Removing weak stability from conditional results for surface, free-by-cyclic, and Baumslag–Solitar groups

Determine whether the conditional approximation statements in Theorems \ref{intro surf the}, \ref{fbc intro}, and \ref{bs intro} can be strengthened from weak stability to full stability; that is, ascertain whether one can approximate a given (ucp) quasi-representation by a genuine representation on the same matrix size without adding an auxiliary block-sum representation, under the respective winding-number vanishing conditions.

Background

The paper proves weak (ucp) stability results for several non-amenable classes of groups—closed surface groups, certain free-by-cyclic groups, and Baumslag–Solitar groups—conditional on natural winding-number vanishing conditions derived from K-theoretic obstructions. Weak stability allows adding an auxiliary representation via block sum.

For 3-manifold groups, the authors show that full (non-weak) stability fails in general (e.g., for Z^3), but they do not know whether the word “weakly” can be removed in their other conditional results. Resolving this would sharpen their approximation theorems.