Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

92 SHINICHI MOCHIZUKI
(v); [AbsTopI], Proposition 4.10, (i)]. Then we observe that for each constituent
morphism D vj → D >,v of the poly-morphism φ Θ v j
, the induced homomorphism
π 1 (D vj ) → π 1 (D >,v ) [well-defined, up to composition with an inner automorphism]
is compatible with the respective outer actions [ofthedomainandcodomainofthis
homomorphism] on π geo
1 (D vj ), π geo
1 (D >,v ) for some [not necessarily unique, but
determined up to finite ambiguity — cf. [SemiAnbd], Theorem 6.4!] outer isomorphism
π geo
1 (D vj ) ∼ → π geo
1 (D >,v ). We shall refer to this fact by saying that “φ Θ v j
is
compatible with the outer actions on the respective geometric [tempered] fundamental
groups”.
(iii) Let v ∈ V good .Foreachj ∈ F l ,write
for the full poly-isomorphism [cf. §0].
φ Θ v j
: D vj
∼
→D>,v
(iv) For each j ∈ F l ,write φ Θ j : D j → D >
for the poly-morphism determined by the collection {φ Θ v j
: D vj →D >,v } v∈V and
φ Θ : D → D >
for the poly-morphism {φ Θ j } j∈F . Thus, whereas the capsule D admits a natural
permutation poly-action by F l
l
, the “labels” — i.e., in effect, elements of
LabCusp(D > ) [cf. Proposition 4.2] — determined by the various collections of
evaluation sections corresponding to a given j ∈ F l
are held fixed by arbitrary
automorphisms of D > [cf. Proposition 4.2].
Example 4.5. Transport of Label Classes of Cusps via Model Base-
Bridges. We continue to use the notation of Examples 4.3, 4.4.
(i) Let j ∈ F l
, v ∈ V. Recall from Example 4.3, (iv), that the data of the
arrow φ NF
j : D j →D ⊚ at v consists of an arrow φ NF
v j
: D vj →D ⊚ .Ifv ∈ V non ,then
induces various outer homomorphisms π 1 (D vj ) → π 1 (D ⊚ ); thus,
φ NF
v j
by considering cuspidal inertia groups of π 1 (D ⊚ ) whose unique index l
subgroup is contained in the image of this homomorphism [cf. Corollary
2.5 when v ∈ V bad ; the discussion of Remark 4.5.1 below],
we conclude that these homomorphisms induce a natural isomorphism of F l -torsors
LabCusp(D ⊚ ) → ∼ LabCusp(D vj ). In a similar vein, if v ∈ V arc , then it follows from
Definition 4.1, (v), that φ NF
v j
consists of certain morphisms of Aut-holomorphic
orbispaces which induce various outer homomorphisms π 1 (D vj ) → π 1 (D ⊚ )from
the [discrete] topological fundamental group π 1 (D vj ) to the profinite group π 1 (D ⊚ );
thus,