FINITELY MAXIMAL FUCHSIAN GROUPS

FINITELY MAXIMAL FUCHSIAN GROUPS DAVID SINGERMAN 1. Introduction By a Fuchsian group F we shall mean a finitely generated discrete subgroup of PSL(2, U), the group of all conformal homeomorphisms of the upper half-plane U. Greenberg [3] defines a Fuchsian group to be finitely maximal if there does not exist another Fuchsian group containing it with finite index. Let R(T) denote the set of all isomorphisms r: F -+ PSL(2, R) with the property that r(F) is discrete and r maps parabolic (hyperbolic boundary) elements to parabolic (hyperbolic boundary) elements. Greenberg showed that for some Fuchsian groups F there exists a Fuchsian group F o containing it withfiniteindex, such that every r e R(T) is just the restriction to F of an r0 eR(T0). In this case r(F) is never finitely maximal. (He also showed that for other groups F, r(F) was " usually "finitelymaximal). Greenberg determined some of these pairs F, F o . In this paper we determine all such pairs. 2. Signatures of Fuchsian groups It is known that every Fuchsian group has a presentation of the following form. Generators: al,b1,...,ag,bg. x1,x2,

(Hyperbolic)

...,xr

(Elliptic)

pu...,ps

(Parabolic)

hu ...,//,

(Hyperbolic Boundary elements)

Relations: x r

= xm2 =

=

ft

Mr=

b

- i h-x

YlxjflPkUh^l.

i=l

j=l

k=l

1=1

We then say that F has signature (g; mu ...,mr; s; t);

(1)

mu ... mr are integers ^ 2 and are called the periods of F. It is sometimes convenient to think of parabolic elements as being elliptic elements of infinite period. When this is the case we will write the signature (1) as [g; ml,...,mu;

t],

(1')

where u = r+sand mr+l = ... = m,,= oo. For any Fuchsian group F we can define L(F), the set of limit points of F [4; p. 86]. L(F) is a subset of the real line of one of the following three types. (a) L(F) has at most two points; Received 4 May, 1971; revised 13 September, 1971. [J. LONDON MATH. SOC. (2), 6 (1972), 29-38]

30

DAVJD SINGERMAN

(b) L(F) = U; (c) L(F) is a perfect nowhere-dense subset of U. Groups of type (a) will not interest us. They are cyclic or have signature (0; 2, 2; 0; 1) and, as Greenberg points out, are never finitely maximal. Groups of type (b) are called groups of the first kind and groups of type (c) of the second kind. We now describe the Riemann-Hurwitz formula. For a group F with signature (1), define

If M(F) > 0, there exists a Fuchsian group with signature (1) and if F is of the first kind M(F) > 0. If Fj c r is a subgroup of finite index, then ^

r,,

Mir,)

(If t = 0 this follows from the fact that 2nM(T) is the hyperbolic measure of a fundamental region for F. If t > 0 it is a result of Maclachlan [7], See Proposition 6.) 3. The space T#(T) Let ylt y2,..., yn be a set of generators for F. We topologize R(T) as a subset of (PSL(2, U))n by associating with each reR(T) the point (KVi), r(y2), •••, Kv,,))We define two isomorphisms rltr2eR(T) to be equivalent (and write rt ~ r2) if there exists an angle-preserving (i.e. conformal or anti-conformal) homeomorphism X : U -> U such that 1

=r2(y)

forallyeF.

Define and give it the quotient topology. It is known that if F is a group with signature (1) then T * (F) is a cell of dimension d(T), where [1,2]. T # (F) is known as the reduced Teichmiiller space and it coincides with the usual Teichmiiller space if and only if F is of thefirstkind [2]. In [3] Greenberg gives the following result. (Also see [6]). PROPOSITION 1. Let Tu F 2 be two Fuchsian groups with a : F x -*• T2 an injective homomorphism such that \T2 : a(F 1 )| is finite. Then a, induces a map [/•]-• [root] #

which is an embedding of T (T2) into T^iTj) whose image is a closed subset of T* (I\). (By [r] we mean the equivalence class of r.)

FINITELY MAXIMAL FUCHSIAN GROUPS

31

We are interested in finding all pairs of groups F, F o such that F £ r 0 with finite index and every reR(T) is just the restriction to F o of an roeR(To), (so that the embedding of Theorem 1 is a surjection). Clearly this property only depends on the signatures of F, F o , as the isomorphisms in R(T) preserve the signature. If a is the signature of F and