Pith. Sign in

RECORD

Strong Rigidity of II$_1$ Factors Arising from Malleable Actions of w-Rigid Groups, I

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Sorin Popa

Operator Algebras · Submitted 21 May 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv IDmath/0305306 (version 13)
Pith IDE47DBJIJ, Pith's permanent record of this paper and its review
CitationSorin Popa. "Strong Rigidity of II$_1$ Factors Arising from Malleable Actions of w-Rigid Groups, I." arXiv:math/0305306 (2003). https://pith.science/paper/E47DBJIJ
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,513 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.16 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v13. Results and the main claim are underlined; .

We consider cross-product II$_1$ factors $M = N\rtimes_{\sigma} G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $\sigma: G \to {\text{\rm Aut}}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable'' and mixing. Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. We use this to calculate the fundamental group $\mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as $G=\Bbb Z^2 \rtimes SL(2, \Bbb Z)$. We deduce that for any countable group $S \subset \Bbb R_+^*$ there exist II$_1$ factors $M$ with $\mycal F(M)=S$, thus bringing new light to a longstanding problem of Murray and von Neumann.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider cross-product II$_1$ factors $M = N\rtimes_{\sigma} G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $\sigma: G \to {\text{\rm Aut}}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable'' and mixing. contribution 0.59 0.23 0.04 asserted 0.73
s1 Examples are the weighted Bernoulli and Bogoliubov shifts. background 0.76 0.16 0.02 asserted 0.96
s2 We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. result 0.94 0.96 0.86 asserted 1.00
s3 We use this to calculate the fundamental group $\mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as $G=\Bbb Z^2 \rtimes SL(2, \Bbb Z)$. result 0.87 0.95 0.11 asserted 1.00
s4 We deduce that for any countable group $S \subset \Bbb R_+^*$ there exist II$_1$ factors $M$ with $\mycal F(M)=S$, thus bringing new light to a longstanding problem of Murray and von Neumann. result 0.64 0.94 0.44 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v13
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We consider cross-product II$_1$ factors $M = N\rtimes_{\sigma} G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $\sigma: G \to {\text{\rm Aut}}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable'' and mixing. Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. We use this to calculate the fundamental group $\mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as $G=\Bbb Z^2 \rtimes SL(2, \Bbb Z)$. We deduce that for any countable group $S \subset \Bbb R_+^*$ there exist II$_1$ factors $M$ with $\mycal F(M)=S$, thus bringing new light to a longstanding problem of Murray and von Neumann.

Human review

How Pith checks papers → · Full audit of this paper →

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.