Bibliographic record
Abstract
The main focus of this thesis is the connection between the L²-Betti numbers of RFRS groups and the existence of epimorphisms to ℤ with kernels having certain desirable properties. In particular, we show that if G is a RFRS group of type FPn(ℚ) for some n ⩾ 0, then G has a finite-index subgroup H ⩽ G admitting an epimorphism H → ℤ with kernel of type FPn(ℚ) if and only if bᵢ⁽²⁾(G) = 0 for all i ⩽ n. A consequence is that the fundamental group of any closed hyperbolic manifold with cubulated fundamental group virtually algebraically fibres with kernel of type FP(ℚ). We also prove that if G is a RFRS group of type FP(ℚ) and with cd_ℚ(G) = n, then G admits a virtual map to ℤ with kernel of rational cohomological dimension n - 1 if and only if bn⁽²⁾(G) = 0. In particular, we show that a finitely generated RFRS group of cohomological dimension two is virtually free-by-cyclic if and only if its second L²-Betti number vanishes (we stress that the free kernel of the free-by-cyclic group is finitely generated if and only if the first L²-Betti number vanishes as well). We obtain more general results in the wider class of residually poly-ℤ groups. We also prove analogues of the results in this and the previous paragraph over fields of positive characteristic, where the L²-Betti numbers must be replaced with suitable positive characteristic analogues. Finally, and in a slightly different direction, we show that any group algebra of a torsion-free 3-manifold group embeds into a division ring, and as a consequence show that group algebras of torsion-free 3-manifold groups satisfy Kaplansky's Zero Divisor Conjecture.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".