Levin conjecture for free products of torsion-free groups

Determine whether every torsion-free group G satisfies n(G * G) 2, as closely related to the Levin conjecture.

Background

The paper relates the OsinThom conjecture to several longstanding questions in group theory and topology. Using the formula for the first -Betti number of a free product, the authors explain that the OsinThom inequality would imply n(G * G) 2 for every torsion-free discrete group G. They note that this statement is closely connected with the Levin conjecture and remains unresolved.

References

The Levin conjecture is closely related to the statement that $n(G\star G)\geqslant 2$ for $G$ torsion-free, which remains a longstanding open problem .

— A note on normal generation and the first $\ell^2$-betti number  (2608.25988 - Fisher et al., 26 Aug 2026) in Section 1, Introduction