Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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40 SHINICHI MOCHIZUKI The following result is the central technical result underlying the theory of the present §2. (Profinite Conjugates of Nontrivial Compact Sub- be a nontrivial Proposition 2.1. groups) In the notation of the above discussion, let Λ ⊆ Π tp G compact subgroup, γ ∈ ̂Π G an element such that γ · Λ · γ −1 ⊆ Π tp G [or, equivalently, Λ ⊆ γ −1 · Π tp G · γ]. Then γ ∈ Πtp G . Proof. Write ̂Γ for the “pro-̂Σ semi-graph” associated to the universal pro-̂Σ étale covering of G [i.e., the covering corresponding to the subgroup {1} ⊆̂Π G ]; Γ tp for the “pro-semi-graph” associated to the universal tempered covering of G [i.e., the covering corresponding to the subgroup {1} ⊆Π tp G ]. Thus,wehaveanaturaldense map Γ tp → ̂Γ. Let us refer to a [“pro-”]vertex of ̂Γ that occurs as the image of a [“pro-”]vertex of Γ tp as tempered. Since Λ, γ · Λ · γ −1 are compact subgroups of Π tp G , it follows from [SemiAnbd], Theorem 3.7, (iii) [cf. also [SemiAnbd], Example 3.10], that there exist verticial subgroups Λ ′ , Λ ′′ ⊆ Π tp G such that Λ ⊆ Λ′ , γ · Λ · γ −1 ⊆ Λ ′′ . Thus, Λ ′ ,Λ ′′ correspond to tempered vertices v ′ , v ′′ of ̂Γ; {1} ̸= γ·Λ·γ −1 ⊆ γ·Λ ′·γ−1 , so (γ · Λ ′ · γ −1 ) ⋂ Λ ′′ ≠ {1}. SinceΛ ′′ , γ · Λ ′ · γ −1 are both verticial subgroups of ̂Π G , it thus follows either from [AbsTopII], Proposition 1.3, (iv), or from [NodNon], Proposition 3.9, (i), that the corresponding vertices v ′′ ,(v ′ ) γ of ̂Γ are either equal or adjacent. In particular, since v ′′ is tempered, we thus conclude that (v ′ ) γ is tempered. Thus,v ′ ,(v ′ ) γ are tempered, so γ ∈ Π tp G , as desired. ○ Next, relative to the notation “C”, “N” and related terminology concerning commensurators and normalizers discussed, for instance, in [SemiAnbd], §0; [CombGC], §0, we have the following result. Proposition 2.2. (Commensurators of Decomposition Subgroups Associated to Sub-semi-graphs) In the notation of the above discussion, ̂Π H (respectively, Π tp H )iscommensurably terminal in ̂Π G (respectively, ̂Π G [hence, also in Π tp G ]). In particular, Πtp G is commensurably terminal in ̂Π G . Proof. First, let us observe that by allowing, in Proposition 2.1, Λ to range over the open subgroups of any verticial [hence, in particular, nontrivial compact!] subgroup of Π tp G , it follows from Proposition 2.1 that Π tp G is commensurably terminal in ̂Π G — cf. Remark 2.2.2 below. In particular, by applying this fact to H [cf. the discussion preceding Proposition 2.1], we conclude that Π tp H is commensurably terminal in ̂Π H . Next, let us observe that it is immediate from the definitions that Π tp H ⊆ C Π tp (Π tp H ) ⊆ ĈΠ (Π tp G G H ) ⊆ ĈΠ (̂Π H ) G [where we think of ̂Π H , ̂Π G , respectively, as the pro-̂Σ completions of Π tp H ,Πtp G ]. On the other hand, by the evident pro-̂Σ analogue of [SemiAnbd], Corollary 2.7, (i),
INTER-UNIVERSAL TEICHMÜLLER THEORY I 41 we have ĈΠG (̂Π H )=̂Π H . Thus, by the commensurable terminality of Π tp H in ̂Π H , we conclude that Π tp H ⊆ ĈΠ G (Π tp H ) ⊆ ĈΠ H (Π tp H )=Πtp H — as desired. ○ Remark 2.2.1. It follows immediately from the theory of [SemiAnbd] [cf., e.g., [SemiAnbd], Corollary 2.7, (i)] that, in fact, Propositions 2.1 and 2.2 can be proven for much more general semi-graphs of anabelioids G than the sort of G that appears in the above discussion. We leave the routine details of such generalizations to the interested reader. Remark 2.2.2. Recall that when ̂Σ =Primes, the fact that Π tp G is normally terminal in ̂Π G may also be derived from the fact that any nonabelian finitely generated free group is normally terminal [cf. [André], Lemma 3.2.1; [SemiAnbd], Lemma 6.1, (i)] in its profinite completion. In particular, the proof of the commensurable terminality of in ̂Π G that is given in the proof of Proposition 2.2 may be thought of as a new proof of this normal terminality that does not require one to invoke [André], Lemma 3.2.1, which is essentially an immediate consequence of the rather difficult conjugacy separability result given in [Stb1], Theorem 1. This relation of Proposition 2.1 to the theory of [Stb1] is interesting in light of the discrete analogue given in Theorem 2.6 below of [the “tempered version of Theorem 2.6” constituted by] Proposition 2.4 [which is essentially a formal consequence of Proposition 2.1]. Π tp G def Now let k be an MLF, k an algebraic closure of k, G k = Gal(k/k), X a hyperbolic curve over k that admits stable reduction over the ring of integers O k of k. Write Π tp X , Δtp X for the respective “̂Σ-tempered” quotients of the tempered fundamental groups π tp 1 (X), π tp 1 (X k ) [relative to suitable basepoints] of X, X def k = X × k k [cf. [André], §4; [Semi- Anbd], Example 3.10]. That is to say, π tp 1 (X k ) ↠ Δtp X is the quotient determined by the intersection of the kernels of all continuous surjections of π tp 1 (X k )ontoextensions of a finite group of order a product [possibly with multiplicities] of primes ∈ ̂Σ by a discrete free group of finite rank; π tp 1 (X) ↠ Πtp X is the quotient of πtp 1 (X) determined by the kernel of the quotient of π tp 1 (X k ) ↠ Δtp X . Write ̂Π X , ̂ΔX for the respective pro-̂Σ [i.e., maximal pro-̂Σ quotients of the profinite] fundamental groups of X, X k . 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], we have natural inclusions Π tp X ↩→ ̂Π X , Δ tp X ↩→ ̂Δ X [cf., e.g., [SemiAnbd], Proposition 3.6, (iii), when ̂Σ =Primes]; ̂Π X , ̂Δ X may be identified with the pro-̂Σ completions of Π tp X ,Δtp X .
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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