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Two splittings may share broad elliptic information yet still fail to fit inside one tree. Compatibility asks for an equivariant common refinement, and the compatibility JSJ construction turns that test into a canonical objective.
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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 compatibility
Route 2 · Work the collapse square
Route 3 · Test refinements and spaces
Route 4 · Check limits and proof evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE COMPATIBILITY
1. Compatibility asks for one tree above two trees
Two simplicial G-trees are compatible when they have a common refinement: there is another G-tree that collapses equivariantly onto each of them. The refined tree records both splittings before selected edge orbits are collapsed.
This is stronger than belonging to the same deformation space. Sharing elliptic subgroups describes which subgroups fix points; compatibility asks whether the particular edge information can coexist in one refinement.
CHAPTER 2 OF 12 · DEFINE COMPATIBILITY
2. Distinguish ellipticity, domination and compatibility
A subgroup is elliptic in a tree if it fixes a point. A tree T dominates T′ when every subgroup elliptic in T is elliptic in T′, equivalently when an equivariant map exists under the usual hypotheses. Compatibility requires a common refinement and is symmetric.
The JSJ-trees guide develops universal ellipticity and domination. Neither word can be substituted for compatibility without checking the definition.
| Relation | Direction | Core test |
|---|---|---|
| elliptic subgroup | subgroup relative to a tree | fixes a point |
| domination | directed between trees | elliptic subgroups transfer |
| compatibility | symmetric between trees | a common refinement exists |
CHAPTER 3 OF 12 · WORK THE COLLAPSE SQUARE
3. Work a collapse-square example
Take a refined quotient graph shaped as a path with two edge orbits, e₁ and e₂. Collapsing e₂ leaves a one-edge splitting T₁; collapsing e₁ leaves another one-edge splitting T₂. The original two-edge tree R maps onto both, so T₁ and T₂ are compatible.
The point is structural, not pictorial. The collapses must be G-equivariant, and the refined vertex and edge stabilisers must define valid actions. Drawing two abstract edges in one picture does not establish a common refinement.
| Tree | Edge orbit retained | Map from R |
|---|---|---|
| T₁ | e₁ | collapse every translate of e₂ |
| T₂ | e₂ | collapse every translate of e₁ |
| R | e₁ and e₂ | identity |
CHAPTER 4 OF 12 · WORK THE COLLAPSE SQUARE
4. Common refinement is not just common subdivision
Subdividing an edge inserts valence-two vertices without adding a new splitting. A common refinement may add an edge orbit that separates vertex-group information so that different collapses recover different trees. It is an operation in the category of G-trees, not merely a finer drawing.
Likewise, taking a product of trees does not automatically give a tree. Compatibility is the special condition that the relevant combined information can still be represented by a one-dimensional tree action.
CHAPTER 5 OF 12 · WORK THE COLLAPSE SQUARE
5. Build the compatibility JSJ objective
A universally compatible tree is compatible with every tree over the allowed class of edge groups. A compatibility JSJ tree seeks a maximal object for this property, analogous in spirit—but not identical—to the usual JSJ objective based on universal ellipticity and domination.
For finitely presented groups under the general framework of Guirardel and Levitt, the compatibility JSJ space exists and yields a preferred compatibility tree. State the group, the allowed family A and the action conventions before invoking the result.
CHAPTER 6 OF 12 · TEST REFINEMENTS AND SPACES
6. Changed context: add a crossing split
Start from the collapse square above. Now propose a third splitting whose edge separates pieces that e₁ and e₂ force to lie on opposite sides in incompatible ways. If no G-tree can carry all required edge partitions, there is no common refinement with R.
The transfer check is not “add a third edge to the picture”. One must test the algebraic and action constraints. Pairwise compatibility can also be subtler than the existence of one simultaneous refinement, so the exact theorem in use matters.
CHAPTER 7 OF 12 · TEST REFINEMENTS AND SPACES
7. Relate compatibility to deformation spaces
Trees in one deformation space have the same elliptic subgroups and can be connected by elementary moves under suitable hypotheses. The deformation-spaces guide shows why a JSJ object may naturally be a family rather than a single graph of groups.
Compatibility provides another selection principle. A preferred compatible tree can be automorphism-invariant even when many ordinary JSJ representatives exist. That conclusion comes from a construction and hypotheses, not from choosing the most symmetric drawing.
CHAPTER 8 OF 12 · TEST REFINEMENTS AND SPACES
8. Use trees of cylinders carefully
Grouping edges by an admissible equivalence relation on stabilisers can produce a tree of cylinders that depends only on a deformation space. In favourable settings it gives a canonical representative with strong compatibility properties. The trees-of-cylinders guide develops that construction.
Universal compatibility is not automatic for every equivalence relation or group class. Cylinder diameter and stabiliser intersections may be needed for acylindrical control, as the acylindricity guide explains.
CHAPTER 9 OF 12 · CHECK LIMITS AND PROOF EVIDENCE
9. Limits and misconceptions
Compatible does not mean isomorphic, in the same deformation space, or mutually dominating. Canonical does not mean unique as an unmarked abstract graph; it refers to the equivariant object and the relevant automorphism action. A common refinement must respect the allowed edge groups.
The collapse-square example suppresses stabiliser labels to isolate the logic. In a real proof those labels are decisive. Results for finitely presented groups, CSA groups or relatively hyperbolic groups should not be transferred to arbitrary groups without checking assumptions.
CHAPTER 10 OF 12 · CHECK LIMITS AND PROOF EVIDENCE
10. Run a proof audit
Write down G, the family A, minimality conventions and whether inversions are excluded. For each proposed tree, list elliptic subgroups and edge stabilisers. Exhibit the refinement and both collapse maps, or identify the obstruction preventing one.
When claiming universality, quantify over all A-trees, not merely the familiar examples. When claiming canonicity, state which automorphisms preserve A. Separate the existence theorem from any later identification of the preferred tree with a tree of cylinders.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Questions readers should ask
Are any two JSJ trees compatible?
Not merely because they are JSJ representatives. Compatibility requires a common refinement; the usual JSJ deformation space is organised by ellipticity and domination.
Does common refinement mean one tree contains the others?
It means there are equivariant collapse maps from the refining tree. Literal subgraph containment is not the definition.
Why seek universal compatibility?
It can select a canonical tree that interacts consistently with every allowed splitting, rather than only a deformation space of alternatives.
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 the JSJ, deformation-space, trees-of-cylinders and acylindricity guides linked above, then Slide Moves, Strongly Slide-Free Trees and Rigidity for a different route from a deformation space to a preferred representative.
Primary sources include Guirardel and Levitt’s JSJ decompositions of groups, their paper on compatibility and acylindricity, and Deformation spaces of trees. The collapse-square and crossing-split examples are original teaching models.
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