Schwarz triangle
A fundamental domain triangle (p q r), with vertex angles π⁄p, π⁄q, and π⁄r, can exist in different spaces depending on the value of the sum of the reciprocals of these integers:
[2] In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods.
The hyperbolic area of Δ equals π – π/a – π/b – π/c, so that The construction of a tessellation will first be carried out for the case when a, b and c are greater than 2.
The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point.
Even in this degenerate case when an angle of π arises, the two collinear edges are still considered as distinct for the purposes of the construction.
So provided that no tiles overlap, the previous argument shows that angles at vertices are no greater than π and hence that Pn is a convex polygon.
Before proving (c) and (b), a Möbius transformation can be applied to map the upper half plane to the unit disk and a fixed point in the interior of Δ to the origin.
Finally it remains to prove that the tiling formed by the union of the triangles covers the whole of the upper half plane.
The previous method for constructing P2, P3, ... is modified by adding an extra triangle each time an angle 3π/2 arises at a vertex.
In 1977 Takeuchi obtained a complete classification of arithmetic triangle groups (there are only finitely many) and determined when two of them are commensurable.
[7] In the case of a Schwarz triangle with one or two cusps, the process of tiling becomes simpler; but it is easier to use a different method going back to Hecke to prove that these exhaust the hyperbolic upper half plane.
It follows that for any z1 in the upper half plane, there is an element g1 in the subgroup Γ1 of Γ generated by T such that w1 = g1(z1) satisfies λ ≤ Re w1 ≤ μ, i.e. this strip is a fundamental domain for the translation group Γ1.
In fact if p, q, r, s are distinct points in the Riemann sphere, then there is a unique complex Möbius transformation g sending p, q and s to 0, ∞ and 1 respectively.
Thus so that It will now be shown that there is a parametrisation of such ideal triangles given by rationals in reduced form with a and c satisfying the "neighbour condition" p2q1 − q2p1 = 1.
To prove that the tiling covers the whole hyperbolic plane, it suffices to show that every rational in [0,1] eventually occurs as an endpoint.
One of the most elementary methods is described in Graham, Knuth & Patashnik (1994) in their development—without the use of continued fractions—of the theory of the Stern-Brocot tree, which codifies the new rational endpoints that appear at the nth stage.
These inequalities force aq – bp ≥ 1 and br – as ≥ 1 and hence, since rp – qs = 1, This puts an upper bound on the sum of the numerators and denominators.
Using a Möbius transformation, it may be assumed to be the unit circle or equivalently the real axis in the upper half plane.
The disjoint union of copies of Δ indexed by elements of Γ with edge identifications has the natural structure of a Riemann surface Σ.
The image P(Σ), i.e. the union of the translates g(Δ), is therefore an open subset of the upper half plane.
[18] In the case of the Lobachevsky or hyperbolic plane, the ideas originate in the nineteenth-century work of Henri Poincaré and Walther von Dyck.
As Joseph Lehner has pointed out in Mathematical Reviews, however, rigorous proofs that reflections of a Schwarz triangle generate a tessellation have often been incomplete, his own 1964 book "Discontinuous Groups and Automorphic Functions", being one example.
[19][20] Carathéodory's elementary treatment in his 1950 textbook Funktiontheorie, translated into English in 1954, and Siegel's 1954 account using the monodromy principle are rigorous proofs.
Let er, es, et be a basis for a 3-dimensional real vector space V with symmetric bilinear form Λ such that
The operators ρ, σ, τ are involutions on V, with respective eigenvectors er, es, et with simple eigenvalue −1.
Similarly define Γr to be the cyclic subgroup of Γ given by the 2-group {1, r}, with analogous definitions for Γs and Γt.
It is well known that, for finite-dimensional real inner product spaces, two orthogonal involutions S and T can be decomposed as an orthogonal direct sum of 2-dimensional or 1-dimensional invariant spaces; for example, this can be deduced from the observation of Paul Halmos and others, that the positive self-adjoint operator (S – T)2 commutes with both S and T. In the case above, however, where the bilinear form Λ is no longer a positive definite inner product, different ad hoc reasoning has to be given.
For g in Γ, ℓ(g) is the number of positive roots made negative by g. Fundamental domain and Tits cone.
The passage from Coxeter groups to tessellation can first be found in the exercises of §4 of Chapter V of Bourbaki (1968), due to Tits, and in Iwahori (1966); currently numerous other equivalent treatments are available, not always directly phrased in terms of symmetric spaces.
The Swiss mathematicians de la Harpe (1991) and Haefliger have provided an introductory account, taking geometric group theory as their starting point.
