Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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44 SHINICHI MOCHIZUKI F↩→ J ∗ v ⋂ J ∗ H ; a similar statement holds when F is replaced by any J v -conjugate of F . Moreover, it follows from the well-known structure of the pro-Σ completion of the fundamental group of a hyperbolic Riemann surface [such as J ∗ v ] that the image of such a group F topologically normally generates a closed subgroup of J ∗ v ⋂ J ∗ H which is infinite and nonabelian. This completes the proof of (†). Next, let us observe that it follows by applying either [AbsTopII], Proposition 1.3, (iv), or [NodNon], Proposition 3.9, (i), to the various ̂Δ X -conjugates in J ∗ of Jv ∗ ⋂ J ∗ H as in (†) that the fact that α commutes with Jv ∗ ⋂ J ∗ H implies that α fixes v. If condition (a) holds, then the fact that conjugation by α on the maximal pro-l quotient of J v [which, as we saw above, is a quotient of Jv ∗ ⋂ J ∗ H ]istrivial implies [cf. the argument concerning the inertia group “I v ⊆ D v ” in the latter portion of the proof of [SemiAnbd], Corollary 3.11] that α not only fixes v, but also acts trivially on the irreducible component of the special fiber of X J determined by v; sinceJ and v as in (†) arearbitrary, we thus conclude that α is the identity element, as desired. Suppose, on the other hand, that condition (b) holds, but condition (a) does not hold. Then since J and v as in (†) arearbitrary, we thus conclude again from [Tama2], Theorem 0.2, (v), that α fixes not only v, but also every closed point on the irreducible component of the special fiber of X J determined by v, hence that α acts trivially on this irreducible component. Again since J and v as in (†) are arbitrary, we thus conclude that α is the identity element, as desired. This completes the proof of assertion (iii). In light of the exact sequences of assertion (iii), assertion (iv) follows immediately from assertion (i). Assertion (vi) follows immediately from [CombGC], Proposition 1.5, (i), by passing to pro-Σ completions. Finally, it follows immediately from the definitions of the various tempered fundamental groups involved that to verify assertion (v), it suffices to verify the following analogue of assertion (v) for a nonabelian finitely generated free discrete group G: for any finitely generated subgroup F ⊆ G, if we use the notation “∧” to denote the profinite completion, then ̂F ⋂ G = F . But to verify this assertion concerning G, it follows immediately from [SemiAnbd], Corollary 1.6, (ii), that we may assume without loss of generality that the inclusion F ⊆ G admits a splitting G ↠ F [i.e., such that the composite F↩→ G ↠ F is the identity on F ], in which case the desired equality “ ̂F ⋂ G = F ” follows immediately. This completes the proof of assertion (v), and hence of Corollary 2.3. ○ Next, we observe the following arithmetic analogue of Proposition 2.1. Proposition 2.4. (Profinite Conjugates of Nontrivial Arithmetic Compact Subgroups) In the notation of the above discussion: (i) Let Λ ⊆ Δ tp X be a nontrivial pro-Σ compact subgroup, γ ∈ ̂Π X an element such that γ · Λ · γ −1 ⊆ Δ tp X [or, equivalently, Λ ⊆ γ−1 · Δ tp X · γ]. Then γ ∈ Π tp X . (ii) Suppose that ̂Σ = Primes. Let Λ ⊆ Π tp X be a [nontrivial] compact subgroup whose image in G k is open, γ ∈ ̂Π X an element such that γ · Λ · γ −1 ⊆ Π tp X [or, equivalently, Λ ⊆ γ−1 · Π tp X · γ]. Then γ ∈ Πtp X .
INTER-UNIVERSAL TEICHMÜLLER THEORY I 45 (iii) Δ tp X (respectively, Πtp X )iscommensurably terminal in ̂Δ X (respectively, ̂Π X ). Proof. First, we consider assertion (i). We begin by observing that since [as is well-known — cf., e.g., [Config], Remark 1.2.2] ̂Δ X is torsion-free, it follows that there exists a finite index characteristic open subgroup J ⊆ Δ tp X [i.e., as in the previous paragraph] such that J ⋂ Λhasnontrivial image in the pro-Σ completion of the abelianization of J, hence in Π tp G J [since, as is well-known, the surjection J ↠ Π tp G J induces an isomorphism between the pro-Σ completions of the respective abelianizations]. Since the quotient Π tp X surjects onto G k,andJ is open of finite index in Δ tp X , we may assume without loss of generality that γ lies in the closure Ĵ of J in ̂Π X .SinceJ ⋂ Λhasnontrivial image in Π tp G J , it thus follows from Proposition 2.1 [applied to G J ] that the image of γ via the natural surjection Ĵ ↠ ̂Π GJ lies in Π tp G J . Since, by allowing J to vary, Π tp X (respectively, ̂ΠX ) may be written as an inverse limit of the topological groups Π tp X /Ker(J ↠ Πtp G J ) (respectively, ̂Π X /Ker(Ĵ ↠ ̂Π GJ )), we thus conclude that [the original] γ lies in Π tp X , as desired. Next, we consider assertion (ii). First, let us observe that it follows from a similar argument to the argument applied to prove Proposition 2.1 — where, instead of applying [SemiAnbd], Theorem 3.7, (iii), we apply its arithmetic analogue, namely, [SemiAnbd], Theorem 5.4, (ii); [SemiAnbd], Example 5.6 — that the image of γ in ̂Π X /Ker( ̂Δ X ↠ ̂Π G ∗) lies in Π tp X /Ker(Δtp X ↠ Πtp G ), where [by invoking the ∗ hypothesis that ̂Σ =Primes] we take G ∗ to be a semi-graph of anabelioids as in [SemiAnbd], Example 5.6, i.e., the semi-graph of anabelioids whose finite étale coverings correspond to arbitrary admissible coverings of the geometric special fiber of the stable model X . Here, we note that when one applies either [AbsTopII], Proposition 1.3, (iv), or [NodNon], Proposition 3.9, (i) — after, say, restricting the outer action of G k on Π tp G to a closed pro-Σ subgroup of the inertia group I ∗ k of G k that maps isomorphically onto the maximal pro-Σ quotient of I k —tothe vertices “v ′′ ”, “(v ′ ) γ ”, one may only conclude that these two vertices either coincide, areadjacent, oradmit a common adjacent vertex; but this is still sufficient to conclude the temperedness of “(v ′ ) γ ”fromthatof“v ′′ ”. Now [just as in the proof of assertion (i)] by applying [the evident analogue of] this observation to the quotients Π tp X ↠ Πtp X /Ker(J ↠ Πtp G )—whereJ ⊆ Δ tp ∗ X is a finite index characteristic J open subgroup, and GJ ∗ is the semi-graph of anabelioids whose finite étale coverings correspond to arbitrary admissible coverings of the geometric special fiber of any stable model of the covering of X determined by J — we conclude that γ ∈ Π tp X , as desired. Finally, we consider assertion (iii). Just as in the proof of Proposition 2.2, the commensurable terminality of Δ tp X in ̂Δ X follows immediately from assertion (i), by allowing, in assertion (i), Λ to range over the open subgroups of a pro-Σ Sylow [hence, in particular, nontrivial pro-Σ compact!] subgroup of a verticial subgroup of Δ tp G . The commensurable terminality of Πtp X in ̂Π X then follows immediately from the commensurable terminality of Δ tp X in ̂Δ X . ○
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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