Topological asymptotic dimension

TL;DR

This paper develops a new notion of asymptotic dimension for locally compact groups, extending classical invariants and establishing finiteness results for broad classes of such groups, including solvable and elementary amenable groups.

Contribution

It introduces the concept of asymptotic dimension for locally compact groups and extends classical results like Hirsch length and Malcev's theorems to the topological setting.

Findings

Asymptotic dimension is finite for many residually compact groups.

Polycyclic-by-compact and certain nilpotent groups are residually compact.

Finite Hirsch length implies finite asymptotic dimension for elementary amenable groups.

Abstract

We initiate a study of asymptotic dimension for locally compact groups. This notion extends the existing invariant for discrete groups and is shown to be finite for a large class of residually compact groups. Along the way, the notion of Hirsch length is extended to topological groups and classical results of Hirsch and Malcev are extended using a topological version of the Poincar\'{e} lemma. We show that polycyclic-by-compact groups and compactly generated, topologically virtually nilpotent groups are residually compact, and that compactly generated nilpotent groups are polycyclic-by-compact. We prove that for compactly generated, solvable-by-compact groups the asymptotic dimension is majorized by the Hirsch length, and equality holds for polycyclic-by-compact groups. We extend the class of elementary amenable groups beyond the discrete case and show that topologically elementary amenable groups with finite Hirsch length have finite asymptotic dimension. We prove that a topologically elementary amenable group of finite Hirsch length with no nontrivial locally elliptic normal closed subgroup is solvable-by-compact. Finally, we show that a totally disconnected, locally compact, second countable [SIN]-group has finite asymptotic dimension, if all of its discrete quotients are so.