Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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38 SHINICHI MOCHIZUKI to conjugacy] by the outer action of Gal(X/C) =Π C /Π X on Π X . This completes the proof of Corollary 1.2. ○ Remark 1.2.1. If follows immediately from Corollary 1.2 that Aut k (X −→ )=Gal(X −→ /C) ( ∼ = Z/2lZ); Aut k ( C −→ )=Gal(C −→ /C) ( ∼ = Z/lZ) [cf. [EtTh], Remark 2.6.1].
INTER-UNIVERSAL TEICHMÜLLER THEORY I 39 Section 2: Complements on Tempered Coverings In the present §2, we discuss certain routine complements — which will be of use in the present series of papers — to the theory of tempered coverings of graphs of anabelioids, as developed in [SemiAnbd], §3 [cf. also the closely related theory of [CombGC]]. Let Σ, ̂Σ benonempty sets of prime numbers such that Σ ⊆ ̂Σ; G a semi-graph of anabelioids of pro-Σ PSC-type [cf. [CombGC], Definition 1.1, (i)], whose underlying graph we denote by G. WriteΠ tp G for the tempered fundamental group of G [cf. the discussion preceding [SemiAnbd], Proposition 3.6] and ̂Π G for the pro-̂Σ [i.e., maximal pro-̂Σ quotient of the profinite] fundamental group of G [cf. the discussion preceding [SemiAnbd], Definition 2.2] — both taken with respect to appropriate choices of basepoints. Thus, since discrete free groups of finite rank inject into their pro-l completions for any prime number l [cf., e.g., [RZ], Proposition 3.3.15], it follows that we have a natural injection [cf. [SemiAnbd], Proposition 3.6, (iii), when ̂Σ =Primes; the proof in the case of arbitrary ̂Σ is entirely similar] that we shall use to regard Π tp G of Π tp G . Next, let Π tp G ↩→ ̂Π G as a subgroup of ̂Π G and ̂Π G as the pro-̂Σ completion H be the semi-graph of anabelioids associated to a connected sub-semi-graph H ⊆ G. One verifies immediately that the underlying graph of anabelioids associated to H coincides with the underlying graph of anabelioids associated to some semi-graph of anabelioids of pro-Σ PSC-type. That is to say, rough speaking, up to the possible omission of some of the cuspidal edges, H “is” a semi-graph of anabelioids of pro- Σ PSC-type. In particular, since the omission of cuspidal edges clearly does not affect either the tempered or pro-̂Σ fundamental groups, we shall apply the notation introduced above for “G” toH. Wethusobtainanatural commutative diagram Π tp H −→ ̂Π H ⏐ ⏐ ↓ ↓ Π tp G −→ ̂Π G of inclusions [cf. [SemiAnbd], Proposition 2.5, (i), when ̂Σ =Primes; the proof in the case of arbitrary ̂Σ is entirely similar] of topological groups, which we shall use to regard all of the groups in the diagram as subgroups of ̂Π G . In particular, one may think of Π tp H (respectively, ̂Π H )asthedecomposition subgroup in Π tp G (respectively, ̂Π G ) associated to the sub-semi-graph H.
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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