Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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42 SHINICHI MOCHIZUKI Now suppose that the residue characteristic p of k is not contained in Σ; that the semi-graph of anabelioids G of the above discussion is the pro-Σ semi-graph of anabelioids associated to the geometric special fiber of the stable model X of X over O k [cf., e.g., [SemiAnbd], Example 3.10]; and that the sub-semi-graph H ⊆ G is stabilized by the natural action of G k on G. Thus,wehavenatural surjections Δ tp X ↠ Πtp G ; ̂ΔX ↠ ̂Π G of topological groups. Corollary 2.3. (Subgroups of Tempered Fundamental Groups Associated to Sub-semi-graphs) In the notation of the above discussion: (i) The closed subgroups Δ tp X,H def = Δ tp X × Π tp G Π tp H ⊆ Δtp X ; ̂ΔX,H def = ̂Δ X ×̂ΠG ̂ΠH ⊆ ̂Δ X are commensurably terminal. In particular, the natural outer actions of G k on Δ tp X , ̂Δ X determine natural outer actions of G k on Δ tp X,H , ̂Δ X,H . (ii) The closure of Δ tp X,H ⊆ Δtp X ⊆ ̂Δ X in ̂Δ X is equal to ̂Δ X,H . (iii) Suppose that [at least] one of the following conditions holds: (a) ̂Σ contains a prime number l/∈ Σ ⋃ {p}; (b)̂Σ =Primes. Then ̂Δ X,H is slim. In particular, the natural outer actions of G k on Δ tp X,H , ̂Δ X,H [cf. (i)] determine natural exact sequences of center-free topological groups [cf. (ii); the slimness of ̂Δ X,H ; [AbsAnab], Theorem 1.1.1, (ii)] 1 → Δ tp X,H → Πtp X,H → G k → 1 1 → ̂Δ X,H → ̂Π X,H → G k → 1 —whereΠ tp X,H , ̂Π X,H are defined so as to render the sequences exact. (iv) Suppose that the hypothesis of (iii) holds. Then the images of the natural inclusions Π tp X,H ↩→ Πtp X , ̂Π X,H ↩→ ̂Π X are commensurably terminal. (v) We have: ̂Δ X,H ⋂ Δ tp X =Δtp X,H ⊆ ̂Δ X . (vi) Let I x ⊆ Δ tp X (respectively, I x ⊆ ̂Δ X ) be an inertia group associated to a cusp x of X. Write ξ for the cusp of the stable model X corresponding to x. Then the following conditions are equivalent: (a) I x lies in a Δ tp X - (respectively, ̂ΔX -) conjugate of Δ tp X,H (respectively, ̂Δ X,H ); (b) ξ meets an irreducible component of the special fiber of X that is contained in H.
INTER-UNIVERSAL TEICHMÜLLER THEORY I 43 Proof. Assertion (i) follows immediately from Proposition 2.2. Assertion (ii) follows immediately from the definitions of the various tempered fundamental groups involved, together with the following elementary observation: IfG ↠ F is a surjection of finitely generated free discrete groups, which induces a surjection Ĝ ↠ ̂F between the respective profinite completions [so, by applying the well-known residual finiteness of free groups [cf., e.g., [SemiAnbd], Corollary 1.7], we think of G and F as subgroups of Ĝ and ̂F , respectively], then H def =Ker(G ↠ F )isdense in =Ker(Ĝ ↠ ̂F ), relative to the profinite topology of Ĝ. Indeed, let ι : F↩→ G be a section of the given surjection G ↠ F [which exists since F is free]. Then if {g i } i∈N is a sequence of elements of G that converges, in the profinite topology of Ĝ, to a given element h ∈ Ĥ, and maps to a sequence of elements {f i} i∈N of F [which necessarily converges, in the profinite topology of ̂F ,totheidentity element 1 ∈ ̂F ], then one verifies immediately that {g i ·ι(f i ) −1 } i∈N is a sequence of elements of H that converges, in the profinite topology of Ĝ, toh. This completes the proof of the observation and hence of assertion (ii). Ĥ def Next, we consider assertion (iii). In the following, we give, in effect, two distinct proofs of the slimness of ̂Δ X,H :oneiselementary, but requires one to assume that condition (a) holds; the other depends on the highly nontrivial theory of [Tama2] and requires one to assume that condition (b) holds. If condition (a) holds, then let us set Σ ∗ def =Σ ⋃ {l}. If condition (b) holds, but condition (a) does not hold [so ̂Σ =Primes =Σ ⋃ {p}], then let us set Σ ∗ def = Σ. Thus, in either case, p ∉ Σ ∗ ⊇ Σ. Let J ⊆ ̂Δ X be an open subgroup. Write J H def = J ⋂ ̂ΔX,H ; J ↠ J ∗ for the maximal pro-Σ ∗ quotient; J ∗ H ⊆ J ∗ for the image of J H in J ∗ . Now suppose that α ∈ ̂Δ X,H commutes with J H .Letv be a vertex of the dual graph of the geometric special fiber of a stable model X J of the covering X J of X k determined by J. Write J v ⊆ J for the decomposition group [well-defined up to conjugation in J] associated to v; J ∗ v ⊆ J ∗ for the image of J v in J ∗ . Then let us observe that (†) there exists an open subgroup J 0 ⊆ ̂Δ X which is independent of J, v, and α such that if J ⊆ J 0 , then for arbitrary v [and α] asabove,itholds that J ∗ v ⋂ J ∗ H (⊆ J ∗ )isinfinite and nonabelian. Indeed, if condition (a) holds, then it follows immediately from the definitions that the image of the homomorphism J v ⊆ J ⊆ ̂Δ X ↠ ̂Π G is pro-Σ; in particular, since l ∉ Σ, and Ker(J v ⊆ J ⊆ ̂Δ X ↠ ̂Π G ) ⊆ J v ⋂ JH , it follows that J v ⋂ JH , hence also J ∗ v ⋂ J ∗ H , surjects onto the maximal pro-l quotient of J v , which is isomorphic to the pro-l completion of the fundamental group of a hyperbolic Riemann surface, hence [as is well-known] is infinite and nonabelian [so we may take J 0 def = ̂Δ X ]. Suppose, on the other hand, that condition (b) holds, but condition (a) does not hold. Then it follows immediately from [Tama2], Theorem 0.2, (v), that, for an appropriate choice of J 0 ,ifJ ⊆ J 0 ,theneveryv corresponds to an irreducible component that either maps to a point in X or contains a node that maps to a smooth point of X . In particular, it follows that for every choice of v, there exists at least one pro-Σ, torsion-free, ⋂ pro-cyclic subgroup F ⊆ J v that lies in Ker(J v ⊆ J ⊆ ̂Δ X ↠ ̂Π G ) ⊆ J v JH and, moreover, maps injectively into J ∗ . Thus, we obtain an injection
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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