Orbifold
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.
The simplest approach (adopted by Haefliger and known also to Thurston) extends the usual notion of loop used in the standard definition of the fundamental group.
In general it is an extension of Γ by π1 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.
For applications in geometric group theory, it is often convenient to have a slightly more general notion of orbifold, due to Haefliger.
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.
In this case each orbispace chart is usually required to be a length space with unique geodesics connecting any two points.
Applying this to constant paths in an orbispace chart, it follows that each local homomorphism is injective and hence: Every orbifold has associated with it an additional combinatorial structure given by 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.
A straightforward inductive argument shows that such an action becomes regular on the barycentric subdivision; in particular 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.
The underlying geometric method of analysing finitely presented groups in terms of metric spaces of non-positive curvature is due to Gromov.
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.
Define K-linear operators on E as follows: The elements ρ, σ, and τ generate a discrete subgroup of GL3(K) which acts properly on the affine Bruhat–Tits building corresponding to SL3(Q2).
This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points.
Mumford also obtains an action simply transitive on the vertices of the building by passing to a subgroup of Γ1 = <ρ, σ, τ, −I>.
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.
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 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".
Let φ be a non-trivial periodic orientation-preserving diffeomorphism of M. Then M admits a φ-invariant hyperbolic or Seifert fibered structure.
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).
For mathematicians, an orbifold is a generalization of the notion of manifold that allows the presence of the points whose neighborhood is diffeomorphic to a quotient of Rn by 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 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.
D-branes propagating on the orbifolds are described, at low energies, by gauge theories defined by the quiver diagrams.
In string theory, gravitational singularities are usually a sign of extra degrees of freedom which are located at a locus point in spacetime.
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.
In superstring theory,[21][22] the construction of realistic 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.
[23] 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.
[25] Beyond their manifold 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[26][27] and later by Dmitri Tymoczko and collaborators.
[32][33][34] Mazzola and Tymoczko have participated in debate regarding their theories documented in a series of commentaries available at their respective web sites.
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.
