Culler–Vogtmann Outer space
E634851
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical object
ⓘ
parameter space ⓘ topological space ⓘ |
| actionType | properly discontinuous action of Out(F_n) ⓘ |
| alsoKnownAs | Outer space NERFINISHED ⓘ |
| analogueOf | Teichmüller space NERFINISHED ⓘ |
| analogyWith | Teichmüller space of a surface NERFINISHED ⓘ |
| baseGraph | rose with n petals ⓘ |
| boundaryConcept | space of very small F_n-actions on R-trees ⓘ |
| contains | open simplices corresponding to fixed graph combinatorial types ⓘ |
| definedFor | free group F_n ⓘ |
| dimension | 3n-4 for rank n ≥ 2 ⓘ |
| equivalenceRelation | change of marking by graph isometry preserving edge lengths ⓘ |
| field |
geometric group theory
ⓘ
group theory ⓘ topology ⓘ |
| generalizationOf | space of metric graphs of rank n with markings ⓘ |
| graphCondition | graphs are finite, connected, with no vertices of valence 1 ⓘ |
| graphRankCondition | fundamental group of each graph is free of rank n ⓘ |
| groupActsOn | Out(F_n) NERFINISHED ⓘ |
| hasPoint | equivalence class of a marked metric graph of rank n ⓘ |
| hasProperty |
Out(F_n)-invariant
ⓘ
contractible ⓘ locally finite-dimensional ⓘ not locally compact ⓘ |
| hasStructure | simplicial complex up to missing faces ⓘ |
| hasSubspace | spine of Outer space ⓘ |
| introducedBy |
Karen Vogtmann
NERFINISHED
ⓘ
Marc Culler NERFINISHED ⓘ |
| introducedInYear | 1986 ⓘ |
| markingDefinition | homotopy equivalence from a fixed reference rose to the graph ⓘ |
| metricCondition | edge lengths are positive real numbers ⓘ |
| normalizationCondition | total volume of each graph is 1 ⓘ |
| parametrizes |
free, minimal, isometric actions of free groups on metric graphs
ⓘ
marked metric graphs ⓘ |
| quotientBy | Out(F_n) NERFINISHED ⓘ |
| quotientInterpretedAs | analogue of moduli space of graphs ⓘ |
| rankParameter | n ≥ 2 ⓘ |
| relatedConcept |
Outer space boundary
ⓘ
curve complex ⓘ free factor complex ⓘ |
| spineProperty | spine is a contractible simplicial complex with cocompact Out(F_n)-action ⓘ |
| topologyInducedBy | simplicial coordinates of edge lengths ⓘ |
| usedFor |
constructing invariants of Out(F_n)
ⓘ
defining train track representatives of automorphisms ⓘ studying dynamics of automorphisms of free groups ⓘ |
| usedToStudy |
Out(F_n)
NERFINISHED
ⓘ
outer automorphism group of a free group ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.