Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 49
conclude that any abelian subgroup J of G is equal to the union of the groups
that appear in some chain G 1 ⊆ G 2 ⊆ ... ⊆ G of cyclic subgroups of G. On
the other hand, by applying assertion (ii) to some generator of G 1 , it follows that
there exists a finite index subgroup G 0 and a positive integer n such that G n j ⊆ G 0
for all j =1, 2,..., and, moreover, G n 1 has nontrivial image in G ab
0 . Thus, by
considering the image in [the finitely generated abelian group] G ab
0 of the chain
of cyclic subgroups G n 1 ⊆ G n 2 ⊆ ..., we conclude that this chain, hence also the
original chain G 1 ⊆ G 2 ⊆ ...,mustterminate. Thus, J is cyclic, as desired. This
completes the proof of assertion (iv).
Finally, we consider assertion (v). By considering the surjection Ĝ ↠ Ẑ,
we conclude immediately that the normalizer NĜ( ̂T )of ̂T in Ĝ is equal to the
centralizer ZĜ( ̂T )of ̂T in Ĝ. If ZĜ( ̂T ) ≠ ̂T , then it follows immediately that,
for some prime number l, there exists a closed [abelian] subgroup ̂T 1 ⊆ ZĜ( ̂T )
containing the pro-l portion of ̂T such that there exists a continuous surjection
Z l × Z l ↠ ̂T 1 whose kernel lies in l · (Z l × Z l ). In particular, one computes easily
that the l-cohomological dimension of ̂T 1 is ≥ 2. On the other hand, since ̂T 1 is
of infinite index in Ĝ, it follows immediately that there exists an open subgroup
Ĝ 1 ⊆ Ĝ of Ĝ such that ̂T 1 ⊆ Ĝ1, and, moreover, there exists a continuous surjection
φ : Ĝ1 ↠ Z l whose kernel Ker(φ) contains ̂T 1 . In particular, since the cohomology
of ̂T 1 may be computed as the direct limit of the cohomologies of open subgroups
of Ĝ containing ̂T 1 , it follows immediately from the existence of φ, together with
the well-known structure of the cohomology of open subgroups of Ĝ, that the l-
cohomological dimension of ̂T 1 is 1, a contradiction. This completes the proof of
assertion (v). ○
Corollary 2.8. (Subgroups of Topological Fundamental Groups of Complex
Hyperbolic Curves) Let Z be a hyperbolic curve over C. Write Π Z for
the usual topological fundamental group of Z; ̂Π Z for the profinite completion of
Π Z . Let H ⊆ Π Z be an infinite subgroup [such as a cuspidal inertia group!];
γ ∈ ̂Π Z an element such that
γ · H · γ −1 ⊆ Π Z
[or, equivalently, H ⊆ γ −1 · Π Z · γ].
Then γ ∈ Π Z ·N̂ΠZ
(H), i.e., γ ·H ·γ −1 = δ ·H ·δ −1 ,forsomeδ ∈ Π Z . If, moreover,
H is nonabelian, thenγ ∈ Π Z .
Remark 2.8.1. Corollary 2.8 is an immediate consequence of Theorem 2.6. In
fact, in the present series of papers, we shall only apply Corollary 2.8 in the case
where Z is non-proper, andH is a cuspidal inertia group. Inthiscase,theproof
of Theorem 2.6 may be simplified somewhat, but we chose to include the general
version given here, for the sake of completeness.