Vol. 6 (2000)
Limit constructions over Riemann surfaces
201
Remark. We avoid genus 2 for the following reason. Let Y = X be a surface
of genus two. We take p and q to be the hyperelliptic involution and the identity
homeomorphism, respectively. Then, in fact, the map T (p) coincides with T (q).
This is the well known non-effectiveness of the action of the mapping class group
in genus 2 (see [N, Section 2.3.7]). But, in the context of compact hyperbolic
Riemann surfaces, that is the one and only case when a non-trivial mapping class
group element induces the identity on the Teichm u¨ ller space.
Proof of Theorem 3.14. First let us note the crucial fact that the limit constructions
we are pursuing are independent of the genus of the base surface. In fact, if
-
α : Xα
X is a covering in I(X), then α sets up a natural isomorphism between
the pairs :
ꢀ
ꢁ
ꢀ
ꢁ
T∞(X), MC∞(X) and
T∞(Xα), MC∞(Xα) .
(3.20)
Therefore, to understand the action of the universal commensurability mapping
class group, we may, and therefore do take X to be of genus greater than or equal
to three.
+
In view of the description of MC∞(X) as the group Vaut (Γ) given in Proposi-
tion 3.5, a copy of the group Γ itself sits embedded inside MC∞(X). Indeed, each
element of Γ determines a virtual automorphism of Γ by inner conjugation. Let us
+
first take care of these elements of Vaut (Γ), which play a rather special role.
Given any non-identity element γ ∈ Γ, we utilize the residual finiteness of the
surface group Γ to find a finite index subgroup H ⊂ Γ so that γ is not in H. Then
in the direct limit construction of T∞(Γ), it follows easily that the automorphism
of T∞(Γ) arising from γ will already act non-trivially on the stratum T (H).
It is also clear by a similar argument that every non-identity mapping class like
element of MC∞(X) (see the Remark following (3.10), and [BN1]), will act non-
trivially on T∞(X). We have already disposed of members of Γ itself. Therefore,
let the element under scrutiny be given by σ ∈ Vnormq.s.(Γ)\Γ. By assumption σ is
mapping class like. One therefore sees easily that it must preserve some appropriate
stratum T (Xα) (as a set), and will act as the standard modular transformation on
that Teichm u¨ ller space. But the classical genus g mapping class group, say MCg,
is known to act effectively, (see [B2], [N, Chapter 2.3]), on the genus g Teichm u¨ ller
space Tg, for every g ≥ 3. That takes care of σ. Actually, by essentially the
1
above argument, we can see that every member of Vnormq.s.(Γ)∩M o¨ bius (S ) acts
non-neutrally on T∞(X).
We now come to the interesting case when the element of MC∞(X) being
investigated is not of the above types. Take therefore a non-trivial element of
MC∞(X) determined by a self correspondence (p, q), namely by the two coverings p
and q from (Y, ∗) onto (X, ∗), as in (3.3). The condition on the element of MC∞(X)
so determined implies that the hypothesis of Lemma 3.15 may be assumed satisfied.
Let t ∈ T∞(X) be a point that is represented as a Riemann surface Xµ, (µ
being a complex structure on X), in the base stratum T (X). Remember that
in the direct limit construction of T∞(X) (over the directed set I(X)), there are