mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a cusp neighborhood is defined as a set of points near a cusp singularity.

Cusp neighborhood for a Riemann surface

The cusp neighborhood for a hyperbolic

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

can be defined in terms of its

Fuchsian model In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface ''R'' as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazar ...

. Suppose that the

Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or co ...

''G'' contains a parabolic element g. For example, the element ''t'' ∈ SL(2,Z) where :

t(z)=\begin 1 & 1 \\ 0 & 1 \end:z = \frac = z+1

is a parabolic element. Note that all parabolic elements of SL(2,C) are conjugate to this element. That is, if ''g'' ∈ SL(2,Z) is parabolic, then

g=h^th

for some ''h'' ∈ SL(2,Z). The set :

U=\

where H is the

upper half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

has :

\gamma(U) \cap U = \emptyset

for any

\gamma \in G - \langle g \rangle

where

\langle g \rangle

is understood to mean the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

generated by ''g''. That is, γ acts

properly discontinuously In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under funct ...

on ''U''. Because of this, it can be seen that the projection of ''U'' onto H/''G'' is thus :

E = U/ \langle g \rangle

. Here, ''E'' is called the neighborhood of the cusp corresponding to g. Note that the hyperbolic area of ''E'' is exactly 1, when computed using the canonical

Poincaré metric In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry ...

. This is most easily seen by example: consider the intersection of ''U'' defined above with the

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each ...

\left\

of the

modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...

, as would be appropriate for the choice of ''T'' as the parabolic element. When integrated over the

volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form \ma ...

d\mu=\frac

the result is trivially 1. Areas of all cusp neighborhoods are equal to this, by the invariance of the area under conjugation.

References

{{reflist Hyperbolic geometry Riemann surfaces

Cusp neighborhood for a Riemann surface

See also

References