Orbifold

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 manifoldead,"we held a vote, and" orbifold "won.

Thurston (1978–1981,p. 300, section 13.2) explaining the origin of the word "orbifold"

23star Orbifold Example

In the mathematical disciplines oftopologyandgeometry,anorbifold(for "orbit-manifold" ) is a generalization of amanifold.Roughly speaking, an orbifold is atopological spacewhich is locally afinite groupquotient of aEuclidean space.

Definitions of orbifold have been given several times: byIchirō Satakein the context ofautomorphic formsin the 1950s under the nameV-manifold;byWilliam Thurstonin the context of the geometry of3-manifoldsin the 1970swhen he coined the nameorbifold,after a vote by his students; and byAndré Haefligerin the 1980s in the context ofMikhail Gromov's programme onCAT(k) spacesunder the nameorbihedron.

Historically, orbifolds arose first as surfaces withsingular pointslong before they were formally defined.One of the first classical examples arose in the theory ofmodular formswith the action of themodular group$\mathrm {SL} (2,\mathbb {Z} )$on theupper half-plane:a version of theRiemann–Roch theoremholds after the quotient is compactified by the addition of two orbifold cusp points. In3-manifoldtheory, the theory ofSeifert fiber spaces,initiated byHerbert Seifert,can be phrased in terms of 2-dimensional orbifolds.Ingeometric group theory,post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces.

Instring theory,the word "orbifold" has a slightly different meaning,discussed in detail below. Intwo-dimensional conformal field theory,it refers to the theory attached to the fixed point subalgebra of avertex algebraunder the action of a finite group ofautomorphisms.

The main example of underlying space is a quotient space of a manifold under theproperly discontinuousaction of a possibly infinitegroupofdiffeomorphismswith finiteisotropy subgroups.In particular this applies to any action of afinite group;thus amanifold with boundarycarries a natural orbifold structure, since it is the quotient of itsdoubleby an action of$\mathbb {Z} _{2}$.

One topological space can carry different orbifold structures. For example, consider the orbifoldOassociated with a quotient space of the 2-sphere along a rotation by$\pi$;it ishomeomorphicto 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, theorbifoldfundamental groupofOis$\mathbb {Z} _{2}$and itsorbifoldEuler characteristicis 1.

Formal definitions

Definition using orbifold atlas

Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled onopen subsetsof$\mathbb {R} ^{n}$,an orbifold is locally modelled on quotients of open subsets of$\mathbb {R} ^{n}$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 theisotropy subgroups.

An$n$-dimensionalorbifoldis aHausdorff topological space$X$,called theunderlying space,with a covering by a collection of open sets$U_{i}$,closed under finite intersection. For each$U_{i}$,there is

• an open subset$V_{i}$of$\mathbb {R} ^{n}$,invariant under afaithfullinear action of a finite group$\Gamma _{i}$;
• a continuous map$\varphi _{i}$of$V_{i}$onto$U_{i}$invariant under$\Gamma _{i}$,called anorbifold chart,which defines a homeomorphism between$V_{i}/\Gamma _{i}$and$U_{i}$.

The collection of orbifold charts is called anorbifold atlasif the following properties are satisfied:

• for each inclusion$U_{i}\subset U_{j}$there is aninjectivegroup homomorphism$f_{ij}:\Gamma _{i}\rightarrow \Gamma _{j}$.
• for each inclusion$U_{i}\subset U_{j}$there is a$\Gamma _{i}$-equivarianthomeomorphism$\psi _{ij}$,called agluing map,of$V_{i}$onto an open subset of$V_{j}$.
• the gluing maps are compatible with the charts, i.e.$\varphi _{j}\circ \psi _{ij}=\varphi _{i}$.
• the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from$V_{i}$to$V_{j}$has the form$g\circ \psi _{ij}$for a unique$g\in \Gamma _{j}$.

As foratlases on manifolds,two orbifold atlases of$X$are equivalent if they can be consistently combined to give a larger orbifold atlas. Anorbifold structureis 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. IfU_i$\subset$U_j$\subset$U_k,then there is a uniquetransition elementg_ijkin Γ_ksuch that

g_ijk·ψ_ik=ψ_jk·ψ_ij

These transition elements satisfy

(Adg_ijk)·f_ik=f_jk·f_ij

as well as thecocycle relation(guaranteeing associativity)

f_km(g_ijk)·g_ikm=g_ijm·g_jkm.

More generally, attached to an open covering of an orbifold by orbifold charts, there is the combinatorial data of a so-calledcomplex 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 adifferentiable orbifold.It will be aRiemannian orbifoldif in addition there are invariantRiemannian metricson the orbifold charts and the gluing maps areisometries.

Definition using Lie groupoids

Recall that agroupoidconsists of a set of objects$G_{0}$,a set of arrows$G_{1}$,and structural maps including the source and the target maps$s,t:G_{1}\to G_{0}$and other maps allowing arrows to be composed and inverted. It is called aLie groupoidif both$G_{0}$and$G_{1}$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$x\in G_{0}$,i.e. the set$(G_{1})_{x}:=s^{-1}(x)\cap t^{-1}(x)$,is theLie groupcalled theisotropy groupof$G_{1}$at$x$.A Lie groupoid is calledproperif the map$(s,t):G_{1}\to G_{0}\times G_{0}$is aproper map,andétaleif both the source and the target maps arelocal diffeomorphisms.

Anorbifold groupoidis given by one of the following equivalent definitions:

• a proper étale Lie groupoid;
• a proper Lie groupoid whose isotropies arediscrete spaces.

Since the isotropy groups of proper groupoids are automaticallycompact,the discreteness condition implies that the isotropies must be actuallyfinite groups.

Orbifold groupoids play the same role as orbifold atlases in the definition above. Indeed, anorbifold structureon a Hausdorff topological space$X$is defined as theMorita equivalenceclass of an orbifold groupoid$G\rightrightarrows M$together with a homeomorphism$|M/G|\simeq X$,where$|M/G|$is the orbit space of the Lie groupoid$G$(i.e. the quotient of$M$by the equivalent relation when$x\sim y$if there is a$g\in G$with$s(g)=x$and$t(g)=y$). This definition shows that orbifolds are a particular kind ofdifferentiable stack.

Relation between the two definitions

Given an orbifold atlas on a space$X$,one can build apseudogroupmade up by all diffeomorphisms between open sets of$X$which preserve the transition functions$\varphi _{i}$.In turn, the space$G_{X}$of germs of its elements is an orbifold groupoid. Moreover, since by definition of orbifold atlas each finite group$\Gamma _{i}$acts faithfully on$V_{i}$,the groupoid$G_{X}$is automatically effective, i.e. the map$g\in (G_{X})_{x}\mapsto \mathrm {germ} _{x}(t\circ s^{-1})$is injective for every$x\in X$.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 aclassical orbifold) is a special kind of orbifold structure according to the second definition.

Conversely, given an orbifold groupoid$G\rightrightarrows M$,there is a canonical orbifold atlas over its orbit space, whose associated effective orbifold groupoid is Morita equivalent to$G$.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.

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_iis thetrivial group.Equivalently, it corresponds to the Morita equivalence class of the unit groupoid.
• IfNis a compactmanifold with boundary,itsdoubleMcan be formed by gluing together a copy ofNand its mirror image along their common boundary. There is naturalreflectionaction ofZ₂on the manifoldMfi xing the common boundary; the quotient space can be identified withN,so thatNhas a natural orbifold structure.
• IfMis a Riemanniann-manifold with acocompactproperisometric action of a discrete group Γ, then the orbit spaceX=M/Γ has natural orbifold structure: for eachxinXtake a representativeminMand an open neighbourhoodV_mofminvariant under the stabiliser Γ_m,identified equivariantly with a Γ_m-subset ofT_mMunder the exponential map atm;finitely many neighbourhoods coverXand each of their finite intersections, if non-empty, is covered by an intersection of Γ-translatesg_m·V_mwith corresponding groupg_mΓg_m⁻¹.Orbifolds that arise in this way are calleddevelopableorgood.
• A classical theorem ofHenri PoincaréconstructsFuchsian groupsas hyperbolicreflection groupsgenerated by reflections in the edges of ageodesic trianglein thehyperbolic planefor thePoincaré metric.If the triangle has anglesπ/n_ifor positive integersn_i,the triangle is afundamental domainand naturally a 2-dimensional orbifold. The corresponding group is an example of a hyperbolictriangle group.Poincaré also gave a 3-dimensional version of this result forKleinian groups:in this case the Kleinian group Γ is generated by hyperbolic reflections and the orbifold isH³/ Γ.
• IfMis a closed 2-manifold, new orbifold structures can be defined onMi by removing finitely many disjoint closed discs fromMand gluing back copies of discsD/ Γ_iwhereDis the closedunit discand Γ_iis a finite cyclic group of rotations. This generalises Poincaré's construction.

Orbifold fundamental group

There are several ways to define theorbifold fundamental group.More sophisticated approaches use orbifoldcovering spacesorclassifying spacesofgroupoids.The simplest approach (adopted by Haefliger and known also to Thurston) extends the usual notion ofloopused in the standard definition of thefundamental group.

Anorbifold pathis 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 anorbifold 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 byhomotopy classesof orbifold loops.

If the orbifold arises as the quotient of asimply connectedmanifoldMby a proper rigid action of a discrete group Γ, the orbifold fundamental group can be identified with Γ. In general it is anextensionof Γ byπ₁M.

The orbifold is said to bedevelopableorgoodif it arises as the quotient by a group action; otherwise it is calledbad.Auniversal covering orbifoldcan be constructed for an orbifold by direct analogy with the construction of theuniversal covering spaceof 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 acontractibleopen subset corresponds to a group Γ, then there is a naturallocal homomorphismof 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 ofdiffeologyand have been proved to be equivalenttoIchirô Satake's original definition:

Definition:An orbifold is a diffeological space locally diffeomorphic at each point to some$\mathbb {R} ^{n}/G$,where$n$is an integer and$G$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$\mathbb {R} ^{n}$.
• 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$\mathbb {R} ^{n}/G$with$G$a finite linear group.
• This definition is equivalentwith Haefliger orbifolds.
• {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:"The notion of$C^{\infty }$-map thus defined is inconvenient in the point that a composite of two$C^{\infty }$-map defined in a different choice of defining families is not always a$C^{\infty }$-map."Indeed, there are smooth maps between orbifolds that do not lift locally as equivariant maps.

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 groupoidand its isotropy groups.

Orbispaces

For applications ingeometric group theory,it is often convenient to have a slightly more general notion of orbifold, due to Haefliger. Anorbispaceis 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 alocally compactspace with arigidaction 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 properlinear subspace.) It is also useful to considermetric spacestructures on an orbispace, given by invariantmetricson the orbispace charts for which the gluing maps preserve distance. In this case each orbispace chart is usually required to be alength spacewith uniquegeodesicsconnecting any two points.

LetXbe 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 oforbispace fundamental groupanduniversal 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 isnon-positively curved,then theBirkhoff curve shortening argumentcan 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 acomplex of groups.

Definition

Acomplex of groups(Y,f,g) on anabstract simplicial complexYis given by

• a finite group Γ_σfor each simplex σ ofY
• an injective homomorphismf_στ:Γ_τ$\rightarrow$Γ_σwhenever σ$\subset$τ
• for every inclusion ρ$\subset$σ$\subset$τ, a group elementg_ρστin Γ_ρsuch that (Adg_ρστ)·f_ρτ=f_ρσ·f_στ(here Ad denotes theadjoint actionby conjugation)

The group elements must in addition satisfy the cocycle condition

f_πρ(g_ρστ)g_πρτ=g_πστg_πρσ

for every chain of simplices$\pi \subset \rho \subset \sigma \subset \tau.$(This condition is vacuous ifYhas dimension 2 or less.)

Any choice of elementsh_στin Γ_σyields anequivalentcomplex of groups by defining

• f'_στ= (Adh_στ)·f_στ
• g'_ρστ=h_ρσ·f_ρσ(h_στ)·g_ρστ·h_ρτ⁻¹

A complex of groups is calledsimplewheneverg_ρστ= 1 everywhere.

• An easy inductive argument shows that every complex of groups on asimplexis equivalent to a complex of groups withg_ρστ= 1 everywhere.

It is often more convenient and conceptually appealing to pass to thebarycentric subdivisionofY.The vertices of this subdivision correspond to the simplices ofY,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

IfXis an orbifold (or orbispace), choose a covering by open subsets from amongst the orbifold chartsf_i:V_i$\rightarrow$U_i.LetYbe the abstract simplicial complex given by thenerve of the covering:its vertices are the sets of the cover and itsn-simplices correspond tonon-emptyintersectionsU_α=U_i1$\cap$···$\cap$U_in.For each such simplex there is an associated group Γ_αand the homomorphismsf_ijbecome the homomorphismsf_στ.For every triple ρ$\subset$σ$\subset$τ corresponding to intersections

$U_{i}\supset U_{i}\cap U_{j}\supset U_{i}\cap U_{j}\cap U_{k}$

there are chartsφ_i:V_i$\rightarrow$U_i,φ_ij:V_ij$\rightarrow$U_i$\cap$U_jand φ_ijk:V_ijk$\rightarrow$U_i$\cap$U_j$\cap$U_kand gluing maps ψ:V_ij$\rightarrow$V_i,ψ':V_ijk$\rightarrow$V_ijand ψ ":V_ijk$\rightarrow$V_i.

There is a unique transition elementg_ρστin Γ_isuch thatg_ρστ·ψ"=ψ·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-commutativesheaf theoryandgerbes,the complex of groups in this case arises as asheaf of groupsassociated to the coveringU_i;the datag_ρστis a 2-cocycle in non-commutativesheaf cohomologyand the datah_στgives a 2-coboundary perturbation.

Edge-path group

Theedge-path groupof a complex of groups can be defined as a natural generalisation of theedge path groupof a simplicial complex. In the barycentric subdivision ofY,take generatorse_ijcorresponding to edges fromitojwherei$\rightarrow$j,so that there is an injection ψ_ij:Γ_i$\rightarrow$Γ_j.Let Γ be the group generated by thee_ijand Γ_kwith relations

e_ij⁻¹·g·e_ij= ψ_ij(g)

forgin Γ_iand

e_ik=e_jk·e_ij·g_ijk

ifi$\rightarrow$j$\rightarrow$k.

For a fixed vertexi₀,the edge-path group Γ(i₀) is defined to be the subgroup of Γ generated by all products

g₀· e_i0i1·g₁· e_i1i2· ··· ·g_n· e_ini0

wherei₀,i₁,...,i_n,i₀ is an edge-path,g_klies in Γ_ikande_ji=e_ij⁻¹ifi$\rightarrow$j.

Developable complexes

A simplicialproper actionof a discrete group Γ on asimplicial complexXwith finite quotient is said to beregularif it satisfies one of the following equivalent conditions:

• Xadmits a finite subcomplex asfundamental domain;
• the quotientY=X/Γ has a natural simplicial structure;
• the quotient simplicial structure on orbit-representatives of vertices is consistent;
• if (v₀,...,v_k) and (g₀·v₀,...,g_k·v_k) are simplices, theng·v_i=g_i·v_ifor somegin Γ.

The fundamental domain and quotientY=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 groupsYis said to bedevelopableif it arises in this way.

• A complex of groups is developable if and only if the homomorphisms of Γ_σinto 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_σfrom Γ_σinto a fixed discrete group Γ such that θ_τ·f_στ= θ_σ.In this case the simplicial complexXis canonically defined: it hask-simplices (σ, xΓ_σ) where σ is ak-simplex ofYandxruns over Γ / Γ_σ.Consistency can be checked using the fact that the restriction of the complex of groups to asimplexis equivalent to one with trivial cocycleg_ρστ.

The action of Γ on the barycentric subdivisionX' ofXalways satisfies the following condition, weaker than regularity:

• whenever σ andg·σ are subsimplices of some simplex τ, they are equal, i.e. σ =g·σ

Indeed, simplices inX' correspond to chains of simplices inX,so that a subsimplices, given by subchains of simplices, is uniquely determined by thesizesof the simplices in the subchain. When an action satisfies this condition, thengnecessarily 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 subdivisionX"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 inX".

There is in fact no need to pass to athirdbarycentric subdivision: as Haefliger observes using the language ofcategory theory,in this case the 3-skeleton of the fundamental domain ofX"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 ofX"has the same structure as the barycentric subdivisionY' of a complex of groupsY,namely:

• a finite 2-dimensional simplicial complexZ;
• an orientation for all edgesi$\rightarrow$j;
• ifi$\rightarrow$jandj$\rightarrow$kare edges, theni$\rightarrow$kis 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 subdivisionZ' and its edge-path group is isomorphic to that ofZ.

Orbihedra

If a countable discrete group acts by aregularsimplicialproper actionon asimplicial 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

LetXbe a finite simplicial complex with barycentric subdivisionX'. Anorbihedronstructure consists of:

• for each vertexiofX', a simplicial complexL_i' endowed with a rigid simplicial action of a finite group Γ_i.
• a simplicial map φ_iofL_i' onto thelinkL_iofiinX', identifying the quotientL_i' / Γ_iwithL_i.

This action of Γ_ionL_i' extends to a simplicial action on the simplicial coneC_ioverL_i' (the simplicial join ofiandL_i'), fi xing the centreiof the cone. The map φ_iextends to a simplicial map of C_ionto thestarSt(i) ofi,carrying the centre ontoi;thus φ_iidentifiesC_i/ Γ_i,the quotient of the star ofiinC_i,with St(i) and gives anorbihedron chartati.

• for each directed edgei$\rightarrow$jofX', an injective homomorphismf_ijof Γ_iinto Γ_j.
• for each directed edgei$\rightarrow$j,a Γ_iequivariant simplicialgluing mapψ_ijofC_iintoC_j.
• the gluing maps are compatible with the charts, i.e. φ_j·ψ_ij= φ_i.
• the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map fromV_itoV_jhas the formg·ψ_ijfor a uniquegin Γ_j.

Ifi$\rightarrow$j$\rightarrow$k,then there is a uniquetransition elementg_ijkin Γ_ksuch that

g_ijk·ψ_ik= ψ_jk·ψ_ij

These transition elements satisfy

(Adg_ijk)·f_ik=f_jk·f_ij

as well as the cocycle relation

ψ_km(g_ijk)·g_ikm=g_ijm·g_jkm.

Main properties

• The group theoretic data of an orbihedron gives a complex of groups onX,because the verticesiof the barycentric subdivisionX' correspond to the simplices inX.
• Every complex of groups onXis associated with an essentially unique orbihedron structure onX.This key fact follows by noting that the star and link of a vertexiofX', corresponding to a simplex σ ofX,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 inXand the link of the barycentre of σ in σ'. Restricting the complex of groups to the link of σ inX,all the groups Γ_τcome with injective homomorphisms into Γ_σ.Since the link ofiinX' is canonically covered by a simplicial complex on which Γ_σacts, this defines an orbihedron structure onX.
• 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 thesimplicial approximation theoremto segments of an orbispace path lying in an orbispace chart: it is a straightforward variant of the classical proof that thefundamental groupof apolyhedroncan be identified with itsedge-path group.
• The orbispace associated to an orbihedron has acanonical 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 inhyperbolic space,with simplices identified isometrically along common boundaries.
• The orbispace associated to an orbihedron isnon-positively curvedif and only if the link in each orbihedron chart hasgirthgreater 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 ofHadamard 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 ingeometric group theoryhas been totriangles of groups.This is the simplest 2-dimensional example generalising the 1-dimensional "interval of groups" discussed inSerre's lectures on trees, whereamalgamated free productsare studied in terms of actions on trees. Such triangles of groups arise any time a discrete group acts simply transitively on the triangles in theaffine Bruhat–Tits buildingforSL₃(Q_p); in 1979Mumforddiscovered the first example forp= 2 (see below) as a step in producing analgebraic surfacenot isomorphic toprojective space,but having the sameBetti numbers.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.

Atriangle of groupsis asimplecomplex of groups consisting of a triangle with verticesA,B,C.There are groups

• Γ_A,Γ_B,Γ_Cat each vertex
• Γ_BC,Γ_CA,Γ_ABfor each edge
• Γ_ABCfor the triangle itself.

There is an injective homomorphisms of Γ_ABCinto all the other groups and of an edge group Γ_XYinto Γ_Xand Γ_Y.The three ways of mapping Γ_ABCinto a vertex group all agree. (Often Γ_ABCis 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 vertexA,say, as the length of the smallest word in the kernel of the natural homomorphism into Γ_Aof theamalgamated free productover Γ_ABCof the edge groups Γ_ABand Γ_AC:

$\Gamma _{AB}\star _{\,\Gamma _{ABC}}\Gamma _{AC}\rightarrow \Gamma _{A}.$

The result using the Euclidean metric structure is not optimal. Angles α, β, γ at the verticesA,BandCwere 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 thehyperbolic planewith thePoincaré 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,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 thecolimitof the triangle of groups, is infinite;
• the homomorphisms of the vertex groups into the edge-path group are injections.

Mumford's example

The Fano plane

Letα=${\sqrt {-7}}$be given by thebinomial expansionof (1 − 8)^1/2inQ₂and setK=Q(α)$\subset$Q₂.Let

ζ= exp 2πi/7

λ= (α− 1)/2 =ζ+ζ²+ζ⁴

μ=λ/λ*.

LetE=Q(ζ), a 3-dimensional vector space overKwith basis 1,ζ,andζ².DefineK-linear operators onEas follows:

• σis the generator of theGalois groupofEoverK,an element of order 3 given by σ(ζ) = ζ²
• τis the operator of multiplication byζonE,an element of order 7
• ρis the operator given byρ(ζ) = 1,ρ(ζ²) =ζandρ(1) =μ·ζ²,so thatρ³is scalar multiplication byμ.

The elementsρ,σ,andτgenerate a discrete subgroup ofGL₃(K) which actsproperlyon theaffine Bruhat–Tits buildingcorresponding toSL₃(Q₂). This group actstransitivelyon all vertices, edges and triangles in the building. Let

σ₁=σ,σ₂=ρσρ⁻¹,σ₃=ρ²σρ⁻².

Then

• σ₁,σ₂andσ₃generate a subgroup Γ ofSL₃(K).
• Γ is the smallest subgroup generated byσandτ,invariant under conjugation byρ.
• Γ actssimply transitivelyon 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 theσ_i's.
• The stabiliser of a vertices of Δ is theFrobenius groupof order 21 generated by the two order 3 elements stabilising the edges meeting at the vertex.
• The stabiliser of Δ is trivial.

The elementsσandτgenerate the stabiliser of a vertex. Thelinkof this vertex can be identified with the spherical building ofSL₃(F₂) and the stabiliser can be identified with thecollineation groupof theFano planegenerated by a 3-fold symmetry σ fi xing a point and a cyclic permutation τ of all 7 points, satisfyingστ=τ²σ.IdentifyingF₈* with the Fano plane, σ can be taken to be the restriction of theFrobenius automorphismσ(x) =x²²ofF₈and τ to be multiplication by any element not in theprime fieldF₂,i.e. an order 7 generator of thecyclic multiplicative groupofF₈.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 onEthus "lift" the formulas onF₈.

Mumford also obtains an actionsimply transitiveon the vertices of the building by passing to a subgroup of Γ₁= <ρ,σ,τ,−I>. The group Γ₁preserves theQ(α)-valued Hermitian form

f(x,y) =xy* +σ(xy*) +σ²(xy*)

onQ(ζ) and can be identified withU₃(f)$\cap$GL₃(S) whereS=Z[α,⁠1/2⁠]. SinceS/(α) =F₇,there is a homomorphism of the group Γ₁intoGL₃(F₇). This action leaves invariant a 2-dimensional subspace inF₇³and hence gives rise to a homomorphismΨof Γ₁intoSL₂(F₇), a group of order 16·3·7. On the other hand, the stabiliser of a vertex is a subgroup of order 21 andΨis injective on this subgroup. Thus if thecongruence subgroupΓ₀is defined as theinverse imageunderΨof the 2-Sylow subgroupofSL₂(F₇), the action of Γ₀on 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 aresimply transitive on vertices.Each such action produces a bijection (or modified duality) between the pointsxand linesx* in theflag complexof a finiteprojective planeand a collection of oriented triangles of points (x,y,z), invariant under cyclic permutation, such thatxlies onz*,ylies onx* andzlies ony* and any two points uniquely determine the third. The groups produced have generatorsx,labelled by points, and relationsxyz= 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 setScontaining inverses, but not the identity;
• a set of relationsghk= 1, invariant under cyclic permutation.

The elementsginSlabel the verticesg·vin the link of a fixed vertexv;and the relations correspond to edges (g⁻¹·v,h·v) in that link. The graph with verticesSand edges (g,h), forg⁻¹hinS,must have girth at least 6. The original simplicial complex can be reconstructed using complexes of groups and the second barycentric subdivision.

Thebipartite Heawood graph

Further examples of non-positively curved 2-dimensional complexes of groups have been constructed by Swiatkowski based on actionssimply transitive on oriented edgesand 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 groupHwith a symmetric set of generatorsS,not containing the identity, such that the correspondingCayley graphhas girth at least 6. The associated group is generated byHand an involution τ subject to (τg)³= 1 for eachginS.

In fact, if Γ acts in this way, fi xing an edge (v,w), there is an involution τ interchangingvandw.The link ofvis made up of verticesg·wforgin a symmetric subsetSofH= Γ_v,generatingHif the link is connected. The assumption on triangles implies that

τ·(g·w) =g⁻¹·w

forginS.Thus, if σ = τgandu=g⁻¹·w,then

σ(v) =w,σ(w) =u,σ(u) =w.

By simple transitivity on the triangle (v,w,u), it follows that σ³= 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 spaceS/~ obtained by identifying inverses inS.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 stabilisersHand <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 ofSare involutions, none of the triangles need to be doubled. IfHis taken to be thedihedral groupD₇of order 14, generated by an involutionaand an elementbof order 7 such that

ab=b⁻¹a,

thenHis generated by the 3 involutionsa,abandab⁵.The link of each vertex is given by the corresponding Cayley graph, so is just thebipartite Heawood graph,i.e. exactly the same as in the affine building forSL₃(Q₂). This link structure implies that the corresponding simplicial complex is necessarily aEuclidean 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 Γ₁(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 orgyration pointof ordern,such as the origin ofR²quotiented out by a cyclic group of ordernof rotations.
• A corner reflector of ordern:the origin ofR²quotiented out by a dihedral group of order 2n.

A compact 2-dimensional orbifold has anEuler characteristic$\chi$ given by

$\chi =\chi (X_{0})-\sum _{i}(1-1/n_{i})/2-\sum _{i}(1-1/m_{i})$,

where$\chi (X_{0})$is the Euler characteristic of the underlying topological manifold$X_{0}$,and$n_{i}$are the orders of the corner reflectors, and$m_{i}$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 eitherbador 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 17wallpaper groups.

Type	Euler characteristic	Underlying 2-manifold	Orders of elliptic points	Orders of corner reflectors
Bad	1 + 1/n	Sphere	n> 1
	1/m+ 1/n	Sphere	n>m> 1
	1/2 + 1/2n	Disk		n> 1
	1/2m+ 1/2n	Disk		n>m> 1
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
	1/2n	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
	1/2n	Disc	2	n
	1/12	Disc	3	2
	1	Projective plane
	1/n	Projective plane	n
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

A 3-manifold is said to besmallif it is closed, irreducible and does not contain any incompressible surfaces.

Orbifold Theorem.LetMbe a small 3-manifold. Let φ be a non-trivial periodic orientation-preserving diffeomorphism ofM.ThenMadmits a φ-invariant hyperbolic or Seifert fibered structure.

This theorem is a special case of Thurston'sorbifold theorem,announced without proof in 1981; it forms part ofhis geometrization conjecture for 3-manifolds.In particular it implies that ifXis a compact, connected, orientable, irreducible, atoroidal 3-orbifold with non-empty singular locus, thenMhas a geometric structure (in the sense of orbifolds). A complete proof of the theorem was published by Boileau, Leeb & Porti in 2005.

Applications

Orbifolds in string theory

Instring theory,the word "orbifold" has a slightly new meaning. For mathematicians, an orbifold is a generalization of the notion ofmanifoldthat allows the presence of the points whose neighborhood isdiffeomorphicto a quotient ofRⁿby a finite group, i.e.Rⁿ/Γ.In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit spaceM/GwhereMis a manifold (or a theory), andGis 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.

Aquantum field theorydefined on an orbifold becomes singular near the fixed points ofG.However string theory requires us to add new parts of theclosed stringHilbert space— namely the twisted sectors where the fields defined on the closed strings are periodic up to an action fromG.Orbifolding is therefore a general procedure of string theory to derive a new string theory from an old string theory in which the elements ofGhave been identified with the identity. Such a procedure reduces the number of states because the states must be invariant underG,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-branespropagating on the orbifolds are described, at low energies, by gauge theories defined by thequiver diagrams.Open strings attached to theseD-braneshave no twisted sector, and so the number of open string states is reduced by the orbifolding procedure.

More specifically, when the orbifold groupGis 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 calledwinding states.

When the orbifold group G is a discrete subgroup of spacetime isometries, and it has fixed points, then these usually haveconical singularities,becauseRⁿ/Z_khas such a singularity at the fixed point ofZ_k.In string theory, gravitational singularities are usually a sign of extradegrees of freedomwhich are located at a locus point in spacetime. In the case of the orbifold thesedegrees of freedomare the twisted states, which are strings "stuck" at the fixed points. When the fields related with these twisted states acquire a non-zerovacuum 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 theEguchi–Hansonspacetime.

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-zerovacuum 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[1],[2].

Calabi–Yau manifolds

Main article:Calabi–Yau manifold

Insuperstring theory, the construction of realisticphenomenological modelsrequiresdimensional reductionbecause the strings naturally propagate in a 10-dimensional space whilst the observed dimension ofspace-timeof the universe is 4. Formal constraints on the theories nevertheless place restrictions on thecompactified spacein which the extra "hidden" variables live: when looking for realistic 4-dimensional models withsupersymmetry,the auxiliary compactified space must be a 6-dimensionalCalabi–Yau manifold.

There are a large number of possible Calabi–Yau manifolds (tens of thousands), hence the use of the term "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 theirsingular points,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 complexK3 surfaces:

• 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 themoduli spaceof K3 surfaces and corresponds to the orbifold$T^{4}/\mathbb {Z} _{2}\,$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 ofmirror symmetryin 1988. The role of orbifolds was first pointed out by Dixon, Harvey, Vafa and Witten around the same time.

Music theory

Beyond their manifold and various applications in mathematics and physics, orbifolds have been applied tomusic theoryat least as early as 1985 in the work ofGuerino Mazzolaand later byDmitri Tymoczkoand collaborators.One of the papers of Tymoczko was the first music theory paper published by the journalScience.Mazzola and Tymoczko have participated in debate regarding their theories documented in a series of commentaries available at their respective web sites.

Animated slices of the three-dimensional orbifold$T^{3}/S_{3}$.Slices of cubes standing on end (with their long diagonals perpendicular to the plane of the image) form coloredVoronoiregions (colored by chord type) which represent the three-note chords at their centers, withaugmented triadsat the very center, surrounded by major and minortriads(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 ofnnotes, which are not necessarily distinct, as points in the orbifold$T^{n}/S_{n}$– the space ofnunordered points (not necessarily distinct) in the circle, realized as the quotient of then-torus$T^{n}$(the space ofnorderedpoints on the circle) by thesymmetric group$S_{n}$(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⁺.
• Musical tones that differ by an octave (a doubling of frequency) are considered the same tone – this corresponds to taking thelogarithmbase 2 of frequencies (yielding the real numbers, as$\mathbf {R} =\log _{2}\mathbf {R} ^{+}$), then quotienting by the integers (corresponding to differing by some number of octaves), yielding a circle (as$S^{1}=\mathbf {R} /\mathbf {Z}$).
• Chords correspond to multiple tones without respect to order – thustnotes (with order) correspond totordered points on the circle, or equivalently a single point on thet-torus$T^{t}:=S^{1}\times \cdots \times S^{1},$and omitting order corresponds to taking the quotient by$S_{t},$yielding an orbifold.

Fordyads(two tones), this yields the closedMöbius strip;fortriads(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 (a⁠1/3⁠twist) – 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 oft) – the open set consists of distinct tones (the partition$t=1+1+\cdots +1$), while there is a 1-dimensional singular set consisting of all tones being the same (the partition$t=t$), 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$3=2+1$), 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 underequal temperament– spacing of 4/4/4 between tones), corresponding to theaugmented triads(thought of asmusical sets) C♯FA, DF♯A♯, D♯GB, and EG♯C (then they cycle: FAC♯ = C♯FA), with the 12major chordsand 12minor chordsbeing 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)

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