Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 43
Proof. Assertion (i) follows immediately from Proposition 2.2. Assertion (ii) follows
immediately from the definitions of the various tempered fundamental groups
involved, together with the following elementary observation: IfG ↠ F is a surjection
of finitely generated free discrete groups, which induces a surjection Ĝ ↠ ̂F
between the respective profinite completions [so, by applying the well-known residual
finiteness of free groups [cf., e.g., [SemiAnbd], Corollary 1.7], we think of G
and F as subgroups of Ĝ and ̂F , respectively], then H def
=Ker(G ↠ F )isdense in
=Ker(Ĝ ↠ ̂F ), relative to the profinite topology of Ĝ. Indeed, let ι : F↩→ G
be a section of the given surjection G ↠ F [which exists since F is free]. Then if
{g i } i∈N is a sequence of elements of G that converges, in the profinite topology of
Ĝ, to a given element h ∈ Ĥ, and maps to a sequence of elements {f i} i∈N of F
[which necessarily converges, in the profinite topology of ̂F ,totheidentity element
1 ∈ ̂F ], then one verifies immediately that {g i ·ι(f i ) −1 } i∈N is a sequence of elements
of H that converges, in the profinite topology of Ĝ, toh. This completes the proof
of the observation and hence of assertion (ii).
Ĥ def
Next, we consider assertion (iii). In the following, we give, in effect, two distinct
proofs of the slimness of ̂Δ X,H :oneiselementary, but requires one to assume that
condition (a) holds; the other depends on the highly nontrivial theory of [Tama2]
and requires one to assume that condition (b) holds. If condition (a) holds, then
let us set Σ ∗ def
=Σ ⋃ {l}. If condition (b) holds, but condition (a) does not hold [so
̂Σ =Primes =Σ ⋃ {p}], then let us set Σ ∗ def
= Σ. Thus, in either case, p ∉ Σ ∗ ⊇ Σ.
Let J ⊆ ̂Δ X be an open subgroup. Write J H
def
= J ⋂ ̂ΔX,H ; J ↠ J ∗ for the
maximal pro-Σ ∗ quotient; J ∗ H ⊆ J ∗ for the image of J H in J ∗ . Now suppose that
α ∈ ̂Δ X,H commutes with J H .Letv be a vertex of the dual graph of the geometric
special fiber of a stable model X J of the covering X J of X k
determined by J. Write
J v ⊆ J for the decomposition group [well-defined up to conjugation in J] associated
to v; J ∗ v ⊆ J ∗ for the image of J v in J ∗ . Then let us observe that
(†) there exists an open subgroup J 0 ⊆ ̂Δ X which is independent of J, v,
and α such that if J ⊆ J 0 , then for arbitrary v [and α] asabove,itholds
that J ∗ v
⋂ J
∗
H (⊆ J ∗ )isinfinite and nonabelian.
Indeed, if condition (a) holds, then it follows immediately from the definitions that
the image of the homomorphism J v ⊆ J ⊆ ̂Δ X ↠ ̂Π G is pro-Σ; in particular, since
l ∉ Σ, and Ker(J v ⊆ J ⊆ ̂Δ X ↠ ̂Π G ) ⊆ J v
⋂
JH , it follows that J v
⋂
JH , hence also
J ∗ v
⋂ J
∗
H , surjects onto the maximal pro-l quotient of J v , which is isomorphic to the
pro-l completion of the fundamental group of a hyperbolic Riemann surface, hence
[as is well-known] is infinite and nonabelian [so we may take J 0
def
= ̂Δ X ]. Suppose,
on the other hand, that condition (b) holds, but condition (a) does not hold. Then
it follows immediately from [Tama2], Theorem 0.2, (v), that, for an appropriate
choice of J 0 ,ifJ ⊆ J 0 ,theneveryv corresponds to an irreducible component that
either maps to a point in X or contains a node that maps to a smooth point of X .
In particular, it follows that for every choice of v, there exists at least one pro-Σ,
torsion-free,
⋂
pro-cyclic subgroup F ⊆ J v that lies in Ker(J v ⊆ J ⊆ ̂Δ X ↠ ̂Π G ) ⊆
J v JH and, moreover, maps injectively into J ∗ . Thus, we obtain an injection