Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

126 SHINICHI MOCHIZUKI
Section 6: Additive Combinatorial Teichmüller Theory
In the present §6, we discuss the additive analogue — i.e., which revolves
around the “functorial dynamics” that arise from labels
∈ F l
—ofthe“multiplicative combinatorial Teichmüller theory” developed in §4 for
labels ∈ F l
. These considerations lead naturally to certain enhancements of the
various Hodge theaters considered in §5. On the other hand, despite the resemblance
of the theory of the present §6 tothetheoryof§4, §5, the theory of the present
§6 is, in certain respects — especially those respects that form the analogue of the
theory of §5 — substantially technically simpler.
In the following, we fix a collection of initial Θ-data
(F/F, X F , l, C K , V, V bad
mod, ɛ)
as in Definition 3.1; also, we shall use the various notations introduced in Definition
3.1 for various objects associated to this initial Θ-data.
Definition 6.1.
(i) We shall write
F ⋊±
l
def
= F l ⋊ {±1}
for the group determined by forming the semi-direct product with respect to the
natural inclusion {±1} ↩→ F × l
and refer to an element of F ⋊±
l
that maps to +1
(respectively, −1) via the natural surjection F ⋊±
l
↠ {±1} as positive (respectively,
negative). We shall refer to as an F ± l
-group any set E equipped with a {±1}-orbit
of bijections E → ∼ F l . Thus, any F ± l -group E is equipped with a natural F l-module
structure. We shall refer to as an F ± l
-torsor any set T equipped with an F⋊±
l
-orbit
of bijections T → ∼ F l [relative to the action of F ⋊±
l
on F l by automorphisms of
the form F l ∋ z ↦→ ±z + λ ∈ F l ,forλ ∈ F l ]. Thus, if T is an F ± l
-torsor, then the
abelian group of automorphisms of the underlying set of F l given by the translations
F l ∋ z ↦→ z + λ ∈ F l ,forλ ∈ F l , determines an abelian group
Aut + (T )
of “positive automorphisms” of the underlying set of T . Moreover, Aut + (T )is
equipped with a natural structure of F ± l
-group [such that the abelian group structure
of Aut + (T ) coincides with the F l -module structure of Aut + (T ) induced by this
F ± l -group structure]. Finally, if T is an F± l
-torsor, then we shall write
Aut ± (T )
for the group of automorphisms of the underlying set of T determined [relative to
the F ± l
-torsor structure on T ] by the group of automorphisms of the underlying set
of F l given by F ⋊±
l
[so Aut + (T ) ⊆ Aut ± (T ) is the unique subgroup of index 2].