Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
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Bibliographic record
Abstract
Abstract Given a group automorphism ϕ : Γ → Γ, one has an action of Γ on itself by ϕ -twisted conjugacy, namely, g.x = gxϕ ( g -1). The orbits of this action are called ϕ -twisted conjugacy classes. One says that Γ has the R ∞ -property if there are infinitely many ϕ -twisted conjugacy classes for every automorphism ϕ of Γ. In this paper we show that SL(n; Z) and its congruence subgroups have the R 8 -property. Further we show that any (countable) abelian extension of Γ has the R 8 -property where Γ is a torsion free non-elementary hyperbolic group, or SL( n ; Z); Sp(2 n ; Z) or a principal congruence subgroup of SL( n ; Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.020 | 0.003 |
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 it