My last post introduced the modular group, and linked them to shape rotation. I used the group operations over matrices describing such, to establish a correspondence to numerical elements. Whilst I started in 2D, I also suggested possible isomorphisms to generalise this to all dimensions. Left open (deliberately) however is the question: what does the modular group have to do with shape rotating? Welcome to my next post!
By the way, the more perspicacious in my audience will have picked up on the fact I only defined the group operations associated with the modular group via a set of matrices that can produce linear transformations. I never established what such linear transformations actually are, and how they correspond to a group. I guess I like to keep readers hooked as if they are reading a story.
Let X be a matrix with elements α,β,γ,δ in SL_2(Z). Let f_A(z) = (αz+β)/(γz+δ) be a linear transformation for γz+δ =/= 0, where z is any complex number. This is called a Möbius transformation. I will also now introduce two particular and important matrices A and B within SL_2(Z):
Now f_A(z) = z+1 and f_B(z) = -1/z. The latter rotates any point in R^2 by 90°counterclockwise, and is particularly fascinating to me (more later). Yet for now, these two operations describe the symmetries that constrain the possible transformations or operations within our group.
We can multiply any vector in R^2 by these matrices to get to any other point corresponding to the modular group elements. The associated Möbius transformations do the same using complex numbers, rather than vectors (then multiplied by matrices), as the input. In other words, A and B are said to be generators for the modular group. Algebraically, we get linear transformations with vectors and matrices. Geometrically, we get points corresponding to numbers when we switch between Euclidean R^2 and C.
Then if we let A=I and B=-I we get f_I(z)=f_-I(z)=z so this group is also defined up to isomorphism. In fact, for every A in SL_2(Z) we get f_A(z)=f_-A(z); a particularly restrictive form of isomorphic symmetry between A vs -A. In fact, A = -A under these group operations, so in fact this is an automorphism. Such elements corresponding to this automorphic group operation gives us the modular group Γ: PSL_2(Z) = SL_2(Z)/{I,-I}.
There are just a few more steps I would like to make. Let A=T and B=S, so S^2=I=e generates our identity e. This is a relation, in this case an identity derived our generators producing some isomorphism that here is a defining feature of the group. Notice I have changed the notation to make it equivalent to a sphere. The motivation for such will become clearer. For now though:
f_T(z) = z+1,
f_S(f_T(z)) = -1/(z+1) = ST(z),
f_T(ST(z)) = 1-1/(z+1) = z/(z+1),
f_S(z/(z+1)) = -(z+1)/z = (ST)^2(z),
f_T((ST)^2(z)) = 1-(z+1)/z = -1/z
f_S(-1/z) = z = (ST)^3(z)
So (ST)^3 = e. ST has order 3 and S order 2. From this, we can now complete our description of the modular group (its group presentation): PSL_2(Z) is the product of S and T such that S^2=e and (ST)^3=e. This is a free product of two cyclic groups of orders 2 and 3 respectively.
Also, from the sphere, whose surface can be described by a finite yet unbounded Euclidean space R^2, we get a smooth and compact object in non-Euclidean R^3. This relation between S^2 and (ST)^3 in the modular group, linked via an identity giving rise to an isomorphism, is one algebraic structure producing a smooth manifold. The smooth manifold is both a topological and algebraic space. The modular group necessarily produces this smooth manifold, yet the topological space does not necessarily have to be described via modular group arithmetic.
So is algebra also geometry, or can geometry exist without algebra (in which case it is a topology)? What algebras describe a given topological object? Is every topological object described by, and derived from, a unique algebra? These are metaphorical questions, yet again, the distinction is blurred.
How then can we link modular group algebra to circles. The generator B also rotates every object in R^2 by 90°counterclockwise, by angle 1/2π given that rotation in R^2 is defined by, for any angle θ:
Apply B enough times, and you return to where you started. Move point to the next closest point by R(θ), then enough times you trace a circle.
These rotations in the R^2 plane are a Lie algebra describing a Lie group, homeomorphic and diffeomorphic smooth manifolds. A circle is an infinite line in R, looped around on itself in R^2. A sphere is the plane produced by the circle loop in R^2, looped on itself to produce a smooth 3D object. All such manifolds possess Lie algebras and are by definition Lie groups. The modular group describes one set of symmetries linking these manifolds in different dimensions together. Yet their arithmetic is only one possible Lie algebra, with Γ one anongst many Lie groups. Technically, Γ is the Lie group of dimension zero, and is a discrete subgroup in the set of Lie groups.
Hence the modular group arithmetic describes transformations on a circle constrained by a restrictive form of symmetry. Again, the circle contains an algebra, and that algebra does not comprehensively describe the full circle (and we invoke another algebra involving polynomial rings to trace a circle). We consider the circle a geometry, also a topology, yet that can be consistent with multiple algebraic structures. So again, where is the link between algebra and topology that geometry implies?
UPDATE: It can get weirder. Let H be the upper half-plane of C. The transformations we covered form an isomorphic f:H→H in this region. This has an associated polynomial ring, R[x,y] in C, where z=x+iy and y>0.
To see why, for a general matrix in SL_2(Z) with elements α to δ, its determinant is αδ-βγ=1. So this means by performing the transformation associated with such matrix to z, we get a scaling (you can calculate this by hand yet incredibly tedious!) that means we remain in the complex upper-half plane1.
Indeed, the modular group relates all the points in H to another point in H. Every complex number with a positive imaginary part is isomorphic to another complex number in that plane! Again, algebra or geometry?
The symmetries of the modular group also act as structure for modular forms. Such an f:H→C of weight k satisfies f(z+1)=f(z) and f(-1/z)=(z^k)f(z). This produces a periodic wave repeating itself across an interval, which can then be decomposed into sines and cosines by Fourier expansion.
We have basically linked circles to music! When listening to Led Zeppelin or Eminem, do you think they are performing acts of geometry? Mathematically, it is unclear whether we can say no to this question…
UPDATE II: Note that for a given point (x,y) = (cos(θ),sin(θ)) in 2D Euclidean space, we get an isomorphism with z=e^(iθ)=cos(θ)+isin(θ), which is also a diffeomorphism (invertible and smooth) and a homeomorphism (continuous, including when applying the inverse), as the correspondance is bijective (provided the former is also Riemannian). Therefore we can get from algebra, points, and shapes to numbers. Both are topologies, yet the former carries abstract elements and the latter numerical elements. When we describe the former in terms of points and direction (not necessary distance), then we get geometry.
Therefore I am inclined to consider geometry as algebraic space defined by its topological elements or characteristics (only the set of elements necessary for a topology like points or lines). This makes sense. Topology is a necessary condition for algebra, and algebra is sufficient (yet not necessary) for topology. Yet when algebra is described in terms of topology, we get geometry, and we should not discuss topologies as geometry if they lack algebraic structure.
So linear transformations or functions representing modular arithmetic are a subset of modular forms f:H→C. Their eigenvalues, if expressed in primes (prime-indexed) with the Fourier coefficient α1 normalised to 1, yield a set of operations called Hecke operators. These are interesting as for two Hecke operators Tp and Tl for primes p and l respectively, TpTl=TlTp, so you get a symmetry linking primes. Again, are prime numbers described by algebra or geometry? Their eigenforms also yield Euler products: dirichlet-L functions, and a Hecke L-function is any modular form transformed via Hecke operators.