%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% G. JONES Tex file of "Dessins d'enfants"  SLC 35
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\magnification=1200
\hsize=12cm
\centerline{\bf Dessins d'enfants: bipartite maps and Galois groups}
\medskip
\centerline{\sl Gareth Jones, Southampton, U. K.}
\bigskip

\midinsert\narrower
\noindent{\sl Abstract.} Bely\u\i's Theorem implies that the Riemann
surfaces defined over the field of algebraic numbers are
precisely those which support bipartite maps; this provides a faithful
representation of the Galois group of this field on these combinatorial
objects.
\endinsert

\bigskip

My aim in this note is to show how combinatorics can play a central role in
uniting such topics as Galois theory, algebraic number theory, Riemann surfaces,
group theory and hyperbolic geometry. The relevant combinatorial objects are
maps on surfaces, often called {\it dessins d'enfants} in view of the rather
na\"\i ve appearance of some of the most common examples. For simplicity I
will restrict attention to bipartite maps, though triangulations and
hypermaps also play an important role in this theory. More detailed surveys
can be found in [2, 5, 7], and for recent progress see [8].

\medskip

A {\sl bipartite map} $\cal B$ consists of a bipartite graph ${\cal G}$ imbedded
(without crossings) in a compact, connected, oriented surface $X$, so that the
faces (connected components of $X\setminus{\cal G}$) are simply connected. One
can describe $\cal B$ by a pair of permutations $g_0$ and $g_1$ of its edge-set
$E$: the vertices can be coloured black or white, so that each edge joins a
black and a white vertex; the orientation of $X$ then determines a
cyclic ordering of the edges around each black or white vertex, and these
are the disjoint cycles of $g_0$ and $g_1$ respectively. These two permutations
generate a subgroup $G=\langle g_0, g_1\rangle$ of the symmetric group $S^E$ of
all permutations of $E$, called the {\sl monodromy group} of $\cal B$; the
topological hypotheses imply that ${\cal G}$ has to be connected, so $G$ acts
transitively on $E$. Conversely, every $2$-generator transitive group arises in
this way from some bipartite map $\cal B$: the edges are the symbols permuted,
the black and white vertices correspond to the cycles of the two generators
$g_0$ and $g_1$, and the faces correspond to the cycles of
$g_{\infty}=(g_0g_1)^{-1}$. Isomorphism of maps (preserving orientation and
vertex-colours) corresponds to conjugacy of pairs $(g_0, g_1)$ in $S^E$, and the
automorphism group of $\cal B$ can be identified with the centraliser of $G$ in
$S^E$, that is, the group of all permutations which commute with $G$.

\medskip

(Historical note: A slight modification of these ideas allows one to
describe any
oriented map, whether bipartite or not, by a pair of permutations [6].
Although generally regarded as a modern development, this use of permutations
can be traced back at least as far as Hamilton's construction of what we now
call Hamiltonian cycles in the icosahedron [4].)

\medskip

Every bipartite map $\cal B$ is a quotient of the {\sl universal bipartite map}
$\hat{\cal B}$, drawn on the upper half-plane ${\cal
U}=\{\,z\in{\bf C}\mid {\rm Im}(z)>0\,\}$; the vertices of $\hat{\cal B}$
are the
rational numbers $r/s$ (in reduced form) with $s$ odd, coloured black or
white as
$r$ is even or odd, and the edges are the hyperbolic geodesics (euclidean
semicircles) joining vertices $r/s$ and $x/y$ with $ry-sx=\pm 1$. The
automorphisms of $\hat{\cal B}$ are the M\"obius transformations
$$z\mapsto{az+b\over cz+d}\qquad(a, b, c, d\in {\bf Z},\; ad-bc=1)\eqno(*)$$
such that $a\equiv d\equiv 1$ and $b\equiv c\equiv 0$ mod $(2)$; these form a
normal subgroup $\Gamma(2)$ of index $6$ in the modular group $\Gamma=PSL_2({\bf
Z})$ of all transformations $(*)$, called the {\sl principal congruence
subgroup} of level $2$ in $\Gamma$. Now $\Gamma(2)$ is a free group of rank
$2$, generated by the transformations
$$T_0:z\mapsto{z\over -2z+1}\qquad{\rm and}\qquad T_1:z\mapsto
{z-2\over 2z-3}\,,$$
so there is an epimorphism $\theta:\Gamma(2)\to G,\; T_i\mapsto g_i$,
which gives a transitive permutation representation of $\Gamma(2)$ on $E$.
If $B$
denotes the subgroup $\theta^{-1}(G_e)$ of $\Gamma(2)$ fixing an edge $e$ of
$\cal B$ then $B$ acts as a group of automorphisms of $\hat{\cal B}$, and one
can show that ${\cal B}\cong{\hat{\cal B}}/B$. The underlying surface of
${\hat{\cal B}}/B$ is now a compact Riemann surface $\overline{{\cal
U}/B}=({\cal
U}\cup{\bf Q}\cup\{\infty\})/B$, formed by compactifying ${\cal U}/B$
with finitely many points, corresponding to the orbits of $B$ on the extended
rationals ${\bf Q}\cup\{\infty\}$. One can regard ${\hat{\cal B}}/B$ as a
rigid, conformal model of $\cal B$, with a complex structure induced from
that of
$\cal U$: for instance, the edges are geodesics, the angles between edges around
any vertex are equal, and the automorphisms of $\cal B$ are conformal
isometries of the Riemann surface.

\medskip

Riemann showed that a Riemann surface $X$ is compact if and only if it is
isomorphic to the Riemann surface of an algebraic curve $f(x,y)=0$ for some
polynomial $f(x,y)\in{\bf C}[x,y]$. Computationally and theoretically, the most
satisfactory polynomials are those with coefficients in the field $\overline{\bf
Q}$ of algebraic numbers; results of Bely\u\i\/ [1] and Weil [9] imply that the
Riemann surfaces corresponding to such polynomials $f(x,y)\in\overline{\bf
Q}[x,y]$ are those obtained from bipartite maps by the above method.
More precisely, Bely\u\i\/ showed that a compact Riemann surface $X$ is defined
over $\overline{\bf Q}$ if and only if there is a Bely\u\i\/ function $\beta$
from $X$ to the Riemann sphere $\Sigma={\bf C}\cup\{\infty\}$, that is, a
meromorphic function on $X$ which is unbranched over
$\Sigma\setminus\{0,1,\infty\}$. In these circumstances, $X$ is the underlying
surface of the bipartite map ${\cal B}=\beta^{-1}({\cal B}_1)$, where ${\cal
B}_1$ is the trivial bipartite map $\hat{\cal B}/\Gamma(2)$ on $\Sigma$ with a
black vertex at $0$, a white vertex at $1$, and a single edge along the unit
interval $[0,1]$. If each edge of $\cal B$ is identified with the sheet of the
covering $\beta:X\to\Sigma$ which contains it, then the monodromy group $G$ of
$\cal B$ coincides with the monodromy group of $\beta$, regarded as a group of
permutations of the sheets; in particular, the elements $g_0, g_1, g_{\infty}\in
G$ describe how the sheets are permuted by lifting small loops in $\Sigma$
around
the branch-points $0, 1$ and $\infty$. Similarly, the automorphism group of
$\cal B$ is identified with the group of covering transformations of $\beta$.

\medskip

Since $\overline{\bf Q}$ is the union of the Galois (finite normal) extensions
$K\geq{\bf Q}$ in $\bf C$, it follows that the {\sl absolute Galois group} ${\bf
G}={\rm Gal}\,(\overline{\bf Q}/{\bf Q})$ of $\overline{\bf Q}$ over $\bf Q$ is
the projective limit of the finite Galois groups ${\rm Gal}\,(K/{\bf Q})$ of
these algebraic number fields; as such, it is an uncountable profinite group,
which embodies the whole of classical Galois theory over $\bf Q$. This group
$\bf G$ is of fundamental importance in several areas of mathematics: for
instance, the representation theory of $\bf G$ played a crucial role in Wiles's
proof of Fermat's Last Theorem [10], and the Inverse Galois Problem (Hilbert's
still unproved conjecture that every finite group is a Galois group over
$\bf Q$)
is equivalent to showing that every finite group is an epimorphic image of $\bf
G$. Fortunately, Bely\u\i's Theorem provides us with an explicit realisation of
$\bf G$ in terms of bipartite maps, which is beginning to add to our rather
meagre knowledge of this complicated group.

\medskip

In [3], Grothendieck showed that the natural action of $\bf G$ on polynomials
over $\overline{\bf Q}$ induces an action of $\bf G$ on bipartite maps (and on
other similar combinatorial objects, generally known as {\it dessins
d'enfants}),
through the above correspondence between maps and polynomials.
Although $\bf G$ preserves such properties of a map as its genus,
the numbers and valencies of its black and white vertices, its monodromy group
and its automorphism group, this action of $\bf G$ is nevertheless faithful, in
the sense that each non-identity element of $\bf G$ sends some bipartite
map to a
non-isomorphic bipartite map. Moreover, this action remains faithful even when
restricted to such simple objects as plane trees (maps on the sphere
with one face). One therefore has a combinatorial approach to Galois theory,
which is attracting interest from a wide spectrum of mathematicians and
theoretical physicists (for whom maps are an effective discrete approximation to
the compact Riemann surfaces which play a major role in quantum gravity).

\bigskip\bigskip

\centerline{\sl References}
\bigskip
\frenchspacing

\item{[1]} G.~V.~BELY\u I, `On Galois extensions of a maximal cyclotomic
field', {\sl Izv. Akad. Nauk SSSR} 43 (1979) 269--276 (Russian); {\sl
Math. USSR Izvestiya} 14 (1980), 247--256 (English translation).
\medskip

\item{[2]} P.~B.~COHEN, C.~ITZYKSON and J.~WOLFART, `Fuchsian
triangle groups and Grothendieck dessins. Variations on a theme of Belyi', {\sl
Comm. Math. Phys.} 163 (1994), 605--627.
\medskip

\item{[3]} A.~GROTHENDIECK, `Esquisse d'un programme', preprint, Montpellier,
1984.
\medskip

\item{[4]} W.~R.~HAMILTON, Letter to John T.~Graves `On the Icosian' (17th
October 1856), in {\sl Mathematical papers, Vol. III, Algebra} (eds.
H.~Halberstam and R.~E.~Ingram), Cambridge University Press, Cambridge,
1967, pp.
612--625.
\medskip

\item{[5]} G.~A.~JONES, `Maps on surfaces and Galois groups', submitted.
\medskip

\item{[6]} G.~A.~JONES and D.~SINGERMAN, `Theory of maps on orientable
surfaces', {\sl Proc. London Math. Soc.} (3)  37 (1978), 273--307.
\medskip

\item{[7]} G.~A.~JONES and D.~SINGERMAN, `Bely\u\i\/ functions, hypermaps
and Galois groups', {\sl Bull.~London Math.~Soc.}, to appear.
\medskip

\item{[8]} L.~SCHNEPS (ed.), {\sl The Grothendieck Theory of Dessins
d'Enfants}, London Math. Soc.~Lecture Note Series 200, Cambridge University
Press, Cambridge, 1994.
\medskip

\item{[9]} A.~WEIL, `The field of definition of a variety', {\sl
Amer.~J.~Math.} 78 (1956), 509--524.
\medskip

\item{[10]} A.~WILES, `Modular elliptic curves and Fermat's Last Theorem',
{\sl Ann. Math.} 141 (1995), 443--551.

\bigskip\bigskip

\line{Department of Mathematics\hfil}
\line{University of Southampton\hfil}
\line{Southampton SO17 1BJ\hfil}
\line{United Kingdom\hfil}

\bigskip

\line{e-mail: gaj@maths.soton.ac.uk\hfil}

\bye