## Crib Sheet on Mathematical Groups

**general linear group**of degree

*n*is the set of

*n*×

*n*invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

**R**(the set of real numbers) is the group of

*n*×

*n*invertible matrices of real numbers, and is denoted by GL

_{n}(

**R**) or GL(

*n*,

**R**).

**special linear group**, written SL(

*n*,

*F*) or SL

_{n}(

*F*), is the subgroup of GL(

*n*,

*F*) consisting of matrices with a determinant of 1, with the group operations of ordinary matrix multiplication and matrix inversion. Matrices of this type form a group since the determinant of the product of two matrices is the product of the determinants of each matrix. The special linear group SL(

*n*,

**R**) can be characterized as the group of

*volume and orientation preserving*linear transformations of

**R**

^{n}; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.

**SL(2,R)**or

**SL**is the group of 2 × 2 real matrices with determinant one. SL(2,

_{2}(R)**R**) acts on the complex upper half-plane by fractional linear transformations

**SL(2,R)**on C, which takes R U ∞ to itself and stabilizes the upper and lower half-planes. Intuition: It takes R to itself because the entries are real. R U ∞ is the boundary of the upper half-plane. The transformation is orientation preserving; as you progress along the real line in the positive direction, the upper half-plane is on your left. This relationship is preserved by any Möbius transformation. So we require that the real line is not only mapped to itself, but it is mapped to itself in the same direction.

**Möbius group**, which is the projective linear group PGL(2,

**C**).

**PGL(2,C)**consists of 2×2 matrices with non-zero determinant: Möbius transformations are defined on the extended complex plane,

**C***. Stereographic projection identifies

**C***with a sphere, which is then called the Riemann sphere; alternatively,

**C***can be thought of as the complex projective line

**CP1**. The Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they are the automorphisms of

**CP1**as an algebraic variety. Therefore, the set of all Möbius transformations forms a group under composition. This group is called the Möbius group, and is sometimes denoted,

**Aut(C*)**.

*c*which we shall consider as a point (kind of a coordinate) in CP¹. Thus there is a one-to-one correspondence between points in CP¹ and lines in C².

**C**embeds naturally in the complex projective line

**CP¹**, the set of complex lines (one-dimensional subspaces) of

**CP¹**. The embedding maps a point z in

**C**to the complex line spanned by (z, 1), seen as a point in

**CP¹**; we call

*z*the

*inhomogeneous coordinate*for this point, while any pair

*(tz, t) in*

**C²**, with t in

**C*=C**\{0}, is called a set of

*homogeneous coordinates*for it.

**C²**give the same projective transformation of

**CP¹**if and only if one is a scalar multiple of the other. Thus, identifying together scalar multiples in the linear group GL(2,

**C**) give the group of projective transformations of

**CP¹**which we naturally denote by PGL(2,

**C**) = GL(2,

**C**)/

**C***. This group is also known as PSL(2,

**C**), because it can be obtained by identifying together scalar multiples in the special linear group SL(2,

**C**), consisting of linear transformations of

**C²**with unit determinant.

**CP¹**.

**Kleinian Group**– is a discrete subgroup of PSL(2,

**C**). (Note that PSL(2,C) ≅ PGL(2,C) but PSL(2,R) is not ≅ to PGL(2,R). I’m not sure why. Thurston mentions it above and there are articles in math overflow about it). Discreteness implies points in

*B*

^{3}have finite stabilizers, and discrete orbits under the group

*G*. But the orbit

*Gp*of a point

*p*will typically accumulate on the boundary of the closed ball

**B3**.

*T*be a periodictessellation of hyperbolic 3-space. The group of symmetries of the tessellation is a Kleinian group

**Fuchsian Group**– Any Fuchsian group (a discrete subgroup of SL(2,

**R**)) is a Kleinian group, and conversely any Kleinian group preserving the real line (in its action on the Riemann sphere) is a Fuchsian group. More generally, any Kleinian group preserving a circle or straight line in the Riemann sphere is conjugate to a Fuchsian group. “Indra’s Pearls” says that any group which can be conjugated to a group meeting the above requirements is also a Fuschian group, which seems reasonable. In other words, the traces of Fuscian matrices must be real; traces don’t change under conjugation.

**quasi-Fuchsian group**is a Kleinian group whose limit set is contained in an invariant Jordan curve. In other words the limit set divides the ordinary set into an inside and an outside. The limit set can be crinkly. The special case when the Jordan curve is a circle or line is called a Fuchsian group.

**modular group**is the projective special linear group

*PSL*(2,

**Z**) of 2 x 2 matrices with integer coefficients and unit determinant. It acts on the upper half of the complex plane because Z is real. Some authors

*define*the modular group to be PSL(2,

**Z**), and still others define the modular group to be the larger group SL(2,

**Z**).

**H**(the upper half plane).

*S*and

*T*. Geometrically,

*S*represents inversion in the unit circle followed by reflection with respect to the imaginary axis, while

*T*represents a unit translation to the right.

The matrices [0,1,-1,0] and [1,1,1,0] generate SL(2Z). Both matrices have determinant = 1. That means they are volume and orientation preserving.

The analogous Möbius transforms:

Γ(*N*) is a normal subgroup of the modular group Γ. The group Γ(*N*) is given as the set of all modular transformations

for which *a* ≡ *d* ≡ ±1 (mod *N*) and *b* ≡ *c* ≡ 0 (mod *N*).

The principal congruence subgroup of level 2, Γ(2), is also called the **modular group Λ**. Since PSL(2, **Z**/2**Z**) is isomorphic to *S*_{3}, Λ is a subgroup of index 6. The group Λ consists of all modular transformations for which *a* and *d* are odd and *b* and *c* are even.

The subgroups , sometimes called the *Hecke congruence subgroup* of level n, is defined as the preimage by of the group of upper triangular matrices. That is,

- .

The subgroups are the preimage of the subgroup of unipotent matrices:

The *theta subgroup* is the congruence subgroup of defined as the preimage of the cyclic group of order two generated by in . It is of index 3 and is explicitly described by:^{[2]}

- .

It is also called the Modular Group Lambda. It is the set of transforms where a and d are odd and b and c are even. Same as above.

The group of 2 by 2 matrices with determinant 1, SL_{2}(Z) acts on the upper half complex plane:

_{2}(Z). These triangles are used to construct the Fundamental domains for other subgroups of SL

_{2}(Z). For this nomenclature:

_{2}(Z) of the following forms respectively:

from [3]

**SU(2)**is the group of unit quaternions. They can be used to define regular polyhedra. A M ̈obius transformation is an isometry of the Riemann sphere (for the chordal metric) if, and only if, it is represented by a 2×2 matrix that is a member of

**SU(2)**. This chordal metric is nothing more than the Euclidean metric between the two points that z1 and z2 are mapped to on the unit sphere by stereographic projection.

By Arkady L Kholodenko, page 360.