The swap conjecture
Examples abound of groups with many, even infinitely many, Nielsen classes of generating sets. The other way round, the authors are unaware of any group with more than one equivalence class with respect to ``swap equivalence'' (``the swap conjecture''), where an ``elementary swap'' changes one generating set to another by changing a single entry. In this paper, these notions are related to relation modules and ``the relation space groups''. The topological proofs of several known related results are given. The swap conjecture is verified for free groups, free abelian groups, surface groups, and for some concrete one-relator groups.
scientific article; zbMATH DE number 168045
scientific article