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{{short description|Conformal mappings in complex analysis}}

[[File:Schwarz triangle function.svg|thumb|The upper half-plane, and the image of the upper half-plane transformed by the Schwarz triangle function with various parameters.]]

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In [[complex analysis]], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|conformally maps]] the [[upper half plane]] to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a [[Schwarz triangle]], although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a [[Möbius triangle]], the inverse of the Schwarz triangle function is a [[single-valued]] [[automorphic function]] for that triangle's [[triangle group]]. More specifically, it is a [[modular function]].

In [[mathematics]], the '''Schwarz triangle tessellation''' was introduced by [[H. A. Schwarz]] as a way of [[tessellation|tessellating]] the hyperbolic [[upper half plane]] by a [[Schwarz triangle]], i.e. a [[hyperbolic triangle|geodesic triangle]] in the upper half plane with angles which are either 0 or of the form {{pi}} over a positive integer greater than one. Through successive hyperbolic reflections in its sides, such a triangle generates a [[tessellation]] of the upper half plane (or the unit disk after composition with the [[Cayley transform]]). The hyperbolic refections generates a [[discrete group]], with an orientation-preserving normal subgroup of index two. Each such tessellation yields a [[conformal mapping]] of the upper half plane onto the interior of the geodesic triangle, generalizing the [[Schwarz–Christoffel mapping]], with the upper half plane replacing the complex plane. The corresponding inverse function, first defined by Schwarz, solved the problem of uniformization a hyperbolic triangle and is called the '''Schwarz triangle function'''.

==Formula==

Through the theory of complex [[ordinary differential equation]]s with [[regular singular point]]s and the [[Schwarzian derivative]], the triangle function can be expressed as the quotient of two solutions of a [[hypergeometric differential equation]] with real coefficients and singular points at 0, 1 and ∞. By the [[Schwarz reflection principle]], the reflection group induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two-dimensional representation corresponds to the [[monodromy]] of the ordinary differential equation and induces a group of [[Möbius transformation]]s on quotients of [[hypergeometric function]]s.

Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle in [[radians]]. Each of ''α'', ''β'', and ''γ'' may take values between 0 and 1 inclusive. Following Nehari,{{sfn|Nehari|1975|page=309}} these angles are in clockwise order, with the vertex having angle ''πα'' at the origin and the vertex having angle ''πγ'' lying on the real line. The Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:

:<math>s(\alpha, \beta, \gamma; z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>

As the inverse function of such a quotient, the triangle function is an [[automorphic function]] for this discrete group of Möbius transformations. This is a special case of a general scheme of [[Henri Poincaré]] that associates automorphic forms with ordinary differential equations with regular singular points. In the special case of [[ideal triangle]]s, where all the angles are zero, the tessellation corresponds to the [[Farey series|Farey tessellation]] and the triangle function yields the [[modular lambda function]].

where

:''a'' = (1−α−β−γ)/2,

:''b'' = (1−α+β−γ)/2,

:''c'' = 1−α,

:''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2,

:''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and

:''c''′ = 2 − ''c'' = 1 + α.

This function maps the upper half-plane to a [[spherical triangle]] if α + β + γ > 1, or a [[hyperbolic triangle]] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a''&thinsp;=&thinsp;0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the [[Schwarz–Christoffel transformation]].

<!--In [[complex analysis]], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|conformally maps]] the [[upper half plane]] to a triangle in the upper half plane having lines or circular arcs for edges. Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle. If any of ''α, β'', and ''γ'' are greater than zero, then the Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:

===Derivation===

:<math>s(z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>

Through the theory of complex [[ordinary differential equation]]s with [[regular singular point]]s and the [[Schwarzian derivative]], the triangle function can be expressed as the quotient of two solutions of a [[hypergeometric differential equation]] with real coefficients and singular points at 0, 1 and ∞. By the [[Schwarz reflection principle]], the reflection group induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two-dimensional representation corresponds to the [[monodromy]] of the ordinary differential equation and induces a group of [[Möbius transformation]]s on quotients of [[hypergeometric function]]s.{{sfn|Nehari|1975|pp=198-208}}

== Singular points ==

where ''a'' = (1−α−β−γ)/2, ''b'' = (1−α+β−γ)/2, ''c'' = 1−α, ''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2, ''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and ''c''′ = 2 − ''c'' = 1 + α. This mapping has singular points at ''z'' = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,

This mapping has [[regular singular points]] at ''z'' = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,{{sfn|Nehari|1975|pages=315−316}}

:<math>\begin{alignedalign}

s(0) &= 0, \\[5mu4mu]

s(1) &= \frac

{\Gamma(1-a')\Gamma(1-b')\Gamma(c')}

{\Gamma(1-a)\Gamma(1-b)\Gamma(c)},\,\text{and} \\[5mu8mu]

s(\infty) &= \exp\left(i \pi \alpha \right)\frac

{\Gamma(1-a')\Gamma(b)\Gamma(c')}

{\Gamma(1-a)\Gamma(b')\Gamma(c)}.,

\end{alignedalign}</math>

where <math display=inline>\Gamma(x)</math> is the [[gamma function]].

This formula can be derived using the [[Schwarzian derivative]].

Near each singular point, the function may be approximated as

This function can be used to map the upper half-plane to a [[spherical triangle]] on the [[Riemann sphere]] if α + β + γ > 1, or a [[hyperbolic triangle]] on the [[Poincaré disk]] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a''&thinsp;=&thinsp;0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the [[Schwarz–Christoffel transformation]]. In the special case of [[ideal triangle]]s, where all the angles are zero, the triangle function yields the [[modular lambda function]].

:<math>\begin{align}

This function was introduced by [[H. A. Schwarz]] as the inverse function of the [[conformal mapping]] uniformizing a [[Schwarz triangle]]. Applying successive hyperbolic reflections in its sides, such a triangle generates a [[tessellation]] of the upper half plane (or the unit disk after composition with the [[Cayley transform]]). The conformal mapping of the upper half plane onto the interior of the [[geodesic triangle]] generalizes the [[Schwarz–Christoffel transformation]]. By the [[Schwarz reflection principle]], the discrete group generated by hyperbolic reflections in the sides of the triangle induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two dimensional representation corresponds to the [[monodromy]] of the ordinary differential equation and induces a group of [[Möbius transformation]]s on quotients of solutions. Since the triangle function is the inverse function of such a quotient, it is therefore an [[automorphic function]] for this discrete group of Möbius transformations. This is a special case of a general method of [[Henri Poincaré]] that associates automorphic forms with [[ordinary differential equation]]s with [[regular singular point]]s.-->

s_0(z) &= z^\alpha (1+O(z)), \\[6mu]

s_1(z) &= (1-z)^\gamma (1+O(1-z)), \\[6mu]

s_\infty(z) &= z^\beta (1+O(1/z)),

\end{align}</math>

where <math>O(x)</math> is [[big O notation]].

==Hyperboloid and Beltrami-Klein models==

[[File:Relation5models.png|thumb|upright=0.85|Geometric relations between Poincaré disk model, hyperboloid model and Beltrami-Klein model]]

== Inverse ==

In this section two different models are given for hyperbolic geometry on the unit disk or equivalently the upper half plane.<ref>See:

When ''α, β'', and ''γ'' are rational, the triangle is a Schwarz triangle. When each of ''α, β'', and ''γ'' are either the reciprocal of an integer or zero, the triangle is a [[Möbius triangle]], i.e. a non-overlapping Schwarz triangle. For a Möbius triangle, the inverse is a [[modular function]].

*{{harvnb|Busemann|1955}}

*{{harvnb|Magnus|1974}}

*{{harvnb|Helgason|2000}}

*{{harvnb|Wolf|2011}}</ref>

In the spherical case, that modular function is a [[rational function]]. For Euclidean triangles, the inverse can be expressed using [[elliptical function]]s.<ref name=Lee />

The group ''G'' = SU(1,1) is formed of matrices

== Ideal triangles ==

:<math> g = \begin{pmatrix} \alpha & \beta \\ \overline{\beta} & \overline{\alpha}\end{pmatrix}

When ''α'' = 0 the triangle is degenerate, lying entirely on the real line. If either of ''β'' or ''γ'' are non-zero, the angles can be permuted so that the positive value is ''α'', but that is not an option for an [[ideal triangle]] having all angles zero.

</math>

Instead, a mapping to an ideal triangle with vertices at 0, 1, and ∞ is given by in terms of the [[complete elliptic integral of the first kind]]:

with

:<math>i\frac{K(1-z)}{K(z)}</math>.

This expression is the inverse of the [[modular lambda function]].{{sfn|Nehari|1975|pp=316-318}}

== Extensions ==

:<math> |\alpha|^2 -|\beta|^2=1.</math>

The [[Schwarz–Christoffel transformation]] gives the mapping from the upper half-plane to any Euclidean polygon.

The methodology used to derive the Schwarz triangle function earlier can be applied more generally to arc-edged polygons. However, for an ''n''-sided polygon, the solution has ''n'' − 3 additional parameters, which are difficult to determine in practice.{{sfn|Nehari|1975|p=202}} See {{slink|Schwarzian derivative#Conformal mapping of circular arc polygons}} for more details.

It is a subgroup of ''G''_''c'' = SL(2,'''C'''), the group of complex 2&nbsp;&times;&nbsp;2 matrices with determinant&nbsp;1.

The group ''G''_''c'' acts by Möbius transformations on the extended complex plane. The subgroup ''G'' acts as automorphisms of the unit disk ''D'' and the subgroup ''G''₁ = SL(2,'''R''') acts as automorphisms of the [[upper half plane]]. If

== Applications ==

:<math>C=\begin{pmatrix}1 & i \\ i & 1\end{pmatrix}</math>

[[L. P. Lee]] used Schwarz triangle functions to derive [[conformal map projection]]s onto [[polyhedral map projection|polyhedral]] surfaces.<ref name=Lee>{{cite book | last = Lee | first = L. P. | author-link = Laurence Patrick Lee | year = 1976 | title = Conformal Projections Based on Elliptic Functions | ___location = Toronto | publisher = B. V. Gutsell, York University | series = Cartographica Monographs | volume = 16 | url = https://archive.org/details/conformalproject0000leel | url-access = limited | isbn = 0-919870-16-3}} Supplement No. 1 to [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer'' '''13'''].</ref>

then

:<math>G=CG_1C^{-1},</math>

since the Möbius transformation corresponding ''M'' is the [[Cayley transform]] carrying the upper half plane onto the unit disk and the real line onto the unit circle.

The Lie algebra <math>\mathfrak{g}</math> of SU(1,1) consists of matrices

:<math> X= \begin{pmatrix} ix & w \\ \overline{w} & -ix \end{pmatrix}</math>

with ''x'' real. Note that ''X''² = (|''w''|² – ''x''²) ''I'' and

:<math> \det X = x^2 -|w|^2 =-\tfrac12 \operatorname{Tr} X^2.</math>

The hyperboloid <math>\mathfrak{H}</math> in <math>\mathfrak{g}</math> is defined by two conditions. The first is that det ''X'' = 1 or equivalently Tr ''X''² = –2.<ref>Note that the [[Killing form]] ''B''(''X,''Y'') = Tr<math>{}_{\mathfrak g}</math> ad ''X'' ad ''Y'' = 4 Tr ''XY''.{{harvnb|Helgason|1978}}</ref> By definition this condition is preserved under [[Conjugacy class|conjugation]] by ''G''. Since ''G'' is connected it leaves the two components with ''x'' > 0 and ''x'' < 0 invariant. The second condition is that ''x'' > 0. For brevity, write ''X'' = (''x'',''w'').

The group ''G'' acts transitively on ''D'' and <math>\mathfrak{H}</math> and the points 0 and (1,0) have [[Stabilizer subgroup|stabiliser]] ''K'' consisting of matrices

:<math>\begin{pmatrix}\zeta & 0 \\ 0 & \overline{\zeta}\end{pmatrix}</math>

with |ζ| = 1. Polar decomposition on ''D'' implies the Cartan decomposition ''G'' = ''KAK'' where ''A'' is the group of matrices

:<math>a_t = \begin{pmatrix}\cosh t & \sinh t \\ \sinh t & \cosh t\end{pmatrix}.</math>

Both spaces can therefore be identified with the homogeneous space ''G''/''K'' and there is a ''G''-equivariant map ''f'' of <math>\mathfrak{H}</math> onto ''D'' sending (1,0) to 0. To work out the formula for this map and its inverse it suffices to compute ''g''(1,0) and

''g''(0) where ''g'' is as above. Thus ''g''(0) = β/{{overline|α}} and

:<math>g(1,0)=(|\alpha|^2 +|\beta|^2,-2i\alpha\beta),</math>

so that

:<math>\frac\beta\overline{\alpha} = \frac{\alpha\beta} {|\alpha|{}^2} = \frac{2\alpha\beta}{|\alpha|{}^2 + |\beta|{}^2 + 1},</math>

recovering the formula

:<math>f(x,w) = \frac{iw}{x + 1}.</math>

Conversely if ''z'' = ''iw''/(''x'' + 1), then |''z''|² = (''x'' – 1)/(''x'' + 1), giving the inverse formula

:<math> f^{-1}(z) = (x,w) = \left( \frac{1 + |z|{}^2}{1 - |z|{}^2}, \frac{-2iz}{1 - |z|{}^2} \right).</math>

This correspondence extends to one between geometric properties of ''D'' and <math>\mathfrak{H}</math>. Without entering into the correspondence of ''G''-invariant [[Riemannian metric]]s,{{efn|1=The Poincaré metric on the disk corresponds to the restriction of the ''G''-invariant pseudo-Riemannian metric ''dx''² – ''dw''² to the hyperboloid.}} each geodesic circle in ''D'' corresponds to the intersection of 2-planes through the origin, given by equations Tr ''XY'' = 0, with <math>\mathfrak{H}</math>. Indeed, this is obvious for rays arg ''z'' = θ through the origin in ''D''—which correspond to the 2-planes arg ''w'' = θ—and follows in general by ''G''-equivariance.

The [[Beltrami-Klein model]] is obtained by using the map ''F''(''x'',''w'') = ''w''/''x'' as the correspondence between <math>\mathfrak{H}</math> and ''D''. Identifying this disk with (1,''v'') with |''v''| < 1, intersections of 2-planes with <math>\mathfrak{H}</math> correspond to intersections of the same 2-planes with this disk and so give straight lines. The Poincaré-Klein map given by

:<math>K(z) = iF \circ f^{-1}(z) = \frac{2z}{1 + |z|{}^2}</math>

thus gives a diffeomorphism from the unit disk onto itself such that Poincaré geodesic circles are carried into straight lines. This diffeomorphism does not preserve angles but preserves orientation and, like all diffeomorphisms, takes smooth curves through a point making an angle less than {{pi}} (measured anticlockwise) into a similar pair of curves.{{efn|1=The condition on tangent vectors '''x''', '''y''' is given by det ('''x''','''y''') ≥ 0 and is preserved because the determinant of the Jacobian is positive.}} In the limiting case, when the angle is {{pi}}, the curves are tangent and this again is preserved under a diffeomorphism. The map ''K'' yields the ''Beltrami-Klein model'' of hyperbolic geometry. The map extends to a homeomorphism of the unit disk onto itself which is the identity on the unit circle. Thus by continuity the map ''K'' extends to the endpoints of geodesics, so carries the arc of the circle in the disc cutting the unit circle orthogonally at two given points on to the straight line segment joining those two points. The use of the Beltrami-Klein model corresponds to [[projective geometry]] and [[cross ratio]]s. (Note that on the unit circle the radial derivative of ''K'' vanishes, so that the condition on angles no longer applies there.)<ref>{{harvnb|Iversen|1992}}</ref><ref>{{harvnb|Magnus|1974}}</ref>

The group ''G''₁ = SL(2,'''R''') is formed of real matrices

:<math> g = \begin{pmatrix} a & b \\ c & d\end{pmatrix}

</math>

with <math>ad -bc =1.</math> The action is by Möbius transformations on the upper half-plane. The Lie algebra of ''G''₁ is <math>\mathfrak{g}_1</math>, the space of 2 x 2 real matrices of trace zero,

:<math> X = \begin{pmatrix} x & y-t \\ y+t & -x\end{pmatrix}.</math>

By transport of construction — conjugating by ''C'' — the symmetric bilinear form Tr ad(''X'')ad(''Y'') = 4 Tr ''XY'' is invariant under conjugation and has signature (2,1). As above ''X''² = (''x''² + ''y''² – ''t''²) ''I'' and

:<math> \det X = t^2 - x^2 - y^2 =-\tfrac12 \operatorname{Tr} X^2.</math>

By [[Sylvester's law of inertia]],<ref>{{harvnb|Ratcliffe|2019|pages=52–56}}</ref> the [[Killing form]] (or [[Cartan-Killing form]]) is, up to equivalence, the unique symmetric bilinear form of signature (2,1) with corresponding quadratic form –''x''² –''y''² +''t''²; and ''G'' / {±''I''} or equivalently ''G''₁ / {±''I''} can be identified with SO(2,1). <!--The group SO(2,1) acts transitively on the two components of the non-zero part of the light cone ''x''² + ''y''² = ''t''². On the interior of the two time-like components, ''x''² + ''y''² < ''t''² with ''t'' strictly positive or negative, it is a disjoint union of orbits, namely the hyperboloids ''x''² + ''y''² + ''a''² = ''t''². For fixed ''a'', say ''a'' = 1, the Beltrami-Klein model gives a equivariant homeomorphism ''f'' of the hyperboloid ''x''² + ''y''² + 1 = ''t''² onto the open unit disk, ''f''(''x'',''y'',''t'') = (''x'' + ''iy'')/''t''. Compactifying the hyperboloid by adding a circle at infinity — the rays of the light cone — ''f'' extends to a homeomorphism onto the closed unit disk.-->

==Tessellation by Schwarz triangles==

In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For triangles without "cusps"—angles equal to zero or equivalently vertices on the real axis—the elementary approach of {{harvtxt|Carathéodory|1954}} will be followed. For triangles with one or two cusps, elementary arguments of {{harvtxt|Evans|1973}}, simplifying the approach of {{harvtxt|Hecke|1935}}, will be used: in the case of a Schwarz triangle with one angle zero and another a right angle, the orientation-preserving subgroup of the reflection group of the triangle is a [[Hecke group]]. For an ideal triangle in which all angles are zero, so that all vertices lie on the real axis, the existence of the tessellation will be established by relating it to the [[Farey series]] described in {{harvtxt|Hardy|Wright|2008}} and {{harvtxt|Series|2015}}. In this case the tessellation can be considered as that associated with three touching circles on the [[Riemann sphere]], a limiting case of configurations associated with three disjoint non-nested circles and their reflection groups, the so-called "[[Schottky group]]s", described in detail in {{harvtxt|Mumford|Series|Wright|2015}}. Alternatively—by dividing the ideal triangle into six triangles with angles 0, {{pi}}/2 and {{pi}}/3—the tessellation by ideal triangles can be understood in terms of tessellations by triangles with one or two cusps.

===Triangles with one or two cusps===

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 [[Erich Hecke|Hecke]] to prove that these exhaust the hyperbolic upper half plane.

In the case of one cusp and non-zero angles {{pi}}/''a'', {{pi}}/''b'' with ''a'', ''b'' integers greater than one, the tiling can be envisaged in the unit disk with the vertex having angle {{pi}}/''a'' at the origin. The tiling starts by adding 2''a'' – 1 copies of the triangle at the origin by successive reflections. This results in a polygon ''P''₁ with 2''a'' cusps and between each two 2''a'' vertices each with an angle {{pi}}/''b''. The polygon is therefore convex. For each non-ideal vertex of ''P''₁, the unique triangle with that vertex can be similar reflected around that vertex, thus adding 2''b'' – 1 new triangles, 2''b'' – 1 new ideal points and 2 ''b'' – 1 new vertices with angle {{pi}}/''a''. The resulting polygon ''P''₂ is thus made up of 2''a''(2''b'' – 1) cusps and the same number of vertices each with an angle of {{pi}}/''a'', so is convex. The process can be continued in this way to obtain convex polygons ''P''₃, ''P''₄, and so on. The polygon ''P''_''n'' will have vertices having angles alternating between 0 and {{pi}}/''a'' for ''n'' even and between 0 and {{pi}}/''b'' for ''n'' odd. By construction the triangles only overlap at edges or vertices, so form a tiling.<ref>{{harvnb|Carathéodory|1954|page=183}}</ref>

{{gallery|width=300px

|File:H2checkers 35i.png|<small>Tessellation by triangle with angles 0, {{pi}}/3, {{pi}}/5</small>

|File:H2checkers 25i.png|<small>Tessellation by triangle with angles 0, {{pi}}/5, {{pi}}/2</small>

|File:H2checkers 5ii.png|<small>Tessellation by triangle with angles 0, 0, {{pi}}/5</small>}}

The case where the triangle has two cusps and one non-zero angle {{pi}}/''a'' can be reduced to the case of one cusp by observing that the trinale is the double of a triangle with one cusp and non-zero angles {{pi}}/''a'' and {{pi}}/''b'' with ''b'' = 2. The tiling then proceeds as before.<ref>{{harvnb|Carathéodory|1954|page=184}}</ref>

To prove that these give tessellations, it is more convenient to work in the upper half plane. Both cases can be treated simultaneously, since the case of two cusps is obtained by doubling a triangle with one cusp and non-zero angles {{pi}}/''a'' and {{pi}}/2. So consider the geodesic triangle in the upper half plane with angles 0, {{pi}}/''a'', {{pi}}/''b'' with ''a'', ''b'' integers greater than one. The interior of such a triangle can be realised as the region ''X'' in the upper half plane lying outside the unit disk |''z''| ≤ 1 and between two lines parallel to the imaginary axis through points ''u'' and ''v'' on the unit circle. Let Γ be the triangle group generated by the three reflections in the sides of the triangle.

To prove that the successive reflections of the triangle cover the upper half plane, it suffices to show that for any ''z'' in the upper half plane there is a ''g'' in Γ such that ''g''(''z'') lies in {{overline|''X''}}. This follows by an argument of {{harvtxt|Evans|1973}}, simplified from the theory of [[Hecke group]]s. Let λ = Re ''a'' and μ = Re ''b'' so that, without loss of generality, λ < 0 ≤ μ. The three reflections in the sides are given by

:<math>R_1(z) = \frac1\overline{z},\ R_2(z) = -\overline{z} + \lambda,\ R_3(z)= -\overline{z} + \mu.</math>

Thus ''T'' = ''R''₃∘''R''₂ is translation by μ − λ. It follows that for any ''z''₁ in the upper half plane, there is an element ''g''₁ in the subgroup Γ₁ of Γ generated by ''T'' such that ''w''₁ = ''g''₁(''z''₁) satisfies λ ≤ Re ''w''₁ ≤ μ, i.e. this strip is a [[fundamental ___domain]] for the translation group Γ₁. If |''w''₁| ≥ 1, then ''w''₁ lies in ''X'' and the result is proved. Otherwise let ''z''₂ = ''R''₁(''w''₁) and find ''g''₂

Γ₁ such that ''w''₂ = ''g''₂(''z''₂) satisfies λ ≤ Re ''w''₂ ≤ μ. If |''w''₂| ≥ 1 then the result is proved. Continuing in this way, either some ''w''_''n'' satisfies |''w''_''n''| ≥ 1, in which case the result is proved; or |''w''_''n''| < 1 for all ''n''. Now since ''g''_{''n'' + 1} lies in Γ₁ and |''w''_''n''| < 1,

:<math>

\operatorname{Im} g_{n+1}(z_{n+1})

= \operatorname{Im} z_{n+1}

= \operatorname{Im} \frac{w_n}{|w_n|{}^2}

= \frac{\operatorname{Im} w_n}{|w_n|{}^2}.

</math>

In particular

:<math>\operatorname{Im} w_{n+1} \ge \operatorname{Im} w_n</math>

and

:<math>\frac{\operatorname{Im} w_{n+1}}{\operatorname{Im} w_n} = |w_n|^{-2} \ge 1.</math>

Thus, from the inequality above, the points (''w''_''n'') lies in the compact set |''z''| ≤ 1, λ ≤ Re ''z'' ≤ μ and Im ''z'' ≥ Im ''w''₁. It follows that |''w''_''n''| tends to 1; for if not, then there would be an ''r'' < 1 such that |''w''_''m''| ≤ ''r'' for inifitely many ''m'' and then the last equation above would imply that Im ''w''_''n'' tends to infinity, a contradiction.

Let ''w'' be a limit point of the ''w''_''n'', so that |''w''| = 1. Thus ''w'' lies on the arc of the unit circle between ''u'' and ''v''. If ''w'' ≠ ''u'', ''v'', then ''R''₁ ''w''_''n'' would lie in ''X'' for ''n'' sufficiently large, contrary to assumption. Hence ''w'' =''u'' or ''v''. Hence for ''n'' sufficiently large ''w''_''n'' lies close to ''u'' or ''v'' and therefore must lie in one of the reflections of the triangle about the vertex ''u'' or ''v'', since these fill out neighborhoods of ''u'' and ''v''. Thus there is an element ''g'' in Γ such that ''g''(''w''_''n'') lies in {{overline|''X''}}. Since by construction ''w''_''n'' is in the Γ-orbit of ''z''₁, it follows that there is a point in this orbit lying in {{overline|''X''}}, as required.<ref>See:

*{{harvnb|Evans|1973|pages=108−109}}

*{{harvnb|Berndt|Knopp|2008|pages=16−17}}</ref>

===Ideal triangles===

The tessellation for an [[ideal triangle]] with all its vertices on the unit circle and all its angles 0 can be considered as a special case of the tessellation for a triangle with one cusp and two now zero angles {{pi}}/3 and {{pi}}/2. Indeed, the ideal triangle is made of six copies one-cusped triangle obtained by reflecting the smaller triangle about the vertex with angle {{pi}}/3.

{{gallery|width=300px

|File:H2checkers_23i.png|<small>Tessellation for triangle with angles 0, {{pi}}/3 and {{pi}}/2</small>

|File:H2chess_23ib.png|<small>Tessellation for ideal triangle</small>

|File:Ideal-triangle hyperbolic tiling.svg|<small>Second realisation of tessellation for ideal triangle</small>

|File:Ideal-triangle hyperbolic tiling line-drawing.svg|<small>Line drawing of tessellation by ideal triangles</small>}}

[[File:Pappusharmonic.svg|thumb|upright=0.8|''D'' is the harmonic conjugate of ''C'' with respect to ''A'' and ''B'']]

[[File:Ideal-Triangle-Reflection.jpg|thumb|Reflection of an ideal triangle in one of its sides]]

Each step of the tiling, however, is uniquely determined by the positions of the new cusps on the circle, or equivalently the real axis; and these points can be understood directly in terms of [[Farey series]] following {{harvtxt|Series|2015}}, {{harvtxt|Hatcher|2013|pages=20–32}} and {{harvtxt|Hardy|Wright|2008|pages=23–31}}. This starts from the basic step that generates the tessellation, the reflection of an ideal triangle in one of its sides. Reflection corresponds to the process of inversion in projective geometry and taking the [[projective harmonic conjugate]], which can be defined in terms of the [[cross ratio]]. 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. The cross ratio (''p'', ''q''; ''r'', ''s'') is defined to be ''g''(''r'') and is given by the formula

:<math>(p, q; r, s) = \frac{(p-r)(q-s)}{(p-s)(q-r)}.</math>

By definition it is invariant under Möbius transformations. If ''a'', ''b'' lie on the real axis, the harmonic conjugate of ''c'' with respect to ''a'' and ''b'' is defined to be the unique real number ''d'' such that (''a'', ''b''; ''c'', ''d'') = −1. So for example if ''a'' = 1 and ''b'' = –1, the conjugate of ''r'' is 1/''r''. In general Möbius invariance can be used to obtain an explicit formula for ''d'' in terms of ''a'', ''b'' and ''c''. Indeed, translating the centre ''t'' = (''a'' + ''b'')/2 of the circle with diameter having endpoints ''a'' and ''b'' to 0, ''d'' – ''t'' is the harmonic conjugate of ''c'' – ''t'' with respect to ''a'' - ''t'' and ''b'' – ''t''. The radius of the circle is ρ = (''b'' – ''a'')/2 so (''d'' - ''t'')/ρ is the harmonic conjugate of {{math|(''c'' – ''t'')/ρ}} with respect to 1 and -1. Thus

:<math>\frac{d-t}\rho = \frac\rho{c-t}</math>

so that

:<math>d = \frac{\rho^2}{r-t} + t = \frac{(c-a)b + (c-b)a}{(c-a) + (c-b)}.</math>

It will now be shown that there is a parametrisation of such ideal triangles given by rationals in reduced form

:<math>a = \frac{p_1}{q_1},\ b = \frac{p_1 + p_2}{q_1 + q_2},\ c = \frac{p_2}{q_2}</math>

with ''a'' and ''c'' satisfying the "neighbour condition" ''p''₂''q''₁ − ''q''₂''p''₁ = 1.

The middle term ''b'' is called the ''Farey sum'' or ''[[mediant (mathematics)|mediant]]'' of the outer terms and written

:<math>b = a \oplus c.</math>

The formula for the reflected triangle gives

:<math>d = \frac{p_1 + 2p_2}{q_1 + 2q_2} = a \oplus b.</math>

Similarly the reflected triangle in the second semicircle gives a new vertex ''b'' ⊕ ''c''. It is immediately verified that ''a'' and ''b'' satisfy the neighbour condition, as do ''b'' and ''c''.

Now this procedure can be used to keep track of the triangles obtained by successively reflecting the basic triangle Δ with vertices 0, 1 and ∞. It suffices to consider the strip with 0 ≤ Re z ≤ 1, since the same picture is reproduced in parallel strips by applying reflections in the lines Re ''z'' = 0 and 1. The ideal triangle with vertices 0, 1, ∞ reflects in the semicircle with base [0,1] into the triangle with vertices ''a'' = 0, ''b'' = 1/2, ''c'' = 1. Thus ''a'' = 0/1 and ''c'' = 1/1 are neighbours and ''b'' = ''a'' ⊕ ''c''. The semicircle is split up into two smaller semicircles with bases [''a'',''b''] and [''b'',''c'']. Each of these intervals splits up into two intervals by the same process, resulting in 4 intervals. Continuing in this way, results into subdivisions into 8, 16, 32 intervals, and so on. At the ''n''th stage, there are 2^''n'' adjacent intervals with 2^''n'' + 1 endpoints. The construction above shows that successive endpoints satisfy the neighbour condition so that new endpoints resulting from reflection are given by the Farey sum formula.

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. There are several ways to see this. One of the most elementary methods is described in {{harvtxt|Graham|Knuth|Patashnik|1994}} in their development—without the use of [[continued fraction]]s—of the theory of the [[Stern-Brocot tree]], which codifies the new rational endpoints that appear at the ''n''th stage. They give [[Stern-Brocot tree#Mediants and binary search|a direct proof]] that every rational appears. Indeed, starting with {0/1,1/1}, successive endpoints are introduced at level ''n''+1 by adding Farey sums or mediants {{math|(''p''+''r'')/(''q''+''s'')}} between all consecutive terms {{math|''p''/''q''}}, {{math|''r''/''s''}} at the ''n''th level (as described above). Let {{math|1=''x'' = ''a''/''b''}} be a rational lying between 0 and 1 with {{math|''a''}} and {{math|''b''}} coprime. Suppose that at some level {{math|''x''}} is sandwiched between successive terms {{math|''p''/''q'' < ''x'' < ''r''/''s''}}. These inequalities force {{math|''aq'' – ''bp'' ≥ 1}} and

{{math|''br'' – ''as'' ≥ 1}} and hence, since {{math|1=''rp'' – ''qs'' = 1}},

:<math>a + b = (r+s)(ap-bq) + (p+q)(br -as) \ge p+q+r+s.</math>

This puts an upper bound on the sum of the numerators and denominators. On the other hand, the mediant {{math|(''p''+''r'')/(''q''+''s'')}} can be introduced and either equals {{math|''x''}}, in which case the rational {{math|''x''}} appears at this level; or the mediant provides a new interval containing {{math|''x''}} with strictly larger numerator-and-denominator sum. The process must therefore terminate after at most {{math|''a'' + ''b''}} steps, thus proving that {{math|''x''}} appears.<ref>{{harvnb|Graham|Knuth|Patashnik|1994|page=118}}</ref>

A second approach relies on the [[modular group]] ''G'' = SL(2,'''Z''').<ref>{{harvnb|Series|2015}}</ref> The Euclidean algorithm implies that this group is generated by the matrices

:<math>S=\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix},\,\,\, T=\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}.</math>

In fact let ''H'' be the subgroup of ''G'' generated by ''S'' and ''T''. Let

:<math>g=\begin{pmatrix} a & b\\ c & d\end {pmatrix}</math>

be an element of SL(2,'''Z'''). Thus ''ad'' − ''cb'' = 1, so that ''a'' and ''c'' are coprime. Let

:<math>v=\begin{pmatrix}a\\ c\end{pmatrix},\,\,\, u= \begin{pmatrix}1\\ 0\end{pmatrix}.</math>

Applying ''S'' if necessary, it can be assumed that |''a''| > |''c''|

(equality is not possible by coprimeness). We write ''a'' = ''mc'' + ''r'' with

0 ≤ ''r'' ≤ |''c''|. But then

:<math>T^{-m}\begin{pmatrix} a\\ c \end{pmatrix} = \begin{pmatrix} r\\ c\end{pmatrix}.</math>

This process can be continued until one of the entries is 0, in which case the other is necessarily ±1. Applying a power of ''S'' if necessary, it follows that ''v'' = ''h'' ''u'' for some ''h'' in ''H''. Hence

:<math>h^{-1}g=\begin{pmatrix}1 & p\\ 0 & q\end{pmatrix}</math>

with ''p'', ''q'' integers. Clearly ''p'' = 1, so that ''h''⁻¹''g'' = ''T''^''q''. Thus ''g'' = ''h'' ''T''^''q'' lies in ''H'' as required.

To prove that all rationals in [0,1] occur, it suffices to show that ''G'' carries Δ onto triangles in the tessellation. This follows by first noting that ''S'' and ''T'' carry Δ on to such a triangle: indeed as Möbius transformations, ''S''(''z'') = –1/''z'' and ''T''(''z'') = ''z'' + 1, so these give reflections of Δ in two of its sides. But then ''S'' and ''T'' conjugate the reflections in the sides of Δ into reflections in the sides of ''S''Δ and ''T''Δ, which lie in Γ. Thus ''G'' normalizes Γ. Since triangles in the tessellation are exactly those of the form ''g''Δ with ''g'' in Γ, it follows that ''S'' and ''T'', and hence all elements of ''G'', permute triangles in the tessellation. Since every rational is of the form ''g''(0) for ''g'' in ''G'', every rational in [0,1] is the vertex of a triangle in the tessellation.

The reflection group and tessellation for an ideal triangle can also be regarded as a limiting case of the [[Schottky group]] for three disjoint unnested circles on the Riemann sphere. Again this group is generated by hyperbolic reflections in the three circles. In both cases the three circles have a common circle which cuts them orthogonally. Using a Möbius transformation, it may be assumed to be the unit circle or equivalently the real axis in the upper half plane.<ref>See:

*{{harvnb|McMullen|1998}}

*{{harvnb|Mumford|Series|Wright|2015}}</ref>

=== Approach of Siegel ===

In this subsection the approach of [[Carl Ludwig Siegel]] to the tessellation theorem for triangles is outlined. Siegel's less elementary approach does not use convexity, instead relying on the theory of [[Riemann surface]]s, [[covering space]]s and a version of the [[monodromy theorem]] for coverings. It has been generalized to give proofs of the more general Poincaré polygon theorem. (Note that the special case of tiling by regular ''n''-gons with interior angles 2{{pi}}/''n'' is an immediate consequence of the tessellation by Schwarz triangles with angles {{pi}}/''n'', {{pi}}/''n'' and {{pi}}/2.)<ref>{{harvnb|Siegel|1971|pages=85–87}}</ref><ref>For proofs of Poincaré's polygon theorem, see

*{{harvnb|Maskit|1971}}

*{{harvnb|de Rham|1971}}

*{{harvnb|Beardon|1983|pages=242–249}}

*{{harvnb|Iversen|1992|pages=200–208}}

*{{harvnb|Epstein|Petronio|1994}}

*{{harvnb|Berger|2010|pages=616–617}}</ref>

Let Γ be the [[free product]] '''Z'''₂ ∗ '''Z'''₂ ∗ '''Z'''₂. If Δ = ''ABC'' is a Schwarz triangle with angles {{pi}}/''a'', {{pi}}/''b'' and {{pi}}/''c'', where ''a'', ''b'', ''c'' ≥ 2, then there is a natural map of Γ onto the group generated by reflections in the sides of Δ. Elements of Γ are described by a product of the three generators where no two adjacent generators are equal. At the vertices ''A'', ''B'' and ''C'' the product of reflections in the sides meeting at the vertex define rotations by angles 2{{pi}}/''a'', 2{{pi}}/''b'' and 2{{pi}}/''c''; Let ''g''_''A'', ''g''_''B'' and ''g''_''C'' be the corresponding products of generators of Γ = '''Z'''₂ ∗ '''Z'''₂ ∗ '''Z'''₂. Let Γ₀ be the normal subgroup of index 2 of Γ, consisting of elements that are the product of an even number of generators; and let Γ₁ be the normal subgroup of Γ generated by (''g''_''A'')^''a'', (''g''_''B'')^''b'' and (''g''_''C'')^''c''. These act trivially on Δ. Let {{overline|Γ}} = Γ/Γ₁ and {{overline|Γ}}₀ = Γ₀/Γ₁.

The disjoint union of copies of {{overline|Δ}} indexed by elements of {{overline|Γ}} with edge identifications has the natural structure of a Riemann surface Σ. At an interior point of a triangle there is an obvious chart. As a point of the interior of an edge the chart is obtained by reflecting the triangle across the edge. At a vertex of a triangle with interior angle {{pi}}/''n'', the chart is obtained from the 2''n'' copies of the triangle obtained by reflecting it successively around that vertex. The group {{overline|Γ}} acts by deck transformations of Σ, with elements in {{overline|Γ}}₀ acting as holomorphic mappings and elements not in {{overline|Γ}}₀ acting as antiholomorphic mappings.

There is a natural map ''P'' of Σ into the hyperbolic plane. The interior of the triangle with label ''g'' in {{overline|Γ}} is taken onto ''g''(Δ), edges are taken to edges and vertices to vertices. It is also easy to verify that a neighbourhood of an interior point of an edge is taken into a neighbourhood of the image; and similarly for vertices. Thus ''P'' is locally a homeomorphism and so takes open sets to open sets. The image ''P''(Σ), i.e. the union of the translates ''g''({{overline|Δ}}), is therefore an open subset of the upper half plane. On the other hand, this set is also closed. Indeed, if a point is sufficiently close to {{overline|Δ}} it must be in a translate of {{overline|Δ}}. Indeed, a neighbourhood of each vertex is filled out the reflections of {{overline|Δ}} and if a point lies outside these three neighbourhoods but is still close to {{overline|Δ}} it must lie on the three reflections of {{overline|Δ}} in its sides. Thus there is δ > 0 such that if ''z'' lies within a distance less than δ from {{overline|Δ}}, then ''z'' lies in a {{overline|Γ}}-translate of {{overline|Δ}}. Since the hyperbolic distance is {{overline|Γ}}-invariant, it follows that if ''z'' lies within a distance less than δ from Γ({{overline|Δ}}) it actually lies in Γ({{overline|Δ}}), so this union is closed. By connectivity it follows that ''P''(Σ) is the whole upper half plane.

On the other hand, ''P'' is a local homeomorphism, so a covering map. Since the upper half plane is simply connected, it follows that ''P'' is one-one and hence the translates of Δ tessellate the upper half plane. This is a consequence of the following version of the monodromy theorem for coverings of Riemann surfaces: if ''Q'' is a covering map between Riemann surfaces Σ₁ and Σ₂, then any path in Σ₂ can be lifted to a path in Σ₁ and any two homotopic paths with the same end points lift to homotopic paths

with the same end points; an immediate corollary is that if Σ₂ is simply connected, ''Q'' must be a homeomorphism.<ref>{{harvnb|Beardon|1984|pages=106–107, 110–111}}</ref> To apply this, let Σ₁ = Σ, let Σ₂ be the upper half plane and let ''Q'' = ''P''. By the corollary of the monodromy theorem, ''P'' must be one-one.

It also follows that ''g''(Δ) = Δ if and only if ''g'' lies in Γ₁, so that the homomorphism of {{overline|Γ}}₀ into the Möbius group is faithful.

=== Hyperbolic reflection groups ===

{{See also|Uniform tilings in hyperbolic plane}}

<!--{{under construction|notready=true}}-->

The tessellation of the Schwarz triangles can be viewed as a generalization of the theory of infinite [[Coxeter group]]s, following the theory of [[hyperbolic reflection group]]s developed algebraically by [[Jacques Tits]]<ref>See:

*{{harvnb|Tits|2013}}

*{{harvnb|Bourbaki|1981}}

*{{harvnb|Humphreys|1990}}</ref> and geometrically by [[Ernest Vinberg]].<ref>See:

*{{harvnb|Vinberg|1971}}

*{{harvnb|Vinberg|1985}}

*{{harvnb|Shvartsman|Vinberg}}</ref> In the case of the [[Lobachevsky plane|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 Forms"'', being one example.<ref>{{harvnb|Lehner|1964}}</ref><ref>{{harvnb|Maskit|1971}}</ref> 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. The approach using Coxeter groups will be summarised here, within the general framework of classification of hyperbolic reflection groups.<ref> See:

*{{harvnb|Brown|1989}}

*{{harvnb|Humphreys|1990}}

*{{harvnb|Abremko|Brown|2007}}

*{{harvnb|Davis|2008}}</ref>

Let {{mvar|''r''}}, {{mvar|''s''}} and {{mvar|''t''}} be symbols and let {{mvar|''a''}}, {{mvar|''b''}}, {{mvar|''c''}} ≥ 2 be integers, possibly ∞, with

:<math>{1 \over a} + {1 \over b} + {1 \over c} < 1.</math>

Define {{mvar|Γ}} to be the [[presentation of a group|group with presentation]] having generators {{mvar|''r''}}, {{mvar|''s''}} and {{mvar|''t''}} that are all [[involution (mathematics)|involutions]] and satisfy ({{mvar|''st''}})^{{{mvar|''a''}}} = 1,

({{mvar|''tr''}})^{{{mvar|''b''}}} = 1 and ({{mvar|''rs''}})^{{{mvar|''c''}}} = 1. If one of the integers is infinite, then the product has infinite order. The generators {{mvar|''r''}}, {{mvar|''s''}} and {{mvar|''t''}} are called the ''simple reflections''.

Set {{mvar|''A''}} = cos {{pi}} / {{mvar|''a''}} if {{mvar|''a''}} ≥ 2 is finite and cosh {{mvar|''x''}} with {{mvar|''x''}} > 0 otherwise; similarly set {{mvar|''B''}} = cos {{pi}} / {{mvar|''b''}} or cosh {{mvar|''y''}} and {{mvar|''C''}} = cos {{pi}} / {{mvar|''c''}} or cosh {{mvar|''z''}}.{{sfn|Heckman|2017}} Let '''e'''_{{{var|''r''}}}, '''e'''_{{{mvar|''s''}}} and '''e'''_{{{mvar|''t''}}} be a basis for a 3-dimensional real vector space {{mvar|''V''}} with symmetric bilinear form {{mvar|Λ}} such that {{mvar|Λ}}('''e'''_{{{mvar|''s''}}},'''e'''_{{{mvar|''t''}}}) = − {{mvar|''A''}}, {{mvar|Λ}}('''e'''_{{{mvar|''t''}}},'''e'''_{{{mvar|''r''}}}) = − {{mvar|''B''}} and {{mvar|Λ}}('''e'''_{{{mvar|''r''}}},'''e'''_{{{mvar|''s''}}}) = − {{mvar|''C''}}, with the three diagonal entries equal to one. The symmetric bilinear form {{mvar|Λ}} is non-degenerate with signature (2,1). Define {{mvar|ρ}}('''v''') = '''v''' − 2 {{mvar|Λ}}('''v''','''e'''_{{{mvar|''r''}}}) '''e'''_{{{mvar|''r''}}}, {{mvar|σ}}('''v''') = '''v''' − 2 {{mvar|Λ}}('''v''','''e'''_{{{mvar|''s''}}}) '''e'''_{{{mvar|''s''}}} and

{{mvar|τ}}('''v''') = '''v''' − 2 {{mvar|Λ}}('''v''','''e'''_{{{mvar|''t''}}}) '''e'''_{{{mvar|''t''}}}.

'''Theorem (geometric representation).''' ''The operators'' {{mvar|ρ}}, {{mvar|σ}} ''and'' {{mvar|τ}} ''are involutions on'' {{mvar|''V''}}, ''with respective eigenvectors'' '''e'''_{{{mvar|''r''}}},

'''e'''_{{{mvar|''s''}}} ''and'' '''e'''_{{{mvar|''t''}}} ''with simple eigenvalue'' −1. ''The products of the operators have orders corresponding to the presentation above (so'' {{mvar|στ}} ''has order'' {{mvar|''a''}}, ''etc). The operators'' {{mvar|ρ}}, {{mvar|σ}} ''and'' {{mvar|τ}} ''induce a representation of'' {{mvar|Γ}} ''on'' {{mvar|''V''}} ''which preserves'' {{mvar|Λ}}.

The bilinear form {{mvar|Λ}} for the basis has matrix

:<math>M = \begin{pmatrix}

1 & -C & -B\\

-C & 1 & -A\\

-B & -A & 1\\

\end{pmatrix},</math>

so has determinant 1−{{mvar|''A''}}²−{{mvar|''B''}}²−{{mvar|''C''}}²−2{{mvar|''ABC''}}. If {{mvar|''c''}} = 2, say, then the eigenvalues of the matrix are 1 and 1 ± ({{mvar|''A''}}²+{{mvar|''B''}}²)^½. The condition {{mvar|''a''}}⁻¹ + {{mvar|''b''}}⁻¹ < ½ immediately forces {{mvar|''A''}}²+{{mvar|''B''}}² > 1, so that Λ must have signature (2,1). So in general {{mvar|''a''}}, {{var|''b''}}, {{mvar|''c''}} ≥ 3. Clearly the case where all are equal to 3 is impossible. But then the determinant of the matrix is negative while its trace is positive. As a result two eigenvalues are positive and one negative, i.e. Λ has signature (2,1). Manifestly {{mvar|ρ}}, {{mvar|σ}} and {{mvar|τ}} are involutions, preserving {{mvar|Λ}} with the given −1 eigenvectors.

To check the order of the products like {{mvar|''στ''}}, it suffices to note that:

# the reflections {{mvar|''σ''}} and {{mvar|''τ''}} generate a finite or infinite [[dihedral group]];

# the 2-dimensional linear span {{mvar|''U''}} of '''e'''_{{{mvar|''s''}}} and '''e'''_{{{mvar|''t''}}} is invariant under {{mvar|''σ''}} and {{mvar|''τ''}}, with the restriction of {{mvar|Λ}} positive-definite;

# {{mvar|''W''}}, the orthogonal complement of {{mvar|''U''}}, is negative-definite on {{mvar|Λ}}, and {{mvar|''σ''}} and {{mvar|''τ''}} act trivially on {{mvar|''W''}}.

(1) is clear since if {{mvar|''γ''}} = {{mvar|''στ''}} generates a normal subgroup with {{mvar|''σγσ''}}^–1 = {{mvar|''γ''}}^–1. For (2), {{mvar|''U''}} is invariant by definition and the matrix is positive-definite since 0 < cos {{pi}} / {{mvar|''a''}} < 1. Since {{mvar|''Λ''}} has signature (2,1), a non-zero vector {{mvar|''w''}} in {{mvar|''W''}} must satisfy {{mvar|''Λ''}}({{mvar|''w''}},{{mvar|''w''}}) < 0. By definition, {{mvar|''σ''}} has eigenvalues 1 and –1 on {{mvar|''U''}}, so {{mvar|''w''}} must be fixed by {{mvar|''σ''}}. Similarly {{mvar|''w''}} must be fixed by {{mvar|''τ''}}, so that (3) is proved. Finally in (1)

:<math> \sigma ({\mathbf e}_s) = -{\mathbf e}_s,\,\,\,\,\, \sigma({\mathbf e}_t) = 2 \cos(\pi / a) {\mathbf e}_s + {\mathbf e}_t,\,\,\,\,\, \tau({\mathbf e}_s) = 2 \cos(\pi / a) {\mathbf e}_s + {\mathbf e}_t,\,\,\,\,\, \tau({\mathbf e}_t) = -{\mathbf e}_t,</math>

so that, if {{mvar|''a''}} is finite, the eigenvalues of {{mvar|''στ''}} are -1, {{mvar|''ς''}} and {{mvar|''ς''}}^–1, where {{mvar|''ς''}} = exp {{mvar|''2πi''}} / {{mvar|''a''}}; and if {{mvar|''a''}} is infinite, the eigenvalues are -1, {{mvar|''X''}} and {{mvar|''X''}}^–1, where

{{mvar|''X''}} = exp {{mvar|''2x''}}. Moreover a straightforward induction argument shows that if {{mvar|''θ''}} = {{mvar|''π''}} / {{mvar|''a''}} then<ref>{{harvnb|Howlett|1996}}</ref>

:<math>(\sigma\tau)^m({\mathbf e}_s) = [\sin(2m+1)\theta/\sin\theta]{\mathbf e}_s + [\sin 2m\theta/\sin\theta]{\mathbf e}_t, </math>

:<math>\tau(\sigma\tau)^m({\mathbf e}_s) = [\sin(2m+1)\theta/\sin\theta]{\mathbf e}_s + [\sin (2m+2)\theta/\sin\theta]{\mathbf e}_t </math>

and if {{mvar|''x''}} > 0 then

:<math> (\sigma\tau)^m({\mathbf e}_s) = [\sinh(2m+1)x/\sinh x]{\mathbf e}_s + [\sinh 2mx/\sinh x]{\mathbf e}_t,</math>

:<math>\tau(\sigma\tau)^m({\mathbf e}_s) = [\sinh(2m+1)x/\sinh x]{\mathbf e}_s + [\sinh (2m+2)x/\sinh x]{\mathbf e}_t.</math><ref>In the limit as {{mvar|''x''}} tends to 0, ({{mvar|''στ''}})^{{{mvar|''m''}}}('''e'''_{{{mvar|''s''}}}) = ({{mvar|''2m''}}+{{mvar|''1''}})'''e'''_{{{mvar|''s''}}} + {{mvar|''2m''}}'''e'''_{{{mvar|''t''}}} and {{mvar|''τ''}}({{mvar|''στ''}})^{{{mvar|''m''}}}('''e'''_{{{mvar|''s''}}}) = ({{mvar|''2m''}}+{{mvar|''1''}})'''e'''_{{{mvar|''s''}}} + ({{mvar|''2m''}}+{{mvar|''2''}})'''e'''_{{{mvar|''t''}}}.</ref>

Let {{mvar|''Γ''}}_{{{mvar|''a''}}} be the dihedral subgroup of {{mvar|''Γ''}} generated by {{mvar|''s''}} and {{mvar|''t''}}, with analogous definitions for {{mvar|''Γ''}}_{{{mvar|''b''}}} and {{mvar|''Γ''}}_{{{mvar|''c''}}}. Similarly define {{mvar|''Γ''}}_{{{mvar|''r''}}} to be the cyclic subgroup of {{mvar|''Γ''}} given by the 2-group {1,{{mvar|''r''}}}, with analogous definitions for {{mvar|''Γ''}}_{{{mvar|''s''}}} and {{mvar|''Γ''}}_{{{mvar|''t''}}}. From the properties of the geometric representation, all six of these groups act faithfully on {{mvar|''V''}}. In particular

{{mvar|''Γ''}}_{{{mvar|''a''}}} can be identified with the group generated by {{mvar|''σ''}} and {{mvar|''τ''}}; as above it decomposes explicitly as a direct sum of the 2-dimensional irreducible subspace {{mvar|''U''}} and the 1-dimensional subspace {{mvar|''W''}} with a trivial action. Thus there is a unique vector '''w''' = '''e'''_{{{mvar|''r''}}} + {{mvar|''λ''}} '''e'''_{{{mvar|''s''}}} + {{mvar|''μ''}} '''e'''_{{{mvar|''t''}}} in {{mvar|''W''}} satisfying {{mvar|''σ''}}('''w''') = '''w''' and {{mvar|''τ''}}('''w''') = '''w'''. Explicitly {{mvar|''λ''}} = ({{mvar|''C''}} + {{mvar|''AB''}})/(1 – {{mvar|''A''}}²) and {{mvar|''μ''}} = ({{mvar|''B''}} + {{mvar|''AC''}})/(1 – {{mvar|''A''}}²).

'''Remark on representations of dihedral groups.''' It is well known that, for finite-dimensional real inner product spaces, two orthogonal involutions {{mvar|''S''}} and {{mvar|''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 ({{mvar|''S''}} – {{mvar|''T''}})² commutes with both {{math|''S''}} and {{mvar|''T''}}. In the case above, however, where the bilinear form {{mvar|''Λ''}} is no longer a positive definite inner product, different ''ad hoc'' reasoning has to be given.

'''Theorem (Tits).''' ''The geometric representation of the Coxeter group is faithful.''

This result was first proved by Tits in the early 1960s and first published in the text of {{harvtxt|Bourbaki|1968}} with its numerous exercises. In the text, the [[Weyl chamber|fundamental chamber]] was introduced by an inductive argument; exercise 8 in §4 of Chapter V was expanded by [[Vinay Deodhar]] to develop a theory of positive and negative roots and thus shorten the original argument of Tits.<ref>See:

*{{harvnb|Tits|2013}}

*{{harvnb|Bourbaki|1968}}

*{{harvnb|Steinberg|1968}}

*{{harvnb|Hiller|1980}}

*{{harvnb|Deodhar|1982}}, {{harvnb|Deodhar|1986}}

*{{harvnb|Humphreys|1990}}

*{{harvnb|Howlett|1996}}

*{{harvnb|Heckman|2017}}

</ref>

Let '''X''' be the convex cone of sums {{var|''κ''}}'''e'''_{{{mvar|''r''}}} + {{mvar|''λ''}}'''e'''_{{{mvar|''s''}}} + {{mvar|''μ''}}'''e'''_{{{mvar|''t''}}} with real non-negative coefficients, not all of them zero. For {{mvar|''g''}} in the group {{mvar|''Γ''}}, define

ℓ({{mvar|''g''}}), the '''[[word problem for groups|word length]]''' or '''length''', to be the minimum number of reflections from {{mvar|''r''}}, {{mvar|''s''}} and {{mvar|''t''}} required to write {{mvar|''g''}} as an ordered composition of simple reflections. Define a '''positive root''' to be a vector {{mvar|''g''}}'''e'''_{{{mvar|''r''}}}, {{mvar|''g''}}'''e'''_{{{mvar|''s''}}} or {{mvar|''g''}}'''e'''_{{{mvar|''r''}}} lying in '''X''', with {{mvar|''g''}} in {{mvar|''Γ''}}.<ref>Here {{mvar|''Γ''}} is regarded as acting on {{mvar|''V''}} through the geometric representation.</ref>

It is routine to check from the definitions that<ref name="HHH">See:

* {{harvnb|Humphreys|1990}}

* {{harvnb|Howlett|1996}}

* {{harvnb|Heckman|2017}}</ref>

*if |ℓ({{mvar|''gq''}}) – ℓ({{mvar|''g''}})| = 1 for a simple reflection {{mvar|''q''}} and, if {{mvar|''g''}} ≠ 1, there is always a simple reflection {{mvar|''q''}} such that ℓ({{mvar|''g''}}) = ℓ({{mvar|''gq''}}) + 1;

*for {{mvar|''g''}} and {{mvar|''h''}} in {{mvar|''Γ''}}, ℓ({{mvar|''gh''}}) ≤ ℓ({{mvar|''g''}}) + ℓ({{mvar|''h''}}).

'''Proposition.''' ''If'' {{mvar|''g''}} ''is in'' {{mvar|''Γ''}} ''and'' ℓ({{mvar|''gq''}}) = ℓ({{mvar|''g''}}) ± 1 ''for a simple reflection'' {{mvar|''q''}}, ''then'' {{mvar|''g''}}'''e'''_{{{mvar|''q''}}} ''lies in'' ±'''X''', ''and is therefore a positive or negative root, according to the sign.''

Replacing {{mvar|''g''}} by {{mvar|''gq''}}, only the positive sign needs to be considered. The assertion will be proved by induction on ℓ({{mvar|''g''}}) = {{mvar|''m''}}, it being trivial for {{var|''m''}} = 0.

Assume that ℓ({{mvar|''gs''}}) = ℓ({{mvar|''g''}}) + 1. If ℓ({{mvar|''g''}}) = {{mvar|''m''}} > 0, without less of generality it may be assumed that the minimal expression for {{mvar|''g''}} ends with ...{{mvar|''t''}}. Since {{mvar|''s''}} and {{mvar|''t''}} generate the dihedral group {{mvar|''Γ''}}_{{{mvar|''a''}}}, {{mvar|''g''}} can be written as a product {{mvar|''g''}} = {{mvar|''hk''}}, where {{mvar|''k''}} = ({{mvar|''st''}})^{{{var|''n''}}} or {{mvar|''t''}}({{mvar|''st''}})^{{{var|''n''}}} and {{mvar|''h''}} has a minimal expression that ends with ...{{mvar|''r''}}, but never with {{mvar|''s''}} or {{mvar|''t''}}. This implies that ℓ({{mvar|''hs''}}) = ℓ({{mvar|''h''}}) + 1 and ℓ({{mvar|''ht''}}) = ℓ({{mvar|''h''}}) + 1. Since ℓ({{mvar|''h''}}) < {{mvar|''m''}}, the induction hypothesis shows that both {{mvar|''h''}}'''e'''_{{{mvar|''s''}}} and

{{mvar|''h''}}'''e'''_{{{mvar|''t''}}} lie in '''X'''. It therefore suffices to show that {{mvar|''k''}}'''e'''_{{{mvar|''s''}}} has the form {{mvar|λ}}'''e'''_{{{mvar|''s''}}} + {{mvar|μ}}'''e'''_{{{mvar|''t''}}} with {{mvar|λ}}, {{mvar|μ}} ≥ 0, not both 0. But that has already been verified in the formulas above.<ref name="HHH" />

'''Corollary (proof of Tits' theorem).''' ''The geometric representation is faithful.''

It suffices to show that if {{mvar|''g''}} fixes '''e'''_{{{mvar|''r''}}}, '''e'''_{{{mvar|''s''}}} and '''e'''_{{{mvar|''t''}}}, then {{mvar|''g''}} = 1. Considering a minimal expression for {{mvar|''g''}} ≠ 1, the conditions ℓ({{mvar|''gq''}}) = ℓ({{mvar|''g''}}) + 1 clearly cannot be simultaneously satisfied by the three simple reflections {{mvar|''q''}}.

Note that, as a consequence of Tits' theorem, the generators {{mvar|''g''}} = {{mvar|''st''}}, {{mvar|''h''}} = {{mvar|''tr''}} and {{mvar|''k''}} = {{mvar|''rs''}} satisfy {{mvar|''g''^''a''}} = 1, {{mvar|''h''^''b''}} = 1 and {{mvar|''k''^''c''}} = 1 with {{mvar|''ghk''}} = 1. This gives a presentation of the orientation-preserving index 2 normal subgroup {{mvar|Γ₁}} of {{mvar|Γ}}. The presentation corresponds to the fundamental ___domain obtained by reflecting two sides of the geodesic triangle to form a geodesic [[parallelogram]] (a special case of Poincaré's polygon theorem).<ref> See:

* {{harvnb|Magnus|Karrass|Solitar|1976}}

* {{harvnb|Magnus|1974}}

* {{harvnb|Iversen|1992}}

* {{harvnb|Ellis|2019}}</ref>

'''Further consequences.''' ''The roots are the disjoint union of the positive roots and the negative roots. The simple reflection'' {{mvar|''q''}} ''permutes every positive root other than'' '''e'''_{{{mvar|''q''}}}. ''For'' {{mvar|''g''}} ''in'' {{mvar|''Γ''}}, ℓ({{mvar|''g''}}) ''is the number of positive roots made negative by'' {{mvar|''g''}}.

'''Fundamental ___domain and Tits cone.'''<ref>See:

* {{harvnb|Tits|2013}}

* {{harvnb|Bourbaki |1968}}

* {{harvnb|Maxwell| 1982}}

* {{harvnb|Abramenko| Brown | 2007}}

* {{harvnb|Davis|2008}}

* {{harvnb|Heckman| 2017}}</ref>

Let {{mvar|''G''}} be the 3-dimensional closed Lie subgroup of GL({{mvar|''V''}}) preserving {{mvar|''Λ''}}. As {{mvar|''V''}} can be identified with a 3-dimensional Lorentzian or Minkowski space with signature (2,1), the group {{mvar|''G''}} is isomorphic to the [[Lorentz group]] O(2,1) and therefore SL_±(2,'''R''') / {±{{mvar|''I''}}}.<ref>SL_±(2,'''R''') is the subgroup of GL(2,'''R''') with determinant ±1.</ref> Choosing '''e''' to be a positive root vector in '''X''', the stabilizer of '''e''' is a maximal compact subgroup {{mvar|''K''}} of {{mvar|''G''}} isomorphic to O(2). The [[homogeneous space]] {{mvar|''X''}} = {{mvar|''G''}} / {{mvar|''K''}} is a [[symmetric space]] of constant negative curvature, which can be identified with the 2-dimensional [[hyperboloid]] or [[Lobachevsky plane]] <math>\mathfrak{H}^2</math>. The discrete group {{mvar|''Γ''}} acts discontinuously on {{mvar|''G''}} / {{mvar|''K''}}: the quotient space {{mvar|''Γ''}} \ {{mvar|''G''}} / {{mvar|''K''}} is compact if {{mvar|''a''}}, {{mvar|''b''}} and {{mvar|''c''}} are all finite, and of finite area otherwise. Results about the Tits fundamental chamber have a natural interpretation in terms of the corresponding Schwarz triangle, which translate directly into the properties of the tessellation of the geodesic triangle through the hyperbolic reflection group {{mvar|''Γ''}}. The passage from Coxeter groups to tessellation can first be found in the exercises of §4 of Chapter V of {{harvtxt|Bourbaki|1968}}, due to Tits, and in {{harvtxt|Iwahori|1966}}; currently numerous other equivalent treatments are available, not always directly phrased in terms of symmetric spaces.

=== Approach of Maskit, de Rham and Beardon ===

{{harvtxt|Maskit|1971}} gave a general proof of Poincaré's polygon theorem in hyperbolic space; a similar proof was given in {{harvtxt|de Rham|1971}}. Specializing to the hyperbolic plane and Schwarz triangles, this can be used to give a modern approach for establishing that the existence of Schwarz triangle tessellations, as described in {{harvtxt|Beardon|1983}} and {{harvtxt|Maskit|1988}}. The Swiss mathematicians {{harvtxt|de la Harpe|1990}} and [[André Haefliger|Haefliger]] have provided an introductory account, taking [[geometric group theory]] as their starting point.<ref>See:

*{{harvnb|Milnor|1975}}

*{{harvnb|Beardon|1983|pages=242–249}}

*{{harvnb|Iversen|1992|pages=200–208}}

*{{harvnb|Bridson|Haefliger|1999}}

*{{harvnb|Berger|2010|pages=616–617}}</ref>

==Conformal mapping of Schwarz triangles==

In this section Schwarz's explicit conformal mapping from the unit disc or the upper half plane to the interior of a Schwarz triangle will be constructed as the ratio of solutions of a hypergeometric ordinary differential equation, following {{harvtxt|Carathéodory|1954|pages=129–194}}, {{harvtxt|Nehari|1975|pages=198–209, 308–332}} and {{harvtxt|Hille|1976|pages=371–401}}. The classical account of {{harvtxt|Ince|1944|pages=389–395}} has an account of the monodromy of second order complex order differential equations with regular singular points. The limiting case of ideal triangles, the [[Farey series]] and the [[modular lambda function]] is explained in {{harvtxt|Ahlfors|1966|pages=254–274}},

{{harvtxt|Chandrasekharan|1985|pages=108–121}}, {{harvtxt|Mumford|Series|Wright|2015}} and {{harvtxt|Hardy|Wright|2008}}.

The '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|conformally maps]] the [[upper half plane]] to a triangle in the upper half plane having lines or circular arcs for edges. Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle. If any of ''α, β'', and ''γ'' are greater than zero, then the Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:

:<math>s(z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>

where ''a'' = (1−α−β−γ)/2, ''b'' = (1−α+β−γ)/2, ''c'' = 1−α, ''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2, ''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and ''c''′ = 2 − ''c'' = 1 + α. This mapping has singular points at ''z'' = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,

:<math>\begin{aligned}

s(0) &= 0, \\[5mu]

s(1) &= \frac

{\Gamma(1-a')\Gamma(1-b')\Gamma(c')}

{\Gamma(1-a)\Gamma(1-b)\Gamma(c)},\,\text{and} \\[5mu]

s(\infty) &= \exp\left(i \pi \alpha \right)\frac

{\Gamma(1-a')\Gamma(b)\Gamma(c')}

{\Gamma(1-a)\Gamma(b')\Gamma(c)}.

\end{aligned}</math>

This formula can be derived using the [[Schwarzian derivative]].

This function can be used to map the upper half-plane to a [[spherical triangle]] on the [[Riemann sphere]] if α + β + γ > 1, or a [[hyperbolic triangle]] on the [[Poincaré disk]] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a''&thinsp;=&thinsp;0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the [[Schwarz–Christoffel transformation]]. In the special case of [[ideal triangle]]s, where all the angles are zero, the triangle function yields the [[modular lambda function]].

== See also ==

* [[Conformal map projection]]s.<ref name=Lee>{{cite book |last=Lee |first=Laurence |title=Conformal Projections based on Elliptic Functions |year=1976 |publisher=University of Toronto Press |series=Cartographica Monographs |volume=16

|isbn=9780919870161 |url=https://archive.org/details/conformalproject0000leel |url-access=limited }} Chapters also published in [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer''. '''13''' (1). 1976.]</ref>

==Notes==

{{notelist}}

==References==

{{reflist|20em30em}}

==Sources==

{{refbegin|colwidth=30em}}

* {{Cite book |last=Ahlfors |first=Lars V. |author-link=Lars Ahlfors |title=Complex analysis: an introduction to the theory of analytic functions of one complex variable |date=1979 |publisher=McGraw-Hill |isbn=0-07-000657-1 |edition=3 |___location=New York |oclc=4036464}}

* {{cite book|mr = 2439729 | last1= Abramenko| first1 =Peter |last2 =Brown|first2 =Kenneth S.| author2-link= Kenneth S. Brown| title=Buildings: Theory and Applications|series= Graduate Texts in Mathematics|volume= 248| publisher = [[Springer-Verlag]]| year= 2007 | isbn = 978-0-387-78834-0}}

* {{cite book |last=Carathéodory |first=Constantin |author-link=Constantin Carathéodory |title=Theory of functions of a complex variable |volume=2 |translator=F. Steinhardt |publisher=Chelsea |year=1954|url=https://books.google.com/books?id=UTTvAAAAMAAJ|oclc=926250115}}

* {{citation |last=Ahlfors |first=Lars V. |author-link=Lars Ahlfors |title=Complex Analysis |publisher=McGraw Hill |year=1966 |edition=2nd }}

* {{citationCite book |last=BeardonHille |first=AlanEinar F.| author-link =Einar Alan Frank BeardonHille |title=TheOrdinary Geometrydifferential ofequations Discretein Groupsthe |series=Graduatecomplex Texts in Mathematics___domain |volumedate=911997 |publisher=SpringerDover |year=1983Publications |isbn=0-387486-9078869620-20 |pages ___location=Mineola, N.Y. 242–252| chapter oclc= Poincaré's Theorem36225146}}

* {{citationCite book |last=BeardonNehari |first=Alan F.Zeev |author-link =Zeev Alan Frank BeardonNehari |title=AConformal primer on Riemann surfaces |journal=London Mathematical Society Lecture Note Seriesmapping |volumedate=781975 |publisher=CambridgeDover University Press |year=1984Publications |isbn=05212710450-486-61137-X |url-access___location=registrationNew York |urloclc=https://archive.org/details/primeronriemanns0000bear 1504503}}

* {{citationcite book |lastlast1=BergerSansone |firstfirst1=MarcelGiovanni |author-link=MarcelGiovanni BergerSansone |last2=Gerretsen |first2=Johan |title=GeometryLectures revealed.on Athe Jacob'stheory ladderof tofunctions modernof highera geometrycomplex |translator=Lestervariable. SenechalII: Geometric theory |publisher=SpringerWolters-Noordhoff |year=20101969 |isbnoclc=978-3-540-70996-1 245996162}}

* {{citation |last1=Berndt |first1=Bruce C. |author-link=Bruce C. Berndt |last2=Knopp |first2=Marvin I. |author-link2=Marvin Knopp |title=Hecke's theory of modular forms and Dirichlet series |series=Monographs in Number Theory |volume=5 |publisher=World Scientific |year=2008 |isbn=978-981-270-635-5 }}

* {{cite book | mr =2133266 |last1= Björner| first1= Anders| author1-link= Anders Björner |

last2= Brenti| first2= Francesco | title=Combinatorics of Coxeter groups| series= Graduate Texts in Mathematics| volume= 231| publisher=[[Springer-Verlag]]| year= 2005 | isbn = 978-3540-442387}}

* {{cite book| mr =1482800 | last = Borel | first = Armand | author-link = Armand Borel | title = Automorphic forms on SL₂(R) | series = Cambridge Tracts in Mathematics | volume = 130

| publisher = [[Cambridge University Press]] | year = 1997 | isbn = 0-521-58049-8 | url = https://www.cambridge.org/core/books/automorphic-forms-on-sl2-r/20CDEAD276B61040B2B9013A21ACC145

|chapter= I. Basic material on SL₂(R), discrete subgroups, and the upper half-plane}}

* {{cite book| mr =0240238|last=Bourbaki|first= Nicolas|author-link= Nicolas Bourbaki |series=Éléments de mathématique |title=Groupes et algèbres de Lie|chapter= Chapitre IV : Groupes de Coxeter et systèmes de Tits • Chapitre V : Groupes engendrés par des réflexions|publisher= Hermann|___location= Paris|year = 1968 | pages =1–56, 57–141 | lang = fr}} Reprinted by Masson in 1981 as {{isbn|2-225-76076-4}}.

* {{cite book |mr =1744486 | last1=Bridson| first1= Martin R.|author1-link= Martin Bridson|last2= Haefliger| first2 = André | author2-link = André Haefliger | title = Metric spaces of non-positive curvature | series =Grundlehren der mathematischen Wissenschaften | volume = 319| publisher = [[Springer-Verlag]] | year = 1999

| isbn = 3-540-64324-9 | url = https://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf |chapter =I. Basic material on SL₂(R), discrete subgroups, and the upper half-plane}}

* {{cite book|mr = 0969123 | last = Brown | first = Kenneth S. | author-link = Kenneth S. Brown| title=Buildings| publisher = [[Springer-Verlag]] | year = 1989 | isbn = 0-387-96876-8 }}

* {{citation |last=Busemann |first=Herbert |author-link=Herbert Busemann |title=The geometry of geodesics |publisher=Academic Press |year=1955 }}

* {{citation |last=Carathéodory |first=Constantin |author-link=Constantin Carathéodory |title=Theory of functions of a complex variable |volume=2 |translator=F. Steinhardt |publisher=Chelsea |year=1954}}

* {{citation |last=Chandrasekharan |first=K. |author-link=K. S. Chandrasekharan |title=Elliptic functions |series=Grundlehren der Mathematischen Wissenschaften |volume=281 |publisher=Springer |year=1985 |isbn=3-540-15295-4 }}

* {{citation |last=Davis |first=Michael W. |title=The geometry and topology of Coxeter groups |series=London Mathematical Society Monographs |volume=32 |publisher=[[Princeton University Press]] |year=2008 |isbn=978-0-691-13138-2| chapter=Appendix D. The Geometric Representation| pages = 439–447 }}

* {{cite book|mr=1170367| last=de la Harpe|first= Pierre | chapter=An invitation to Coxeter groups|title= Group theory from a geometrical viewpoint (Trieste, 1990)|pages= 193–253|publisher= [[World Scientific]] |year= 1991}}

* {{cite journal| mr =0647210 |last=Deodhar | first= Vinay V. | author-link= Vinay Deodhar| title = On the root system of a Coxeter group |journal = [[Comm. Algebra]]| volume = 10 | year =1982 |issue=6 | pages = 611–630 |

doi=10.1080/00927878208822738}}

* {{cite journal| url = https://www.e-periodica.ch/cntmng?var=true&pid=ens-001:1986:32::54| last=Deodhar | first= Vinay V. | author-link= Vinay Deodhar| journal = [[Enseign. Math.]] | mr = 0850554 | title=Some characterizations of Coxeter groups | volume = 32 | year =1986 | pages = 111–120 }}

* {{cite journal|last=de Rham|first= G.|author-link=Georges de Rham|title= Sur les polygones générateurs de groupes fuchsiens| lang=fr |journal= [[Enseign. Math.]]|volume= 17 |year =1971|pages= 49–61|url= https://www.e-periodica.ch/cntmng?pid=ens-001%3A1971%3A17%3A%3A33}}

* {{cite book | last = Ellis | first = Graham | chapter = Triangle groups | isbn = 978-0-19-883298-0 | mr = 3971587 | pages = 441–444 | publisher = [[Oxford University Press]] | title = An Invitation to Computational Homotopy | year = 2019}}

* {{cite journal|mr = 1279064|last1=Epstein|first1= David B. A.|author1-link=David B. A. Epstein|last2= Petronio|first2= Carlo |title=An exposition of Poincaré's polyhedron theorem|journal=[[Enseign. Math.]]|volume= 40 | year= 1994|pages= 113–170|url= https://www.e-periodica.ch/cntmng?pid=ens-001%3A1994%3A40%3A%3A49}}

* {{citation |last=Evans |first=Ronald |title=A fundamental region for Hecke's modular group |journal=[[Journal of Number Theory]] |volume=5 |year=1973 |issue=2 |pages=108–115 |doi=10.1016/0022-314x(73)90063-2 |bibcode=1973JNT.....5..108E |doi-access=free }}

* {{citation |last1=Fricke |first1=Robert |author-link1=Robert Fricke|last2=Klein |first2=Felix |author-link2=Felix Klein |date=1897 |title=Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen |url=https://archive.org/details/vorlesungenber01fricuoft |publisher=B. G. Teubner |language=de |isbn=978-1-4297-0551-6 |jfm=28.0334.01 }}

* {{citation |last1=Graham |first1=Ronald L. |author-link=Ronald Graham (mathematician) |last2=Knuth |first2=Donald E. |author-link2=Donald Knuth |last3=Patashnik |first3=Oren |author-link3=Oren Patashnik |year=1994 |title=Concrete mathematics |edition=2nd |publisher=Addison-Wesley |isbn=0-201-55802-5 |pages=116–118 }}

* {{citation |last1=Hardy |first1=G. H. |author-link=G. H. Hardy |last2=Wright |first2=E. M. |author-link2=E. M. Wright |title=An introduction to the theory of numbers |edition=6th |publisher=Oxford University Press |year=2008 |isbn=978-0-19-921986-5 }}

* {{cite book |first=Allen |last=Hatcher |author-link=Allen Hatcher |title=Topology of Numbers |publisher=[[Cornell University]] |year=2013 |url=https://pi.math.cornell.edu/~hatcher/TN/TNbook.pdf

|access-date=25 February 2022|chapter= 1. The Farey Diagram }}

* {{citation |title=Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung |language=de |last=Hecke |first=E. |author-link=Erich Hecke |journal=[[Mathematische Annalen]] |volume=112 |year=1935 |pages=664–699 |doi=10.1007/bf01565437 |doi-access=free }}

* {{cite web|last=Heckman| first= Gert J. |title = Coxeter Groups| year=2018|access-date= 3 March 2022| url = https://www.math.ru.nl/~heckman/CoxeterGroups.pdf | work= [[Radboud University Nijmegen]]}}

*{{citation|first=Sigurdur|last=Helgason| author-link= Sigurður Helgason (mathematician)| title=Differential geometry, Lie groups and symmetric spaces|year=1978|publisher=Academic Press|isbn=0-12-338460-5}}

* {{citation |last=Helgason |first=Sigurdur |author-link=Sigurdur Helgason (mathematician) |title=Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions |series=Mathematical Surveys and Monographs |volume=83 |publisher=American Mathematical Society |year=2000 |isbn=0-8218-2673-5 }}

* {{citation |last=Hille |first=Einar |author-link=Einar Hille |title=Ordinary differential equations in the complex ___domain |publisher=Wiley-Interscience |year=1976 }}

* {{cite book|mr = 0649068| last= Hiller | first = Howard | title= Geometry of Coxeter groups| series=Research Notes in Mathematics| volume = 54 | publisher= Pitman | year = 1982 |isbn = 0-273-08517-4 }}

* {{cite book|first=Robert | last = Howlett| author-link= Robert Howlett| chapter= Introduction to Coxeter Groups| year= 1996 | chapter-url = https://www.maths.usyd.edu.au/u/ResearchReports/Algebra/How/1997-6.html |

title = Australian National University Geometric Group Theory Workshop|___location= [[Sydney]] }}

* {{cite book|mr =1066460 |last=Humphreys|first= James E. | author-link =James E. Humphreys| title = Reflection groups and Coxeter groups| series = Cambridge Studies in Advanced Mathematics| volume = 29| publisher = [[Cambridge University Press]]| year = 1990| isbn = 0-521-37510-X}}

* {{citation |last=Ince |first=E. L. |author-link=E. L. Ince |title=Ordinary Differential Equations |publisher=Dover |year=1944 }}

* {{citation |last=Iversen |first=Birger |title=Hyperbolic geometry |series=London Mathematical Society Student Texts |volume=25 |publisher=Cambridge University Press |year=1992 |isbn=0-521-43508-0 }}

* {{cite book| mr = 0217741 | last =Iwahori | first= Nagayoshi| author-link= Nagayoshi Iwahori|chapter =On discrete reflection groups on symmetric Riemannian manifolds| year = 1966| title= Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965)| pages = 57–62| publisher = Nippon Hyoronsha| ___location= Tokyo}}

* {{cite book| mr = 3839612 | last =Johnson | first = Norman W. | title = Geometries and Transformations |

publisher = [[Cambridge University Press]]| year = 2018 | isbn = 978-1-107-10340-5 |

url = https://www.cambridge.org/core/books/geometries-and-transformations/94D1016D7AC64037B39440729CE815AB }}

* {{citation |last=Magnus |first=Wilhelm |author-link=Wilhelm Magnus |title=Noneuclidean tesselations and their groups |series=Pure and Applied Mathematics |volume=61 |publisher=Academic Press |year=1974|url=

https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/61/suppl/C}}

* {{cite book| mr = 0207802 | last1 = Magnus| first1 = Wilhelm| author1-link= Wilhelm Magnus| last2= Karrass| first2 = Abraham| last3 = Solitar | first3= Donald | author3-link = Donald Solitar | title=Combinatorial group theory: Presentations of groups in terms of generators and relations| year =1976|

publisher= [[Dover Books]]| edition= Second revised}}

* {{citation |last=Maskit |first=Bernard |author-link=Bernard Maskit |title=On Poincaré's theorem for fundamental polygons |journal=[[Advances in Mathematics]] |volume=7 |pages=219–230 |year=1971 |issue=3 |doi=10.1016/s0001-8708(71)80003-8 |doi-access=free }}

* {{cite book | mr = 0959135 | last = Maskit| first = Bernard | author-link = Bernard Maskit| title=Kleinian groups| series = Grundlehren der mathematischen Wissenschaften | volume = 287| publisher= [[Springer-Verlag]] | year = 1988 | isbn = 3-540-17746-9| chapter = Poincaré's theorem}}

* {{cite journal| mr = 0679972 | last=Maxwell | first =George|

title = Sphere packings and hyperbolic reflection groups | journal = [[J. Algebra]]| volume = 79 | year=1982 | pages = 78–97| doi=10.1016/0021-8693(82)90318-0 }}

* {{citation |last=McMullen |first=Curtis T. |author-link=Curtis McMullen |title=Hausdorff dimension and conformal dynamics. III. Computation of dimension |journal=American Journal of Mathematics |volume=120 |year=1998 |pages=691–721 |doi=10.1353/ajm.1998.0031 |s2cid=15928775 }}

* {{cite journal|mr =0183818 | last= Matsumoto | first= Hideya| author-link= Hideya Matsumoto| title=Générateurs et relations des groupes de Weyl généralisés|lang = fr|journal=[[C. R. Acad. Sci. Paris]]|volume= 258 |year =1964 | pages= 3419–3422 | url= https://gallica.bnf.fr/ark:/12148/bpt6k4011c/f1029.item}}

* {{cite book|mr=0418127|last= Milnor| first = John| author-link= John Milnor| chapter = On the 3-dimensional Brieskorn manifolds M(p,q,r)| title= Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox)|pages= 175–225|series= Ann. of Math. Studies| volume =84| publisher= [[Princeton University Press]] | year = 1975}}

* {{citation |last1=Mumford |first1=David |author-link1=David Mumford |last2=Series |first2=Caroline |author-link2=Caroline Series |last3=Wright |first3=David |date=2015 |title=Indra's pearls. The vision of Felix Klein |publisher=Cambridge University Press |isbn=978-1-107-56474-9 }}

* {{citation |last=Nehari |first=Zeev |author-link=Zeev Nehari |title=Conformal mapping |publisher=Dover |year=1975 }}

* {{cite book| mr = 1299730| last = Ratcliffe |first = John G. | title =Foundations of hyperbolic manifolds| edition = Third| series = Graduate Texts in Mathematics | volume = 149

| publisher = Springer| year = 2019 | isbn = 978-3-030-31597-9}}

* {{cite book|mr = 3186310 |last= Schlag| first= Wilhelm |author-link = Wilhelm Schlag|title=A course in complex analysis and Riemann surfaces| series= Graduate Studies in Mathematics| volume = 154|

publisher = [[American Mathematical Society]] | year= 2014 |isbn= 978-0-8218-9847-5|chapter= Uniformization

|pages = 305–351}}

* {{citation |last=Series |first=Caroline |author-link=Caroline Series |title=Continued fractions and hyperbolic geometry, Loughborough LMS Summer School |year=2015 |url=http://homepages.warwick.ac.uk/~masbb/HypGeomandCntdFractions-2.pdf |access-date=15 February 2017}}

* {{citation |last=Siegel| first=C. L. |author-link=Carl Ludwig Siegel |title=Topics in complex function theory |volume=II. Automorphic functions and abelian integrals |translator1=A. Shenitzer |translator2=M. Tretkoff |publisher=Wiley-Interscience |year=1971 |isbn=0-471-60843-2 |pages=85–87 }}

* {{cite book| mr = 0230728 | last= Steinberg | first = Robert | author-link = Robert Steinberg | title= Endomorphisms of linear algebraic groups | series = [[Memoirs of the American Mathematical Society]]| volume= 80 | publisher= [[American Mathematical Society]] | year = 1968 | url = https://bookstore.ams.org/memo-1-80/9}}

* {{cite book |mr = 1333415 | last = Szenthe| first= J. | chapter= A significant property of real hyperbolic spaces| title= Proceedings of Symposium in Geometry (Cluj-Napoca and Tîrgu Mureş, 1992)| pages= 153–161| year = 1993}}

* {{citation |last=Takeuchi |first=Kisao |title=Arithmetic triangle groups |journal=Journal of the Mathematical Society of Japan |volume=29 |year=1977a |pages=91–106 |doi=10.2969/jmsj/02910091 |doi-access=free }}

* {{citation |last=Takeuchi |first=Kisao |title=Commensurability classes of arithmetic triangle groups |journal=Journal of the Faculty of Science, the University of Tokyo, Section IA, Mathematics |volume=24 |year=1977b |pages=201–212 }}

* {{citation |last=Threlfall |first=W. |author-link=William Threlfall |url=http://digital.slub-dresden.de/fileadmin/data/304910686/304910686_tif/jpegs/304910686.pdf |title=Gruppenbilder |journal=Abhandlungen der Mathematisch-physischen Klasse der Sachsischen Akademie der Wissenschaften |volume=41 |pages=1–59 |publisher=Hirzel |year=1932 }}

* {{cite book|last=Tits|first= Jacques |author-link = Jacques Tits |title= Œuvres/Collected works, Volume I|lang=fr |editor1 = F. Buekenhout| editor2 = B. M. Mühlherr| editor3 = J-P. Tignol |editor4 = H. Van Maldeghem|series= Heritage of European Mathematics|url = https://www.ems-ph.org/books/book.php?proj_nr=168 |

publisher= [[European Mathematical Society]]|___location = Zürich| year= 2013| isbn= 978-3-03719-126-2|

chapter= Groupes et géométries de Coxeter |pages = 803–817}} This manuscript was the foundational text for the theory of Coxeter groups, used for preparing Chapter IV of Bourbaki's ''Groupes et Algèbres de Lie''; it was first published in 2001.

* {{cite journal| first= Ernest B. | last = Vinberg| author-link = Ernest Vinberg| year=1971|

title = Discrete linear groups generated by reflections | journal = [[Mathematics of the USSR-Izvestiya]]|

pages= 1083–1119 | volume= 5 | issue = 5| doi = 10.1070/IM1971v005n05ABEH001203| bibcode = 1971IzMat...5.1083V| translator= P. Flor |mr = 0302779}}

* {{cite journal|first=Ernest B. |last = Vinberg| author-link = Ernest Vinberg|year= 1985|journal= [[Russian Mathematical Surveys]]|volume= 40|pages= 31–75 |url = https://iopscience.iop.org/article/10.1070/RM1985v040n01ABEH003527/pdf | publisher = [[London Mathematical Society]]| title = Hyperbolic reflection groups| issue=1 | doi=10.1070/RM1985v040n01ABEH003527 | bibcode=1985RuMaS..40...31V |translator = J. Wiegold}}

* {{cite book|mr=1254933|author1-link=Ernest Vinberg|last1=Vinberg|first1= Ernest B.|last2= Shvartsman|first2= O. V. | chapter= Discrete groups of motions of spaces of constant curvature|title = Geometry II: Spaces of Constant Curvature|pages= 139–248| series = Encyclopaedia Math. Sci.| volume = 29| publisher = [[Springer-Verlag]]|year= 1993| isbn = 3-540-52000-7}}

* {{citation |first=Matthias |last=Weber |title=Kepler's small stellated dodecahedron as a Riemann surface |journal=[[Pacific Journal of Mathematics]] |volume=220 |year=2005 |pages=167–182 |url=http://msp.org/pjm/2005/220-1/p09.xhtml |doi=10.2140/pjm.2005.220.167|doi-access=free }}

* {{citation |last=Wolf |first=Joseph A. |author-link=Joseph A. Wolf |title=Spaces of constant curvature |edition=6th |publisher=AMS Chelsea |year=2011 |isbn=978-0-8218-5282-8 }}

* {{cite book|mr=0986252| last = Yoshida | first= Masaaki | title= Fuchsian differential equations, with special emphasis on the Gauss-Schwarz theory|series= Aspects of Mathematics| volume= E11| publisher= Friedr. Vieweg & Sohn| year= 1987| isbn = 3-528-08971-7}}

{{refend}}

== Further reading ==

* {{citation |title=Automorphic Functions |first=Lester R. |last=Ford |author-link=Lester R. Ford |publisher=[[American Mathematical Society]] |year=1951 |isbn=0821837419 |orig-date=1929}}

* {{citation |last=Lehner |first=Joseph |author-link=Joseph Lehner |title=Discontinuous groups and automorphic functions |series=Mathematical Surveys |volume=8 |publisher=American Mathematical Society |year=1964 }}. (Note that Lehner has pointed out that his proof of Poincaré's polygon theorem is incomplete. He has subsequently recommended de Rham's 1971 exposition.)

* {{citation |last1=Sansone |first1=Giovanni |author-link=Giovanni Sansone |last2=Gerretsen |first2=Johan |title=Lectures on the theory of functions of a complex variable. II: Geometric theory |publisher=Wolters-Noordhoff |year=1969 }}

* {{citation |last=Series |first=Caroline |author-link=Caroline Series |title=The modular surface and continued fractions |journal=Journal of the London Mathematical Society |volume=31 |year=1985 |pages=69–80 |doi=10.1112/jlms/s2-31.1.69 }}

* {{citation |last=Thurston |first=William P. |author-link=William Thurston |title=Three-dimensional geometry and topology. Vol. 1. |editor=Silvio Levy |series=Princeton Mathematical Series |volume=35 |publisher=Princeton University Press |year=1997 |isbn=0-691-08304-5 }}

[[Category:Complex analysis]]

[[Category:Hyperbolic geometry]]

[[Category:Conformal mappings]]

[[Category:Ordinary differential equations]]

[[Category:Geometric group theory]]

[[Category:Coxeter groups]]

[[Category:Reflection groups]]

[[Category:Modular forms]]

[[Category:Spherical geometry]]

[[Category:Automorphic forms]]