The technical lemma underlying the 5-dimensional topologicals-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating π1-null immersions of disks. These conjectures are theorems precisely for those fundamental groups (“good groups”) where the π1-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations: (1) extension and (2) direct limit. The finitely generated groups in this class are amenable and no amenable group is known to lie outside this class.
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