Outer space (Culler–Vogtmann Outer space)
E634850
Outer space (Culler–Vogtmann Outer space) is a topological space introduced by Culler and Vogtmann that parametrizes marked metric graphs and serves as an analogue of Teichmüller space for the study of the outer automorphism group of a free group.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
moduli space
ⓘ
topological space ⓘ |
| alsoKnownAs |
CV_n (for rank n free group)
NERFINISHED
ⓘ
Culler–Vogtmann Outer space NERFINISHED ⓘ |
| analogueOf | Teichmüller space NERFINISHED ⓘ |
| generalizes | classical Teichmüller theory ideas to free groups ⓘ |
| hasActionBy | Out(F_n) NERFINISHED ⓘ |
| hasActionType | properly discontinuous action ⓘ |
| hasAnalogyWith | moduli space of Riemann surfaces NERFINISHED ⓘ |
| hasBasePointAnalogue | rose graph with n petals and marking by isomorphism F_n → π_1(rose) ⓘ |
| hasBoundary | space of very small actions of F_n on R-trees ⓘ |
| hasCellDecomposition | open simplices corresponding to marked graphs with fixed combinatorial type ⓘ |
| hasCompactification | simplicial bordification by adding actions on R-trees ⓘ |
| hasDimension | 3n - 4 for rank n ≥ 2 ⓘ |
| hasGeometricRole | classifying space for Out(F_n) up to finite quotient issues ⓘ |
| hasNaturalMetricStructures | various asymmetric and Lipschitz-type metrics ⓘ |
| hasNormalizationCondition | sum of edge lengths equals 1 ⓘ |
| hasPointRepresentedBy |
finite connected graph with no degree-1 vertices
ⓘ
graph with fundamental group isomorphic to a free group F_n ⓘ graph with total edge length normalized to 1 ⓘ |
| hasProperty |
contractible
ⓘ
finite-dimensional ⓘ locally finite-dimensional ⓘ non-compact ⓘ |
| hasRank2CaseHomeomorphicTo | infinite 3-regular tree of simplices glued along faces ⓘ |
| hasRankParameter | free group rank n ≥ 2 ⓘ |
| hasRestriction | graphs have no separating edges in top-dimensional simplices ⓘ |
| hasStructure | simplicial complex structure up to subdivision ⓘ |
| hasTopologyInducedBy | simplicial topology from edge-length coordinates ⓘ |
| introducedBy |
Karen Vogtmann
NERFINISHED
ⓘ
Marc Culler NERFINISHED ⓘ |
| introducedInField |
geometric group theory
ⓘ
low-dimensional topology ⓘ |
| isContractible | true ⓘ |
| isExampleOf | parameter space for group actions on graphs ⓘ |
| parametrizes |
finite connected metric graphs with fundamental group a free group
ⓘ
free, minimal, isometric actions of a free group on metric graphs ⓘ marked metric graphs ⓘ |
| quotientBy | Out(F_n) gives moduli space of metric graphs of rank n NERFINISHED ⓘ |
| relatedTo |
automorphisms of free groups
ⓘ
moduli space of graphs ⓘ |
| usedFor |
constructing invariants of Out(F_n)
ⓘ
studying dynamics of automorphisms of free groups ⓘ |
| usedToDefine |
cohomology classes of Out(F_n)
ⓘ
train track representatives 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.