A new mathematical construction reports a family of countable, torsion-free groups that contradicts the unrestricted Osin–Thom conjecture. For every natural number n, the construction produces a group whose first ℓ2-Betti number is n while its normal rank is 1. In simple terms, the examples make the invariant on the left side of the conjectured bound arbitrarily large while the normal rank remains one.
The result is a disproof of the conjecture in its full torsion-free form, according to the preprint. It does not resolve the version that asks whether the same bound holds for finitely generated torsion-free groups.
The bound under examination
The Osin–Thom conjecture, introduced in 2011, says that the first ℓ2-Betti number of a torsion-free discrete group should be no greater than its normal rank minus one. The conjecture therefore compares a Betti-number invariant with a measure of normal generation, using the latter to set an upper limit on the former.
The reported family breaks that proposed relationship consistently: its normal rank is 1, while its first ℓ2-Betti number is n for each group indexed by a natural number n. The claim concerns an entire constructed family rather than a single example.
A family built from free groups
This was not an experiment and involved no statistical sample, assigned treatment or comparison group. Instead, the authors define each group Γ_n through a directed union of free groups. A directed union assembles a final group from a system of successive stages.
The proof tracks homology through that assembly. For a directed union, the homology of the final group is computed as the directed limit of the homologies at its stages. The argument further establishes that the connecting homomorphism between consecutive stages is injective and concludes that the induced map between those stages is an isomorphism.
To calculate the relevant invariants, the paper expresses the Betti numbers as dimensions of homology over the Linnell ring. It also states that the groups satisfy the Strong Atiyah Conjecture, which in this setting makes the associated ring D_G a division ring. Those algebraic features provide the framework for the calculation through the directed union.
What the construction says about the groups
The groups in the family are locally free and therefore locally indicable. These are structural features of the constructed examples, showing how the groups are organized at the local level. They do not establish that all torsion-free groups share the same properties.
The proof also gives a statement about higher degrees: under the hypotheses of the paper’s Lemma 3.3, the higher Betti numbers vanish for every degree m at least 2. This helps describe the constructed examples, but the counterexample itself comes from the first ℓ2-Betti number and the normal rank.
The unresolved boundary
The central qualification is that the constructed groups are not finitely generated. Because the examples fall outside that class, the preprint does not show that the Osin–Thom conjecture fails for finitely generated torsion-free groups.
The paper leaves open whether the conjecture still holds in the finitely generated class. Its conclusions apply to the specified family of countable torsion-free groups and do not classify all torsion-free groups. The work therefore challenges the unrestricted conjecture without settling the broader question.
The document is a preprint identified as arXiv:2608.25988v1 and dated 26 August 2026; its peer-review status is not reported. The first author reports support from Spanish Ministry grants CEX2023-001347-S and EUR2025-164928, while the second reports support from NSF CAREER award DMS-2552707.
Paper data and sources
Original title: A note on normal generation and the first $\ell^2$-betti number
Authors: Sam P. Fisher, Yash Lodha
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text