On the quasi-isometric classification of locally compact groups

Abstract

This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous negatively curved manifolds. A main conjecture provides a general description; an extended discussion reduces this conjecture to more specific statements. In the course of the paper, we provide statements of quasi-isometric rigidity for general symmetric spaces of noncompact type and also discuss accessibility issues in the realm of compactly generated locally compact groups.

51 pages, 2 figures. v1->v2: significantly expanded version. The title has been changed; v2->v3: added Sections 4I and 5B [now 19.4.9 and 19.5.2]; v3->v4: renumbering of sections and theorems so as to fit with published version

References in corpus (6)

• Metric geometry of locally compact groups
• Dimension of asymptotic cones of Lie groups
• Contracting automorphisms and L^p-cohomology in degree one
• A note on spaces of asymptotic dimension one
• Commability and focal locally compact groups
• Vanishing of the first reduced cohomology with values in an $L^p$-representation

Cited by in corpus (9)

• Metric geometry of locally compact groups
• Commability and focal locally compact groups
• Rational discrete first degree cohomology for totally disconnected locally compact groups
• Cone-equivalent nilpotent groups with different Dehn functions
• Sublinear quasiconformality and the large-scale geometry of Heintze groups
• Large-scale sublinearly Lipschitz geometry of hyperbolic spaces
• Zooming in on the large-scale geometry of locally compact groups
• On quasi-isometric nilpotent Lie groups
• Local-to-Global-rigidity of lattices in $S{L}_n(\mathbb{K})$