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A deformation space may contain several reduced trees with the same elliptic subgroups. Slide moves explain one way representatives can change without leaving that space. A strongly slide-free condition blocks the ambiguous slides that matter for Forester’s rigidity result.
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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 · Name the move and data
Route 2 · Work the slide criterion
Route 3 · Read rigidity correctly
Route 4 · Check hypotheses and limits
Route 5 · Ask and continue
CHAPTER 1 OF 12 · NAME THE MOVE AND DATA
1. A slide changes attachment, not the elliptic data
In Bass–Serre theory, a group G acts on a tree without inversions. Vertices and edges have stabilizer subgroups, and the quotient can be read as a graph of groups. A slide move changes where one edge orbit is attached by moving it across another compatible edge orbit.
The previous guide on deformation spaces of trees explains collapses, expansions and shared elliptic subgroups. A slide is a more focused way that reduced representatives can move inside such a space.
CHAPTER 2 OF 12 · NAME THE MOVE AND DATA
2. The inclusion test makes a slide legal
Suppose oriented edges e and f begin at the same vertex v. In a standard slide configuration, the stabilizer of e must be contained in the stabilizer of f, after choosing the relevant representatives. That inclusion lets the attaching data for e pass across f without breaking equivariance.
The statement is about subgroups and orbits, not a drawing in which one line happens to pass another. Labels must include the vertex and edge groups and the monomorphisms. An unlabeled quotient graph cannot decide legality.
| Object | Data to retain |
|---|---|
| vertex v | stabilizer Gv |
| edges e,f | stabilizers Ge, Gf |
| incidence | chosen origins and edge orbits |
| slide test | Ge contained in Gf up to the stated action |
CHAPTER 3 OF 12 · WORK THE SLIDE CRITERION
3. Work a stabilizer-inclusion example
Use an invented quotient picture with a vertex stabilizer V and two incident edge stabilizers E = 2Z and F = Z, embedded in V so that E is contained in F. The inclusion permits the E-edge orbit to slide across the F-edge orbit in the indicated orientation.
Reverse the labels: if E = Z and F = 2Z, then E is not contained in F. The same-looking drawn move is not licensed by that inclusion. Changing an arrow or choosing conjugate representatives can also change which explicit inclusion must be checked.
This example illustrates the criterion only. It does not specify a full group presentation or prove that every global action with those symbols is a valid reduced G-tree.
CHAPTER 4 OF 12 · WORK THE SLIDE CRITERION
4. Slides explain non-unique reduced representatives
Collapsible edges can be removed, so a reduced tree has no permitted elementary collapse of the relevant kind. Reduced does not always mean unique. A slide can change attachment while leaving the tree reduced and within the same deformation space.
Thus “we reduced the graph of groups” is not yet a canonicality proof. One must know which moves remain and which hypotheses block them. Deformation equivalence preserves selected group-action information, not every combinatorial incidence.
CHAPTER 5 OF 12 · WORK THE SLIDE CRITERION
5. Changed context: remove the inclusion
Return to the example and replace F by 3Z while E remains 2Z inside Z. Neither 2Z is contained in 3Z nor 3Z in 2Z. The proposed slide loses its subgroup-inclusion certificate even though both edge groups are infinite cyclic.
The transfer check shows why rank or isomorphism type is insufficient. Two edge stabilizers can both look like Z abstractly while their images in the vertex group are not nested.
CHAPTER 6 OF 12 · READ RIGIDITY CORRECTLY
6. Strongly slide-free is an orbit condition
Forester defines a strongly slide-free G-tree so that when two edges e1 and e2 have the same origin v and Ge1 is contained in Ge2, the edges lie in the same orbit under Gv. The condition does not merely say “no slide was drawn.” It says any potential nested stabilizers at a vertex are already equivalent in the required local sense.
Check the action, stabilizers and vertex-group orbits. A quotient picture can hide multiple edge representatives or make distinct local configurations appear identical.
CHAPTER 7 OF 12 · READ RIGIDITY CORRECTLY
7. Rigidity identifies the reduced representative
Forester’s rigidity theorem states, under its hypotheses, that a strongly slide-free tree is the unique reduced tree in its deformation space, up to the appropriate G-equivariant isomorphism. Guirardel gives a short proof and describes the role of maximal elliptic subgroups.
Read each qualifier. The claim is not that every G-tree is rigid, every deformation space is a point, or every graph of groups has a unique drawing. It identifies a reduced representative when the strong slide-free condition and the theorem’s action assumptions hold.
CHAPTER 8 OF 12 · READ RIGIDITY CORRECTLY
8. Canonical does not mean label-free
A rigid tree can support an automorphism-invariant splitting when automorphisms preserve the relevant elliptic data, which is important for JSJ-type applications. But an informal relabelling of quotient vertices is not the same as a G-equivariant isomorphism.
State what equivalence is being used: equality, graph-of-groups isomorphism, G-tree isomorphism or membership in one deformation space. Different questions have different uniqueness claims.
CHAPTER 9 OF 12 · CHECK HYPOTHESES AND LIMITS
9. Limits and misconceptions
Do not confuse this rigidity with geometric rigidity of frameworks or Mostow rigidity. The object here is a group action on a simplicial tree and uniqueness inside a deformation space.
Do not infer strongly slide-free from “no obvious slide.” Stabilizer containment can be hidden by conjugacy and orbit choice. Conversely, an edge motion that looks like a slide may fail the algebraic inclusion.
The cyclic-subgroup example is a local teaching model. A proof about a specific splitting must verify minimality, reduction, action conventions and all theorem hypotheses globally.
CHAPTER 10 OF 12 · CHECK HYPOTHESES AND LIMITS
10. A practical proof check
List vertex and edge orbits, choose representatives and record stabilizers with their inclusions into vertex stabilizers. Verify reduction first. For every pair of incident representative edges with nested stabilizers, test whether the vertex stabilizer carries one edge to the other.
Then identify the deformation space through elliptic subgroups or an explicit elementary deformation, as appropriate. Only after these checks invoke rigidity. Keep the map or orbit argument that establishes the claimed G-isomorphism.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Questions readers should ask
Does every deformation between reduced trees use slides?
Not without extra hypotheses. Collapse and expansion describe general elementary deformation; slide descriptions require the applicable setting.
Is a strongly slide-free tree literally the only tree in the deformation space?
The theorem identifies the unique reduced representative up to G-isomorphism under its hypotheses. Non-reduced expansions can still represent the same elliptic data.
Why inspect stabilizer images rather than names?
Because containment is inside the vertex stabilizer. Abstractly isomorphic edge groups can have non-nested images and permit different moves.
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. Build the branch through Stallings Splittings and Bass–Serre Trees, Accessibility and Reduced Graphs of Groups and Deformation Spaces of Trees.
Use Forester’s primary paper Deformation and Rigidity of Simplicial Group Actions on Trees and Guirardel’s A Very Short Proof of Forester’s Rigidity Result. They support the definitions and theorem; the integer-subgroup illustration is an original local example with stated limits.
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