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Two Bass–Serre trees can draw the same group splitting information in visibly different ways. Deformation spaces organise that non-uniqueness: collapse and expansion moves change the representative while preserving which subgroups remain elliptic.
Open the chapter contents · Topology root · Gold Standard of Education · How X Works
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 invariant
Route 2 · Work collapse and expansion
Route 3 · Compare representatives
Route 4 · Test hypotheses and limits
Route 5 · Check and continue
CHAPTER 1 OF 12 · NAME THE INVARIANT
1. A splitting is represented by a group action on a tree
Bass–Serre theory connects a graph of groups with an action of a group G on a simplicial tree T. Vertex and edge stabilisers record the algebra. Different quotient graphs or trees can represent closely related splitting information, so a drawing is not automatically a canonical object.
A subgroup H of G is elliptic in T if it fixes a point. The collection of elliptic subgroups records which parts of G remain confined to vertices. Deformation-space theory uses this collection as the central invariant.
| Object | What may change | What a deformation preserves |
|---|---|---|
| Quotient graph | vertices and edges | elliptic subgroups |
| G-tree | representative geometry | fixed-point family |
| Graph-of-groups presentation | visible decomposition | group G and allowed moves |
CHAPTER 2 OF 12 · NAME THE INVARIANT
2. Collapse and expansion are inverse elementary moves
For a non-loop edge in a graph of groups, suppose the edge group maps isomorphically onto one incident vertex group. That edge is collapsible. Collapsing its orbit in the Bass–Serre tree removes redundant structure without changing which subgroups are elliptic. Expansion is the inverse operation.
An elementary deformation is a finite sequence of collapses and expansions. Forester’s formulation shows, under the relevant cocompact simplicial setting, that trees related this way share the appropriate elliptic-subgroup data, and conversely that equal elliptic subgroups characterise the same deformation relation.
The word “collapse” is restricted. Contracting an arbitrary edge whose inclusion is proper can change the splitting information and is not the same elementary move.
CHAPTER 3 OF 12 · WORK COLLAPSE AND EXPANSION
3. Work the redundant amalgam A star-C C
Let C be a subgroup of A. Build a graph of groups with vertex groups A and C and one edge group C. Map the edge group into A by the given inclusion and into the C vertex by the identity. Its fundamental group is the amalgam A ∗C C, which is naturally isomorphic to A.
Because the edge-to-C map is an isomorphism, the edge is collapsible. After collapse, one vertex labelled A remains. The subgroup C has not disappeared from the group: it sits inside A and remains elliptic because A fixes the remaining vertex.
For a concrete instance, take A = Z and C = 2Z. The expanded graph displays Z and 2Z as two vertices joined by an edge 2Z; collapse recovers the one-vertex Z tree.
CHAPTER 4 OF 12 · WORK COLLAPSE AND EXPANSION
4. Verify the invariant before and after collapse
Before collapse, conjugates of A and C fix vertices. Because C is already a subgroup of A, every subgroup elliptic in the C vertex is elliptic in the A vertex as well. After collapse, conjugates of A fix the single vertex, so the same relevant subgroups remain elliptic.
The quotient graph loses one edge and one vertex, but that count was redundant. This is why an accessibility theorem counts reduced decompositions: expansion could otherwise insert arbitrarily many collapsible pieces.
| Check | Expanded representative | Collapsed representative |
|---|---|---|
| Fundamental group | A ∗C C ≅ A | A |
| C elliptic? | yes, fixes its vertex | yes, contained in A |
| Edge orbit | one collapsible orbit | none |
CHAPTER 5 OF 12 · WORK COLLAPSE AND EXPANSION
5. Changed context: make both inclusions proper
Now replace the identity inclusion at the second vertex by a proper embedding into a larger group B. The graph represents A ∗C B. Neither incident map is an isomorphism, so the edge is not collapsible by the elementary rule.
Contracting it anyway would replace visible vertex stabilisers A and B with a larger object and can change which subgroups fix points. That would move outside the original deformation space.
This transfer check distinguishes a topological-looking contraction from an allowed Bass–Serre collapse. Always inspect stabiliser maps before simplifying a quotient graph.
CHAPTER 6 OF 12 · COMPARE REPRESENTATIVES
6. Reduced trees remove collapsible redundancy
A G-tree is reduced when it has no allowable collapsible edge orbit under the chosen definition. Reduction selects leaner representatives, but it does not always select one unique representative. Different reduced trees can lie in the same deformation space.
Slide moves and related transformations can rearrange incident edge data while preserving elliptic subgroups. Some such moves can be expressed through expansions followed by collapses. Therefore “both reduced” does not imply “isomorphic.”
Uniqueness needs extra rigidity hypotheses. Forester’s strongly slide-free condition provides one route to uniqueness of a reduced representative within a deformation space; it is not automatic for every action.
CHAPTER 7 OF 12 · COMPARE REPRESENTATIVES
7. Equal groups do not imply equal deformation spaces
Two trees can have fundamental group G and still make different subgroups elliptic. Sharing the ambient group is far too weak. Even using the same class of edge groups does not force the same vertex stabilisers or fixed-point family.
To compare representatives, list elliptic vertex stabilisers up to conjugacy and then ask whether every subgroup fixing a point in one tree fixes a point in the other. Length functions and domination relations provide other characterisations under appropriate hypotheses.
A deformation space is therefore not “all splittings of G.” It is a structured family with a specified invariant.
CHAPTER 8 OF 12 · COMPARE REPRESENTATIVES
8. Deformation space is not the same as topological isotopy
The term deformation may suggest continuously bending one embedded tree. That is not the meaning here. The objects are group actions on trees, and elementary moves change combinatorial and stabiliser data in a controlled algebraic way.
Guirardel and Levitt study topology on deformation spaces and prove contractibility results for these complexes, but the elementary definition still begins with elliptic subgroups and moves. Do not import intuition from isotopy of curves or homeomorphism of underlying graphs without checking the action.
Likewise, a graph with the same unlabelled shape can represent a different splitting when its group labels and monomorphisms change.
CHAPTER 9 OF 12 · TEST HYPOTHESES AND LIMITS
9. What deformation equivalence does not establish
Being in one deformation space does not make two quotient graphs identical, equally convenient or canonical. It does not say every edge group has the same embedding, and it does not select a preferred JSJ tree without further structure.
The standard equivalences use hypotheses such as simplicial actions and, for finite move sequences in the common formulation, cocompactness. Moving to non-simplicial real trees, non-cocompact actions or different edge classes requires the matching theory.
Nor does contractibility mean there is one vertex in the deformation space. It means the space can be continuously contracted in its defined topology, not that all representatives are isomorphic.
CHAPTER 10 OF 12 · TEST HYPOTHESES AND LIMITS
10. A practical comparison checklist
Record the G-action, quotient graph of groups, stabilisers and edge monomorphisms. Check minimality and cocompactness assumptions. For every proposed collapse, verify that the edge is not an inverted loop and that the relevant inclusion is an isomorphism.
After each move, reconstruct the fundamental group and compare elliptic subgroups. Mark which edge orbits were collapsed rather than editing only the finite drawing. Reduce both representatives before making complexity comparisons.
If claiming uniqueness, name the rigidity hypothesis and test it. If claiming two splittings are different, show an elliptic subgroup, length or domination property that distinguishes them.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions readers should ask
Does collapse change the group?
An allowed elementary collapse preserves the represented group and elliptic-subgroup family. An arbitrary graph contraction need not.
Are all reduced trees in a deformation space isomorphic?
No. Additional rigidity conditions are required for uniqueness.
Is a deformation space all Bass–Serre trees for G?
No. Its trees share the same elliptic subgroups, equivalently the specified deformation relation under the standard hypotheses.
Why retain edge monomorphisms in a diagram?
Because collapsibility and the fundamental group depend on the embeddings, not only on group names or graph shape.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Continue through the verified topology ecosystem
Begin with the Topology root, the Gold Standard of Education and the How X Works hub. Build the prerequisite chain through Stallings Splittings, Finite Edge Groups and Bass–Serre Trees and Accessibility, Reduced Graphs of Groups and Repeated Finite-Edge Splittings. For the surrounding invariant, revisit Ends of Groups, Cayley Graphs and the Zero–One–Two-or-Infinite Alternative.
For primary technical reading, use Max Forester’s Deformation and Rigidity of Simplicial Group Actions on Trees and Vincent Guirardel and Gilbert Levitt’s Deformation Spaces of Trees. The first develops collapse-and-expansion characterisations and rigidity; the second studies common structure and topology across a deformation space. Apply their precise hypotheses rather than extending the statements to every tree action.
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