Gromov’s monster group- notes Probability, Geometry and Groups seminar, Toronto, 01.03.2013
Bibliographic record
Abstract
Our goal is to present the construction of Gromov’s monster group- a finitely generated group which does not embed coarsely into any Hilbert space. This is perhaps the most prominent example of how random objects can be useful in geometric group theory. We don’t provide all the details and proofs here, since some parts of the construction, involving small cancellation theory and hyperbolicity of random groups, are rather involved. More comprehensive references for the topic include: • Gromov’s original paper [Gro03] (contains lots of ideas, but only sketches of proofs) • Arzhantseva and Delzant’s paper fleshing out Gromov’s ideas [AD08] (somewhat hard to read, but contains all ingredients of the construction, including a general approach to graphical small cancellation theory) • Ollivier’s expository paper [Oll03] and references therein (easier to read than the previous two, sketches a different, combinatorial approach to small cancellation in random groups; see also [Oll05]) The main motivation for constructing groups which do not embed coarsely into Hilbert spaces came from looking for counterexamples to the Baum-Connes conjecture, although
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".