Categories of Hyperbolic Riemann Surfaces

7/27/2019 Categories of Hyperbolic Riemann Surfaces

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CONFORMAL AND QUASICONFORMAL CATEGORICAL

REPRESENTATION OF HYPERBOLIC RIEMANN SURFACES

Shinichi Mochizuki

August 2006

In this paper, we consider various categories of hyperbolic Riemann sur-

faces and show, in various cases, that the conformal or quasiconformalstructure of

the Riemann surface may be reconstructed, up to possible confusion between holo-

morphic and anti-holomorphic structures, in a natural way from such a category. The

theory exposed in the present paper is motivated partly by a classical result concern-

ing the categorical representation of sober topological spaces, partly by previous work

of the author concerning the categorical representation of arithmetic log schemes, and

partly by a certain analogy with p-adic anabelian geometry an analogy which the

theory of the present paper serves to render more explicit.

Contents:

Introduction

0. Notations and Conventions

1. Reconstruction via the Upper Half-Plane Uniformization

2. Categories of Parallelograms, Rectangles, and Squares

Appendix: Quasiconformal Linear Algebra

Introduction

In this paper, we continue our study [cf., [Mzk2], [Mzk10]] of the topic of

representingvarious objects that appear in conventional arithmetic geometry bymeans ofcategories. As discussed in [Mzk2], [Mzk10], this point of view is partiallymotivated by the anabelian philosophy of Grothendieck [cf., e.g., [Mzk3], [Mzk4],[Mzk5]], and, in particular, by the more recent work of the author on absoluteanabelian geometry [cf. [Mzk6], [Mzk7], [Mzk8], [Mzk9], [Mzk11], [Mzk12]].

One way to think about anabelian geometry is that it concerns the issue ofrepresenting schemes by means of categories [i.e., Galois categories] that capturecertain aspects of the [etale] topologyof the scheme [i.e., its fundamental group].From this point of view, another important, albeit elementary, example of the issue

2000 Mathematical Subject Classification. 14H55, 30F60.

Typeset by AMS-TEX

1


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2 SHINICHI MOCHIZUKI

of representing a space by means of a category of topological origin is the well-known example of thecategory of open subsets of a sober topological space[cf., e.g.,[Mzk2], Theorem 1.4; [Mzk10], Proposition 4.1]. In some sense, this example is theexample that motivated the construction of the categories appearing in the presentpaper.

The main resultsof this paper may be summarized as follows:

(1) The holomorphic structure of a hyperbolic Riemann surface of finite typemay be reconstructed, up to possible confusion with the correspondinganti-holomorphic structure, from a certain category of localizationsof theRiemann surface that includes the upper half-plane uniformizationof theRiemann surface, together with its natural P SL2(R)-action [cf. Theorem1.12].

(2) Given a hyperbolic Riemann surface of finite type equipped with a

nonzero logarithmic square differential, one may define certain categoriesof parallelograms, rectangles, or squares associated to this data. Then[isomorphism classes of] equivalences between corresponding categories ofparallelograms (respectively, rectangles; squares) are in natural bijectivecorrespondence with[quasiconformal] Teichmuller mappings(respectively,conformal mappings) between such Riemann surfaces equipped with dif-ferentials, again up to possible confusion between holomorphic and anti-holomorphic structures [cf. Theorem 2.3].

Here, we note that the categories of (2) are especially close to the categories of open

subsets of a sober topological space referred to above i.e., roughly speaking, in-stead of consideringallthe open subsets of the Riemann surface, one restricts oneselfto those which are parallelograms (or, alternatively, rectangles, or squares),in a sense determined by the natural parameters[i.e., of Teichmuller theory cf.,e.g., [Lehto], Chapter IV, 6.1] associated to the given square differential.

On the other hand, from another point of view, the main motivation for the re-sults obtained in this paper came from the analogy with p-adic anabelian geometry.This analogy has been pointed out previously by the author [cf., e.g., [Mzk1], Intro-duction, 0.10; [Mzk5], 3]. In some sense, however, the theory of the present paperallows one to make this analogy more explicit. Indeed, at the level ofobjects under

considerationthe theory of the present paper suggests a certain dictionary, assummarized in Table 1 below.

The first two non-italicized rows of Table 1 are motivated by the fact thatthe datum of a nonzero logarithmic square differentialmay be thought of, in thecontext of Teichmuller theory, as the datum of a geodesicin Teichmuller space. Inparticular, if one thinks of oneself as only knowing the differential up to a nonzerocomplex multiple[cf. Theorem 2.3], then one is, in essence, working with acomplexTeichmuller geodesic. Moreover, just as such a complex geodesic is of holomor-phic dimension one and real/topological dimension two, the spectrum of the ringof integers of ap-adic local fieldKis of algebraic dimension one, while the absolute

Galois groupGKof thep-adic local fieldKis of cohomological dimension two. This


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CATEGORIES OF HYPERBOLIC RIEMANN SURFACES 3

observation also motivates the point of view of the third non-italicized row of Table1, which is also discussed in [Mzk1], Introduction, 0.10. From the point of view ofthis third non-italicized row of Table 1, the conformal structure may be thought ofas the metric, or angular, structureof the S1 acting by rotations locally on thesurface. On the other hand, from the point of view ofp-adic anabelian geometry,

one may completely recover the algebraic structure of the p-adic curve in question[cf. the main result of [Mzk4]], so long as one restricts oneself to working withgeo-metricisomorphisms [i.e., isomorphisms arising from isomorphisms of fields] of theabsolute Galois groups of the p-adic local fields in question. Moreover, as one seesfrom the theory of [Mzk3], this geometricity condition corresponds to the preser-vation of the metric structureof the copy of the units OKinside the abelianizationGabK ofGK [more precisely, the preservation of such metric structures for all opensubgroups ofGK].

complex case p-adic casethe given Riemann surface the logarithmic special fiber

a complex Teichmuller geodesic a lifting of the special fiberoriginating from the given to a hyperbolic curve over a

Riemann surface p-adic local field Kaction ofC on the action of the absolute Galois

surface by rotations (S1 C) group GKon [the Galoisand flows (R C) category associated to]

the profinite geometric

fundamental groupsquares, rectangles, as opposed preservation of the metric

to parallelograms i.e., preservation structure of the copyof the metric structure ofS1 ofOK in G

abK

Table 1: Dictionary of objects under consideration

This dictionary of objects under consideration then suggests a dictionaryof results, as summarized in Table 2 below. The analogy between the p-adicTeichmuller theoryof [Mzk1] [and, in particular, the canonical representation con-structed in this theory] and the upper half-plane uniformization of a hyperbolicRiemann surface of finite type is one of thecornerstonesof the theory of [Mzk1]; inparticular, a lengthy discussion of this analogy may be found in the Introduction to[Mzk1]. Also, relative to the issue of reconstructing the original hyperbolic curveor Riemann surface, it is interesting to note that just as Theorem 1.12 does notrequire the datum of a logarithmic square differential, the absoluteness of canon-ical liftings [cf. [Mzk7], Theorem 3.6] only involves the datum of the logarithmicspecial fiber i.e., there is no choice of a p-adic lifting involved [just as there isno choice of a complex Teichmuller geodesic in Theorem 1.12]. By contrast, justas the results on the left-hand side of the second and third non-italicized rows of

Table 2doinvolve the choice of such a complex Teichmuller geodesic, the hyperbolic


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curves involved on the right-hand side of the second and third non-italicized rowsof Table 2 require thechoiceof a p-adic lifting of the logarithmic special fiber. Assuggested by the dictionary of Table 1, the preservation of the metric structureof the units [i.e.,S1 C or OKG

abK] corresponds to complete reconstruction

of the conformal structure of the Riemann surface or the algebraic structure of

the p-adic curve in the second and third non-italicized rows of Table 2. On theother hand, reconstruction of the quasiconformal structure of the Riemann surface[essentially a topological invariant] corresponds, in the final row of Table 2, to thereconstruction of the dual semi-graph [also essentially atopologicalinvariant] of thelogarithmic special fiber, in theabsenceof the preservation of the metric structureof the units. Also, it is interesting to note that the theory of the first non-italicizedrow of Table 2 is not functorial with respect to ramified coverings of the Riemannsurface/non-admissible coverings of thep-adic hyperbolic curve, whereas the theoryof the latter three non-italicized rows of Table 2 is functorial with respect to suchcoverings.

complex case p-adic casecategorical representation the canonical representationvia the upper half-plane of p-adic Teichmuller theory,

uniformization the absoluteness of canonical liftings[cf. Theorem 1.12] [cf. [Mzk1]; [Mzk7], Theorem 3.6]

conformal structure via relative p-adic profinite versioncategories of rectangles of the Grothendieck Conjecture[cf. Theorem 2.3, (iii)] [cf. [Mzk4], Theorem A]

conformal structure via relative p-adic pro-p versioncategories of squares of the Grothendieck Conjecture

[cf. Theorem 2.3, (iii)] [cf. [Mzk4], Theorem A]quasiconformal structure reconstruction of dual semi-graph

via categories of of logarithmic special fiber viaparallelograms absolute p-adic pro-prime-to-p

[cf. Theorem 2.3, (ii)] anabelian geometry or itstempered analogue

[cf. [Mzk6], Lemma 2.3;[Mzk11], Corollary 3.11]

Table 2: Dictionary of results

Here, we remark that although it is quite possible that the relative p-adicprofinite [or pro-p] versions of the Grothendieck Conjecture proven in [Mzk4] admitabsolutegeneralizations [cf., e.g., [Mzk12], Corollary 2.12], if [as on the right-handside of the fourth non-italicized row of Table 2] one restricts oneself to the pro-prime-to-p portion of the geometric fundamental group, then there is no hope [cf.the unbridgeable gap between conformal and quasiconformal structures!] of recov-ering the generic fiber of the original p-adic curve from the outer Galois action onthe pro-prime-to-p geometric fundamental group, since this outer Galois action is

completely determined by the logarithmic special fiber.


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Acknowledgements:

I would like to thankAkio Tamagawa, Makoto Matsumoto, andSeidai Yasudafor many helpful comments concerning the material presented in this paper.


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Section 0: Notations and Conventions

Numbers:

The notationZ (respectively,R; C) will be used to denote the set of rationalintegers(respectively,real numbers; complex numbers).

Topological Groups:

A homomorphism of topological groups G H will be called dense if theimage ofG is dense in H.

A topological group G will be called tempered [cf. [Mzk11], Definition 3.1, (i)]ifG is isomorphic, as a topological group, to an inverse limit of an inverse system

of surjections of countable discrete topological groups.

Categories:

Let Cbe a category. We shall denote by

Ob(C)

the collection ofobjectsofC. IfA Ob(C) is an objectofC, then we shall denoteby

CA

the category whoseobjectsare morphismsB AofCand whose morphisms (froman object B1 A to an object B2 A) are A-morphisms B1 B2 in C. Thus,we have anatural functor

(jA)!: CA C

(given by forgetting the structure morphism to A).

We shall call an object A Ob(C) terminal if for every object B Ob(C),there exists a unique arrow B A inC.

We shall refer to a natural transformationbetween functors all of whose com-ponent morphisms are isomorphisms as an isomorphism between the functors inquestion. A functor: C1 C2 between categories C1, C2 will be called rigid ifhas no nontrivial automorphisms. A categoryCwill be called slim if the naturalfunctor CA C is rigid, for every A Ob(C).

A diagram of functors between categories will be called 1-commutative if thevarious composite functors in question areisomorphic. When such a diagram com-mutes in the literal sense we shall say that it 0-commutes. Note that when a dia-gram in which the various composite functors are all rigid1-commutes, it followsfrom therigidityhypothesis that any isomorphism between the composite functors

in question is necessarily unique. Thus, to state that such a diagram 1-commutes


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does not result in any loss of information by comparison to the datum of a specificisomorphismbetween the various composites in question.

We shall say that a nonempty [i.e., non-initial] object A Ob(C) is connectedif it is not isomorphic to the coproduct of two nonempty objects ofC. We shall say

that an objectA Ob(C) ismobile(respectively,infinitely mobile) if there exists anobject B Ob(C) such that the set HomC(A, B) hascardinality 2 [i.e., the diag-onal from this set to the product of this set with itself is not bijective] (respectively,infinite cardinality). We shall say that an object A Ob(C) is quasi-connected ifit is either immobile [i.e., not mobile] or connected. Thus, connected objects arealways quasi-connected. If everyobject of a category C is quasi-connected, thenwe shall say that C is a category of quasi-connected objects. We shall say that acategory C is totally(respectively, almost totally) epimorphic if every morphismin C whose domain is arbitrary(respectively, nonempty) and whose codomain isquasi-connectedis an epimorphism.

We shall say that C isof finitely (respectively, countably) connected typeif it isclosed under formation of finite (respectively, countable) coproducts; every object ofCis a coproduct of a finite (respectively, countable) collection of connected objects;and, moreover, all finite (respectively, countable) coproducts

Ai in the category

satisfy the condition that the natural map HomC(B, Ai) HomC(B,

Ai)

is bijective, for all connected B Ob(C). IfC is of finitely or countably connectedtype, then every nonempty object ofC is mobile; in particular, a nonempty objectofCis connected if and only ifit is quasi-connected.

If a mobileobject A Ob(C) satisfies the condition that every morphism in Cwhose domain is nonempty and whose codomain is A is an epimorphism, then Ais connected. [Indeed, C1

C2

A, where C1, C2 are nonempty, implies that the

composite map

HomC(A, B) HomC(A, B) HomC(A, B) HomC(C1, B) HomC(C2, B)

= HomC(C1

C2, B) HomC(A, B)

is bijective, for all B Ob(C).]

IfC is a category of finitely or countably connected type, then we shall write

C0 C

for thefull subcategoryof connected objects. [Note, however, that in general, objectsofC0 are not necessarily connected or even quasi-connected as objects ofC0!]On the other hand, if, in addition, C isalmost totally epimorphic, thenC0 istotallyepimorphic, and, moreover, an object ofC0 isconnected[as an object ofC0!] if andonly if [cf. the argument of the preceding paragraph!] it is mobile [as an object ofC0]; in particular, [assuming still that Cis almost totally epimorphic!] every object

ofC0 isquasi-connected[as an object ofC0].


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IfC is a category, then we shall write

C (respectively,C)

for the category formed from Cby taking arbitrary formal [possibly empty] finite

(respectively, countable) coproducts of objects in C. That is to say, we define theHom ofC (respectively,C) by the formula

Hom(i

Ai,j

Bj)def=i

j

HomC(Ai, Bj)

[where the Ai, Bj are objects ofC]. Thus, C (respectively,C) is a category of

finitely connected type(respectively, category of countably connected type). Notethat objects ofCdefineconnectedobjects ofC orC. Moreover, there are natural[up to isomorphism] equivalences of categories

(C)0 C; (C)0

C; (D0)

D; (E0)

E

if D (respectively, E) is a category of finitely connected type(respectively, cate-gory of countably connected type). If C is a totally epimorphic category of quasi-connected objects, then C (respectively,C) is an almost totally epimorphic cate-gory of finitely (respectively, countably) connected type.

In particular, the operations 0, (respectively, ) define one-to-onecorrespondences [up to equivalence] between the totally epimorphic categories ofquasi-connected objectsand the almost totally epimorphic categories of finitely (re-

spectively, countably) connected type.


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Section 1: Reconstruction via the Upper Half-Plane Uniformization

In this Section, we show that the conformal structureof a hyperbolic Riemannsurface may be functorially reconstructed by applying the well-known geometry

of the upper half-plane uniformization of the Riemann surface from a certaincategory of localizations naturally associated to the Riemann surface. These cate-gories of localizations are intended to be reminiscent of i.e., a sort ofarchimedeananalogue of the categories of localizations of [Mzk11], 4.

In the following discussion, we shall denote the [Riemann surface constitutedby the] upper half-planeby the notation H. Next, we introduce some terminology:

Definition 1.1.

(i) We shall refer to a smooth Hausdorff complex analytic stack which admits

an open dense subset isomorphic to a complex manifold and [for simplicity] whoseuniversal covering is a complex manifold as a complex orbifold.

(ii) We shall refer to a one-dimensional complex orbifold with at most countablymany connected components as aRiemann orbisurface. We shall refer to a Riemannorbisurface which is a complex manifold [i.e., whose orbifold structure is trivial]as a Riemann surface.

(iii) We shall refer to a Riemann orbisurface as being of finite type(respectively,of almost finite type) if it may be obtained as the complement of a finite subset(respectively, [possibly infinite] discrete subset) in a compact Riemann orbisurface(respectively, a Riemann orbisurface of finite type).

(iv) We shall refer to a connected Riemann orbisurface X (respectively, arbi-trary Riemann orbisurface X) as being an H-domain if there exists a finite [i.e.,proper], surjective etale covering X Xsuch that X admits an etale [i.e., withderivative everywhere nonzero] holomorphic map X H (respectively, if everyconnected component ofX is an H-domain).

(v) We shall refer to as an RC-orbifold [i.e., real complex orbifold] a pairX = (X, X), whereX is a complex orbifold, and X is an anti-holomorphic invo-

lution [i.e., automorphism of order 2]; we shall refer to X as the

complexificationof the RC-orbifold X [cf. Remark 1.3.1 below]. Moreover, we shall append theprefixRC- to the beginning of any of the terms introduced in (i) (iv) to referto RC-orbifolds X = (X, X) for which Xsatisfies the conditions of the term inquestion.

(vi) An RC-holomorphic map

X Y

between complex orbifolds X, Y is a map which is either holomorphic or anti-

holomorphic at each point ofX.


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(vii) Amorphism between RC-orbifolds

X = (X, X) Y = (Y, Y)

whereX is connected [i.e., X acts transitivelyon the set of connected compo-

nents ofX] is an equivalence class of RC-holomorphic mapsX Y compatiblewith X , Y, where we consider two RC-holomorphic maps equivalent if they differby composition with X [or, equivalently, Y]. A morphism between RC-orbifolds

X = (X, X) Y = (Y, Y)

where X is not necessarily connected is the datum of a morphism of RC-orbifolds from each connected component ofX to Y.

Remark 1.1.1. Note that a Riemann orbisurface of finite typeadmits a unique

algebraic structureoverC. We refer to Lemma 1.3, (iii), for the RC analogue ofthis statement.

Remark 1.1.2. IfXis an H-domain, andY Xis anetale morphism of complexorbifolds, then it is immediate from the definitions that Y is also an H-domain.

Remark 1.1.3. If Y X is a finite etale covering of connected Riemannorbisurfaces, then the symmetric functions in the various conjugates [i.e., withrespect to the finite covering Y X] of any bounded holomorphic function on

Y [e.g., a function arising from a morphism Y H] give rise to various boundedholomorphic functions on X which determine, up to a finite indeterminacy, theoriginal bounded holomorphic function on Y.

Remark 1.1.4. For any morphism of RC-orbifolds

:X = (X, X) Y = (Y, Y)

there exists a unique holomorphic map : X Y lying in the equivalence classthat constitutes . Indeed, we may assume without loss of generality thatX is

connected. Then if1 :X Y isanyRC-holomorphic map lying in , then [sinceX but not necessarily X! is connected] 1 is either holomorphic or anti-holomorphic. If1 is holomorphic (respectively, anti-holomorphic), then we take

def= 1 (respectively,

def= Y 1 = 1 X).

Proposition 1.2. (Complex Orbifolds as RC-Orbifolds)

(i) LetXbe acomplex orbifold; writeXc for itscomplex conjugate [i.e.,holomorphic functions onXc are anti-holomorphic functions onX]. Then

R :X (X

Xc

, R(X))


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where R(X) switchesX, Xc via the [anti-holomorphic!] identification of their

underlying real analytic stacks determines afully faithful functor Rfrom thecategory of complex orbifolds and RC-holomorphic mapsinto thecategoryof RC-orbifolds [and morphisms of RC-orbifolds].

(ii) Let X = (X, X) be anRC-orbifold. Then there is a natural mor-phism of RC-orbifolds

R(X) X

which isfinite etaleof degree2 given by mappingX X

Xc (respectively,Xc X

Xc) to X via the identity map (respectively, X).

Proof. Immediate from the definitions.

Lemma 1.3. (Removable Singularities)

(i) No H-domain is a Riemann orbisurface ofalmost finite type.

(ii) A connectedH-domainis necessarilyhyperbolic [i.e., its universal cov-ering is biholomorphic to H].

(iii) Any finite etale RC-holomorphic map XY between Riemann orbisur-faces X, Y of finite type [each of which, by Remark 1.1.1, admits a unique al-gebraic structure overC] is necessarilyalgebraic overR. In particular, everyRC-Riemann orbisurface of finite type admits a unique algebraic structure overR.

Proof. Assertion (i) follows immediately [cf. Remark 1.1.3] from the observationthat every bounded holomorphic function on a Riemann orbisurface of almost finitetype extends to a bounded holomorphic function on a Riemann orbisurface of finitetype, hence to a bounded holomorphic function on a compact Riemann orbisurface,which is necessarily constant. Assertion (ii) follows from the same fact, applied tothe case where the Riemann orbisurface of finite type in question is the complexplane. Assertion (iii) follows by observing that the properness [i.e., finiteness] as-sumption implies that this map X Y extends to the one-point compactificationsof X, Y which possess a natural structure of [the stack-theoretic version of]

complex analytic space [i.e., the point at infinity may be singular!] and thenapplying the well-known fact that holomorphic [hence also RC-holomorphic] mapsbetween algebrizable compact complex analytic spaces are necessarily algebrizable.

Remark 1.3.1. Thus, just as complex manifolds are an analytic analogue ofsmooth schemes over C, RC-manifolds [i.e., RC-orbifolds whose stack structure istrivial] are intended to be ananalytic analogueofsmooth schemes overR. Relativeto this analogy, the functor R of Proposition 1.2, (i), is the analogue of the functor

(XC C) (XC R)


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that maps a smooth scheme XC overC to the underlyingR-scheme. Similarly, thefirst datum X of an RC-complex manifold X = (X, X), is the analogue, for asmooth scheme XR over R, of the associated smooth C-schemeXR RC, and theetale double cover of Proposition 1.2, (ii), is the analogue of the etale double coverof smoothR-schemes

XR R C XR

(given by projection to the first factor).

Remark 1.3.2. Note that it follows immediately from Lemma 1.3, (iii), thatevery Riemann orbisurface of finite type Xadmits a canonical compactificationbya compact Riemann orbisurface X X whose stack structure is trivial nearX\X. A similar statement holds for RC-Riemann orbisurfaces.

Definition 1.4. Let X

= (X, X) be an RC-orbifold. Then:

(i) We shall refer to the set X(C) of points ofX [i.e., points of the coarsecomplex analytic space associated to the stack X] as the set ofcomplex pointsofX.

(ii) We shall refer to the set X(R) X(C) of complex points fixed byX asthe set ofreal pointsofX.

(iii) We shall refer to the set X[C] def= X(C)/X ofX -orbits of complex pointsofX as the set ofRC-pointsofX.

(iv) We shall refer to H def= R(H) as the RC-upper half-plane. We shall refer

to an RC-H-domain [i.e., the RC version of an H-domain] as an H-domain.

Remark 1.4.1. If X = (X, X) is a connected RC-orbifold, then one verifieseasily that X(R) admits a natural structure ofreal analytic orbifoldwhose realdimension is equal to the complex dimension ofX.

Let X = (X, X) be an RC-orbifold. Then note that one may consider the

notion of a covering morphism [of RC-orbifolds] Y

= (Y, Y) (X, X) [i.e.,Y X is a covering morphism, in the usual sense of algebraic topology]. Inparticular, ifX is connected, then, by considering universal covering morphisms,we may define the fundamental group

1(X)

of the RC-orbifoldX.

Proposition 1.5. (Fundamental Groups of RC-orbifolds) Let X =

(X, X) be a connected RC-orbifold.


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(i) IfX arises from acomplex orbifold, i.e., X = R(X0) [cf. Proposition

1.2, (i)], then we have a natural isomorphism1(X0) 1(X

). In this case, weshall say thatX isof complex type.

(ii) IfX isconnected, then we have a natural exact sequence1 1(X)

1(X) Gal(C/R) 1. Here, the surjection1(X) Gal(C/R) correspondsto the double covering of Proposition 1.2, (ii). In this case, we shall say thatX isof real type.

(iii) Suppose thatX is ahyperbolic Riemann orbisurface. ThenX= H

if and only if1(X) = {1}.

Proof. Assertions (i) and (ii), as well as thenecessityportion of assertion (iii), areimmediate from the definitions. As for thesufficiencyportion of (iii), we observethat the condition 1(X

) = {1} implies, by assertion (ii), that X arises from a

connected Riemann orbisurface X0. Thus, since X = X0

Xc0 is hyperbolic, weconclude [from the definition of hyperbolic!] that X0 = H, so X = H, asdesired.

Next, let us assume thatX is aconnected hyperbolic RC-Riemann orbisurfaceof finite type. Write 1(X

) for the profinite completionof1(X). Suppose that

we have been given a quotient

1(X)

of profinite groups. Then we may define a category of (-)localizations ofX

Loc(X)

as follows: IfX = (X, X) isof real type(respectively,of complex type, andX0 Xis a connected component ofX), then theobjects

Y (respectively,Y)

of this category are the RC-Riemann orbisurfaces (respectively, Riemann orbisur-

faces) which are eitherH-domains (respectively, H-domains) orRC-Riemann or-bisurfaces (respectively, Riemann orbisurfaces) of finite type that appear as [notnecessarily connected] finite etale coverings of X (respectively, X0) that factorthrough thequotient. [Here, we recall that by Lemma 1.3, (i), this either-or ismutually exclusive.] Themorphisms

Y1 Y2 (respectively,Y1 Y2)

of this category are arbitrary etale morphisms of RC-orbifolds(respectively, arbi-trary etale holomorphic morphisms) which are, moreover, properand lie over X

(respectively,X0) whenever Y1, Y2 (respectively, Y1, Y2) are of finite type. Thus,


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by Lemma 1.3, (i), [cf. also Remark 1.1.2] the codomain of any arrow with domainof finite type is also of finite type.

To keep the notation and language simple, even when X is of complex type,we shall regard the objects and morphisms of this category as RC-orbifolds and

morphisms of RC-orbifolds, via the fully faithful functor R of Proposition 1.2;moreover, thinking about things in this way renders explicit the independenceofLoc(X

) of the choice ofX0, as the notation suggests.

Lemma 1.6. (Basic Categorical Properties) Let : Y1 Y2 be a

morphism inLoc(X). Then:

(i) If :Z2 Y2 is a morphism inLoc(X

), then the projection morphisms

Y1 Y2 Z2 Z

2 ; Y

1 Y2 Z

2 Y

1

obtained by forming the fibered product of Y1, Z2 over Y

2 in the category of

RC-orbifolds lie inLoc(X).

(ii) is amonomorphism if and only if it factors as the composite of an

isomorphismY1 Y3 with anopen immersionY

3 Y

2, whereY

3 is the object

determined by some open subset ofY2[C].

(iii) IfY1 =, andY2 is aconnected RC-orbifold, then

is anepimor-phism. In particular, the full subcategory ofLoc(X

) consisting of the connectedobjects is atotally epimorphic category of quasi-connected objects[cf. 0].

Proof. Assertion (i) is immediate from the definitions ifY1 and Z2 are of finite

type; if either Y1 or Z2 is an H

-domain, then assertion (i) follows by applying theobservation of Remark 1.1.2. Assertion (ii) may be reduced to the case whereY2is of complex type, by base-changing [cf. assertion (i)] via the double covering ofProposition 1.2, (ii) [applied to Y2]. When Y

2 is of complex type, assertion (ii)

follows immediately from the definitions, by considering various maps H Y2.Finally, assertion (iii) follows from the elementary complex analysis fact that aholomorphic function on a connected domain which vanishes on an open subset isnecessarily identically zero.

Lemma 1.7. (Infinitely Mobile Opens) LetY Ob(Loc(X)). Write

Loc(X)Y Loc(X

)Y

for thefull subcategorydetermined by the objects constituted by arrowsZ Y

which aremonomorphisms. Then:

(i) There is a naturalfully faithful functor

Loc(X

)Y Open(Y

[C])


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[where Open() denotes the category whose objects are open subsets and whosemorphisms are inclusions of the topological space in parentheses cf. [Mzk10],4] given by assigning to a monomorphism Z Y the image of the inducedmap Z[C] Y[C]. This functor is an equivalence if and only if Y is anH-domain.

(ii) IfY isinfinitely mobile [cf. 0] as an object ofLoc(X), thenY is

anH-domain.

Proof. First, let us observe the easily verified e.g., bycardinality considerationsconcerning the set of isomorphism classes of objects ofLoc(X

) which areof finitetype fact that, ifY is of finite type, then there exist open subsetsU Y[C]of the form Y[C]\E, where E Y[C] is a finite set, which do not lie in theessential imageof the functor of assertion (i) [cf. Lemma 1.3, (i)]. In light of thisobservation, assertion (i) is a formal consequence of Lemma 1.6, (ii); Remark 1.1.2.Finally, assertion (ii) is an immediate consequence of the definition of the categoryLoc(X

).

Lemma 1.8. (Category-Theoreticity of the Topological Space ofRC-Points) For i = 1, 2, let Xi be a connected hyperbolic RC-Riemannorbisurface of finite type; 1(X

i) i aquotient. Let

: Loc1(X1 )

Loc2(X

2 )

be an equivalence of categories; Yi Ob(Loci(Xi)); assume that Y2 =(Y1). Then induces ahomeomorphism

Y1[C] Y2[C]

on the topological spaces of RC-points which is functorial in both and theYi .In particular, Y1 isof finite type if and only ifY

2 = (Y

1) is of finite type.

Proof. Note that the infinitely mobile objectsare manifestly preservedby and

that H

is infinitely mobile. In particular, every object ofLoci(X

i) is coveredbyinfinitely mobile opens. Thus, by functoriality [and an evident gluing argument],we may assume, without loss of generality, that the Yi are infinitely mobile. Butthen, since the topological spaces Yi [C] are clearly sober, the existence of a func-torial homeomorphism as desired [as well as the fact that preserves objects offinite type] follows from Lemma 1.7, (i), (ii), together with a well-known result fromtopos theory [i.e., to the effect that a sober topological space may be recoveredfrom the category of sheaves on the space cf., e.g., [Mzk2], Theorem 1.4].

Lemma 1.9. (Category-Theoreticity of the Fundamental Group)For

i= 1, 2, letXi, i,,Yi be as in Lemma 1.8. Then preserves the arrows which


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arecovering morphisms. In particular, preserves isomorphs ofH and, if theYi are connected, induces anisomorphism of groups

1(Y1)

1(Y

2)

well-defined up to composition with aninner automorphism which isfunc-torial in both and the choices of universal covering morphismZi Y

i used to

define the1s.

Proof. Indeed, covering morphisms may be characterized by the existence of localbase-changes over which the given morphismsplitsas adisjoint unionof isomorphsof the base. Thus, the fact that preserves covering morphisms follows fromLemmas 1.6, (i); 1.8. The assertion concerning fundamental groups then followsformally; the assertion concerning isomorphs ofH follows from Proposition 1.5,(iii).

Lemma 1.10. (Category-Theoreticity of the RC-Orbifold Structure)For i= 1, 2, letXi, i, , Y

i be as in Lemma 1.8. Then induces an isomor-

phism of RC-orbifoldsY1

Y2

which is functorial in both and the Yi and compatible with the homeomor-phisms of Lemma 1.8. In particular,X1 (respectively, Y

1) isof real type if and

only ifX2 (respectively, Y2) is.

Proof. Indeed, by functoriality, we may assume, without loss of generality, thatthe Yi are connected. Choose universal coveringsZi Y

i [so Z

i

= H] whichare compatible with [cf. Lemma 1.9]. Note that we have an exact sequence oftopological groups

1 SL2(R)/{1} AutRC-orbifolds(H) Gal(C/R) 1

where the topology on AutRC-orbifolds(H) is that induced by the action of

AutRC-orbifolds(H) on H[C]. In particular, Aut(Zi)

def= AutLoci (Xi)(Z

i) is con-

nectedif and only ifXi isof complex type. Moreover, by Lemmas 1.8, 1.9, inducesa commutative diagram

1(Y1) Aut(Z1 ) 1(Y

2) Aut(Z

2 )

in which thevertical arrowsare isomorphisms of topological groups. Note that sinceAut(Zi) is a real analytic Lie group, we thus conclude [byCartans theorem cf.,

e.g., [Serre], Chapter V, 9, Theorem 2] that the isomorphism Aut(Z1 ) Aut(Z2 )

is, in fact, an isomorphism of real analytic Lie groups.

Next, let us choosemaximal connected compact subgroupsKi Aut(Zi) which

are compatible with . Then ifXi is of complex type [so Aut(Zi) is connected],


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then let us write Aut(Zi)0 def= Aut(Zi). On the other hand, ifX

i is of real type,

then we havenatural exact sequences

1 Aut(Zi)0 Aut(Zi) Gal(C/R) 1

[where the superscript 0 denotes the connected component containing the identityelement] which are compatible with . Whether Xi is of real or complex type,

let us write K0idef= Ki

Aut(Zi)

0; Yi = (Yi, Yi). Note thatYi is of real type if

and only if1(Yi ) Aut(Z

i) has image ={1} in Aut(Z

i)/Aut(Z

i)

0. IfYi is ofreal type, then 1(Yi) 1(Y

i ) may be identified with the kernel of this map to

Aut(Zi)/Aut(Zi)

0, and Yi equipped with its Yi-action isnaturally isomorphicto

Ki\Aut(Zi)/1(Yi)

[where the / is in the sense of stacks!] equipped with the natural action by1(Y

i )/1(Yi)

= Gal(C/R) [from the right]. If Yi is of complex type, then Yi

is naturally isomorphicto the result of applying the functor R to the Riemannorbisurface

K0i \Aut(Zi)

0/1(Yi )

[where the / is in thesense of stacks!]. Thus, we conclude that [for Xi of real or

complex type] induces an isomorphism of RC-orbifoldsY1Y2, as desired.

That this isomorphism iscompatiblewith the homeomorphisms of Lemma 1.8follows by comparing the respective induced maps on points where we notethat in the context of Lemma 1.8 (respectively, the present proof), points of,say,Zi , amount to systems of neighborhoods of an element ofZ

i[C] (respectively,

left cosets of Ki in Aut(Z

i

) or of K0i

in Aut(Zi

)0) by considering the actionofAut(Zi), Ki on such systems of neighborhoods. Finally, the functorialityof the

isomorphismY1 Y2 with respect to (respectively, theY

i ) is clear (respectively,

a consequence of the compatibility with the homeomorphisms of Lemma 1.8).

Corollary 1.11. (Preservation of Like Parity) For i = 1, 2, let Xi, i,, Yi be as in Lemma 1.8; suppose further that the X

i are of real type. Let

Zi Ob(Loci(Xi)); assume that Z

2 = (Z

1 ), and that the Y

i and Z

i are all

connected. Suppose that we are given two morphisms

i, i: Zi Y

i

inLoci(Xi) such that 2 = (1); 2 = (1). Then 1, 1 have the same

parity i.e., theirunique holomorphic representatives[cf. Remark 1.1.4]induce the same maps on sets of connected components if and only if2, 2 do.

Proof. Immediate from the functorial isomorphisms of RC-orbifoldsof Lemma1.10.

Theorem 1.12. (Categorical Reconstruction of Hyperbolic RC-Rie-mann Orbisurfaces) For i = 1, 2, let Xi be a connected hyperbolic RC-

Riemann orbisurface of finite type; 1(Xi) i aquotient. Then the


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categoriesLoci(Xi) areslim [cf. 0], and, moreover, anyequivalence of cate-

gories : Loc1(X

1 )

Loc2(X

2 )

is [uniquely] isomorphic [as a functor] to the equivalence induced by a unique

isomorphism of RC-orbifolds X1

X2 . That is to say, the natural map

IsomR((X1 , 1), (X2 , 2)) Isom(Loc1(X

1 ),Loc2(X

2 ))

from isomorphisms of RC-orbifolds X1 X2 which admit [uniquely determined,

up to inner automorphisms arising from1(Xi) cf. Lemma 1.9] compatible iso-

morphisms1 2 to isomorphism classes of equivalences between the categories

Loci(Xi) isbijective.

Proof. Indeed, slimnessfollows, for instance, by considering the functorial home-

omorphismsof Lemma 1.8, while the asserted bijectivityfollows formally from thefunctorial isomorphisms of RC-orbifolds of Lemma 1.10. Here, we note that theobject Xi ofLoci(X

i) may be characterized, up to isomorphism, as the object of

finite type [cf. Lemma 1.8] which forms aterminal object in the full subcategory ofLoci(X

i) determined by the objects of finite type.

Corollary 1.13. (Induced Isomorphisms of Quotients of Profinite Fun-damental Groups) In the notation of Theorem 1.12, the isomorphism

1 2

induced by [well-defined up to composition with an inner automorphism of i]isindependent of the choice of, up to thegeometrically-induced automor-phisms of i i.e., the automorphisms arising from the automorphisms of theRC-orbifoldXi that preserve the quotient1(X

i) i.

Proof. A formal consequence of Theorem 1.12.


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Section 2: Categories of Parallelograms, Rectangles, and Squares

In this Section, we show that thequasiconformal(respectively,conformal;con-formal) structureof a connected hyperbolic RC-Riemann orbisurface of finite type

may be functorially reconstructed from a certain category ofparallelogram(respec-tively, rectangle; square) localizations. Although, just as was the case with thecategories of1, these categories of localizations are intended to be reminiscent ofthe categories of localizations of [Mzk11], 4, they differ from the categories of1in the following crucial way: They admit terminal objects [cf. the categories of[Mzk11], 4, which also, essentially, admit terminal objects, up to finitely manyautomorphisms, or, alternatively, the categories called temperoidsof [Mzk11], 3].

Definition 2.1.

(i) We shall refer to a connected hyperbolic Riemann (respectively, RC-Riemann)orbisurface as a punctured torus(respectively, punctured RC-torus) if it (respec-tively, each connected component of its complexification) arises as the complementof a finite, nonempty subset of a one-dimensional complex torus [i.e., the Riemannsurface associated to an elliptic curve over C]. If this finite subset is a translate ofa subgroup of the complex torus (respectively, is of cardinality one), then we shallrefer to the punctured torus (respectively, punctured RC-torus) as being of torsiontype(respectively, once-punctured).

(ii) Let Ybe a compact connected Riemann orbisurface; Y Y the Riemannorbisurface of finite type obtained by removing some finite set Sof points from Y.[Thus, by Lemma 1.3, (iii),Y is completely determined byY.] Then we shall referto as a logarithmic square differentialon Y a section over Y of the line bundle2Y [whereY is the holomorphic line bundle of differentials on Y] which extends

to a section over Y of the line bundle 2Y

(S) [where Y is the holomorphic line

bundle of differentials on Y; we use the notation Sto denote the reduced effectivedivisor on Ydetermined by the set S]. Thenoncritical locus

Ynon Y

of a logarithmic square differential on Yis defined to be the Riemann orbisurfaceof points at which = 0; the universalizationof a logarithmic square differential on Y is defined to be the universal covering Ynon Ynon of the noncritical locusYnon of. As is well-known [cf., e.g., [Lehto], Chapter IV,6.1], if0 [i.e., isnot identically zero], then thepath integral of the square root of over Ynon

determines a natural parameter

z:Ynon C


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on Ynon, which is independent of the choice of square root and the choice of abasepoint for the integral, up to multiplication by1 and addition of a constant.In particular, it makes sense to define a -parallelogram(respectively, -rectangle;-square) of Ynon to be an open subset or Ynon [or, by abuse of terminology,the associated Riemann surface] that maps bijectively via z onto a parallelogram

(respectively, rectangle; square) ofC, in the sense of Definition A.3, (i), (ii), of theAppendix. We shall refer to a -parallelogram as pre-compactif it is contained ina compact subset ofYnon.

(iii) Alogarithmic square differential on a connected RC-Riemann orbisur-face of finite type X is defined to be a logarithmic square differential on [eachconnected component of] the complexification ofX which is preserved by the anti-holomorphic involution ofX. Given a logarithmic square differential on X,the noncritical locus (respectively, universalization; natural parameters [whenever 0]) associated to the corresponding logarithmic square differential on the com-

plexification of X

thus determine a noncritical locus X

non X

(respectively,universalizationXnon Xnon; natural parametersz : X

non[C] C) associated

to . Here, any two natural parameters z , z are related to one another as

follows: z is equal to either z+ , for some C, or the complex conjugateof this expression. In particular, we obtain a notion of-parallelograms(respec-tively, -rectangles; -squares; pre-compact -parallelograms) associated to

[all of which are to be regarded as RC-Riemann surfaces over Xnon].

(iv) Let Y, Zbe Riemann orbisurfaces of finite type. IfY, Zare connected,then we shall refer to a map Y Z as anti-quasiconformal(respectively, anti-Teichmuller) if it is quasiconformal (respectively, a Teichmuller mapping cf.

Remark 2.1.1 below) with respect to the holomorphic structure on Y given bythe holomorphic functions and the holomorphic structure on Zgiven by the anti-holomorphic functions. IfY,Zare not necessarily connected, then we shall refer toa mapY Zas RC-quasiconformal(respectively,RC-Teichmuller) if its restrictionto each connected component ofY determines a map to some connected componentof Z that is either quasiconformal or anti-quasiconformal (respectively, either aTeichmuller mapping or an anti-Teichmuller mapping).

(v) Let Y = (Y, Y),Z = (Z, Z) be connected RC-Riemann orbisurfaces of fi-

nite type. Then we shall refer to as an RC-quasiconformal morphism(respectively,RC-Teichmuller morphism) Y Z an equivalence classof RC-quasiconformal(respectively, RC-Teichmuller) maps Y Z compatible with Y, Z, where weconsider two such mapsequivalentif they differ by composition with Y [or, equiv-alently,Z]. If Y, Z are tempered topological groups, and

1(Y) Y; 1(Z

) Z

are dense [cf. 0] morphisms of tempered [cf. 0] topological groups [i.e., we thinkof1(Y

), 1(Z) as being equipped with the discrete topology, so 1(Y

),1(Z)

are tempered topological groups], then we shall say that an RC-quasiconformalmorphism Y Z is (Y, Z)-compatible if there exists a [necessarily unique,

by the dense-ness assumption] isomorphism Y

Z that is compatible [in


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the evident sense] with the outer isomorphism 1(Y)

1(Z

) induced by theRC-quasiconformal morphismY Z.

(vi) A Teichmuller pair(X, ) (respectively, RC-Teichmuller pair(X, )) isdefined to be a pair consisting of a connected hyperbolic Riemann (respectively,

RC-Riemann) orbisurface of finite type X(respectively, X

) and a non-identicallyzero logarithmic square differential (respectively, ) on X(respectively, X).

Remark 2.1.1. We refer to [Lehto], Chapter V,7, 8, for more on the theoryofTeichmuller mappingsbetween Riemann orbisurfaces of finite type. Note thatalthough the theory of Teichmuller mappings is typically only developed forcompactRiemann surfaces, it extends immediately to the case of an arbitrary Riemannorbisurface of finite typeYby passing to an appropriateGalois finite etale coveringZ Y which extends to a ramified covering of compact Riemann orbisurfacesZ Y, where Zis a Riemann surface, and Z Y isramifiedat every point ofZ\Z.

[Indeed, the ramification condition implies that a logarithmic square differential onYpulls back to a logarithmic square differential on Zwhich extends to a squaredifferential withoutpoles on Z.]

Remark 2.1.2. Let : Y Z be an RC-quasiconformal morphism (re-spectively, RC-Teichmuller morphism), as in Definition 2.1, (v) [so Y, Z areconnected]. Then [cf. Remark 1.1.4] there exists a unique quasiconformal map (re-spectively, Teichmuller mapping) : Y Z lying in the equivalence class thatconstitutes .

Remark 2.1.3. One important example of anRC-Teichmuller pair

(X, )

is the case where X admits a finite etale covering Y X such that Y is apunctured RC-torus of complex type, and the square differential |Y extends to asquare differential on thecanonical compactification[cf. Remark 1.3.2] ofY. Notethat in this case, iscompletely determined, up to anonzero constant multiple. Inthe following, we shall refer to such a pair as toral. Note that ifZ X is also afinite etale covering ofX by a punctured RC-torus of complex type Z such that|Z extends to a square differential on the canonical compactification ofZ

in

which case we shall say thatZ X istoralizing then one verifies immediately[by considering thenatural parametersassociated to ] that there exists a toralizingfinite etale covering W X that dominatesthe coverings Y X, Z X.In particular, it follows that there exists a unique [up to not necessarily uniqueisomorphism] minimal toralizing finite etale covering YminX

[i.e., such thatevery other toralizing finite etale coveringY X factorsthrough Ymin X

].

Let (X = (X, X), ) be anRC-Teichmuller pair. Suppose that we have been

given a tempered topological group and a dense morphism

1(X

)


22/38


of tempered topological groups. Thus, for every open subgroup H , the inducedmorphism 1(X

) /H is surjective. Let us refer to a connected covering ofX

as being a -covering if it appears as a subcovering of the covering determined bysuch a quotient 1(X

) /H. In the following, we shall also make the followingtwo assumptions on :

(1) is totally ramified at infinityin the sense that there existGalois finite-coverings of X which are ramified over every point of the canonicalcompactification [cf. Remark 1.3.2] X

X which is not contained in

X.

(2) is stack-resolving in the sense that there exist Galois finite -coveringsofX which are of complex type and whose stack structure istrivial.

Now we define the category of parallelogram (-)localizations of(X, )

LocP (X

, )

as follows: TheobjectsZ

of this category are the RC-Riemann orbisurfaces which are either pre-compact-parallelograms of the universalization Xnon orRC-Riemann orbisurfaces thatappear as connected [but not necessarily finite] -coverings ofX. Objects of theformer type will be referred to asparallelogram objects; objects of the latter type will

be referred to as complete objects. A parallelogram object defined by a

-rectangle(respectively,-square) will be referred to as a rectangle object(respectively,squareobject). A complete object that arises from a finite covering ofX will be referredto as a finite object. Themorphisms

Z1 Z2

of this category arearbitrary etale morphisms of RC-orbifolds overX which, more-over, satisfy the property that ifZ1 is a parallelogram object, then eitherthe givenarrowZ1 Z

2 is an isomorphism of RC-orbifoldsorthe given arrowZ

1 Z

2 has

pre-compact image [i.e., the image ofZ1 [C] lies inside a compactsubset ofZ2 [C]].

Similarly, we define thecategory of rectangle (-)localizations of(X, )

LocR (X, )

to be the full subcategory ofLocP(X, ) determined by the objects which areei-

thercomplete objectsorrectangle objects, and thecategory of square (-)localizationsof(X, )

LocS (X, )

to be the full subcategory ofLocP (X, ) determined by the objects which are

eithercomplete objectsorsquare objects.


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Observe that whenX isof complex type, and we think of the objectsZ X

ofLocP (X, ) as being endowed with the holomorphic structure determined

by a connected component X0 X, then all of the morphisms Z1 Z2 of

LocP (X

, ) induce holomorphicmorphisms between the connected componentsof the complexifications ofZ1 , Z

2 lying over X0 [cf. Remark 1.1.4]. Put another

way, in this case, the category LocP(X, ) may be thought of as the image via the

fully faithful functor R of Proposition 1.2 of a certain category ofholomorphic mor-phisms between Riemann orbisurfaces. A similar statement holds for LocR (X

, ),

LocS (X, ).

Proposition 2.2. (Basic Categorical Properties)Let be either P, R,or S. Then:

(i) The result of applying to the full subcategory ofLoc(X, ) deter-

mined by thecomplete objects is aconnected temperoid [cf. [Mzk11], Defini-

tion 3.1, (ii)], with tempered fundamental group isomorphic to . In particular, itmakes sense to speak of complete objects as beingGalois [cf. [Mzk11], Definition3.1, (iv)].

(ii) Thecodomain of any arrow ofLoc(X, ) with complete domain is

also complete.

(iii) An objectZ ofLoc(X, )iscompleteif and only if every monomor-

phismZ W [inLoc(X, )] is an isomorphism.

(iv) The object ofLoc(X, ) determined by X is a terminal object of

the categoryLoc(X, ).

(v) The categoryLoc(X, ) is atotally epimorphic category of quasi-

connected objects [cf. 0].

(vi) The automorphism groupAut(Z)of a complete objectZ ofLoc(X, )

is isomorphic to a subquotient of a group of the form /H, where H is anopen subgroup.

(vii) IfZ is aparallelogram object ofLoc(X, ), then everyendomor-

phism ofZ [inLoc(X, )] is anautomorphism, and, moreover, the auto-

morphism group Aut(Z) [ofZ as a object ofLoc(X, )] isfinite.

(viii) Every morphismZ1 Z2 betweenparallelogramobjects ofLoc

(X

, )is amonomorphism.

(ix) Everymonomorphism Z1 Z2 ofLoc

(X

, ) which isnot an iso-morphism factors as a compositeZ1 Z

3 Z

2 ofnon-isomorphismsZ

1 Z

3 ,

Z3 Z2 , whereZ

1 , Z

3 areparallelogram objects.

Proof. Assertions (i), (iv), (v), (vi) are immediate from the definitions [cf. also

the proof of Lemma 1.6, (iii), in the case of assertion (v)]. To prove assertion


24/38


(ii), let Z Y be an arrow such that Z is complete, but Y is not complete.Thus,Y is a parallelogram object, and the morphismZ Y isoverX, henceover Xnon. In particular, we conclude that X

non = X

. Note, moreover, that

Z X is a covering morphism which [outside of the category Loc(X, )] is a

subcovering of the coveringXnon X. In particular, if webase-changeoverX by

Xnon X, we obtain [since Y issimply connected] a morphismXnon Y overXnon, which is absurd [since, for instance, Y

[C], unlike Xnon[C], has pre-compactimage in Xnon[C]]. In light of assertion (ii), assertion (iii) is immediate from ourpre-compactness assumption in the definition of the morphisms of Loc(X

, )withparallelogram domain[together with the observation that morphisms betweencomplete objects are always covering morphisms, hence are monomorphisms if andonly if they are isomorphisms].

Next, we considerendomorphisms of parallelogram objects, i.e., assertion (vii).First, let us observe that pulling back the standard volume form on Cvia anatural

parameteryields avolume formXnon onX

non[C] that is compatible with theaffinelinear structureon Xnon[C] determined by the natural parameters, and, moreover,is held fixedby Gal(Xnon/X

non) [since automorphisms of Gal(X

non/X

non) fix

,hence map natural parameters associated to to natural parameters associated to

]. In particular, since all morphisms ofLoc(X, ) areoverX, it follows that

Xnon

(respectively, the affine linear structure on Xnon[C]) determines a volumeform Z (respectively, affine linear structure) on Z

[C] that is compatible withall endomorphisms of Z. Thus, the fact that every endomorphism of Z is anautomorphism follows immediately from the [easily verified, elementary] fact thatevery volume-preserving, affine linear automorphism ofC that maps a parallelogramofXnon into itself necessarily induces a bijection of this parallelogram onto itself.

Moreover, it is immediate [for instance, by considering the induced bijections ofedges and vertices of the closure of the parallelogram] that the group of affinelinear automorphisms of this parallelogram that arise in this fashion is finite.

Next, we consider assertion (viii). First, observe that any two morphismsZi

Xnon [wherei= 1, 2] that arise from lifting morphismsZi X

ofLoc(X, )

differ by composition with an element of Gal(Xnon/Xnon), and that it is immediate

from the definitions that there existsuch morphisms Zi Xnon which are open

immersions. In particular, it follows thatevery morphismZi Xnon that arises

from lifting a morphism Zi X of Loc(X

, ), hence, in particular, every

composite Z

1 Z

2 X

non of such a lifted morphism Z

2 X

non with anarbitrary morphism Z1 Z2 of Loc

(X

, ) is an open immersion. Thus, itfollows immediately that any morphism Z1 Z

2 is a monomorphism, as desired.

Finally, we consider assertion (ix). First, we recall that it is immediate fromthe definition of a connected temperoid [cf. [Mzk11], Definition 3.1, (ii)] that anymonomorphism between connected objects of a connected temperoid is, in fact, anisomorphism. Thus, it follows from assertion (i) that Z1 is a parallelogramobject.If Z2 is also a parallelogram object, then it follows immediately from our pre-

compactnessassumption in the definition of the morphisms ofLoc(X, ) with

parallelogram domain that Z1 Z2 admits a factorization of the desired type.

If, on the other hand, Z2 is complete, then [as discussed above], the morphism


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Z1 Z2 factors as a composite Z

1 X

non Z

2 . Now since the image of

the morphism Z1 Xnon is [by the definition of the parallelogram objects of

Loc(X, )] pre-compact, it follows immediately that the morphism Z1 X

non

factors as a composite Z1 Z3 X

non, where Z

1 Z

3 is a non-isomorphism of

Loc

(X, ) between parallelogram objects, and Z

3

X

non

is an open immersion.Thus, by composing the arrow Z3 X

non with the arrow X

non Z

2 , we obtain

a factorization Z1 Z3 Z

2 of the desired type. This completes the proof of

assertion (ix).

Theorem 2.3. (Categorical Reconstruction of the Quasiconformal orConformal Structure of an RC-Teichmuller pair) For i = 1, 2, let(Xi,

i )

be anRC-Teichmuller pair; i atempered topological group;

1(Xi) i

a dense [cf. 0] morphism of tempered [cf. 0] topological groups suchthati istotally ramified at infinity andstack-resolving [cf. the abovediscussion]. Then:

(i) The categoriesLocPi(Xi,

i ), Loc

Ri(X

i,

i ), Loc

Si(X

i,

i ) areslim [cf.

0].

(ii) There is a naturalbijectionbetween isomorphism classes ofequivalencesof categories

: LocP1(X1 ,

1) LocP2(X

2 ,

2)

and(1, 2)-compatible RC-Teich-muller morphisms

X1 X2

that map 1 to a nonzero complex multiple of 2 [i.e.,

1 (respectively, some

nonzero complex multiple of2) is the initial (respectively, terminal) dif-ferentialof the RC-Teichmuller morphism cf., e.g., [Lehto], Chapter V, The-orem 8.1]. Moreover, this bijection is obtained by considering the equivalence of

categories naturally induced by such an RC-Teichmuller morphismX1 X2 .

(iii) There is a natural bijection between isomorphism classes of equiva-

lences of categories

: LocR1(X1 ,

1) LocR2(X

2 ,

2)

and(1, 2)-compatible isomorphisms of RC-orbisurfaces

X1 X2

that map 1 to a nonzero complex multiple of 2. Moreover, this bijection is ob-

tained by considering the equivalence of categories naturally induced by such an iso-morphism of RC-orbisurfacesX1

X2 . A similar statement holds when Loc

R

is replaced by LocS

.


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Proof. First, let us observe that it is immediate from the definitions that anisomorphismX1 X

2 of the type stated in assertions (ii), (iii), induces an equiva-

lence of categories between the respective categories Loc [where is P, R,or S]. [In the case of RC-Teichmuller morphisms, this follows immediately fromthe manifestly affine linear explicit local form of a Teichmuller mapping cf., e.g.,[Lehto], Chapter V, Theorem 8.1.] In particular, we note that the definition of eachof these categories is unaffected by multiplying by a nonzero complex number.

Next, let us suppose that we have been given an equivalence between the

respective categories Loc. WriteC def= Loci(X

i,

i ). Let us refer to an ordered

set which is isomorphic, as an ordered set, to the set of natural numbers [equippedwith its usual ordering] as a naturally ordered set. IfW Ob(C), then let us referto as a P-system [i.e., a system of parallelograms] over W a projective systemZ= {Zj }jJ

. . . Zj . . . Zj . . .

inCW , indexed by a naturally ordered setJ, such that: (a) each object Zj W

of this system is an arrow ofCwhose domain Zj is a parallelogram; (b) no arrowZj Z

j is an isomorphism. Recall from Proposition 2.2, (viii), that every arrow

Zj Zj is a monomorphism. There is an evident notion ofmorphisms between

P-systems over W [i.e., morphisms of projective systems]. We shall call a P-system Z over W minimal if every morphism ofP-systems [over W] Z Z isan isomorphism.

Let Z = {Zj }jJ be a P-system over W. Then it follows from our pre-

compactnessassumption in the definition of the morphisms of Loc with paral-

lelogram domainthat if we denote the closure of the subset

Zjdef= Im(Zj [C]) W

def= W[C]

by Kj W, then Kj is compact; moreover, we have an equality

Zdef=jJ

Zj =jJ

Kj W

of subsets ofW. Now suppose that for each j J,zj Zj ; letz W be acluster

pointof the set {zj}jJ [i.e., some subsequence of the sequence constituted by thezj converges toz]. [Note that since the Kj arecompact, such a cluster point alwaysexists.] Then I claim that z Z. Indeed, we may assume [by replacing J by

a cofinal subset ofJ] that zj z. Then if we write Ajdef= {zj}jj

{z}, then

Aj Kj , so

zjJ

Aj jJ

Kj = Z

as desired. In particular, since theZj are nonempty, it follows thatZisnonempty.

Now Iclaimthat Z={Zj }jJ is minimal if and only if the cardinality |Z|

of the set Z is equal to 1. Indeed, if|Z| >1, then it is immediate that one can


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construct a morphism of P-systems Z Zsuch that Z Z [where Z is the

analogue of Z forZ], soZ Z isnotan isomorphism of P-systems. On theother hand, suppose that |Z|= 1. Now since the topological space W is clearlymetrizable, let us assume that it is equipped with a metricd(, ). Let

Z = {Zj}jJ Z= {Zj }jJ

be a morphism of P-systems over W. Thus, Z = Z. Write Z ={z}. Thenobserve that for every real >0, there exists a j0 Jsuch that for all j j0, Zj

is contained in the set B(z, )def= {w W | d(z, w)< }. Indeed, if this were false,

then it would follow that for every [sufficiently large, hence every] j J, there existaj , bj Zj such that d(aj , bj) . Moreover, by choosing the aj, bj appropriately,we may assume that aj a, bj b, for some a, b W. But by our discussion ofcluster points in the preceding paragraph, it thus follows that a = b = z, hence that d(aj , bj) d(a, b) = 0, which is absurd. Thus, we conclude thatZj B (z, )

for sufficiently large j J. On the other hand, since, given a j

J

, there existsan > 0 such that B(z, ) Zj , it thus follows immediately that Z

Z is anisomorphism, thus proving the asserted minimalityofZ.

Thus, in summary, we conclude that:

There is a natural bijective correspondence between the set W[C]non[where the subscript non denotes the open subset determined by thenoncricital locus] and theset of isomorphism classes of minimal P-systemsoverW.

In particular, since, by Proposition 2.2, (iii), preserves parallelogram objects, weconclude that inducesnatural bijections

W1 [C]non W2 [C]non

[where, for i= 1, 2,Wi Ob(LocPi

(Xi, i )), (W

1 ) =W

2 ] which arefunctorialin

theWi. Moreover, since the images of parallelograms in Wi[C] clearly form abasis

for the topology ofWi[C], we conclude [by considering collections of isomorphismclasses of P-systems over Wi that factor through some given fixed parallelogramover Wi ] that these bijections are, in fact, homeomorphisms.

Note that these functorial homeomorphismsare already sufficient to concludethat the category C is slim [cf. the proof of slimness in Theorem 1.12 via Lemma1.8]. This completes the proof of assertion (i).

Next, let us observe that it follows from our assumption that i is stack-resolving that there exist finite Galois [cf. Proposition 2.2, (i)] Wi such that(W1 ) =W

2 , and, moreover,W

i isof complex type, withtrivialstack structure.

Thus, it follows, by applying Proposition A.4 [of the Appendix] to sufficiently smallpre-compact parallelogram neighborhoods ofWi[C], that, in the case of assertion(ii) (respectively, (iii)), the functorial homeomorphism

W1 [C]non

W2 [C]non


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istoral [cf. Remark 2.1.3] (respectively, andW1 isof torsion type) if and onlyif the same is true of(W2 ,

2|W2 ) (respectively, andW

2 ).

Proof. All of these assertions follow formally from Theorem 2.3, (ii), (iii). Here,

we note that in the genus 1 case, torality is [easily verified to be] equivalent tothe condition that the natural parameters [arising from the affine linear structure]extend to neighborhoods of the points at infinity of the canonical compactifica-tion. Once one has established torality, the property of being of torsion typeis completely determined by the affine linear structuredetermined by the naturalparameters.

Next, we define a somewhat different type of category of localizations, namely,a category of finite etale localizations [cf. the categories Loc(), Lock() of[Mzk7], 2]

FELoc(X, )

associated to an RC-Teichmuller pair (X, ). The objects of this category areRC-Teichmuller pairs (Y, ), where Y admits a finite etale morphism [of RC-Riemann orbisurfaces]Y X such that is the pull-back to Y of . Themorphisms

(Y1, 1) (Y

2,

2)

are finite etale morphisms[of RC-Riemann orbisurfaces] Y1 Y2 [which are not

necessarily overX!] with respect to which2 pulls back to 1 . Similarly, ifX

is of complex type, then one may define a similar category

FELocC(X, )

by taking the objects to be the objects of FELoc(X, ) and the morphisms tobe the holomorphic morphisms, i.e., the morphisms (Y1,

1) (Y

2,

2) of

FELoc(X, ) that induce holomorphicmaps from each connected component ofthe complexification ofY1 lying over some fixed connected component X0 of thecomplexification ofX to some connected component of the complexification ofY2lying over X0.

Definition 2.5. We shall refer to the RC-Teichmuller pair (X, ) as a core(respectively, C-core) if X is of real or complex type (respectively, of complextype), and, moreover, the object of FELoc(X, ) (respectively, FELocC(X

, ))determined by (X, ) forms a terminal object of FELoc(X, ) (respectively,FELocC(X

, )) [cf. [Mzk7], Definition 2.1, (ii); [Mzk7], Remark 2.1.1].

Corollary 2.6. (Extension of Equivalences of Categories)In the notationof Theorem 2.3, suppose further that, fori = 3, 4,(Xi,

i )is anRC-Teichmuller

pair, and that, for i = 1, 2, 3, 4, the morphism 1(Xi) i is the identity

morphism on1(Xi). [Thus, it is immediate thati is both totally ramified at


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infinity and stack-resolving.] Moreover, fori= 1, 2, let us assume that we havebeen given an equivalence of categories

: Loc1(X1 ,

1) Loc2(X

2 ,

2)

[where is P, R, or S], as well as a finite etale morphism of RC-Riemannorbisurfaces

(Xi, i ) (X

i+2,

i+2)

with respect to whichi+2 pulls back to i . Then:

(i) The morphism(Xi, i ) (X

i+2,

i+2) induces a naturalequivalence of

categories

Loc(Xi, i ) Loc(Xi+2,

i+2)(Xi ,i )

[where i = 1, 2; is P, R, or S; we omit the subscripted is]. Inparticular, we obtain anatural functor

Loc(Xi, i ) Loc

(Xi+2, i+2)

[i.e., by composing the natural functorLoc(Xi+2, i+2)(Xi ,i ) Loc

(Xi+2, i+2)

with the above equivalence].

(ii) Suppose that is R or S, and that, fori = 1, 2,(Xi+2, i+2)is either

acore or aC-core. Then there exists a1-commutative diagram

Loc(X1 , 1) Loc

(X1+2, 1+2)

Loc(X2 , 2) Loc

(X2+2, 2+2)

in which the vertical arrows areequivalences of categories; the horizontal ar-rows are the natural functors of (i); is uniquely determined, up to uniqueisomorphism, by the condition that the diagram1-commute.

(iii) Suppose that = P, and that, for i = 1, 2, there exists a cartesiancommutative diagram of finite etale morphisms of RC-orbifolds

Yi Yi+2

Xi Xi+2

in which the lower horizontal arrow arises from the morphism(Xi, i ) (X

i+2,

i+2)

given above; Yi , Yi+2 arepunctured RC-tori of complex type; Y

i+2 isonce-

punctured [which implies that Yi is of torsion type, and that (Yi , i |Yi ),


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Appendix: Quasiconformal Linear Algebra

In this Appendix, we review various well-known facts concerning the geom-etry and linear algebra of the euclidean plane that are relevant to the theory of

quasiconformal maps.Write

GL>02 (R), GL02 (R)

by mappingC a + ib

a b

b a

. In the following discussion, we shall often

identifyC with its image under this immersion and write C GL>02 (R). Thesubgroup C GL>02 (R) is normalized by the matrix

def=

0 1

1 0

conjugation by which induces complex conjugationonC.

IfM GL2(R), then we shall write

fM : C C

for the associated map from C to itself. Also, we shall often think of GL2(R)as acting on the upper half-planeH in the standard fashion, via linear fractional

transformations, i.e., ifz is the standard coordinate on H, then M =

a b

c d

GL2(R) acts via the transformation

zaz+ b

cz+ d

ifM GL>02 (R), and via the transformation

zaz+ b

cz+ d

ifM GL


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Proposition A.1. (The Dilatation of a Quasiconformal Map)

(i) The map

t t 0

0 1

[where t R1def= {s R | s 1}] determines a structure of one-dimensional

manifold with boundary [i.e., {1} is the boundary ofR1] on the double coset space

C\GL>02 (R)/C=SO(2)\SL2(R)/SO(2)

where = denotes the bijection induced by the natural inclusion SL2(R) GL>02 (R).

(ii) The map

M Dil(M) def=fM/z

fM/z

determines an isomorphism of manifolds with boundary

C\GL>02 (R)/C [0, 1)

which is given, relative to the bijection withR1 appearing in (i), by the map

t t 1

t + 1

[where t R1]. Alternatively, if we apply the bijection ofH with theopen unitdisk given byz iz+1

iz1 , then the subset [0, 1) of the open unit disk determines a

parametrization ofC\GL>02 (R)/C relative to which the map M Dil(M) is

given by the identity.

Proof. First, we consider assertion (i). It is immediate from the definitions that thenatural inclusion SL2(R) GL>02 (R) induces a homeomorphism of coset spacesC\GL>02 (R)/C

=SO(2)\SL2(R)/SO(2). Moreover, if we apply the homeomor-

phismSL2(R)/SO(2)

H given by lettingSL2(R) act on the pointi H, followedby the homeomorphism discussed in assertion (ii) ofH with the open unit disk, thenthe parametrization of assertion (i) is clearly mapped onto the inverval [0, 1), whichmay be identified with thequotient of the unit disk by the action of the unit circleS1 C. This completes the proof of assertion (i).

To verify assertion (ii), let us first observe that we may write fM(z) =c1z+c2z,wherec1 =a1+ib1,c2 = a2+ib2;a1, a2, b1, b2 R; Dil(M) = |c2|/|c1|. This descrip-tion offM, Dil(M) renders evident the fact that M Dil(M) depends only on theimage ofM inC\GL>02 (R)/C

. Now applying Dil() to the parametrization ofassertion (i) yields the function t1

t+1 [since 2(ta+ib) = (t+1)(a +ib)+(t1)(aib)].

This completes the proof of assertion (ii).


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Proposition A.2. (Dictionary between Function Theory and LinearAlgebra) LetM GL2(R). Then:

(i) The subgroupC GL2(R) is equal to the set of matricesGL2(R) thatcommute with the matrix determined by i C.

(ii) M lies inC (respectively, GL>02 (R); C ; GL02 (R); C

; GL


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or affine ortho-linear [i.e., either affine conformal linear or affine anti-conformallinear] is by applying the following result:

Proposition A.4. (Squares, Rectangles, and Parallelograms)LetB R2

be aconnected open subset; let

h: B R2

be a map that determines a homeomorphism of B onto a parallelogram ofR2.Then:

(i) Suppose thath maps pre-compactparallelograms inB to parallelogramsinR2. Thenh is [the restriction to B of a map R2 R2 that is]affine linear.

(ii) Suppose that h maps pre-compact rectangles in B to rectangles inR2.

Thenh is [the restriction to B of a mapR2

R2

that is] affine ortho-linear.

(iii) Suppose thath maps pre-compactsquares inB to squares inR2. Thenh is [the restriction to B of a mapR2 R2 that is] affine ortho-linear.

Proof. First, we observe that, by considering squares with edges parallel to thecoordinate axes contained inB and applying an appropriate affine ortho-linear mapto B , we may assume without loss of generality thatB itself is a square with edgesparallel to the coordinate axes centered at the originthat contains the points (a, b),wherea, b R,|a| = |b| = 1.

Next, we consider assertion (i). Define an edge-segment of a pre-compactparallelogramP Bto be an infinite set of the formK

K, whereKis the closure

ofP; K is the closure of another pre-compact parallelogram P; and P

P =.Consider the equivalence relation on edge-segments of P generated by the pre-equivalence relation that two edge-segments E1, E2 are pre-equivalent if theintersection E1

E2 is infinite. Then observe that the edges of a pre-compact

parallelogram P B are in natural bijective correspondence with the equivalenceclasses of edge-segments of P, and that, under this bijective correspondence, anedge ofP is given by the unionof edge-segments that belong to the correspondingequivalence class of edge-segments. The vertices of P may then be recovered as

the nonempty intersections of pairs of edges. Thus, the affine linear structureof B may be recovered by considering the combinatorics of intersections amongthe various edges of the pre-compact parallelograms ofB [i.e., in the notation ofDefinition A.3, (i), this combinatorial data encodes precisely the information thatif one takes w as the origin, then the sumof the points w+u, w+v is equal tow+u+v]. Since this description of the affine linear structure ofB is preservedbyh, we thus conclude that h isaffine linear, as desired.

Next, we consider assertion (ii). By composing h with an appropriate affineortho-linear map R2 R2, we may assume, without loss of generality, that hfixesthe points (0, 0) and (1, 1). Next, let us observe that the [rectangle-theoretic

analogue of the parallelogram-theoretic] topological description of vertices and


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edges given in the preceding paragraph [i.e., where P, P are assumed to berectangles] implies thathpreserves line segments. Since, moreover, a square may becharacterized as a rectanglePsuch that the line segments given by thediagonalsofthe rectangle are orthogonal[i.e., admit sub-line segments that appear as adjacentedges of some rectangle], we conclude that h preserves squares. Thus, to complete

the proof of assertion (ii), it suffices to verify assertion (iii).

Finally, we consider assertion (iii). By composing h with an appropriate affineortho-linear mapR2 R2, we may assume, without loss of generality, that hfixesthe points (0, 0) and (1, 1). Since [as one verifies immediately] there is preciselyone square S R2 that has the points (0, 0) as (1, 1) as opposite vertices, oneconcludes from the topological description of vertices and edges given above thathpreservesthis squareS. Thus, by possibly composing h with areflectionabout thediagonal ofS[which is manifestly an affine ortho-linear map], we may assume thathinduces theidentitymorphism on the set of edges ofS. Moreover, the topologicaldescription of the vertices and edges applied above also implies that h maps line

segments in B that are parallel to one of the two coordinate axes ofR2 [i.e., toone of the edges ofS] to line segments in R2 that are parallel to one of the twocoordinate axes ofR2. On the other hand, this last property implies [in light of thefact that h induces the identitymorphism on the set of edges ofS] that h may bewritten in the form

h((a, b)) = (f(a), g(b))

[where f, g are real-valued continuous functions on some open interval I R suchthat 0 IandIis preserved by multiplication by 1]. Since, moreover,h preservessquares, it follows that f=g .

Next, let us observe that for a, b Isuch that a, b = 0, ab I, the fact that hpreserves line segments [cf. the argument applied in the discussion of assertion (ii)]implies that f(ab)/f(a) is independent ofa, hence [since f(1) = 1] that f(ab) =f(a) f(b). Since f(0) = 0, we thus conclude that for all a, b Isuch that ab I,we have f(ab) = f(a) f(b). Thus, sinceR is a real analytic Lie group, we thusconclude [by Cartans theorem cf., e.g., [Serre], Chapter V, 9, Theorem 2] thatthere exists a positive real such that

f(x) = |x| (x/|x|)

for all nonzero x I. On the other hand, since, for sufficiently small > 0, the

function x f(x + ) f()

satisfies similar hypothesestof, we conclude that this function may be written, atleast for, say, x J I, where J is some open inverval of positive real numbers,in the form x c x

, for some c, >0. That is to say, we obtain the relation

(x + ) =c x

[for x J]. Thus, by, say, differentiating this relation with respect to x, taking thenatural logarithm, and then differentiating again with respect to x, we obtain that

( 1)x= (

1)(x + )


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[Mzk1] S. Mochizuki, Foundations of p-adic Teichmuller Theory, AMS/IP Studiesin Advanced Mathematics 11, American Mathematical Society/InternationalPress (1999).

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[Mzk8] S. Mochizuki, The Geometry of Anabelioids,Publ. Res. Inst. Math. Sci. 40

(2004), pp. 819-881.[Mzk9] S. Mochizuki, Galois Sections in Absolute Anabelian Geometry,Nagoya Math.

J. 179 (2005), pp. 17-45.

[Mzk10] S. Mochizuki, Categories of log schemes with Archimedean Structures,J. Math.Kyoto Univ. 44 (2004), pp. 891-909.

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Research Institute for Mathematical Sciences

Kyoto University

Kyoto 606-8502, Japan

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