## 7-2-66. Felix Klein to H. Poincaré, English translation

Leipzig, 7 May 1882

Sophienstraße 10

Dear Sir,

I recently read your note in the *Comptes rendus* of 10 April
1882.^{1}^{1}endnote:
^{1}
Poincaré (1882); reed. Nörlund and Lebon (1916, 41–43).
The note was all the more interesting to me, as I believe your current
considerations, and also your methods, to be closely related to
mine. I prove my theorems using *continuity*, relying on the the
following two lemmas: 1) to every “groupe discontinu” there always belongs
a Riemann surface, and 2) to each suitably cut Riemann surface there
always* belongs just *one* such group (if any group at all belongs to
it).

Up to now I have not tried to derive the series developments
that you establish. How in fact do you prove the existence of the
number $m$ for which $\sum\frac{1}{(\gamma_{i}\eta+\delta_{i})^{m}}$
converges absolutely? And do you have an *exact* lower bound or
an approximate one?

As for me, in the meantime I have been able to give the theorems in
question an even more general form, but I shall have to write to you
about that later, as I am pressed for time and need to prepare a note
for the *Annalen*. In the case of my first theorem, the whole of
the closed sphere $\eta$ with infinitely many points removed is
covered by the images of the fundamental region. In the case of the
second theorem, the interior of a disc, but of *one only*,
remains uncovered. I have now noted the existence of representations
(which for each Riemann surface are always unique) for which the case
of *infinitely many discs* is excluded. In this direction, I
formulate here only the the simplest theorem (in which it is assumed
always that one has an unbranched representation of the Riemann
surface).

Let $p=\mu_{1}+\mu_{2}+\ldots+\mu_{m}$, where none
of the $\mu=1$. Choose $m$ points $O_{1}$, …, $O_{m}$ on the
Riemann surface, and starting from $O_{1}$ make, in the usual way,
$2\mu_{1}$ cuts
$A_{1}$, $B_{1}$;
$A_{2}$, $B_{2}$;
…;
$A_{\mu_{1}}$, $B_{\mu_{1}}$;
from $O_{2}$ [make] $2\mu_{2}$ cuts, etc. On the other hand, on the sphere $\eta$,
draw $m$ pairwise disjoint circles and in the interior of the space
bounded by the latter taken together a polygon bounded by $4\mu_{1}$
circular arcs orthogonal to the first fundamental circle, then one
bounded by $4\mu_{2}$ circular arcs orthogonal to the second fundamental
circle, etc. (so that each polygon of circular arcs is $m$-fold
connected). The bounding circles are ordered in pairs in accordance
with the familiar sequence
$A_{1}$, $B_{1}$, $A_{1}^{-1}$, $B_{1}^{-1}$, $A_{2}$, $B_{2}$, …, that is, by means of
linear substitutions acting on $\eta$ in each case leaving the
fundamental circle invariant. Furthermore, assume the product of the
linear transformations in question – for example
$A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}\ldots A_{\mu_{1}}^{-1}B_{\mu_{1}}^{-1}$ – is
always the identity. *Then there is always one and only one
analytic function mapping the sectioned Riemann surface onto one of
the given polygons of circular arcs.* The case when one of the $\mu$
is equal to 1 differs only in that then the corresponding fundamental
circle reduces to a point and the corresponding linear substitutions
become “parabolic”, and fix that point. So, enough for today. Would
it be impossible to obtain a complete collection of your relevant
offprints? After Pentecost I will be giving a series of lectures in my
seminar on single-valued functions admitting linear transformations,
and would, if possible, like to make available such a collection to my
auditors.

Very respectfully yours,

F. Klein

* i.e., under the constraints of the theorem in question.

PTrL. Slightly emended from the translation by R. Burns of the original German (§ 4-47-21) in Saint-Gervais (2016). See also the French translation (§ 7-2-40).

Time-stamp: "20.09.2023 18:45"

## References

- Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. link1 Cited by: endnote 1.
- Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 94, pp. 1038–1040. link1 Cited by: endnote 1.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-66. Felix Klein to H. Poincaré, English translation.