Uniform property Γ and the small boundary property
Bibliographic record
Abstract
We prove that, for a free action <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha colon upper G clockwise top semicircle arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi> α </mml:mi> <mml:mo> : </mml:mo> <mml:mi>G</mml:mi> <mml:mo> ↷ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \colon G \curvearrowright X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Cartan subalgebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper C left-parenthesis upper X right-parenthesis subset-of-or-equal-to upper C left-parenthesis upper X right-parenthesis right-normal-factor-semidirect-product Subscript alpha Baseline upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⊆ </mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo> ⋊ </mml:mo> <mml:mi> α </mml:mi> </mml:msub> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(C(X) \subseteq C(X) \rtimes _\alpha G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The reverse implication has been demonstrated by Kerr and Szabó, from which we obtain that these two conditions are equivalent. We moreover show that, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also minimal, then almost finiteness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is implied by tracial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -stability of the subalgebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper C left-parenthesis upper X right-parenthesis subset-of-or-equal-to upper C left-parenthesis upper X right-parenthesis right-normal-factor-semidirect-product Subscript alpha Baseline upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⊆ </mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo> ⋊ </mml:mo> <mml:mi> α </mml:mi> </mml:msub> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(C(X) \subseteq C(X) \rtimes _\alpha G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha colon upper G clockwise top semicircle arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi> α </mml:mi> <mml:mo> : </mml:mo> <mml:mi>G</mml:mi> <mml:mo> ↷ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \colon G \curvearrowright X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta colon upper H clockwise top semicircle arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo> : </mml:mo> <mml:mi>H</mml:mi> <mml:mo> ↷ </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta \colon H \curvearrowright Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".