View source for Orbifold

{{Short description|Generalized manifold}}
{{Use dmy dates|date=April 2014}}
{{quote box
|align=right
|width=33%
|quote=
This terminology should not be blamed on me. It was obtained by a democratic process in my course of 1976–77. An orbifold is something with many folds; unfortunately, the word "manifold" already has a different definition. I tried "foldamani", which was quickly displaced by the suggestion of "manifolded". After two months of patiently saying "no, not a manifold, a manifol''dead''," we held a vote, and "orbifold" won.
|source={{harvtxt|Thurston|1978–1981|p=300|loc=section 13.2}} explaining the origin of the word "orbifold"
}}
[[File:23star orbifold example.png|alt=Hyperbolic symmetry comparison to Euclidean symmetry|thumb|23star Orbifold Example]]
In the mathematical disciplines of [[topology]] and [[geometry]], an '''orbifold''' (for "orbit-manifold") is a generalization of a [[manifold]]. Roughly speaking, an orbifold is a [[topological space]] that is locally a [[finite group]] quotient of a [[Euclidean space]].

Definitions of orbifold have been given several times: by [[Ichirō Satake]] in the context of [[automorphic form]]s in the 1950s under the name ''V-manifold'';{{sfn|Satake|1956}} by [[William Thurston]] in the context of the geometry of [[3-manifold]]s in the 1970s{{sfn|Thurston|1978–1981|loc=Chapter 13}} when he coined the name ''orbifold'', after a vote by his students; and by [[André Haefliger]] in the 1980s in the context of [[Mikhail Gromov (mathematician)|Mikhail Gromov]]'s programme on [[CAT(k) space]]s under the name ''orbihedron''.{{sfn|Haefliger|1990}}

Historically, orbifolds arose first as surfaces with [[Singularity (mathematics)|singular points]] long before they were formally defined.{{sfn|Poincaré|1985}} One of the first classical examples arose in the theory of [[modular forms]]{{sfn|Serre|1970}} with the action of the [[modular group]] <math>\mathrm{SL}(2,\Z)</math> on the [[upper half-plane]]: a version of the [[Riemann–Roch theorem]] holds after the quotient is compactified by the addition of two orbifold cusp points. In [[3-manifold]] theory, the theory of [[Seifert fiber spaces]], initiated by [[Herbert Seifert]], can be phrased in terms of 2-dimensional orbifolds.{{sfn|Scott|1983}} In [[geometric group theory]], post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces.{{sfn|Bridson|Haefliger|1999}}

In [[string theory]], the word "orbifold" has a slightly different meaning,{{sfn|Di Francesco|Mathieu|Sénéchal|1997}} discussed in detail below. In [[two-dimensional conformal field theory]], it refers to the theory attached to the fixed point subalgebra of a [[vertex algebra]] under the action of a finite group of [[automorphism]]s.

The main example of underlying space is a quotient space of a manifold under the [[properly discontinuous]] action of a possibly infinite [[group (mathematics)|group]] of [[diffeomorphism]]s with finite [[isotropy subgroup]]s.{{sfn|Bredon|1972}} In particular this applies to any action of a [[finite group]]; thus a [[manifold with boundary]] carries a natural orbifold structure, since it is the quotient of its [[Glossary of differential geometry and topology#D|double]] by an action of <math>\Z_2</math>.

One topological space can carry different orbifold structures. For example, consider the orbifold <math>O</math> associated with a quotient space of the 2-sphere along a rotation by <math>\pi</math> ; it is [[Homeomorphism|homeomorphic]] to the 2-sphere, but the natural orbifold structure is different. It is possible to adopt most of the characteristics of manifolds to orbifolds and these characteristics are usually different from correspondent characteristics of underlying space. In the above example, the ''orbifold [[fundamental group]]'' of <math>O</math> is <math>\Z_2</math> and its ''orbifold [[Euler characteristic]]'' is 1.

==Formal definitions==

=== Definition using orbifold atlas ===
Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on [[Open set|open subsets]] of <math>\R^n</math>, an orbifold is locally modelled on quotients of open subsets of <math>\R^n</math> by finite group actions. The structure of an orbifold encodes not only that of the underlying quotient space, which need not be a manifold, but also that of the [[isotropy subgroup]]s.

An <math>n</math>-dimensional '''orbifold''' is a [[Hausdorff space|Hausdorff topological space]] <math>X</math>, called the '''underlying space''', with a covering by a collection of open sets <math>U_i</math>, closed under finite intersection. For each <math>U_i</math>, there is
* an open subset <math>V_i</math> of <math>\R^n</math>, invariant under a [[Group action (mathematics)#Remarkable properties of actions|faithful]] linear action of a finite group <math>\Gamma_i</math>;
* a continuous map <math>\varphi_i</math> of <math>V_i</math> onto <math>U_i</math> invariant under <math>\Gamma_i</math>, called an '''orbifold chart''', which defines a homeomorphism between <math>V_i/\Gamma_i</math> and <math>U_i</math>.

The collection of orbifold charts is called an '''orbifold atlas''' if the following properties are satisfied:
* for each inclusion <math>U_i\subset U_j</math> there is an [[injective]] [[group homomorphism]] <math>f_{ij}:\Gamma_i\rightarrow\Gamma_j</math>.
* for each inclusion <math>U_i\subset U_j</math> there is a <math>\Gamma_i</math>-[[equivariant]] homeomorphism <math>\psi_{ij}</math>, called a '''gluing map''', of <math>V_i</math> onto an open subset of ''<math>V_j</math>''.
* the gluing maps are compatible with the charts, i.e. <math>\varphi_j\circ\psi_{ij} = \varphi_i</math>.
* the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from <math>V_i</math> to ''<math>V_j</math>'' has the form <math>g\circ\psi_{ij}</math> for a unique <math>g\in\Gamma_j</math>.

As for [[Atlas (topology)|atlases on manifolds]], two orbifold atlases of ''<math>X</math>'' are equivalent if they can be consistently combined to give a larger orbifold atlas. An '''orbifold structure''' is therefore an equivalence class of orbifold atlases.

Note that the orbifold structure determines the isotropy subgroup of any point of the orbifold up to isomorphism: it can be computed as the stabilizer of the point in any orbifold chart. If ''U''<sub>''i''</sub> <math> \subset </math> ''U''<sub>''j''</sub> <math> \subset </math> ''U''<sub>''k''</sub>, then there is a unique ''transition element'' ''g''<sub>ijk</sub> in Γ<sub>''k''</sub> such that

:''g''<sub>''ijk''</sub>·''ψ''<sub>''ik''</sub> = ''ψ''<sub>''jk''</sub>·''ψ''<sub>''ij''</sub>

These transition elements satisfy

:(Ad ''g''<sub>''ijk''</sub>)·''f''<sub>''ik''</sub> = ''f''<sub>''jk''</sub>·''f''<sub>''ij''</sub>

as well as the ''cocycle relation'' (guaranteeing associativity)

: ''f''<sub>''km''</sub>(''g''<sub>''ijk''</sub>)·''g''<sub>''ikm''</sub>   = ''g''<sub>''ijm''</sub>·''g''<sub>''jkm''</sub>.

More generally, attached to an open covering of an orbifold by orbifold charts, there is the combinatorial data of a so-called ''complex of groups'' (see below).

Exactly as in the case of manifolds, differentiability conditions can be imposed on the gluing maps to give a definition of a '''differentiable orbifold'''. It will be a ''Riemannian orbifold'' if in addition there are invariant [[Riemannian metric]]s on the orbifold charts and the gluing maps are [[Riemannian manifold|isometries]].

===Definition using Lie groupoids===

Recall that a '''[[groupoid]]''' consists of a set of objects <math>G_0</math>, a set of arrows <math>G_1</math>, and structural maps including the source and the target maps <math>s, t: G_1 \to G_0</math> and other maps allowing arrows to be composed and inverted. It is called a '''[[Lie groupoid]]''' if both <math>G_0</math> and <math>G_1</math> are smooth manifolds, all structural maps are smooth, and both the source and the target maps are submersions. The intersection of the source and the target fiber at a given point <math>x \in G_0</math>, i.e. the set <math>(G_1)_x := s^{-1}(x) \cap t^{-1}(x)</math>, is the [[Lie group]] called the '''isotropy group''' of <math>G_1</math> at <math>x</math>. A Lie groupoid is called '''proper''' if the map <math>(s,t): G_1 \to G_0 \times G_0</math> is a [[proper map]], and '''étale''' if both the source and the target maps are [[local diffeomorphism]]s.

An '''orbifold groupoid''' is given by one of the following equivalent definitions:

* a proper étale Lie groupoid;
* a proper Lie groupoid whose isotropies are [[discrete space]]s.

Since the isotropy groups of proper groupoids are automatically [[Compact space|compact]], the discreteness condition implies that the isotropies must be actually [[finite group]]s.<ref>{{Cite conference |last=Moerdijk |first=Ieke |author-link=Ieke Moerdijk |date=2002 |title=Orbifolds as Groupoids: an Introduction |url=https://bookstore.ams.org/conm-310 |conference=Orbifolds in mathematics and physics |series=Contemporary Mathematics |publisher=[[American Mathematical Society]] |volume=310 |pages=205–222 |arxiv=math/0203100 |isbn=978-0-8218-2990-5}}</ref>

Orbifold groupoids play the same role as orbifold atlases in the definition above. Indeed, an '''orbifold structure''' on a Hausdorff topological space <math>X</math> is defined as the [[Lie groupoid#Morita equivalence|Morita equivalence]] class of an orbifold groupoid <math>G \rightrightarrows M</math> together with a homeomorphism <math>|M/G| \simeq X</math>, where <math>|M/G|</math> is the orbit space of the Lie groupoid <math>G</math> (i.e. the quotient of <math>M</math> by the equivalent relation when <math>x \sim y</math> if there is a <math>g \in G</math> with <math>s(g)=x</math> and <math>t(g)=y</math>). This definition shows that orbifolds are a particular kind of [[differentiable stack]].

== Relation between the two definitions ==
Given an orbifold atlas on a space <math>X</math>, one can build a [[pseudogroup]] made up by all diffeomorphisms between open sets of <math>X</math> which preserve the transition functions <math>\varphi_i</math>. In turn, the space <math>G_X</math> of germs of its elements is an orbifold groupoid. Moreover, since by definition of orbifold atlas each finite group <math>\Gamma_i</math> acts faithfully on <math>V_i</math>, the groupoid <math>G_X</math> is automatically effective, i.e. the map <math>g \in (G_X)_x \mapsto \mathrm{germ}_x (t \circ s^{-1})</math> is injective for every <math>x \in X</math>. Two different orbifold atlases give rise to the same orbifold structure if and only if their associated orbifold groupoids are Morita equivalent. Therefore, any orbifold structure according to the first definition (also called a '''classical orbifold''') is a special kind of orbifold structure according to the second definition.

Conversely, given an orbifold groupoid <math>G \rightrightarrows M</math>, there is a canonical orbifold atlas over its orbit space, whose associated effective orbifold groupoid is Morita equivalent to <math>G</math>. Since the orbit spaces of Morita equivalent groupoids are homeomorphic, an orbifold structure according to the second definition reduces an orbifold structure according to the first definition in the effective case.<ref>{{Cite book |last1=Moerdijk |first1=Ieke |url=https://www.cambridge.org/core/books/introduction-to-foliations-and-lie-groupoids/75BBFED277FF39A56594731202789016 |title=Introduction to Foliations and Lie Groupoids |last2=Mrcun |first2=Janez |date=2003 |publisher=[[Cambridge University Press]] |isbn=978-0-521-83197-0 |series=Cambridge Studies in Advanced Mathematics |pages=140–144 |doi=10.1017/cbo9780511615450 |author-link1=Ieke Moerdijk}}</ref>

Accordingly, while the notion of orbifold atlas is simpler and more commonly present in the literature, the notion of orbifold groupoid is particularly useful when discussing non-effective orbifolds and maps between orbifolds. For example, a map between orbifolds can be described by a homomorphism between groupoids, which carries more information than the underlying continuous map between the underlying topological spaces.

==Examples==
* Any manifold without boundary is trivially an orbifold, where each of the groups Γ<sub>''i''</sub> is the [[trivial group]]. Equivalently, it corresponds to the Morita equivalence class of the unit groupoid.
* If ''N'' is a compact [[manifold with boundary]], its '''double''' ''M'' can be formed by gluing together a copy of ''N'' and its mirror image along their common boundary. There is natural ''reflection'' action of '''Z'''<sub>2</sub> on the manifold ''M'' fixing the common boundary; the quotient space can be identified with ''N'', so that ''N''  has a natural orbifold structure.
* If ''M'' is a Riemannian ''n''-manifold with a [[cocompact group action|cocompact]] [[Group action#proper|proper]] isometric action of a discrete group Γ, then the orbit space ''X'' = ''M''/Γ has natural orbifold structure: for each ''x'' in ''X'' take a representative ''m'' in ''M'' and an open neighbourhood ''V''<sub>''m''</sub> of ''m'' invariant under the stabiliser Γ<sub>''m''</sub>, identified equivariantly with a Γ<sub>''m''</sub>-subset of ''T''<sub>''m''</sub>''M'' under the exponential map at ''m''; finitely many neighbourhoods cover ''X'' and each of their finite intersections, if non-empty, is covered by an intersection of Γ-translates ''g''<sub>''m''</sub>·''V''<sub>''m''</sub> with corresponding group ''g''<sub>''m''</sub> Γ ''g''<sub>''m''</sub><sup>−1</sup>. Orbifolds that arise in this way are called ''developable'' or ''good''.
* A classical theorem of [[Henri Poincaré]] constructs [[Fuchsian group]]s as hyperbolic [[reflection group]]s generated by reflections in the edges of a [[geodesic triangle]] in the [[hyperbolic geometry|hyperbolic plane]] for the [[Poincaré metric]]. If the triangle has angles {{pi}}/''n''<sub>''i''</sub> for positive integers ''n''<sub>''i''</sub>, the triangle is a [[fundamental domain]] and naturally a 2-dimensional orbifold. The corresponding group is an example of a hyperbolic [[triangle group]]. Poincaré also gave a 3-dimensional version of this result for [[Kleinian group]]s: in this case the Kleinian group Γ is generated by hyperbolic reflections and the orbifold is '''H'''<sup>3</sup> / Γ.
* If ''M'' is a closed 2-manifold, new orbifold structures can be defined on ''M'' by removing finitely many disjoint closed discs from ''M'' and gluing back copies of discs ''D''/ Γ<sub>''i''</sub> where ''D'' is the closed [[unit disc]] and Γ<sub>''i''</sub> is a finite cyclic group of rotations. This generalises Poincaré's construction.

==Orbifold fundamental group==
There are several ways to define the '''orbifold fundamental group'''. More sophisticated approaches use orbifold [[covering space]]s or [[classifying space]]s of [[groupoid]]s. The simplest approach (adopted by Haefliger and known also to Thurston) extends the usual notion of [[fundamental group|loop]] used in the standard definition of the [[fundamental group]].

An '''orbifold path''' is a path in the underlying space provided with an explicit piecewise lift of path segments to orbifold charts and explicit group elements identifying paths in overlapping charts; if the underlying path is a loop, it is called an '''orbifold loop'''. Two orbifold paths are identified if they are related through multiplication by group elements in orbifold charts. The orbifold fundamental group is the group formed by [[homotopy class]]es of orbifold loops.

If the orbifold arises as the quotient of a [[simply connected]] manifold ''M'' by a proper rigid action of a discrete group Γ, the orbifold fundamental group can be identified with Γ. In general it is an [[group extension|extension]] of Γ by {{pi}}<sub>1</sub> ''M''.

The orbifold is said to be ''developable'' or ''good'' if it arises as the quotient by a group action; otherwise it is called ''bad''. A ''universal covering orbifold'' can be constructed for an orbifold by direct analogy with the construction of the [[fundamental group|universal covering space]] of a topological space, namely as the space of pairs consisting of points of the orbifold and homotopy classes of orbifold paths joining them to the basepoint. This space is naturally an orbifold.

Note that if an orbifold chart on a [[contractible]] open subset corresponds to a group Γ, then there is a natural ''local homomorphism'' of Γ into the orbifold fundamental group.

In fact the following conditions are equivalent:
* The orbifold is developable.
* The orbifold structure on the universal covering orbifold is trivial.
* The local homomorphisms are all injective for a covering by contractible open sets.

=== Orbifolds as diffeologies ===

Orbifolds can be defined in the general framework of [[diffeology]]{{sfn|Iglesias-Zemmour|2013}} and have been proved to be equivalent{{sfn|Iglesias|Karshon|Zadka|4=2010}} to [[Ichirô Satake]]'s original definition:{{sfn|Satake|1956}}

'''Definition:''' An orbifold is a diffeological space locally diffeomorphic at each point to some <math>\R^n/G</math>, where <math>n</math> is an integer and <math>G</math> is a finite linear group which may change from point to point.

This definition calls a few remarks:

* This definition mimics the definition of a manifold in diffeology, which is a diffeological space locally diffeomorphic at each point to <math>\R^n</math>.
* An orbifold is regarded first as a diffeological space, a set equipped with a diffeology. Then, the diffeology is tested to be locally diffeomorphic at each point to a quotient <math>\R^n/G</math> with <math>G</math> a finite linear group.
* This definition is equivalent{{sfn|Iglesias|Karshon|Zadka|5=2010|loc=Theorem 46}} with Haefliger orbifolds.{{sfn| Haefliger | 1984}}
* {Orbifolds} makes a subcategory of the category {Diffeology} whose objects are diffeological spaces and morphisms smooth maps. A smooth map between orbifolds is any map which is smooth for their diffeologies. This resolves, in the context of Satake's definition, his remark:{{sfn|Satake|1957|loc=Footnote p.469}} "''The notion of <math>C^\infty</math>-map thus defined is inconvenient in the point that a composite of two <math>C^\infty</math>-map defined in a different choice of defining families is not always a <math>C^\infty</math>-map.''" Indeed, there are smooth maps between orbifolds that do not lift locally as equivariant maps.{{sfn|Iglesias|Karshon|Zadka|5=2010|loc=Example 25}}

Note that the fundamental group of an orbifold as a diffeological space is not the same as the fundamental group as defined above. That last one is related to the structure groupoid{{sfn|Iglesias-Zemmour|Laffineur|3=2017}} and its isotropy groups.

==Orbispaces==

For applications in [[geometric group theory]], it is often convenient to have a slightly more general notion of orbifold, due to Haefliger. An '''orbispace''' is to topological spaces what an orbifold is to manifolds. An orbispace is a topological generalization of the orbifold concept. It is defined by replacing the model for the orbifold charts by a [[locally compact]] space with a ''rigid'' action of a finite group, i.e. one for which points with trivial isotropy are dense. (This condition is automatically satisfied by faithful linear actions, because the points fixed by any non-trivial group element form a proper [[linear subspace]].) It is also useful to consider [[metric space]] structures on an orbispace, given by invariant [[metric space|metrics]] on the orbispace charts for which the gluing maps preserve distance. In this case each orbispace chart is usually required to be a [[Intrinsic metric#Definitions|length space]] with unique [[Intrinsic metric|geodesics]] connecting any two points.

Let ''X'' be an orbispace endowed with a metric space structure for which the charts are geodesic length spaces. The preceding definitions and results for orbifolds can be generalized to give definitions of ''orbispace fundamental group'' and ''universal covering orbispace'', with analogous criteria for developability. The distance functions on the orbispace charts can be used to define the length of an orbispace path in the universal covering orbispace. If the distance function in each chart is [[CAT(k) space|non-positively curved]], then the [[Surface (topology)#Surfaces of non-positive curvature|Birkhoff curve shortening argument]] can be used to prove that any orbispace path with fixed endpoints is homotopic to a unique geodesic. Applying this to constant paths in an orbispace chart, it follows that each local homomorphism is injective and hence:
* every non-positively curved orbispace is developable (i.e. ''good'').

==Complexes of groups==
Every orbifold has associated with it an additional combinatorial structure given by a ''complex of groups''.

===Definition===
A '''complex of groups''' (''Y'',''f'',''g'') on an [[abstract simplicial complex]] ''Y'' is given by
* a finite group Γ<sub>σ</sub> for each simplex σ of ''Y''
* an injective homomorphism ''f''<sub>στ</sub> : Γ<sub>τ</sub> <math>\rightarrow</math> Γ<sub>σ</sub> whenever σ <math>\subset</math> τ
* for every inclusion ρ <math>\subset</math> σ <math>\subset</math> τ, a group element ''g''<sub>ρστ</sub> in Γ<sub>ρ</sub> such that (Ad ''g''<sub>ρστ</sub>)·''f''<sub>ρτ</sub> = ''f''<sub>ρσ</sub>·''f''<sub>στ</sub> (here Ad denotes the [[adjoint representation of a Lie group|adjoint action]] by conjugation)

The group elements must in addition satisfy the cocycle condition

:''f''<sub>{{pi}}''ρ''</sub>(''g''<sub>ρστ</sub>) ''g''<sub>πρτ</sub> = ''g''<sub>{{pi}}''στ''</sub> ''g''<sub>{{pi}}''ρσ''</sub>

for every chain of simplices <math>\pi \subset \rho\subset \sigma \subset \tau.</math> (This condition is vacuous if ''Y'' has dimension 2 or less.)

Any choice of elements ''h''<sub>''στ''</sub> in Γ<sub>''σ''</sub> yields an ''equivalent'' complex of groups by defining
* ''f'''<sub>''στ''</sub> = (Ad ''h''<sub>''στ''</sub>)·''f''<sub>''στ''</sub>
* ''g'''<sub>''ρστ''</sub> =    ''h''<sub>ρσ</sub>·''f''<sub>''ρσ''</sub>(''h''<sub>''στ''</sub>)·''g''<sub>''ρστ''</sub>·''h''<sub>''ρτ''</sub><sup>−1</sup>

A complex of groups is called '''simple''' whenever ''g''<sub>''ρστ''</sub> = 1 everywhere.
* An easy inductive argument shows that every complex of groups on a ''simplex'' is equivalent to a complex of groups with ''g''<sub>''ρστ''</sub> = 1 everywhere.

It is often more convenient and conceptually appealing to pass to the [[barycentric subdivision]] of ''Y''. The vertices of this subdivision correspond to the simplices of ''Y'', so that each vertex has a group attached to it. The edges of the barycentric subdivision are naturally oriented (corresponding to inclusions of simplices) and each directed edge gives an inclusion of groups. Each triangle has a transition element attached to it belonging to the group of exactly one vertex; and the tetrahedra, if there are any, give cocycle relations for the transition elements. Thus a complex of groups involves only the 3-skeleton of the barycentric subdivision; and only the 2-skeleton if it is simple.

===Example===
If ''X'' is an orbifold (or orbispace), choose a covering by open subsets from amongst the orbifold charts ''f''<sub>''i'' </sub>: ''V''<sub>''i''</sub> <math>\rightarrow</math> ''U''<sub>''i''</sub>. Let ''Y'' be the abstract simplicial complex given by the [[nerve of an open covering|nerve of the covering]]: its vertices are the sets of the cover and its ''n''-simplices correspond to ''non-empty'' intersections ''U''<sub>''α''</sub> = ''U''<sub>''i''<sub>1</sub></sub> <math>\cap</math> ··· <math>\cap</math> ''U''<sub>''i''<sub>''n''</sub></sub>. For each such simplex there is an associated group Γ<sub>α</sub> and the homomorphisms ''f''<sub>''ij''</sub> become the homomorphisms ''f''<sub>στ</sub>. For every triple ρ <math> \subset </math> σ  <math> \subset </math> τ corresponding to intersections

: <math> U_i \supset U_i \cap U_j \supset U_i \cap U_j \cap U_k</math>

there are charts ''φ''<sub>''i''</sub> : ''V''<sub>''i''</sub> <math>\rightarrow</math> ''U''<sub>''i''</sub>, ''φ''<sub>''ij''</sub> : ''V''<sub>''ij''</sub> <math>\rightarrow</math> ''U''<sub>''i''</sub> <math>\cap</math> ''U''<sub>''j''</sub> and φ<sub>''ijk''</sub> : ''V''<sub>''ijk''</sub> <math>\rightarrow</math> ''U''<sub>''i''</sub> <math>\cap</math> ''U''<sub>''j''</sub> <math>\cap</math> ''U''<sub>''k''</sub> and gluing maps ψ : ''V''<sub> ''ij''</sub> <math>\rightarrow </math> ''V''<sub>''i''</sub>, ψ' : ''V''<sub> ''ijk''</sub> <math>\rightarrow </math> ''V''<sub>''ij''</sub> and ψ" : ''V''<sub> ''ijk''</sub> <math>\rightarrow </math> ''V''<sub>''i''</sub>.

There is a unique transition element ''g''<sub>ρστ</sub> in Γ<sub>''i''</sub> such that ''g''<sub>''ρστ''</sub>·''ψ''" = ''ψ''·''{{prime|ψ}}''. The relations satisfied by the transition elements of an orbifold imply those required for a complex of groups. In this way a complex of groups can be canonically associated to the nerve of an open covering by orbifold (or orbispace) charts. In the language of non-commutative [[sheaf theory]] and [[gerbe]]s, the complex of groups in this case arises as a [[sheaf theory|sheaf of groups]] associated to the covering ''U''<sub>''i''</sub>; the data ''g''<sub>ρστ</sub> is a 2-cocycle in non-commutative [[sheaf cohomology]] and the data ''h''<sub>στ</sub> gives a 2-coboundary perturbation.

===Edge-path group===
The '''edge-path group''' of a complex of groups can be defined as a natural generalisation of the [[Fundamental group#Edge-path group of a simplicial complex|edge path group]] of a simplicial complex. In the barycentric subdivision of ''Y'', take generators ''e''<sub>''ij''</sub>  corresponding to edges from ''i'' to ''j'' where ''i'' <math>\rightarrow </math> ''j'', so that there is an injection ψ<sub>''ij''</sub> : Γ<sub>''i''</sub> <math> \rightarrow</math> Γ<sub>''j''</sub>. Let Γ be the group generated by the ''e''<sub>''ij''</sub> and Γ<sub>''k''</sub> with relations

:''e''<sub>''ij''</sub> <sup>−1</sup> · ''g'' · ''e''<sub>''ij''</sub> = ψ<sub>''ij''</sub>(''g'')

for ''g'' in Γ<sub>''i''</sub>  and

:''e''<sub>''ik''</sub>  = ''e''<sub>''jk''</sub>·''e''<sub>''ij''</sub>·''g''<sub>''ijk''</sub>

if ''i'' <math>\rightarrow </math> ''j'' <math>\rightarrow</math> ''k''.

For a fixed vertex ''i''<sub>0</sub>, the edge-path group Γ(''i''<sub>0</sub>) is defined to be the subgroup of Γ  generated by all products

:''g''<sub>0</sub> · e<sub>''i''<sub>0</sub> ''i''<sub>1</sub></sub> · ''g''<sub>1</sub> · e<sub>''i''<sub>1</sub> ''i''<sub>2</sub></sub> · ··· · ''g''<sub>''n''</sub>  · e<sub>''i''<sub>''n''</sub>''i''<sub> 0</sub> </sub>

where ''i''<sub>0</sub>, ''i''<sub>1</sub>, ..., ''i''<sub>''n''</sub>, ''i''<sub>0</sub>
is an edge-path, ''g''<sub>''k''</sub> lies in Γ<sub>''i''<sub>''k''</sub></sub> and ''e''<sub>''ji''</sub>=''e''<sub>''ij''</sub><sup>−1</sup> if ''i'' <math>\rightarrow </math> ''j''.

===Developable complexes===
A simplicial [[proper action]] of a discrete group Γ on a [[simplicial complex]] ''X'' with finite quotient is said to be '''regular''' if it
satisfies one of the following equivalent conditions:{{sfn|Bredon|1972}}
* ''X'' admits a finite subcomplex as [[fundamental domain]];
* the quotient ''Y'' = ''X''/Γ has a natural simplicial structure;
* the quotient simplicial structure on orbit-representatives of vertices is consistent;
* if (''v''<sub>0</sub>, ..., ''v''<sub>''k''</sub>) and (''g''<sub>0</sub>·''v''<sub>0</sub>, ..., ''g''<sub>''k''</sub>·''v''<sub>''k''</sub>) are simplices, then ''g''·''v''<sub>''i''</sub> = ''g''<sub>''i''</sub>·''v''<sub>''i''</sub> for some ''g'' in Γ.

The fundamental domain and quotient ''Y'' = ''X'' / Γ can naturally be identified as simplicial complexes in this case, given by the stabilisers of the simplices in the fundamental domain. A complex of groups ''Y'' is said to be '''developable''' if it arises in this way.
* A complex of groups is developable if and only if the homomorphisms of Γ<sub>σ</sub> into the edge-path group are injective.
* A complex of groups is developable if and only if for each simplex σ there is an injective homomorphism θ<sub>σ</sub>  from Γ<sub>σ</sub> into a fixed discrete group Γ such that θ<sub>τ</sub>·''f''<sub>στ</sub> = θ<sub>σ</sub>. In this case the simplicial complex ''X'' is canonically defined: it has ''k''-simplices (σ, xΓ<sub>σ</sub>) where σ is a ''k''-simplex of ''Y'' and ''x'' runs over Γ / Γ<sub>σ</sub>. Consistency can be checked using the fact that the restriction of the complex of groups to a ''simplex'' is equivalent to one with trivial cocycle ''g''<sub>ρστ</sub>.

The action of Γ on the barycentric subdivision ''X'' ' of ''X'' always satisfies the following condition, weaker than regularity:
* whenever σ and ''g''·σ are subsimplices of some simplex τ, they are equal, i.e. σ = ''g''·σ

Indeed, simplices in ''X'' ' correspond to chains of simplices in ''X'', so that a subsimplices, given by subchains of simplices, is uniquely determined by the ''sizes'' of the simplices in the subchain. When an action satisfies this condition, then ''g'' necessarily fixes all the vertices of σ. A straightforward inductive argument shows that such an action becomes regular on the barycentric subdivision; in particular
* the action on the second barycentric subdivision ''X''" is regular;
* Γ is naturally isomorphic to the edge-path group defined using edge-paths and vertex stabilisers for the barycentric subdivision of the fundamental domain in ''X''".

There is in fact no need to pass to a ''third'' barycentric subdivision: as Haefliger observes using the language of [[category theory]], in this case the 3-skeleton of the fundamental domain of ''X''" already carries all the necessary data – including transition elements for triangles – to define an edge-path group isomorphic to Γ.

In two dimensions this is particularly simple to describe. The fundamental domain of ''X''" has the same structure as the barycentric subdivision ''Y'' ' of a complex of groups ''Y'', namely:
* a finite 2-dimensional simplicial complex ''Z'';
* an orientation for all edges ''i'' <math>\rightarrow </math> ''j'';
* if ''i'' <math>\rightarrow </math> ''j'' and ''j'' <math>\rightarrow </math> ''k'' are edges, then ''i'' <math>\rightarrow </math> ''k'' is an edge and (''i'', ''j'', ''k'') is a triangle;
* finite groups attached to vertices, inclusions to edges and transition elements, describing compatibility, to triangles.

An edge-path group can then be defined. A similar structure is inherited by the barycentric subdivision ''Z'' ' and its edge-path group is isomorphic to that of ''Z''.

==Orbihedra==
If a countable discrete group acts by a ''regular'' ''simplicial'' [[proper action]] on a [[simplicial complex]], the quotient can be given not only the structure of a complex of groups, but also that of an orbispace. This leads more generally to the definition of "orbihedron", the simplicial analogue of an orbifold.

===Definition===
Let ''X'' be a finite simplicial complex with barycentric subdivision ''X'' '. An '''orbihedron''' structure consists of:
* for each vertex ''i'' of ''X'' ', a simplicial complex ''L''<sub>''i''</sub>' endowed with a rigid simplicial action of a finite group Γ<sub>''i''</sub>.
* a simplicial map φ<sub>''i''</sub> of ''L''<sub>''i''</sub>' onto the [[link (geometry)|link]] ''L''<sub>''i''</sub> of ''i'' in ''X'' ', identifying the quotient ''L''<sub>''i''</sub>' / Γ<sub>''i''</sub> with ''L''<sub>''i''</sub>.

This action of Γ<sub>''i''</sub> on ''L''<sub>''i''</sub>' extends to a simplicial action on the simplicial cone ''C''<sub>''i''</sub> over ''L''<sub>''i''</sub>' (the simplicial join of ''i'' and ''L''<sub>''i''</sub>'), fixing the centre ''i'' of the cone. The map φ<sub>''i''</sub> extends to a simplicial map of
''C''<sub>''i''</sub> onto the [[simplicial complex|star]] St(''i'') of ''i'', carrying the centre onto ''i''; thus φ<sub>''i''</sub> identifies ''C''<sub>''i''</sub> / Γ<sub>''i''</sub>, the quotient of the star of ''i'' in ''C''<sub>''i''</sub>, with St(''i'') and gives an ''orbihedron chart'' at ''i''.
* for each directed edge ''i'' <math>\rightarrow</math> ''j'' of ''X'' ', an injective homomorphism ''f''<sub>''ij''</sub> of Γ<sub>''i''</sub> into Γ<sub>''j''</sub>.
* for each directed edge ''i'' <math>\rightarrow</math> ''j'', a Γ<sub>''i''</sub> equivariant simplicial ''gluing map'' ψ<sub>''ij''</sub> of ''C''<sub>''i''</sub> into ''C''<sub>''j''</sub>.
* the gluing maps are compatible with the charts, i.e. φ<sub>''j''</sub>·ψ<sub>''ij''</sub> = φ<sub>''i''</sub>.
* the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from ''V''<sub>''i''</sub> to ''V''<sub>''j''</sub> has the form ''g''·ψ<sub>''ij''</sub> for a unique ''g'' in Γ<sub>''j''</sub>.

If ''i''<math> \rightarrow </math> ''j'' <math> \rightarrow </math> ''k'', then there is a unique ''transition element'' ''g''<sub>ijk</sub> in Γ<sub>''k''</sub> such that

:''g''<sub>''ijk''</sub>·ψ<sub>''ik''</sub> = ψ<sub>''jk''</sub>·ψ<sub>''ij''</sub>

These transition elements satisfy

:(Ad ''g''<sub>''ijk''</sub>)·''f''<sub>''ik''</sub> = ''f''<sub>''jk''</sub>·''f''<sub>''ij''</sub>

as well as the cocycle relation

:ψ<sub>''km''</sub>(''g''<sub>''ijk''</sub>)·''g''<sub>''ikm''</sub>   = ''g''<sub>''ijm''</sub>·''g''<sub>''jkm''</sub>.

===Main properties===
* The group theoretic data of an orbihedron gives a complex of groups on ''X'', because the vertices ''i'' of the barycentric subdivision ''X'' ' correspond to the simplices in ''X''.
* Every complex of groups on ''X'' is associated with an essentially unique orbihedron structure on ''X''. This key fact follows by noting that the star and link of a vertex ''i'' of ''X'' ', corresponding to a simplex σ of ''X'', have natural decompositions: the star is isomorphic to the abstract simplicial complex given by the join of σ and the barycentric subdivision σ' of σ; and the link is isomorphic to join of the link of σ in ''X'' and the link of the barycentre of σ in σ'. Restricting the complex of groups to the link of σ in ''X'', all the groups Γ<sub>τ</sub> come with injective homomorphisms into Γ<sub>σ</sub>. Since the link of ''i'' in ''X'' ' is canonically covered by a simplicial complex on which Γ<sub>σ</sub> acts, this defines an orbihedron structure on ''X''.
* The orbihedron fundamental group is (tautologically) just the edge-path group of the associated complex of groups.
* Every orbihedron is also naturally an orbispace: indeed in the geometric realization of the simplicial complex, orbispace charts can be defined using the interiors of stars.
* The orbihedron fundamental group can be naturally identified with the orbispace fundamental group of the associated orbispace. This follows by applying the [[simplicial approximation theorem]] to segments of an orbispace path lying in an orbispace chart: it is a straightforward variant of the classical proof that the [[fundamental group]] of a [[simplicial complex#Algebraic topology|polyhedron]] can be identified with its [[Fundamental group#Edge-path group|edge-path group]].
* The orbispace associated to an orbihedron has a ''canonical metric structure'', coming locally from the length metric in the standard geometric realization in Euclidean space, with vertices mapped to an orthonormal basis. Other metric structures are also used, involving length metrics obtained by realizing the simplices in [[hyperbolic space]], with simplices identified isometrically along common boundaries.
* The orbispace associated to an orbihedron is [[non-positively curved space|non-positively curved]] if and only if the link in each orbihedron chart has [[girth (graph theory)|girth]] greater than or equal to 6, i.e. any closed circuit in the link has length at least 6. This condition, well known from the theory of [[CAT(k) space|Hadamard spaces]], depends only on the underlying complex of groups.
* When the universal covering orbihedron is non-positively curved the fundamental group is infinite and is generated by isomorphic copies of the isotropy groups. This follows from the corresponding result for orbispaces.

==Triangles of groups==
Historically one of the most important applications of orbifolds in [[geometric group theory]] has been to ''triangles of groups''. This is the simplest 2-dimensional example generalising the 1-dimensional "interval of groups" discussed in [[Jean-Pierre Serre|Serre]]'s lectures on trees, where [[free product|amalgamated free products]] are studied in terms of actions on trees. Such triangles of groups arise any time a discrete group acts simply transitively on the triangles in the [[Building (mathematics)#Spherical and affine buildings for SLn|affine Bruhat–Tits building]] for ''SL''<sub>3</sub>('''Q'''<sub>p</sub>); in 1979 [[David Mumford|Mumford]] discovered the first example for ''p'' = 2 (see below) as a step in producing an [[algebraic surface]] not isomorphic to [[projective space]], but having the same [[Betti number]]s. Triangles of groups were worked out in detail by Gersten and Stallings, while the more general case of complexes of groups, described above, was developed independently by Haefliger. The underlying geometric method of analysing finitely presented groups in terms of metric spaces of non-positive curvature is due to Gromov. In this context triangles of groups correspond to non-positively curved 2-dimensional simplicial complexes with the regular action of a group, '''''transitive on triangles'''''.

[[File:Triangle.Centroid.svg|thumb|right]]
A '''triangle of groups''' is a ''simple'' complex of groups consisting of a triangle with vertices '''A''', '''B''', '''C'''. There are groups
* Γ<sub>A</sub>, Γ<sub>B</sub>, Γ<sub>C</sub> at each vertex
* Γ<sub>BC</sub>, Γ<sub>CA</sub>, Γ<sub>AB</sub> for each edge
* Γ<sub>ABC</sub> for the triangle itself.

There are injective homomorphisms of Γ<sub>ABC</sub> into all the other groups and of an edge group Γ<sub>XY</sub> into Γ<sub>X</sub> and Γ<sub>Y</sub>. The three ways of mapping Γ<sub>ABC</sub> into a vertex group all agree. (Often Γ<sub>ABC</sub> is the trivial group.) The Euclidean metric structure on the corresponding orbispace is non-positively curved if and only if the link of each of the vertices in the orbihedron chart has girth at least 6.

This girth at each vertex is always even and, as observed by Stallings, can be described at a vertex '''A''', say, as the length of the smallest word in the kernel of the natural homomorphism into Γ<sub>A</sub>  of the [[free product|amalgamated free product]] over Γ<sub>ABC</sub> of the edge groups Γ<sub>AB</sub> and Γ<sub>AC</sub>:

:<math> \Gamma_{AB} \star_{\,\Gamma_{ABC}} \Gamma_{AC} \rightarrow \Gamma_A.</math>

The result using the Euclidean metric structure is not optimal. Angles α, β, γ at the vertices '''A''', '''B''' and '''C''' were defined by Stallings as 2π divided by the girth. In the Euclidean case α, β, γ ≤ π/3. However, if it is only required that α + β + γ ≤ π, it is possible to identify the
triangle with the corresponding geodesic triangle in the [[hyperbolic geometry|hyperbolic plane]] with the [[Poincaré metric]] (or the Euclidean plane if equality holds). It is a classical result from hyperbolic geometry that the hyperbolic medians intersect in the hyperbolic barycentre,<ref>[http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/pmedian.html Theorem of the hyperbolic medians]</ref> just as in the familiar Euclidean case. The barycentric subdivision and metric from this model yield a non-positively curved metric structure on the corresponding orbispace. Thus, if α+β+γ≤π,
* the orbispace of the triangle of groups is developable;
* the corresponding edge-path group, which can also be described as the [[colimit]] of the triangle of groups, is infinite;
* the homomorphisms of the vertex groups into the edge-path group are injections.

===Mumford's example===
[[File:Fano plane.svg|thumb|right|[[Fano plane|The Fano plane]]]]
Let ''α'' = <math>\sqrt{-7}</math> be given by the [[binomial expansion]] of (1&nbsp;−&nbsp;8)<sup>1/2</sup> in '''Q'''<sub>2</sub> and set ''K'' = '''Q'''(''α'') <math>\subset </math> '''Q'''<sub>2</sub>. Let

: ''ζ'' = exp 2{{pi}}''i''/7

: ''λ'' = (''α'' − 1)/2 = ''ζ'' + ''ζ''<sup>2</sup>  + ''ζ''<sup>4</sup>

: ''μ'' = ''λ''/''λ''*.

Let ''E'' = '''Q'''(''ζ''), a 3-dimensional vector space over ''K'' with basis 1, ''ζ'', and ''ζ''<sup>2</sup>. Define ''K''-linear operators on ''E'' as follows:
* ''σ'' is the generator of the [[Galois group]] of ''E'' over ''K'', an element of order 3 given by σ(ζ) = ζ<sup>2</sup>
* ''τ'' is the operator of multiplication by ''ζ'' on ''E'', an element of order 7
* ''ρ'' is the operator given by ''ρ''(''ζ'') = 1, ''ρ''(''ζ''<sup>2</sup>) = ''ζ'' and ''ρ''(1) = ''μ''·''ζ''<sup>2</sup>, so that ''ρ''<sup>3</sup> is scalar multiplication by&nbsp;''μ''.

The elements ''ρ'', ''σ'', and ''τ'' generate a discrete subgroup of ''GL''<sub>3</sub>(''K'') which acts [[proper action|properly]] on the [[Building (mathematics)#Spherical and affine buildings for SLn|affine Bruhat–Tits building]] corresponding to ''SL''<sub>3</sub>('''Q'''<sub>2</sub>). This group acts ''transitively'' on all vertices, edges and triangles in the building. Let

: ''σ''<sub>1</sub> = ''σ'', ''σ''<sub>2</sub> = ''ρσρ''<sup>−1</sup>, ''σ''<sub>3</sub> = ''ρ''<sup>2</sup>''σρ''<sup>−2</sup>.

Then
* ''σ''<sub>1</sub>, ''σ''<sub>2</sub> and ''σ''<sub>3</sub> generate a subgroup Γ of ''SL''<sub>3</sub>(''K'').
* Γ is the smallest subgroup generated by ''σ'' and ''τ'', invariant under conjugation by ''ρ''.
* Γ acts [[Group action (mathematics)#Remarkable properties of actions|simply transitively]] on the triangles in the building.
* There is a triangle Δ such that the stabiliser of its edges are the subgroups of order 3 generated by the ''σ''<sub>''i''</sub>'s.
* The stabiliser of a vertices of Δ is the [[Frobenius group]] of order 21 generated by the two order 3 elements stabilising the edges meeting at the vertex.
* The stabiliser of Δ is trivial.

The elements ''σ'' and ''τ'' generate the stabiliser of a vertex. The [[link (geometry)|link]] of this vertex can be identified with the spherical building of ''SL''<sub>3</sub>('''F'''<sub>2</sub>) and the stabiliser can be identified with the [[collineation|collineation group]] of the [[Fano plane]] generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying ''στ'' = ''τ''<sup>2</sup>''σ''. Identifying '''F'''<sub>8</sub>* with the Fano plane, σ can be taken to be the restriction of the [[Frobenius automorphism]] ''σ''(''x'') = ''x''<sup>22</sup> of '''F'''<sub>8</sub> and τ to be multiplication by any element not in the [[Characteristic (algebra)#Case of fields|prime field]] '''F'''<sub>2</sub>, i.e. an order 7 generator of the [[finite field#Applications|cyclic multiplicative group]] of '''F'''<sub>8</sub>. This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points. The formulas for σ and τ on ''E'' thus "lift" the formulas on '''F'''<sub>8</sub>.

Mumford also obtains an action [[Group action (mathematics)#Remarkable properties of actions|simply transitive]] on the vertices of the building by passing to a subgroup of Γ<sub>1</sub> = <''ρ'', ''σ'', ''τ'', −''I''>. The group Γ<sub>1</sub> preserves the '''Q'''(''α'')-valued Hermitian form

: ''f''(''x'',''y'') = ''xy''* + ''σ''(''xy''*) + ''σ''<sup>2</sup>(''xy''*)

on '''Q'''(ζ) and can be identified with ''U''<sub>3</sub>(f) <math>\cap</math> ''GL''<sub>3</sub>(''S'') where ''S'' = '''Z'''[''α'',{{sfrac|1|2}}]. Since ''S''/(''α'') = '''F'''<sub>7</sub>, there is a homomorphism of the group Γ<sub>1</sub> into ''GL''<sub>3</sub>('''F'''<sub>7</sub>). This action leaves invariant a 2-dimensional subspace in  '''F'''<sub>7</sub><sup>3</sup> and hence gives rise to a homomorphism ''Ψ'' of Γ<sub>1</sub> into ''SL''<sub>2</sub>('''F'''<sub>7</sub>), a group of order 16·3·7. On the other hand, the stabiliser of a vertex is a subgroup of order 21 and ''Ψ'' is injective on this subgroup. Thus if the [[congruence subgroup]] Γ<sub>0</sub> is defined as the [[inverse image]] under ''Ψ'' of the 2-[[Sylow subgroup]] of ''SL''<sub>2</sub>('''F'''<sub>7</sub>), the action of
Γ<sub>0</sub> on vertices must be simply transitive.

===Generalizations===
Other examples of triangles or 2-dimensional complexes of groups can be constructed by variations of the above example.

Cartwright et al. consider actions on buildings that are '''''simply transitive on vertices'''''. Each such action produces a bijection (or modified duality) between the points ''x'' and lines ''x''* in the [[flag (linear algebra)|flag complex]] of a finite [[projective plane]] and a collection of oriented triangles of points (''x'',''y'',''z''), invariant under cyclic permutation, such that ''x'' lies on ''z''*, ''y'' lies on ''x''* and ''z'' lies on ''y''* and any two points uniquely determine the third. The groups produced have generators ''x'', labelled by points, and relations ''xyz'' = 1 for each triangle. Generically this construction will not correspond to an action on a classical affine building.

More generally, as shown by Ballmann and Brin, similar algebraic data encodes all actions that are simply transitively on the vertices of a non-positively curved 2-dimensional simplicial complex, provided the link of each vertex has girth at least 6. This data consists of:
* a generating set ''S'' containing inverses, but not the identity;
*  a set of relations ''g'' ''h'' ''k'' = 1, invariant under cyclic permutation.

The elements ''g'' in ''S'' label the vertices ''g''·''v'' in the link of a fixed vertex ''v''; and the relations correspond to edges (''g''<sup>−1</sup>·''v'', ''h''·''v'') in that link. The graph with vertices ''S'' and edges (''g'', ''h''), for ''g''<sup>−1</sup>''h'' in ''S'', must have girth at least 6. The original simplicial complex can be reconstructed using complexes of groups and the second barycentric subdivision.

[[File:Heawood graph bipartite.png|thumb|right|The [[Heawood graph|bipartite Heawood graph]]]]
Further examples of non-positively curved 2-dimensional complexes of groups have been constructed by Swiatkowski based on actions '''''simply transitive on oriented edges''''' and inducing a 3-fold symmetry on each triangle; in this case too the complex of groups is obtained from the regular action on the second barycentric subdivision. The simplest example, discovered earlier with Ballmann, starts from a finite group ''H'' with a symmetric set of generators ''S'', not containing the identity, such that the corresponding [[Cayley graph]] has girth at least 6. The associated group is generated by ''H'' and an involution τ subject to (τg)<sup>3</sup> = 1 for each ''g'' in ''S''.

In fact, if Γ acts in this way, fixing an edge (''v'', ''w''), there is an involution τ interchanging ''v'' and ''w''. The link of ''v'' is made up of vertices ''g''·''w'' for ''g'' in a symmetric subset ''S'' of ''H'' =  Γ<sub>''v''</sub>, generating ''H'' if the link is connected. The assumption on triangles implies that

:τ·(''g''·''w'') = ''g''<sup>−1</sup>·''w''

for ''g'' in ''S''. Thus, if σ = τ''g'' and ''u'' = ''g''<sup>−1</sup>·''w'', then

:σ(''v'') = ''w'', σ(''w'') = ''u'', σ(''u'') = ''w''.

By simple transitivity on the triangle (''v'', ''w'', ''u''), it follows that σ<sup>3</sup> = 1.

The second barycentric subdivision gives a complex of groups consisting of singletons or pairs of barycentrically subdivided triangles joined along their large sides: these pairs are indexed by the quotient space ''S''/~ obtained by identifying inverses in ''S''. The single or "coupled" triangles are in turn joined along one common "spine". All stabilisers of simplices are trivial except for the two vertices at the ends of the spine, with stabilisers ''H'' and <τ>, and the remaining vertices of the large triangles, with stabiliser generated by an appropriate σ. Three of the smaller triangles in each large triangle contain transition elements.

When all the elements of ''S'' are involutions, none of the triangles need to be doubled. If ''H'' is taken to be the [[dihedral group]] ''D''<sub>7</sub> of order 14, generated by an involution ''a'' and an element ''b'' of order 7 such that

:''ab''= ''b''<sup>−1</sup>''a'',

then ''H'' is generated by the 3 involutions ''a'', ''ab'' and ''ab''<sup>5</sup>. The link of each vertex is given by the corresponding Cayley graph, so is just the [[Heawood graph|bipartite Heawood graph]], i.e. exactly the same as in the affine building for ''SL''<sub>3</sub>('''Q'''<sub>2</sub>). This link structure implies that the corresponding simplicial complex is necessarily a [[building (mathematics)|Euclidean building]]. At present, however, it seems to be unknown whether any of these types of action can in fact be realised on a classical affine building: Mumford's group Γ<sub>1</sub> (modulo scalars) is only simply transitive on edges, not on oriented edges.

==Two-dimensional orbifolds==

Two-dimensional orbifolds have the following three types of singular points:

* A boundary point
* An elliptic point or [[gyration point]] of order ''n'', such as the origin of '''R'''<sup>2</sup> quotiented out by a cyclic group of order ''n'' of rotations.
* A corner reflector of order ''n'': the origin of '''R'''<sup>2</sup> quotiented out by a dihedral group of order 2''n''.

A compact 2-dimensional orbifold has an '''[[Euler characteristic]]''' <math>\chi</math>
given by
:<math> \chi= \chi(X_0) - \sum_{i}(1 - 1/n_i)/2 - \sum_{i} (1 - 1/m_i )</math>,
where <math>\chi(X_0)</math> is the Euler characteristic of the underlying topological manifold <math>X_0</math>, and <math>n_i</math> are the orders of the corner reflectors, and <math>m_i</math> are the orders of the elliptic points.

A 2-dimensional compact connected orbifold has a hyperbolic structure if its Euler characteristic is less than 0, a Euclidean structure if it is 0, and if its Euler characteristic is positive it is either '''bad''' or has an elliptic structure (an orbifold is called bad if it does not have a manifold as a covering space). In other words, its universal covering space has a hyperbolic, Euclidean, or spherical structure.

The compact 2-dimensional connected orbifolds that are not hyperbolic are listed in the table below. The 17 parabolic orbifolds are the quotients of the plane by the 17 [[wallpaper group]]s.

{| class="wikitable"
|-----
! Type
! Euler characteristic
! Underlying 2-manifold
! Orders of elliptic points
! Orders of corner reflectors
|-----
| rowspan=4| Bad
| 1 + 1/''n''
| Sphere
| ''n'' &gt; 1
|
|-----
| 1/''m'' + 1/''n''
| Sphere
| ''n'' &gt; ''m'' &gt; 1
|
|-----
| 1/2 + 1/2''n''
| Disk
|
| ''n'' &gt; 1
|-----
| 1/2''m'' + 1/2''n''
| Disk
|
| ''n'' &gt; ''m'' &gt; 1
|-----
| rowspan=17 | Elliptic
| 2
| Sphere
|
|
|-----
| 2/''n''
| Sphere
| ''n'', ''n''
|
|-----
| 1/''n''
| Sphere
| 2, 2, ''n''
|
|-----
| 1/6
| Sphere
| 2, 3, 3
|
|-----
| 1/12
| Sphere
| 2, 3, 4
|
|-----
| 1/30
| Sphere
| 2, 3, 5
|
|-----
| 1
| Disc
|
|
|-----
| 1/''n''
| Disc
|
| ''n'', ''n''
|-----
| 1/2''n''
| Disc
|
| 2, 2, ''n''
|-----
| 1/12
| Disc
|
| 2, 3, 3
|-----
| 1/24
| Disc
|
| 2, 3, 4
|-----
| 1/60
| Disc
|
| 2, 3, 5
|-----
| 1/''n''
| Disc
| ''n''
|
|-----
| 1/2''n''
| Disc
| 2
| ''n''
|-----
| 1/12
| Disc
| 3
| 2
|-----
| 1
| Projective plane
|
|
|-----
| 1/''n''
| Projective plane
| ''n''
|
|-----
| rowspan=17 | Parabolic
| 0
| Sphere
| 2, 3, 6
|
|-----
| 0
| Sphere
| 2, 4, 4
|
|-----
| 0
| Sphere
| 3, 3, 3
|
|-----
| 0
| Sphere
| 2, 2, 2, 2
|
|-----
| 0
| Disk
|
| 2, 3, 6
|-----
| 0
| Disk
|
| 2, 4, 4
|-----
| 0
| Disk
|
| 3, 3, 3
|-----
| 0
| Disk
|
| 2, 2, 2, 2
|-----
| 0
| Disk
| 2
| 2, 2
|-----
| 0
| Disk
| 3
| 3
|-----
| 0
| Disk
| 4
| 2
|-----
| 0
| Disk
| 2, 2
|
|-----
| 0
| Projective plane
| 2, 2
|
|-----
| 0
| Torus
|
|
|-----
| 0
| Klein bottle
|
|
|-----
| 0
| Annulus
|
|
|-----
| 0
| Moebius band
|
|
|}

==3-dimensional orbifolds==
{{Expand section|date=July 2008}}
A 3-manifold is said to be ''small'' if it is closed, irreducible and does not contain any incompressible surfaces.

'''Orbifold Theorem.'''  Let ''M'' be a small 3-manifold. Let φ be a non-trivial periodic orientation-preserving diffeomorphism of ''M''. Then ''M'' admits a φ-invariant hyperbolic or Seifert fibered structure.

This theorem is a special case of Thurston's [[orbifold theorem]], announced without proof in 1981; it forms part of [[Thurston's geometrization conjecture|his geometrization conjecture for 3-manifolds]]. In particular it implies that if ''X'' is a compact, connected, orientable, irreducible, atoroidal 3-orbifold with non-empty singular locus, then ''M'' has a geometric structure (in the sense of orbifolds). A complete proof of the theorem was published by Boileau, Leeb & Porti in 2005.<ref>General introductions to this material can be found in Peter Scott's 1983 notes and the expositions of Boileau, Maillot & Porti and Cooper, Hodgson & Kerckhoff.</ref>

==Applications==

===Orbifolds in string theory===
In [[string theory]], the word "orbifold" has a slightly new meaning. For mathematicians, an orbifold is a generalization of the notion of [[manifold]] that allows the presence of the points whose neighborhood is [[Diffeomorphism|diffeomorphic]] to a quotient of '''R'''<sup>''n''</sup> by a finite group, i.e. '''R'''<sup>''n''</sup>/''Γ''. In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit space ''M''/''G'' where ''M'' is a manifold (or a theory), and ''G'' is a group of its isometries (or symmetries) — not necessarily all of them. In string theory, these symmetries do not have to have a geometric interpretation.

A [[quantum field theory]] defined on an orbifold becomes singular near the fixed points of ''G''. However string theory requires us to add new parts of the [[closed string]] [[Hilbert space]] — namely the twisted sectors where the fields defined on the closed strings are periodic up to an action from ''G''. Orbifolding is therefore a general procedure of string theory to derive a new string theory from an old string theory in which the elements of ''G'' have been identified with the identity. Such a procedure reduces the number of states because the states must be invariant under ''G'', but it also increases the number of states because of the extra twisted sectors. The result is usually a perfectly smooth, new string theory.

[[D-branes]] propagating on the orbifolds are described, at low energies, by gauge theories defined by the [[quiver diagram]]s. Open strings attached to these [[D-branes]] have no twisted sector, and so the number of open string states is reduced by the orbifolding procedure.

More specifically, when the orbifold group ''G'' is a discrete subgroup of spacetime isometries, then if it has no fixed point, the result is usually a compact smooth space; the twisted sector consists of closed strings wound around the compact dimension, which are called ''winding states''.

When the orbifold group G is a discrete subgroup of spacetime isometries, and it has fixed points, then these usually have [[Gravitational singularity#Conical|conical singularities]], because '''R<sup>''n''</sup>/[[Cyclic group|Z<sub>''k''</sub>]]''' has such a singularity at the fixed point of [[Cyclic group|''Z''<sub>''k''</sub>]]. In string theory, gravitational singularities are usually a sign of extra [[Degrees of freedom (physics and chemistry)|degrees of freedom]] which are located at a locus point in spacetime. In the case of the orbifold these [[Degrees of freedom (physics and chemistry)|degrees of freedom]] are the twisted states, which are strings "stuck" at the fixed points. When the fields related with these twisted states acquire a non-zero [[vacuum expectation value]], the singularity is deformed, i.e. the metric is changed and becomes regular at this point and around it. An example for a resulting geometry is the [[Gravitational instanton#Eguchi–Hanson|Eguchi–Hanson]] spacetime.

From the point of view of D-branes in the vicinity of the fixed points, the effective theory of the open strings attached to these D-branes is a supersymmetric field theory, whose space of vacua has a singular point, where additional massless degrees of freedom exist. The fields related with the closed string twisted sector couple to the open strings in such a way as to add a Fayet–Iliopoulos term to the supersymmetric field theory Lagrangian, so that when such a field acquires a non-zero [[vacuum expectation value]], the Fayet–Iliopoulos term is non-zero, and thereby deforms the theory (i.e. changes it) so that the singularity no longer exists [https://arxiv.org/abs/hep-th/9603167], [http://www-spires.fnal.gov/spires/find/hep/www?j=NUPHA,B342,246].

====Calabi–Yau manifolds====
{{Main article|Calabi–Yau manifold}}
In [[superstring theory]],<ref>M. Green, J. Schwartz and E. Witten, ''Superstring theory'', Vol. 1 and 2, Cambridge University Press, 1987, ISBN
0521357527</ref><ref>J. Polchinski, ''String theory'', Vol. 2, Cambridge University Press, 1999, {{isbn|0-521-63304-4}}</ref>
the construction of realistic [[Phenomenology (particle physics)|phenomenological models]] requires [[dimensional reduction]] because the strings naturally propagate in a 10-dimensional space whilst the observed dimension of [[space-time]] of the universe is 4. Formal constraints on the theories nevertheless place restrictions on the [[Compactification (physics)|compactified space]] in which the extra "hidden" variables live: when looking for realistic 4-dimensional models with [[supersymmetry]], the auxiliary compactified space must be a 6-dimensional [[Calabi–Yau manifold]].<ref>P. Candelas, ''Lectures On Complex Manifolds'', in *Trieste 1987, Proceedings, Superstrings '87* 1-88, 1987</ref>

There are a large number of possible Calabi–Yau manifolds (tens of thousands), hence the use of the term "[[String theory landscape|landscape]]" in the current theoretical physics literature to describe the baffling choice. The general study of Calabi–Yau manifolds is mathematically complex and for a long time examples have been hard to construct explicitly. Orbifolds have therefore proved very useful since they automatically satisfy the constraints imposed by supersymmetry. They provide degenerate examples of Calabi–Yau manifolds due to their [[singularity (mathematics)|singular points]],<ref>{{citation|title=Basic Concepts of String Theory|series=Theoretical and Mathematical Physics|first1=Ralph|last1=Blumenhagen|first2=Dieter|last2=Lüst|first3=Stefan|last3=Theisen|publisher=Springer|year=2012|isbn=9783642294969|url=https://books.google.com/books?id=-3PNFQn6AzcC&pg=PA487|page=487|bibcode=2013bcst.book.....B|quote=Orbifolds can be viewed as singular limits of smooth Calabi–Yau manifolds}}.</ref> but this is completely acceptable from the point of view of theoretical physics. Such orbifolds are called "supersymmetric": they are technically easier to study than general Calabi–Yau manifolds. It is very often possible to associate a continuous family of non-singular Calabi–Yau manifolds to a singular supersymmetric orbifold. In 4 dimensions this can be illustrated using complex [[K3 surface]]s:

:*Every K3 surface admits 16 cycles of dimension 2 that are topologically equivalent to usual 2-spheres. Making the surface of these spheres tend to zero, the K3 surface develops 16 singularities. This limit represents a point on the boundary of the [[moduli space]] of K3 surfaces and corresponds to the orbifold <math>T^4/\mathbb{Z}_2\,</math> obtained by taking the quotient of the torus by the symmetry of inversion.

The study of Calabi–Yau manifolds in string theory and the duality between different models of string theory (type IIA and IIB) led to the idea of [[mirror symmetry (string theory)|mirror symmetry]] in 1988. The role of orbifolds was first pointed out by Dixon, Harvey, Vafa and Witten around the same time.<ref>{{Cite journal|last1=Dixon|first1=L.|last2=Harvey|first2=J. A.|last3=Vafa|first3=C.|last4=Witten|first4=E.|date=1985-01-01|title=Strings on orbifolds|url=https://dx.doi.org/10.1016/0550-3213%2885%2990593-0|journal=Nuclear Physics B|language=en|volume=261|pages=678–686|doi=10.1016/0550-3213(85)90593-0|bibcode=1985NuPhB.261..678D |issn=0550-3213|url-access=subscription}}</ref>

===Music theory===
Beyond their manifold<!-- pun intended --> and various applications in mathematics and physics, orbifolds have been applied to [[music theory]] at least as early as 1985 in the work of [[Guerino Mazzola]]<ref name="Mazzola1985">{{cite book |last1=Mazzola |first1=Guerino |title=Gruppen und Kategorien in der Musik: Entwurf einer mathematischen Musiktheorie |url=https://books.google.com/books?id=ktPuAAAAMAAJ |access-date=26 February 2012 |year=1985 |publisher=Heldermann |isbn=978-3-88538-210-2}}</ref><ref name="MazzolaMüller2002">{{cite book |last1=Mazzola |first1=Guerino |last2=Müller |first2=Stefan |title=The topos of music: geometric logic of concepts, theory, and performance |url=https://books.google.com/books?id=6I9U9-Rls8oC |access-date=26 February 2012 |year=2002 |publisher=Birkhäuser |isbn=978-3-7643-5731-3}}</ref> and later by [[Dmitri Tymoczko]] and collaborators.{{sfn|Tymoczko|2006}}{{sfn|Callender|Quinn|Tymoczko|2008}}<ref>Dmitri Tymoczko, ''[http://dmitri.mycpanel.princeton.edu/sciencearticle.html The Geometry of Music]'' – links to papers and to visualization software.</ref><ref>''[http://quomodocumque.wordpress.com/2008/02/28/the-moduli-space-of-chords-dmitri-tymoczko-on-geometry-and-music-friday-7-mar-230pm/ The moduli space of chords: Dmitri Tymoczko on "Geometry and Music", Friday 7 Mar, 2:30pm],'' posted 28/Feb/08 – talk abstract and high-level mathematical description.</ref> One of the papers of Tymoczko was the first music theory paper published by the journal ''[[Science (journal)|Science]].''<ref>Michael D. Lemonick, ''[https://web.archive.org/web/20070204041551/http://www.time.com/time/magazine/article/0,9171,1582330,00.html The Geometry of Music],'' ''[[Time (magazine)|Time]],'' 26 January 2007</ref><ref>Elizabeth Gudrais, ''[http://harvardmagazine.com/2007/01/mapping-music.html Mapping Music],'' Harvard Magazine, Jan/Feb 2007</ref><ref>Tony Phillips, ''[https://www.ams.org/mathmedia/archive/10-2006-media.html Tony Phillips' Take on Math in the Media] {{Webarchive|url=https://web.archive.org/web/20081005194933/https://www.ams.org/mathmedia/archive/10-2006-media.html |date=5 October 2008 }},'' [[American Mathematical Society]], October 2006</ref> Mazzola and Tymoczko have participated in debate regarding their theories documented in a series of commentaries available at their respective web sites.<ref>{{cite web| url = http://www.encyclospace.org/special/answer_to_tymoczko.pdf| title=On D. Tymoczko's critique of Mazzola's counterpoint theory|first1=Octavio Alberto |last1=Agustín-Aquino|first2=Guerino|last2= Mazzola|date=June 14, 2011}}</ref><ref>{{cite web|url=http://dmitri.tymoczko.com/files/publications/mazzola.pdf|title=Mazzola's Counterpoint Theory|first=Dmitri|last=Tymoczko|access-date=27 February 2012|archive-date=17 October 2016|archive-url=https://web.archive.org/web/20161017195221/http://dmitri.tymoczko.com/files/publications/mazzola.pdf|url-status=dead}}</ref>

<!-- Deleted image removed: [[File:Topological III by Robert R. Wilson, at Harvard University.JPG|upright|thumb|Musical triads can be modeled as points in a certain 3-dimensional orbifold, which may be realized as a solid torus with triangular cross-section and a twist.]] -->
[[File:3-orbifold-voronoi-animated.gif|thumbnail|Animated slices of the three-dimensional orbifold <math>T^3 / S_3</math>.

Slices of cubes standing on end (with their long diagonals perpendicular to the plane of the image) form colored [[Voronoi diagram|Voronoi]] regions (colored by chord type) which represent the three-note chords at their centers, with [[augmented triad]]s at the very center, surrounded by major and minor [[Triad (music)|triads]] (lime green and navy blue). The white regions are degenerate trichords (one-note repeated three times), with the three lines (representing two note chords) connecting their centers forming the walls of the twisted triangular prism, 2D planes perpendicular to plane of the image acting as mirrors.]]

Tymoczko models musical chords consisting of ''n'' notes, which are not necessarily distinct, as points in the orbifold <math>T^n/S_n</math> – the space of ''n'' unordered points (not necessarily distinct) in the circle, realized as the quotient of the ''n''-[[torus]] <math>T^n</math> (the space of ''n'' ''ordered'' points on the circle) by the [[symmetric group]] <math>S_n</math> (corresponding from moving from an ordered set to an unordered set).

Musically, this is explained as follows:
* Musical tones depend on the frequency (pitch) of their fundamental, and thus are parametrized by the positive real numbers, '''R'''<sup>+</sup>.
* Musical tones that differ by an octave (a doubling of frequency) are considered the same tone – this corresponds to taking the [[logarithm]] base 2 of frequencies (yielding the real numbers, as <math>\mathbf{R} = \log_2 \mathbf{R}^+</math>), then quotienting by the integers (corresponding to differing by some number of octaves), yielding a circle (as <math>S^1 = \mathbf{R}/\mathbf{Z}</math>).
* Chords correspond to multiple tones without respect to order – thus ''t'' notes (with order) correspond to ''t'' ordered points on the circle, or equivalently a single point on the ''t''-torus <math>T^t := S^1 \times \cdots \times S^1,</math> and omitting order corresponds to taking the quotient by <math>S_t,</math> yielding an orbifold.

For [[dyad (music)|dyads]] (two tones), this yields the closed [[Möbius strip]]; for [[triad (music)|triads]] (three tones), this yields an orbifold that can be described as a triangular prism with the top and bottom triangular faces identified with a 120° twist (a {{sfrac|1|3}} twist) – equivalently, as a solid torus in 3 dimensions with a cross-section an equilateral triangle and such a twist.

The resulting orbifold is naturally stratified by repeated tones (properly, by integer partitions of ''t'') – the open set consists of distinct tones (the partition <math>t = 1 + 1 + \cdots + 1</math>), while there is a 1-dimensional singular set consisting of all tones being the same (the partition <math>t = t</math>), which topologically is a circle, and various intermediate partitions. There is also a notable circle which runs through the center of the open set consisting of equally spaced points. In the case of triads, the three side faces of the prism correspond to two tones being the same and the third different (the partition <math>3 = 2 + 1</math>), while the three edges of the prism correspond to the 1-dimensional singular set. The top and bottom faces are part of the open set, and only appear because the orbifold has been cut – if viewed as a triangular torus with a twist, these artifacts disappear.

Tymoczko argues that chords close to the center (with tones equally or almost equally spaced) form the basis of much of traditional Western harmony, and that visualizing them in this way assists in analysis. There are 4 chords on the center (equally spaced under [[equal temperament]] – spacing of 4/4/4 between tones), corresponding to the [[augmented triad]]s (thought of as [[Set (music)|musical sets]]) C♯FA, DF♯A♯, D♯GB, and EG♯C (then they cycle: FAC♯ = C♯FA), with the 12 [[major chord]]s and 12 [[minor chord]]s being the points next to but not on the center – almost evenly spaced but not quite. Major chords correspond to 4/3/5 (or equivalently, 5/4/3) spacing, while minor chords correspond to 3/4/5 spacing. Key changes then correspond to movement between these points in the orbifold, with smoother changes effected by movement between nearby points.

==See also==
* [[Branched covering]]
* [[Euler characteristic of an orbifold]]
* [[Geometric quotient]]
* [[Kawasaki's Riemann–Roch formula]]
* [[Orbifold notation]]
* [[Orientifold]]
* [[Ring of modular forms]]
* [[Stack (mathematics)]]

==Notes==
{{Reflist|30em}}

==References==
{{refbegin}}
* {{cite book |first1=Alejandro |last1=Adem |first2=Johann |last2=Leida |first3=Yongbin |last3=Ruan |title=Orbifolds and Stringy Topology |series=Cambridge Tracts in Mathematics |volume=171 |publisher=[[Cambridge University Press]] |date=2007 |doi=10.1017/CBO9780511543081 |isbn=9780521870047 }}
* {{cite book |first=Werner |last=Ballmann |editor1-first=Étienne |editor1-last=Ghys |editor1-link=Étienne Ghys |editor2-first=Pierre |editor2-last=de La Harpe |date=1990 |chapter=Singular spaces of non-positive curvature |pages=189–201 |title=Sur les groupes hyperboliques d'après Mikhael Gromov |series=Progress in Mathematics |volume=83 |publisher=Birkhäuser |publication-place=Boston |isbn=0-8176-3508-4 }}
* {{cite journal |doi=10.1007/BF01265309 |doi-access=free |last1=Ballmann |first1=Werner |last2=Brin |first2=Michael |year=1994 |title=Polygonal complexes and combinatorial group theory |journal=[[Geometriae Dedicata]] |volume=50 |issue=2 |pages=165–191 |s2cid=119617693 }}
* {{cite web |first=Michel |last=Boileau |url=http://www.crm.es/Publications/Quaderns/Quadern25-1.pdf |title=Geometrizations of 3-manifolds with symmetries |access-date=6 December 2007 |archive-url=https://web.archive.org/web/20110930020447/http://www.crm.es/Publications/Quaderns/Quadern25-1.pdf |archive-date=30 September 2011 |url-status=dead }}
* {{cite book |first1=Michel |last1=Boileau |first2=Sylvain |last2=Maillot |first3=Joan |last3=Porti |date=2003 |title=Three-dimensional orbifolds and their geometric structures |series=Panoramas and Syntheses |volume=15 |publisher=Société Mathématique de France|publication-place=Paris |isbn=2-85629-152-X |oclc=56349823 }}.
* {{cite journal | doi = 10.4007/annals.2005.162.195 | last1 = Boileau | first1 = Michel | last2 = Leeb | first2 = Bernhard | last3 = Porti | first3 = Joan | year = 2005 | title = Geometrization of 3-dimensional orbifolds | journal =  Annals of Mathematics | volume = 162 | pages = 195–290 | arxiv = math/0010185 | s2cid = 119624092 }}
* {{cite book |first=Glen |last=Bredon |author-link=Glen Bredon |title=Introduction to Compact Transformation Groups |publisher=[[Academic Press]] |year=1972 |isbn=0-12-128850-1 }}
* {{cite book |last1=Bridson |first1=Martin |last2=Haefliger |first2=André |title=Metric Spaces of Non-Positive Curvature |series=Grundlehren der mathematischen Wissenschaften |volume=319 |date=1999 |publisher=Springer |isbn=3-540-64324-9 |doi=10.1007/978-3-662-12494-9 |doi-access=free}}
* {{cite arXiv |first=Matthew |last=Brin |date=2007 |title=Seifert Fibered Spaces: Notes for a course given in the Spring of 1993 |class=math.GT |eprint=0711.1346 }}
* {{Cite journal | title = Generalized Voice-Leading Spaces | first1 = Clifton | last1 = Callender | first2 = Ian | last2 = Quinn | first3 = Dmitri | last3 = Tymoczko | journal = [[Science (journal)|Science]] | volume = 320 | issue = 5874 | pages = 346–348 | date = 18 April 2008 | url = http://www.music.princeton.edu/~dmitri/voiceleading.pdf | doi = 10.1126/science.1153021 | pmid = 18420928 | bibcode = 2008Sci...320..346C | s2cid = 35229232 | archive-date = 17 May 2008 | access-date = 24 January 2010 | archive-url = https://web.archive.org/web/20080517002520/http://music.princeton.edu/~dmitri/voiceleading.pdf | url-status = dead }}
* {{cite journal | last1=Cartwright | first1=Donald | last2=Mantero | first2=Anna Maria | last3=Steger | first3=Tim | last4=Zappa | first4=Anna | date=1993 | title=Groups acting simply transitively on the vertices of a building of type Ã{{sub|2}}, I | journal=[[Geometriae Dedicata]] | volume=47 | issue=2 | pages=143–166 | doi=10.1007/BF01266617 | doi-access=free}}
* {{cite book |first1=Daryl |last1=Cooper |first2=Craig |last2=Hodgson |first3=Steven |last3=Kerckhoff |date=2000 |title=Three-dimensional orbifolds and cone-manifolds |series=MSJ Memoirs |volume=5 |publisher=Mathematical Society of Japan |publication-place=Tokyo |isbn=4-931469-05-1 }}.
* {{cite conference |first=Pierre |last=de la Harpe |title=An invitation to Coxeter group |pages=193–253 |conference=Group theory from a geometrical viewpoint, 26 March - 6 April 1990, ICPT, Trieste, Italy (proceedings) |editor-last1=Ghys |editor-first1=Étienne |editor-link1=Étienne Ghys |date=1991 |page= |publisher=World Scientific |publication-place=Singapore |isbn=981-02-0442-6 |doi=10.1142/1235 }}.
* {{cite book |last1=Di Francesco |first1=Philippe |last2=Mathieu |first2=Pierre
|last3=Sénéchal |first3=David |title=Conformal field theory |series=Graduate Texts in Contemporary Physics |publisher=Springer-Verlag |date=1997 |isbn=0-387-94785-X |doi=10.1007/978-1-4612-2256-9 |doi-access=free}}.

* {{cite book |last=Haefliger |first=André |title=Groupoides d'holonomie et classifiants |series=Astérisque |publisher=Société Mathématique de France |publication-place=Paris |volume=116 |date=1984 |pages=70–97 }}
* {{cite book |last1=Haefliger |first1=André |chapter=Orbi-espaces |pages=203–213 |title=Sur les groupes hyperboliques d'après Mikhael Gromov |editor-last1=Ghys |editor-first1=Étienne |editor-link1=Étienne Ghys |editor-last2=de la Harpe |editor-first2=Pierre |series=Progress in Mathematics |volume=83 |date=1990 |publisher=Birkhäuser |isbn=0-8176-3508-4 |doi=10.1007/978-1-4684-9167-8_11 |doi-access=free }}
* {{cite conference |first=André |last=Haefliger |title=Complexes of groups and orbihedra |pages=504–540 |conference=Group theory from a geometrical viewpoint, 26 March - 6 April 1990, ICPT, Trieste, Italy (proceedings) |editor-last1=Ghys |editor-first1=Étienne |editor-link1=Étienne Ghys |date=1991 |page= |publisher=World Scientific |publication-place=Singapore |isbn=981-02-0442-6 |doi=10.1142/1235 }}
* {{cite journal |first1=Patrick |last1=Iglesias |first2=Yael |last2=Karshon |first3=Moshe |last3=Zadka |title=Orbifolds as Diffeologies |journal=Transactions of the American Mathematical Society |volume=362 |issue=6 |pages=2811–2831 |date=June 2010 |doi=10.1090/S0002-9947-10-05006-3 |s2cid=15210173 |arxiv=math/0501093 }}
* {{cite book |first=Patrick |last=Iglesias-Zemmour |title=Diffeology |publisher=American Mathematical Society |series=Mathematical Surveys and Monographs |date=2013 |isbn= 978-0-8218-9131-5 }}
* {{cite journal |first1=Patrick |last1=Iglesias-Zemmour |first2=Jean-Pierre |last2=Laffineur |title=Noncommutative Geometry & Diffeology: The Case Of Orbifolds | journal=Journal of Noncommutative Geometry |volume=12 |issue=4 |pages= 1551–1572 |date=2017 |doi=10.4171/JNCG/319 |s2cid=126092375 }}
* {{cite book|first=Katsuo |last=Kawakubo |title=The Theory of Transformation Groups |publisher=[[Oxford University Press]] |date=1991 |isbn=0-19-853212-1}}
* {{cite journal |last1=Köhler |first1=Peter |last2=Meixner |first2=Thomas |last3=Wester |first3=Michael |date=1985 |title=The 2-adic affine building of type Ã{{sub|2}} and its finite projections |journal=[[Journal of Combinatorial Theory]] |series=Series A |volume=38 |issue=2 |pages=203–209 |doi=10.1016/0097-3165(85)90070-6 |doi-access=free }}
* {{cite journal |first=David |last=Mumford |date=1979 |title=An algebraic surface with K ample, (K<sup>2</sup>) = 9, p<sub>g</sub> = q = 0 |journal=American Journal of Mathematics |volume=101 |issue=1 |pages=233–244 |doi=10.2307/2373947 |jstor=2373947 |url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:3612860 }}
* {{cite book|last1=Poincaré |first1=Henri |author-link=Henri Poincaré |title=Papers on Fuchsian functions |translator-last1=Stillwell |translator-first1=John |translator-link1=John Stillwell |publisher=Springer |date=1985 |isbn=3-540-96215-8 }}
* {{cite journal |last=Satake |first=Ichirô |author-link=Ichirô Satake |date=1956 |title=On a generalization of the notion of manifold |journal=[[Proceedings of the National Academy of Sciences of the United States of America]] |volume=42 |issue=6 |pages=359–363 |pmc=528292 |bibcode=1956PNAS...42..359S | doi=10.1073/pnas.42.6.359 |doi-access=free |pmid=16578464 }}
* {{cite journal |last=Satake |first=Ichirô |author-link=Ichirô Satake |date=1957 |title=The Gauss-Bonnet theorem for V-manifolds |journal=Journal of the Mathematical Society of Japan |volume=9 |issue=4| pages=464–492 |doi=10.2969/jmsj/00940464 |doi-access=free }}
* {{cite journal |last1=Scott |first1=Peter |author-link1=G. Peter Scott |title=The geometries of 3-manifolds |journal=Bulletin of the London Mathematical Society |volume=15 |date=1983 |issue=5 |pages=401–487 |doi=10.1112/blms/15.5.401 |hdl=2027.42/135276 |url=http://www.math.lsa.umich.edu/~pscott/8geoms.pdf }} <br> Errata: {{cite web |last1=Scott |first1=Peter |author-link1=G. Peter Scott |title=Errata for "The geometries of 3-manifolds", Bull. London Math. Soc. 15 (1983), 401-487 |url=http://www.math.lsa.umich.edu/~pscott/errata8geoms.pdf }}
* {{cite book |first=Jean-Pierre |last=Serre |author-link=Jean-Pierre Serre |title=Cours d'arithmétique |publisher=Presse Universitaire de France |date=1970 }}
* {{cite book |first=Jean-Pierre |last=Serre |author-link=Jean-Pierre Serre |title=Trees |publisher=Springer |publication-place=Berlin |date=2003 |translator-first=John |translator-last=Stillwell |translator-link=John Stillwell |doi=10.1007/978-3-642-61856-7 |isbn=978-3-642-61858-1 }} <br> English translation of: {{cite book |first=Jean-Pierre |last=Serre |author-link=Jean-Pierre Serre |title=Arbres, amalgames, SL<sub>2</sub> |edition=3rd |series=Astérisque |publisher=Société Mathématique de France |publication-place=Paris |volume=46 |date=1983 }}
* {{cite conference |first=John |last=Stallings |title=Triangles of groups |pages=491–503 |conference=Group theory from a geometrical viewpoint, 26 March - 6 April 1990, ICPT, Trieste, Italy (proceedings) |editor-last1=Ghys |editor-first1=Étienne |editor-link1=Étienne Ghys |date=1991 |page= |publisher=World Scientific |publication-place=Singapore |isbn=981-02-0442-6 |doi=10.1142/1235 }}
* {{cite journal |doi=10.1093/qjmath/52.2.231 |last1=Świątkowski |first1=Jacek |date=2001 |title=A class of automorphism groups of polygonal complexes |journal= [[Quarterly Journal of Mathematics]] |volume=52 |issue=2 |pages=231–247 }}
* {{cite book |first=William |last=Thurston |author-link=William Thurston |url=http://msri.org/publications/books/gt3m/ |title=The Geometry and Topology of Three-Manifolds |series=Princeton University lecture notes |date=1978–1981 }}
* {{cite journal |doi=10.1090/S0273-0979-1982-15003-0 |last=Thurston |first=William |author-link=William Thurston |date=1982 |title=Three-dimensional manifolds, Kleinian groups and hyperbolic geometry |journal=[[Bulletin of the American Mathematical Society]] |volume=6 |issue=3 |pages=357–381 |doi-access=free }}
* {{Cite journal |first=Dmitri |last=Tymoczko |journal=[[Science (journal)|Science]] |date=7 July 2006 |volume=313 |issue=5783 |pages=72–74 |doi=10.1126/science.1126287 |url=http://www.brainmusic.org/EducationalActivitiesFolder/Tymoczko_chords2006.pdf |title=The Geometry of Musical Chords |pmid=16825563 |bibcode=2006Sci...313...72T |citeseerx=10.1.1.215.7449 |s2cid=2877171 }}
{{refend}}

{{Manifolds}}{{String theory topics |state=collapsed}}
{{Authority control}}

[[Category:Differential topology]]
[[Category:Generalized manifolds]]
[[Category:Group actions]]