Topological asymptotic dimension

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.