OsinThom conjecture for finitely generated torsion-free groups

Determine whether the OsinThom inequality \(\beta^{(2)}_1(G)\leqslant n(G)-1\) holds for every finitely generated torsion-free discrete group G.

Background

The paper constructs countable torsion-free locally free groups with normal rank one and arbitrarily large first -Betti number, thereby disproving the unrestricted OsinThom conjecture. Because the counterexamples are not finitely generated, the authors explicitly leave unresolved whether the conjectured inequality remains valid under the additional hypothesis of finite generation. They emphasize that this finitely generated version retains the major consequences motivating the original conjecture and describe it as an important and difficult open problem.

References

Since our groups are not finitely generated, this leaves open the following question:

Does the Osin--Thom conjecture hold for finitely generated torsion-free groups?

The authors wish to emphasize that the addition of the finite generation hypothesis in the Osin--Thom conjecture makes the problem much more interesting. In particular, the consequences towards the group theory and topology problems described above essentially already follow from the finitely generated version. Indeed, the authors believe that the reformulated problem is an important and difficult open problem which deserves the attention of researchers in the area.

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