Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

46 SHINICHI MOCHIZUKI
Remark 2.4.1. Thus, when ̂Σ =Primes, the proof given above of Proposition
2.4, (iii), yields a new proof of [André], Corollary 6.2.2 [cf. also [SemiAnbd], Lemma
6.1, (ii), (iii)] which is independent of [André], Lemma 3.2.1, hence also of [Stb1],
Theorem 1 [cf. the discussion of Remark 2.2.2].
Corollary 2.5. (Profinite Conjugates of Tempered Decomposition and
Inertia Groups) In the notation of the above discussion, suppose further that
̂Σ =Primes. Then every decomposition group in ̂Π X (respectively, inertia
group in ̂Π X ) associated to a closed point or cusp of X (respectively, to a cusp of
X) is contained in Π tp X if and only if it is a decomposition group in Πtp X (respectively,
inertia group in Π tp X
) associated to a closed point or cusp of X (respectively, to a
cusp of X). Moreover, a ̂Π X -conjugate of Π tp X
contains a decomposition group in
Π tp X
(respectively, inertia group in Πtp X
) associated to a closed point or cusp of X
(respectively, to a cusp of X) if and only if it is equal to Π tp X .
Proof. Let D x ⊆ Π tp X be the decomposition group in Πtp X
associated to a closed
def ⋂
point or cusp x of X; I x = D x Δ
tp
X . Then the decomposition groups of ̂Π X
associated to x are precisely the ̂Π X -conjugates of D x ; the decomposition groups
of Π tp X associated to x are precisely the Πtp X -conjugates of D x.SinceD x is compact
and surjects onto an open subgroup of G k , it thus follows from Proposition 2.4,
(ii), that a ̂Π X -conjugate of D x is contained in Π tp X
ifandonlyifitis,infact,a
Π tp X -conjugate of D x, and that a ̂Π X -conjugate of Π tp X
contains D x if and only if
it is, in fact, equal to Π tp X . In a similar vein, when x is a cusp of X [so I ∼ x = Ẑ],
it follows — i.e., by applying Proposition 2.4, (i), to the unique maximal pro-Σ
subgroup of I x —thatâΠ X -conjugate of I x is contained in Π tp X ifandonlyifitis,
in fact, a Π tp X -conjugate of I x, and that a ̂Π X -conjugate of Π tp X contains I x if and
only if it is, in fact, equal to Π tp X
. This completes the proof of Corollary 2.5. ○
Remark 2.5.1. The content of Corollary 2.5 may be regarded as a sort of [very
weak!] version of the “Section Conjecture” of anabelian geometry — i.e., as the
assertion that certain sections of the tempered fundamental group [namely, those
that arise from geometric sections of the profinite fundamental group] are geometric
as sections of the tempered fundamental group. This point of view is reminiscent
of the point of view of [SemiAnbd], Remark 6.9.1. Perhaps one way of summarizing
this circle of ideas is to state that one may think of
(i) the classification of maximal compact subgroups of tempered fundamental
groups given in [SemiAnbd], Theorem 3.7, (iv); [SemiAnbd], Theorem 5.4,
(ii), or, for that matter,
(ii) the more elementary fact that “any finite group acting on a tree [without
inversion] fixes at least one vertex” [cf. [SemiAnbd], Lemma 1.8, (ii)] from
which these results of [SemiAnbd] are derived
as a sort of combinatorial version of the Section Conjecture.