Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 127
(ii) Let
† D = { † D v } v∈V
be a D-prime-strip [relative to the given initial Θ-data]. Observe [cf. the discussion
of Definition 4.1, (i)] that if v ∈ V non ,thenπ 1 ( † D v ) determines, in a functorial
fashion, a profinite group corresponding to “X v ” [cf. Corollary 1.2 if v ∈ V good ;
[EtTh], Proposition 2.4, if v ∈ V bad ], which contains π 1 ( † D v ) as an open subgroup;
thus, if we write † D ± v for B(−) 0 of this profinite group, then we obtain a natural
morphism † D v → † D ± v [cf. §0]. In a similar vein, if v ∈ V arc ,thensinceX −→v admits a
K v -core, a routine translation into the “language of Aut-holomorphic orbispaces” of
the argument given in the proof of Corollary 1.2 [cf. also [AbsTopIII], Corollary 2.4]
reveals that † D v determines, in a functorial fashion, an Aut-holomorphic orbispace
† D ± v corresponding to “X v ”, together with a natural morphism † D v → † D ± v of
Aut-holomorphic orbispaces. Thus, in summary, one obtains a collection of data
† D ± = { † D ± v } v∈V
completely determined by † D.
(iii) Suppose that we are in the situation of (ii). Then observe [cf. the discussion
of Definition 4.1, (ii)] that by applying the group-theoretic algorithm of
[AbsTopI], Lemma 4.5, to the topological group π 1 ( † D v )whenv ∈ V non ,orby
considering π 0 (−) of a cofinal collection of “neighborhoods of infinity” [i.e., complements
of compact subsets] of the underlying topological space of † D v when v ∈ V arc ,
it makes sense to speak of the set of cusps of † D v ; a similar observation applies to
† D ± v ,forv ∈ V. If v ∈ V, then we define a ±-label class of cusps of † D v to be the
set of cusps of † D v that lie over a single cusp [i.e., corresponding to an arbitrary element
of the quotient “Q” that appears in the definition of a “hyperbolic orbicurve
of type (1,l-tors)” given in [EtTh], Definition 2.1] of † D ± v ;write
LabCusp ± ( † D v )
for the set of ±-label classes of cusps of † D v . Thus, [for any v ∈ V!] LabCusp ± ( † D v )
admits a natural action by F × l
[cf. [EtTh], Definition 2.1], as well as a zero element
† η 0 v ∈ LabCusp± ( † D v )
and a ±-canonical element
† η ± v ∈ LabCusp± ( † D v )
—whichiswell-defined up to multiplication by ±1, and which may be constructed
solely from † D v [cf. Definition 4.1, (ii)] — such that, relative to the natural bijection
{LabCusp ± ( † D v ) \{ † η 0 v } }
/{±1} ∼ → LabCusp( † D v )
[cf. the notation of Definition 4.1, (ii)], we have † η ± v ↦→ † η v
. In particular, we
obtain a natural bijection
LabCusp ± ( † D v )
∼
→
F l