VC-sets and generic compact domination

TL;DR

This paper proves that in certain groups, sets with finite VC-dimension have negligible topological borders, leading to generic compact domination results for definably amenable NIP groups.

Contribution

It establishes that sets with finite VC-dimension have measure-zero borders and extends this to constructible sets under additional conditions, impacting the understanding of NIP groups.

Findings

Sets with finite VC-dimension have Haar measure zero borders.

Under extra conditions, constructible sets also have measure-zero borders.

Results imply generic compact domination for definably amenable NIP groups.

Abstract

Let X be a closed subset of a locally compact second countable group G whose family of translates has finite VC-dimension. We show that the topological border of X has Haar measure 0. Under an extra technical hypothesis, this also holds if X is constructible. We deduce from this generic compact domination for definably amenable NIP groups.