Osin–Thom conjecture on the first ℓ²-Betti number and normal rank

Osin and Thom conjectured that every torsion-free discrete group satisfies the upper bound $\beta^{(2)}_1(G)\leq\operatorname{nrk}(G)-1$. The conjecture was posed in their 2011 preprint and has implications for related normal-generation questions.

Known results

• Osin and Thom (2011): proved the bound for finitely generated groups that are limits of left-orderable amenable groups, via $\beta^{(2)}_1(G)\leq\beta_1(G)-1\leq\operatorname{nrk}(G)-1$.
• Osin and Thom (2011): constructed torsion examples with large $\beta^{(2)}_1$, but explicitly left the torsion-free conjecture open.
• Jaikin-Zapirain (2014): established related inequalities for infinite finitely presented residually $p$-finite groups with prescribed normal generators and finite orders.

August 2026 claimed counterexample

A note titled A note on normal generation and the first $\ell^2$-Betti number claims that, for every natural number $n$, there is a countable torsion-free group with $\beta^{(2)}_1(G)=n$ and $\operatorname{nrk}(G)=1$, contradicting the conjectured bound. The claim is unrefereed and has not been independently verified.

Current status (as of August 2026): the general conjecture has a direct but unverified claimed counterexample; the finitely generated restricted cases remain established, while no verified refutation or proof of the full conjecture is recorded.