On Applications of Topological and Combinatorial Methods to the Theory of Groups
Bibliographic record
Abstract
Graphs are topological objects called 1-dimensional CW complexes, and their fundamental groups are free groups.More generally, any group can be represented by a 2-dimensional CW complex, which is a graph with discs glued along the boundaries of closed paths corresponding to relations in the group.These objects can be studied from the topological viewpoint of covering space theory, introduced by John R. Stallings, which allows us to visualize groups and determine their subgroup structure.Alternatively, graphs can be studied from a combinatorial point of view, developed by Ilya Kapovich and Alexei Myasnikov, which provides simple algorithms that answer questions about free groups.We give an exposition of both approaches and demonstrate how they are used to answer questions about subgroups of free groups and free products.i I'd like to thank my supervisor Professor Inna Bumagin for being a fantastic teacher and patiently guiding me through the writing process.The {a, b}-regular graph ∆ * 1 , as described in the proof of Proposition 8.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Cayley graph of the free product Z 2 * Z 2 = a, b | a 2 , b 2 . .The Cayley complex of the free product Z 2 * Z 2 = a, b | a 2 , b 2 .A quotient complex of the Cayley complex of a free product .A 2-dimensional CW complex with fundamental group Z 2 . .A 2-dimensional CW complex with fundamental group Z 2 * Z 2 The universal cover of the complex in Figure 39. . . . . . . . .A disjoint union of 4 copies of S 2 . . . . . . . . . . . . . . . . .The addition of edges and vertices to Figure 42, as outlined in the proof of Theorem 9.7. . . . . . . . . .
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.005 | 0.005 |
| Science and technology studies | 0.002 | 0.009 |
| Scholarly communication | 0.005 | 0.007 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 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".