Crib Sheet on Mathematical Groups

This is just a bunch of quotes I’ve pulled off of Wikipedia and elsewhere. I tried to get a handle on some of the basic groups used in hyperbolic tilings and complex analysis.
In mathematics, the 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.
To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GLn(R) or GL(nR).
The special linear group, written SL(nF) or SLn(F), is the subgroup of GL(nF) 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(nR) can be characterized as the group of volume and orientation preserving linear transformations of Rn; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.
the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one. SL(2,R) acts on the complex upper half-plane by fractional linear transformations
There is an action of 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.
The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2,C).
The Möbius group 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*).
The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds.
From [2]:
We already know that S² is the same object as CP¹. The result should be understood as follows. Consider CP¹ made out of equivalence classes [z0:z1]=c[z0:z1]. This means that each equivalent class is characterized by a nonzero complex number 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.
Thurston [1] continues:
The complex plane 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 , with t in C*=C\{0}, is called a set of homogeneous coordinates for it.
Two non-singular linear maps of  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  with unit determinant.
A Möbius transformation of S^n-1 can be extended to a unique isometry of H^n. Since PGL(2,C) actor on S² by  Möbius transformations, this action can be extended to all of H³… The group of orientation-preserving isometries of H³ is PGL(2,C), identified via the action on S²=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 B3 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 ballB3.
Let 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.
A 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.
The 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).
It is a subgroup of PSL(2,R) and therefore the modular group is a subgroup of the group of orientation-preserving isometries of (the upper half plane).
Note that any member of the modular group maps the projectively extended real line one-to-one to itself, and furthermore bijectively maps the projectively extended rational line (the rationals with infinity) to itself, the irrationals to the irrationals, the transcendental numbers to the transcendental numbers, the non-real numbers to the non-real numbers, the upper half-plane to the upper half-plane, et cetera.
The modular group can be shown to be generated by the two transformations:
S: z -> -1/z
T: z -> z + 1
so that every element in the modular group can be represented (in a non-unique way) by the composition of powers of 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:

    S: z -> -1/z
    T: z -> z + 1
generate the modular group. They are not volume preserving in the upper half plane but they are in the hyperbolic plane: all tiles are the same size in the hyperbolic plane.

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 ad ≡ ±1 (mod N) and bc ≡ 0 (mod N).

The principal congruence subgroup of level 2, Γ(2), is also called the modular group Λ. Since PSL(2, Z/2Z) is isomorphic to S3, 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 \Gamma _{0}(n), sometimes called the Hecke congruence subgroup of level n, is defined as the preimage by  \pi _{n} of the group of upper triangular matrices. That is,

 {\displaystyle \Gamma _{0}(n)=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \Gamma :c\equiv 0{\pmod {n}}\right\}}.

The subgroups {\displaystyle \Gamma _{1}(n)} are the preimage of the subgroup of unipotent matrices:

 {\displaystyle \Gamma _{1}(n)=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \Gamma :a,d\equiv 1{\pmod {n}},c\equiv 0{\pmod {n}}\right\}}

The theta subgroup  \Lambda is the congruence subgroup of  \Gamma defined as the preimage of the cyclic group of order two generated by {\displaystyle {\bigl (}{}_{1}^{0}\,{}_{0}^{-1}{\bigr )}} in  {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} /2\mathbb {Z} )}. It is of index 3 and is explicitly described by:[2]

{\displaystyle \Lambda =\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \Gamma :ac\equiv 0{\pmod {n}},bd\equiv 0{\pmod {n}}\right\}}.

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, SL2(Z) acts on the upper half complex plane:

[a,b;c,d]z -> (az+b)/(cz+d)
The triangles drawn by this program are Fundamental domains for SL2(Z). These triangles are used to construct the Fundamental domains for other subgroups of SL2(Z). For this nomenclature:

Gamma_0(N), Gamma_1(N), Gamma(N), Gamma^1(N) and Gamma^0(N),
where N is a positive integer. These subgroups consist of matrices in SL2(Z) of the following forms respectively:

[a,b;Nc,d], , [Na+1,Nb;Nc,Nd+1], [Na+1,Nb;Nc+1,d] and [a,Nb;c,d].
Here a,b,c,d are integers.

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.
[1] Three-Dimensional Geometry and Topology. William P. Thurston. Pages 98-99.
[2] Applications of Contact Geometry and Topology in Physics
By Arkady L Kholodenko, page 360.

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