Lie groups
• Superrigidity of actions on finite rank median spaces
• Dense normal subgroups and chief factors in locally compact groups
• Strong distortion in transformation groups
• Borel density for approximate lattices
• Discrete and free two-generated subgroups of $\mathrm{S}{\mathrm{L}}_2$ over non-archimedean local fields
• The Burnside problem for locally compact groups
• Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs
• Bounding the covolume of lattices in products
• The Haagerup property is not invariant under quasi-isometry
• Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces
• Functional boundedness of balleans
• Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups
• Linear repetitivity beyond abelian groups
• Automorphisms of contact graphs of $\mathrm{CAT}(0)$ cube complexes
• Densely related groups
• Exactness of locally compact groups
• Model geometries of finitely generated groups
• Isomorphisms of Poisson systems over locally compact groups
• Intersection spaces and multiple transverse recurrence
• Decomposable Fourier Multipliers and an Operator-Algebraic Characterization of Amenability
• Growing trees from compact subgroups
• A Szemerédi type theorem for sets of positive density in approximate lattices
• A class of well-founded totally disconnected locally compact groups
• Highly irregular separated nets
• Groups acting faithfully on trees and properly on products of trees
• Fundamental groups and group presentations with bounded relator lengths
• A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces
• On profinite subgroups of an algebraic group over a local field
• Notes on unitary theta representations of compact groups
• Simplicity of crossed products of the actions of totally disconnected locally compact groups on their boundaries