Good Groups

Abstract

Good groups are defined in terms of whether capped gropes of height 1.5 contain certain types of immersed discs. The disc embedding theorem holds for 4-manifolds with good fundamental group. It is proven that the infinite cyclic group and finite groups are good, and that extensions and colimits of good groups are good. This shows that all elementary amenable groups are good. The proofs use grope height raising and contraction, together with an analysis of how fundamental group elements behave under these operations. A central open problem in the study of topological 4-manifolds is to determine precisely which groups are good.