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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.

==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>

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* {{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]]