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A Guirardel core can fail to be connected. The augmented core repairs that pathology with prescribed one-dimensional connectors rather than filling every missing rectangle.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Define the missing geometry
Route 2 · Work the repair ledger
Route 3 · Test dimension and invariance
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE MISSING GEOMETRY
1. Begin with the gap a core can leave
The Guirardel core is formed by removing light quadrants from a product of two trees. Under important hypotheses it is connected, but in general it may fail to be connected. The augmented core is a controlled enlargement that repairs this problem while preserving convex fibres.
This is not a generic instruction to fill every visual gap. The repair is organised by twice-light rectangles and their special corners.
CHAPTER 2 OF 12 · DEFINE THE MISSING GEOMETRY
2. Define a twice-light rectangle
Take directions delta one in the first tree and delta two in the second. Their product is a quadrant. A rectangle can be twice light when opposite product quadrants associated with its corners are light in the required configuration. The missing rectangle then records how two pieces of the core can face one another without already being joined.
| Object | Role | Do not confuse it with |
|---|---|---|
| light quadrant | region escaped by the core definition | an arbitrary empty rectangle |
| twice-light rectangle | paired light geometry that exposes a gap | any two overlapping quadrants |
| augmented core | core plus prescribed diagonals | the entire convex hull in the product |
CHAPTER 3 OF 12 · WORK THE REPAIR LEDGER
3. Work a square-and-diagonal ledger
Use a unit square R = [a,b] × [c,d] in a product of simplicial trees. Suppose the lower-left and upper-right product quadrants are light in the twice-light configuration, while the other two corners accumulate core points. The open square is not part of the core.
The augmentation inserts the diagonal from (a,d) to (b,c), joining the two relevant corners. Its one-dimensional length in the product path metric is 2 if it is travelled as one unit in each coordinate, while the square area remains absent. The repair connects; it does not add a two-cell.
| Candidate repair | Connectedness | Added two-dimensional area | Preserves intended economy |
|---|---|---|---|
| add nothing | may remain disconnected | 0 | no |
| add full square | connected locally | 1 square unit | no |
| add prescribed diagonal | connected locally | 0 | yes |
Find the distinguished corners
The corners chosen by the construction are not interchangeable. They are determined by how light quadrants meet and by which corner can lie in the closure of the core. Check directions and closures explicitly before drawing a diagonal.
A picture can be misleading near branch points. State the underlying directions in each tree, not only a planar sketch of their product.
CHAPTER 4 OF 12 · WORK THE REPAIR LEDGER
4. Preserve convex fibres
A subset of the product has convex fibres when every slice over a point in either tree is convex in the other tree. The augmented core is designed to remain compatible with this fibrewise geometry.
This connects directly to Core Slices, Convex Fibres and One-Dimensional Compatibility. Test vertical and horizontal slices after every proposed repair; a shortcut that connects components but breaks convex fibres is the wrong object.
CHAPTER 5 OF 12 · WORK THE REPAIR LEDGER
5. Separate connectedness from contractibility
Connectedness says there is a path inside the set between points. Contractibility says the whole set can be continuously shrunk to a point. Guirardel’s construction uses the augmented core to obtain stronger geometric properties, including contractibility and a CAT(0) statement under the induced path metric.
Do not infer those properties merely because a few diagonals were drawn. They depend on the full invariant construction and its hypotheses.
CHAPTER 6 OF 12 · TEST DIMENSION AND INVARIANCE
6. Changed context: fill the square instead
Return to the unit twice-light rectangle. Replace the prescribed diagonal with the whole square. Both changes may connect the same two local pieces. Only the full square adds area, changing covolume and potentially the intersection-number reading.
This transfer check shows why “repair connectedness” is not enough to specify the object. The dimensional cost of the repair matters.
CHAPTER 7 OF 12 · TEST DIMENSION AND INVARIANCE
7. Check invariance under the group action
If one twice-light rectangle is augmented, its group translates must be treated coherently. Adding a single favourite diagonal in one fundamental-domain picture can destroy invariance.
Record the orbit rule and verify that stabilisers do not give contradictory choices. The augmented set should be defined from the actions, not from coordinates chosen by the illustrator.
CHAPTER 8 OF 12 · TEST DIMENSION AND INVARIANCE
8. Know when augmentation is unnecessary
When the core is already connected, augmentation should not invent extra geometry. Guirardel gives conditions under which connectedness follows, including the absence of a shared non-trivial simplicial refinement in the relevant setting.
Begin with the ordinary core and test its connectedness. The augmented core is a remedy for a specific pathology, not the default replacement for every core.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
The unit-square ledger is a combinatorial teaching model. General real-tree products can have non-simplicial geometry, dense branch structure and actions that are not discrete. “Diagonal” describes the prescribed connector inside a twice-light rectangle; it is not permission to triangulate the whole product.
An augmented core can be connected without being the ordinary core. Added diagonals should not be counted as two-dimensional cells when computing the original core’s intersection covolume.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Build an evidence check
Specify the two group actions, directions, light-quadrant test, ordinary core, every twice-light rectangle, selected corners, diagonal orbits and induced slices. Check non-emptiness, invariance, connectedness and convexity of fibres before invoking stronger results.
For a simplicial example, enumerate square orbits and diagonal orbits separately. Verify that removing the added diagonals recovers the ordinary core and that adding full rectangles would change the two-cell count.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Questions readers should ask
Why not take the ordinary convex hull?
CAT(0) convexity in the product is generally too large for the intersection information the core is meant to retain.
Does every empty rectangle need a diagonal?
No. The construction uses the twice-light condition and distinguished corners.
Does augmentation change the intersection number?
The added connectors are one-dimensional; the original intersection number is read from the two-dimensional covolume of the core.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue through verified routes
Begin with the Topology root, Gold Standard of Education and How X Works. Continue through Guirardel Cores, Heavy Quadrants and Intersection Numbers, Compatibility Fans, Irreducible Trees and Length-Function Sums and Compatibility JSJ Trees, Common Refinements and Universal Compatibility.
The primary source is Vincent Guirardel’s Core and intersection number for group actions on trees, especially its sections on twice-light rectangles, connectedness, the augmented core and CAT(0) geometry. The accessible arXiv version exposes the same section structure. For the broader compatibility setting, continue to Guirardel and Levitt’s JSJ decompositions: definitions, existence, uniqueness. The square-and-diagonal ledger is an original teaching example.
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