Poincaré’s theory of Fuchsian groups

Poincaré’s theory of Fuchsian groups

1

Background

The origin of the Fuchsian group lies in the

hypergeometric equation

studied by Gauss, and later by Riemann, in the 19th century. Riemann’s work, which developed ideas of

analytic continuation

, was extended by Fuchs in the mid to late 19th century to the study of general n

t

⁢

h

order

linear ordinary differential equations

whose solutions have no

essential singularities

[

2

]

. Poincaré’s first encounter with the problem occured when he read Fuchs’ May 1880 paper in Borchardt’s Journal

[

1

]

concerning

second order

linear differential equations

[

4

]

. In this paper, Fuchs studies the quotients of solutions to the equations, noting that if

f

⁢

(

x

)

and

g

⁢

(

x

)

are two such solutions, and

f

⁢

(

x

)

g

⁢

(

x

)

=

z

,

then in certain cases

x

is a

meromorphic function

of

z

. This result allows Fuchs to define

functions

, analogous to the

doubly periodic

elliptic functions

, that would allow him to understand analytic continuation of the solutions around singularities

[

2

]

. By the end of that month, Poincaré had produced an essay, to be entered for the 1880 Académie des sciences prize in mathematical sciences, in which he addresses several issues from within Fuchs’s paper. Most importantly, however, Poincaré provides the geometric insight needed to prove that

inversion

of the quotient leads to a

well-defined

function that is

invariant

under a certain group of

linear fractional transformations

[

4

]

.

Poincaré goes on to give a detailed exposition of these so-called Fuchsian functions and groups in a series of papers published in

Acta Mathematica

in 1882 and 1883, considering them of interest in their own right, and building up an

analogy

to the doubly periodic

inverse functions

of

elliptic integrals

. He begins, in

Theorie des Groupes Fuchsiens

[

3

]

, by studying the properties of Fuchsian groups, showing how they are groups of non-Euclidean

isometries

, and that their

periodicity

can be described by non-Eulidean tesselations.

2

Theorie des Groupes Fuchsiens

The first two chapters of Poincaré’s

Theorie des Groupes Fuchsiens

provide a basic theory of linear fractional transformations, much of which does not differ significantly from a modern account of the subject; familiar

concepts

such as

cross-ratio

preservation and

transitivity

are covered. Most importantly, however, Poincaré’s study of the cross-ratio leads him to investigate

congruence

under groups of real linear fractional transformations, and from here he discovers the crucial fact that this congruence is congruence in the

non-Euclidean geometry

of Lobachevsky.

From here Poincaré begins his study of Fuchsian groups: groups of real linear fractional transformations acting properly discontinuously on the

complex

upper half-plane. He notices the

existence

of a fundamental domain, investigating side-pairing and vertex cycles, and devising methods of transforming a fundamental domain into one that is convex. Poincaré’s investigation into fundamental domains leads him to his famous

theorem

, known as Poincaré’s

polygon

theorem, giving conditions for a non-Euclidean polygon, equipped with side-pairing

transformations

, to be the fundamental domain of a Fuchsian group.

In the later chapters of

Theorie des Groupes Fuchsiens

Poincaré looks at the surfaces formed by identifying paired edges and vertices of a fundamental domain, classifying them by genus using the Euler-Poincaré

formula

. These surfaces are of great interest to Poincaré as the qotients of solutions to

differential equations

originally motivating him become well-defined when viewed as functions on them; Poincaré’s analogy to elliptic functions is

complete

.

3

A Modern Perspective

Various authors on Poincaré’s papers on Fuchsian groups and functions have commented on the sheer speed at which he worked. This

reflects

in his work, which tends to lack rigour, and occasionally assumes non-trivial results without proof. That said, Poincaré’s fundamental ideas were entirely correct, and his work needed little major

development

to turn it into the formal theory studied today. The developments that have occured are generally due to advances in the theories of

Riemann surfaces

that occured after Poincaré’s time; with modern theory of covering spaces it is now possible to prove formally Poincaré’s polygon theorem, and that the quotient produced by identifying the edges of a fundamental domain is indeed a Riemann surface.

Translations

of Poincaré’s key papers can be found in John Stillwell’s

Papers on Fuchsian Functions

.

References

1

Fuchs, L., 1880. Über eine Klasse von Functionen mehrerer Variabeln, welche durch Umkehrung der Integrale von Lösungen der linearen Differentialgleichungen mit rationalen Coefficienten entstehen.

Journal für Mathematik

. 89, pp. 151-169.

2

Gray, J., 1986.

Linear Differential Equations and Group Theory From Riemann to Poincaré

. Birkhäuser.

3

Poincaré, H., 1882. Theorie des Groupes Fuchsiens.

Acta Mathematica

. 1, pp1-63.

4

Poincaré, H., 1985.

Papers on Fuchsian Functions

. Translated from French by J. Stillwell. Springer-Verlag.

Title

Poincaré’s theory of Fuchsian groups

Canonical name

PoincaresTheoryOfFuchsianGroups

Date of creation

2013-05-12 18:24:29

Last modified on

2013-05-12 18:24:29

Owner

mairiwalker (1000235)

Last modified by

unlord (1)

Numerical id

2

Author

mairiwalker (1)

Entry type

Topic

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Classification

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Classification

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