nodally nondegenerate outer representations

7/27/2019 Nodally Nondegenerate Outer Representations

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ON THE COMBINATORIAL ANABELIAN GEOMETRYOF NODALLY NONDEGENERATE OUTER

REPRESENTATIONS

YUICHIRO HOSHI AND SHINICHI MOCHIZUKI

DECEMBER 2011

Abstract. Let be a nonempty set of prime numbers. In thepresent paper, we continue the study, initiated in a previous paperby the second author, of the combinatorial anabelian geometryofsemi-graphs of anabelioids of pro- PSC-type, i.e., roughly speak-ing, semi-graphs of anabelioids associated to pointed stable curves.Our first main resultis a partial generalization of one of the maincombinatorial anabelian results of this previous paper to the caseofnodally nondegenerate outer representations, i.e., roughly speak-ing, a sort of abstract combinatorial group-theoretic generalizationof the scheme-theoretic notion of a family of pointed stable curvesover the spectrum of a discrete valuation ring. We then apply thisresult to obtain a generalization, to the case ofproperhyperboliccurves, of a certain injectivityresult, obtained in another paper by

the second author, concerning outer automorphisms of the pro-fundamental group of a configuration space associated to a hyper-bolic curve, as the dimension of this configuration space is loweredfrom two to one. This injectivity allows one to generalize a certainwell-known injectivity theorem of Matsumototo the case ofproperhyperbolic curves.

Contents

Introduction 20. Notations and Conventions 81. Some complements concerning semi-graphs of anabelioids of

PSC-type 102. Nodally nondegenerate outer representations 223. Group-theoretic aspects of the geometry of the underlying

semi-graphs 36

2000 Mathematics Subject Classification. Primary 14H30; Secondary 14H10.Key words and phrases. hyperbolic curve, outer Galois representation, injec-

tivity, semi-graph of anabelioids, nodally nondegenerate, combinatorial anabeliangeometry, combinatorial cuspidalization.

The first author was supported by Grant-in-Aid for Young Scientists (B)

(20740010).1


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2 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI

4. A combinatorial anabelian theorem for nodallynondegenerate outer representations 51

5. Injectivity via nodally nondegenerate degenerations 546. Consequences of injectivity 61References 68

Introduction

Let be a nonempty set of prime numbers. In the present paper,we continue the study, initiated in [Mzk4] by the second author, ofthe combinatorial anabelian geometryofsemi-graphs of anabelioids ofpro- PSC-type, i.e., roughly speaking, semi-graphs of anabelioids as-sociated to pointed stable curves. In particular, it was shown in [Mzk4](cf. [Mzk4], Corollary 2.7, (iii)) that in the case of a semi-graph of an-abelioids of pro- PSC-type that arises from a stable log curveover alogpoint(i.e., the spectrum of an algebraically closed fieldk of characteris-ticp equipped with the log structure determined by the morphismof monoids N 1 0 k), the semi-graph of anabelioids in questionmay be reconstructed group-theoretically from the outer actionof the pro- logarithmic fundamental group of the log point (which

is noncanonically isomorphic to the maximal pro- quotient

Z of

Z)

on the pro- fundamental group of the semi-graph of anabelioids. Asdiscussed in the introduction to [Mzk4], this result may be regardedas a substantial refinement of the pro-l criterion of Takayuki Oda fora proper hyperbolic curve over a discretely valued field to have goodreduction (i.e., a special fiber whose associated semi-graph consists ofa single vertex and no edges). We shall refer to an outer action ofthe type just described as an outer representation of IPSC-type (cf.Definition 2.4, (i)).

In the present paper, the theory of [Mzk4] is generalized to the caseof nodally nondegenerate outer representations, or outer representa-tions of NN-type, for short (cf. Definition 2.4, (iii)). Indeed, ourfirst

main result(cf. Corollary 4.2; Remark 4.2.1) is the following partialgeneralization of [Mzk4], Corollary 2.7, (iii).

Theorem A (Graphicity of certain group-theoretically cuspi-dal isomorphisms). Let be a nonempty set of prime numbers, GandH semi-graphs of anabelioids of pro- PSC-type (cf. [Mzk4], Def-inition 1.1, (i)),G (respectively,H) the pro- fundamental group ofG (respectively, H), : G

H an isomorphism of profinite groups,

I andJ profinite groups, I: I Aut(G) andJ: J Aut(H) con-tinuous homomorphisms, and : I

J an isomorphism of profinite

groups. Suppose that the following three conditions are satisfied:


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NODALLY NONDEGENERATE OUTER REPRESENTATIONS 3

(i) The diagram

I Out(G)

Out()J Out(H)

where the right-hand vertical arrow is the homomorphisminduced by ; the upper and lower horizontal arrows are thehomomorphisms determined byI andJ, respectively com-mutes.

(ii) I, J areof NN-type (cf. Definition 2.4, (iii)).

(iii) Cusp(G)=, and the isomorphism isgroup-theoreticallycuspidal(i.e., roughly speaking, preserves cuspidal inertia groups cf. [Mzk4], Definition 1.4, (iv)).

Then the isomorphism isgraphic (i.e., roughly speaking, is compat-ible with the respective semi-graph structures cf. [Mzk4], Definition1.4, (i)).

The notion of an outer representation of NN-type may be regardedas a natural outgrowth of the philosophy pursued in [Mzk4] of reducing(various aspects of) the classical pro- scheme-theoretic arithmetic

geometryof stable curves over a discrete valuation ring whose residuecharacteristic is not contained in to a matter ofcombinatorics.Ideally, one would like to reduce the entire profinite classical scheme-theoretic arithmetic geometry of hyperbolic curves over number fieldsor p-adic local fields to a matter of combinatorics, but since this taskappears to be too formidable at the time of writing, we concentrate onthe pro-prime-to-paspects of stable log curves over a log point. On theother hand, whereas the outer representations of IPSC-type studiedin [Mzk4] literally arise from (log) scheme theory (i.e., a stable logcurve over a log point), the outer representations of NN-type studiedin the present paper are defined in purely combinatorial terms, without

reference to any scheme-theoretic family of stable log curves. If onethinks of a stable log curve as a sort of rational point of the modulistack of stable curves, then this point of view may be thought of asa sort ofabandonment of the point of view implicit in the so-calledSection Conjecture: that is to say, instead of concerning oneself withthe issue ofprecisely whichgroup-theoretic objects arise from a scheme-theoretic rational point (as is the case with the Section Conjecture),

one takes the definition of group-theoretic objects viapurelycombinatorial/group-theoreticconditionsi.e., group-theoretic objects which do not necessarily

arise from scheme theory as the starting point of


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ones research, and one regards as the goal of ones re-search the study of the intrinsic combinatorial ge-

ometry of such group-theoretic objects (i.e., withoutregard to the issue of the extent to which these objectsarise from scheme theory).

This point of view may be seen throughout the development of thetheory of the present paper, as well as in the theory of [Mzk6].

On the other hand, from a more concrete point of view, the theoryof the present paper was motivated by the goal of generalizing the in-jectivityportion of [Mzk7], Theorem A, (i), to proper hyperbolic curvesin the case of the homomorphism induced by the projection from two-dimensional to one-dimensional configuration spaces (cf. Theorem B

below). The main injectivity result thatwasproven in [Mzk7] (namely,[Mzk7], Corollary 2.3) was obtained by applying the combinatorial an-abelian resultgiven in [Mzk4], Corollary 2.7, (iii). On the other hand,this result of [Mzk4] is insufficient in the case of proper hyperboliccurves. To see why this is so, we begin by recalling that this result of[Mzk4] is applied in [Mzk7] (cf. the discussion ofcanonical splittingsin the Introduction to [Mzk7]) to study the degenerations of familiesof hyperbolic curves that arise when

(a) a moving point on an affine hyperbolic curve collides with acusp.

On the other hand, since proper hyperbolic curves haveno cusps, in or-der to apply the techniques for proving injectivity involving canon-ical splittings developed in [Mzk7], it is necessary to consider thedegenerations of families of hyperbolic curves that arise when

(b) a moving point on a (not necessarily affine) degenerate hyper-bolic curve (i.e., a stable curve) collides with a node.

Since the local pro- fundamental group in a neighborhood of a cusp ora node i.e., the profinite group that corresponds to the fundamentalgroup of the base space of the degenerating family of hyperbolic curvesunder consideration is isomorphic (in both the cuspidal and nodal

cases!) to the (same!) profinite groupZ, one might at first glancethink that the situation of (b) may also be analyzed via the results of[Mzk7]. Put another way, both (a) and (b) involve a continuous action

of a profinite group isomorphic toZ on a semi-graph of anabelioidsof pro- PSC-type. On the other hand, closer inspection reveals thatthere is a fundamental intrinsic differencebetween the situations of(a) and (b). Indeed, in the situation of (a), we apply the reconstructionalgorithms developed in [Mzk4], which depend in an essential way on acertainpositivity, namely, thepositivity of the period matrix which

implies, in particular, thenondegeneracyof this period matrix of the


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Jacobians of the various coverings of the degenerating family of curvesunder consideration (cf. the proof of [Mzk4], Proposition 2.6). By

contrast, one verifies easily that

the symmetry in a neighborhood of a node inducedbyswitching the two branches of the node impliesthat an analogous positivity of the period matrix ofthe Jacobians of the various coverings of the degenerat-ing family of curves under consideration can only hold inthe sitation of (b) if this positivity satisfies the prop-erty of being invariant with respect to multiplication by1 which is absurd!

In particular, one concludes that the situation of (b) canneverbe ab-stractly group-theoretically isomorphic to the situation of (a). Thiswas what led the second author to seek, in cooperation with the firstauthor, a (partial)generalization(cf. Theorem A) of [Mzk4], Corollary2.7, (iii), to the case of arbitrary nodally nondegenerate outer represen-tations(which includes the situation of (b) cf. Proposition 2.14, asit is applied in the proof of Corollary 5.3).

In passing, we note that the sense in which Theorem A is only apartialgeneralization (cf. Remark 4.2.1) of [Mzk4], Corollary 2.7, (iii),is interesting in light of the above discussion ofpositivity. Indeed, in thecase of [Mzk4], Corollary 2.7, (iii), it isnot necessaryto assume that thesemi-graph of anabelioids of pro- PSC-type under consideration hasany cusps. On the other hand, in the case of Theorem A, it is necessaryto assume that the semi-graph of anabelioids of pro- PSC-type underconsideration hasat least one cusp (cf. condition (iii) of Theorem A).That is to say, this state of affairs suggests that perhaps there is somesort ofgeneral principle underlying these results which, at the timeof writing, the authors have yet to succeed in making explicit thatrequires the existence ofat least one cusp, whether that cusp lie inthe baseof the degenerating family of curves under consideration (cf.

(a); [Mzk4], Corollary 2.7, (iii)) or in the fibersof this degeneratingfamily (cf. (b); condition (iii) of Theorem A).The content of the various sections of the present paper may be

summarized as follows. In 1, we review various well-known aspectsof thecombinatorial group-theoretic geometryof semi-graphs of anabe-lioids of pro- PSC-type i.e., without considering any continuousaction of a profinite group on the semi-graph of anabelioids under con-sideration. In2, we define and develop the basic theory surroundingnodally nondegenerate outer representations. In 3, we discuss vari-ous analogues of the combinatorial group-theoretic geometry reviewedin 1 in the case of nodally nondegenerate outer representations. In

4, we observe that the theory developed in1,2, and 3 is sufficient


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to prove the analogue discussed above (i.e., Theorem A) of the com-binatorial anabelian result given in [Mzk4], Corollary 2.7, (iii), in the

case of nodally nondegenerate outer representations. In 5, we applythis result (i.e., Theorem A) to generalize (cf. the above discussion)[Mzk7], Corollary 2.3, to the case of not necessarily affine curves (cf.Corollary 5.3). Finally, in 6, we discuss various consequences of theinjectivity result proven in5. The first of these is the following partialgeneralization (cf. Theorem 6.1) of [Mzk7], Theorem A.

Theorem B (Partial profinite combinatorial cuspidalization).Let be a set of prime numbers which is either of cardinality oneor equal to the set of all prime numbers, n a positive integer,X a hyperbolic curve of type (g, r) over an algebraically closed field

of characteristic , Xn the n-thconfiguration space of X (i.e.,roughly speaking, the complement of the diagonals in the product ofncopies ofX cf. [MzTa], Definition 2.1, (i)), n the maximal pro-quotient of the fundamental group ofXn, andOut

FC(n) Out(n)the subgroup of the group Out(n) consisting of the outomorphisms(cf. the discussion entitled Topological groups in0) ofn which areFC-admissible (i.e., roughly speaking, preserve fiber subgroups and

cuspidal inertia groups cf. [Mzk7], Definition 1.1, (ii)). Setn0def= 2

if X is affine, i.e., r 1; n0def= 3 if X is proper, i.e., r = 0 (cf.

[Mzk7], Theorem A). Then the natural homomorphism

OutFC(n+1)OutFC(n)

induced by the projectionXn+1 Xn obtained by forgetting the(n+1)-st factor isinjective ifn 1 andbijective ifn n0+ 1. Moreover,the image of the natural inclusion

Sn Out(n)

where we writeSn for the symmetric group onn letters obtainedby permuting the various factors of the configuration spaceXn is con-tained in thecentralizer ZOut(n)(Out

FC(n)).

In Corollary 6.6, we also give a discrete analogue of the profiniteresult constituted by Theorem B.

In passing, we observe that the injectivity portion of the pro-l caseof Theorem B may be derived from the Lie-theoretic versionof Theo-rem B that was obtained (in the mid-1990s!) by Naotake Takao (cf.[Tk], Corollary 2.7). In this context, we note that the point of viewof [Tk] differs quite substantially from the point of view of the presentpaper and is motivated by the goal of completing the proof of a cer-tain conjecture of Takayuki Odaconcerning pro-l outer Galois actionsassociated to various moduli stacks of stable curves. Nevertheless, this

point of view of [Tk] is interesting in light of the point of view discussed


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above to the effect that the content of [Mzk4] and hence also of The-orem A above may be thought of as a sort of substantial refinement

of Odas good reduction criterion.Theorem B allows one to obtain the following generalization (cf.

Corollaries 6.2; 6.3, (i)) to not necessarily affine hyperbolic curves ofa well-known injectivity result of Matsumoto (cf. [Mts], Theorems 2.1,2.2).

Theorem C(Kernels of outer representations arising from hy-perbolic curves). Let be a set of prime numbers which is eitherofcardinality one orequal to the set of all prime numbers, X ahyperbolic curve over a perfect fieldk such that every element of isinvertible ink, k an algebraic closure ofk, n a positive integer, Xnthen-th configuration space ofX, Gk def= Gal(k/k), Xn the maximalpro- quotient of the fundamental group ofXnk k, and P1k\{0,1,}the maximal pro-quotient of the fundamental group ofP1

k\ {0, 1, }.

Then the following hold:

(i) The kernel of the natural outer representation

Xn/k : GkOut(Xn)

isindependent ofn and contained in the kernel of the nat-ural outer representation

P1k\{0,1,}/k

: GkOut(P1k\{0,1,}) .

(ii) Suppose that is the set of all prime numbers. (Thus, kis necessarily of characteristic zero.) WriteQ for the algebraic

closure ofQ determined byk andGQdef= Gal(Q/Q). Then the

kernel of the homomorphismXn/k is contained in the kernelof the outer homomorphism

GkGQ

determined by the natural inclusionQ k.

In particular, ifk is a number field or p-adic local field (cf. thediscussion entitled Numbers in 0), and is the set of all primenumbers, then the outer representation

X/k : GkOut(1(Xkk))

determined by the natural exact sequence

11(Xkk)1(X)Gk1

isinjective.


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Finally, we remark that in [Bg], a result that corresponds to a certainspecial case of Theorem C, (i), is asserted (cf. [Bg], Theorem 2.5). At

the time of writing, the authors of the present paper were not able tofollow the proof of this result given in [Bg]. Nevertheless, in a sequelto the present paper, we hope to discuss in more detail the relationshipbetween the theory of the present paper and the interesting geometricideas of [Bg] concerning the issue ofcanonical liftings of cycles on aRiemann surface.

0. Notations and Conventions

Sets: IfSis a set, then we shall denote by 2S the power setofS andbyS the cardinalityofS.

Numbers: The notation N will be used to denote the set or (additive)monoid of nonnegative rational integers. The notation Z will be usedto denote the set, group, or ring of rational integers. The notation Qwill be used to denote the set, group, or field of rational numbers. The

notationZwill be used to denote the profinite completion ofZ. Ifp isa prime number, then the notation Zp (respectively, Qp) will be usedto denote the p-adic completion ofZ (respectively, Q).

A finite extension field ofQ will be referred to as a number field. Ifpis a prime number, then a finite extension field ofQp will be referredto as a p-adic local field.

Monoids: We shall write Mgp for the groupificationof a monoid M.

Topological groups: Let G be a topological group and H G aclosed subgroup ofG. Then we shall denote by ZG(H) (respectively,NG(H); respectively, CG(H)) the centralizer (respectively, normalizer;respectively, commensurator) ofH in G, i.e.,

ZG(H)def= { g G | ghg1 =h for any h H} ,

NG(H)def= { g G | g H g1 =H} ,

CG(H)def= { g G | HgHg1 is of finite index in H and gHg1 } ;

we shall refer to Z(G) def= ZG(G) as the centerofG. It is immediatefrom the definitions that

ZG(H) NG(H) CG(H) ; HNG(H) .

We shall say that the subgroup H is commensurably terminal in G ifH=CG(H).

We shall say that a profinite group G is slimifZG(H) ={1}for anyopen subgroup HofG.

Let be a set of prime numbers, l a prime number, andGa profinitegroup. Then we shall write G for the maximal pro- quotientofG

and G(l)def

= G{l}.


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We shall write Gab for theabelianizationof a profinite group G, i.e.,the quotient ofGby the closure of the commutator subgroup ofG.

If G is a profinite group, then we shall denote the group of auto-morphisms ofG by Aut(G) and the group of inner automorphisms ofG by Inn(G) Aut(G). Conjugation by elements ofG determines asurjection of groups G Inn(G). Thus, we have a homomorphismof groups G Aut(G) whose image is Inn(G) Aut(G). We shalldenote by Out(G) the quotient of Aut(G) by the normal subgroupInn(G) Aut(G) and refer to an element of Out(G) as an outomor-phism of G. In particular, if G is center-free, then the natural ho-momorphism G Inn(G) is an isomorphism; thus, we have an exactsequence of groups

1G Aut(G)Out(G)1 .If, moreover, G is topologically finitely generated, then one verifies eas-ily that the topology ofG admits a basis ofcharacteristic open sub-groups, which thus induces a profinite topologyon the groups Aut(G)and Out(G) with respect to which the above exact sequence determinesan exact sequence ofprofinite groups. If : JOut(G) is a continuoushomomorphism, then we shall denote by

Gout J

the profinite group obtained by pulling back the above exact sequence

of profinite groups via . Thus, we have a natural exact sequenceofprofinite groups

1G Gout JJ1 .

One verifies easily (cf. [Hsh], Lemma 4.10) that if an automorphism

ofGout Jpreserves the subgroup G G

out Jand induces the identity

automorphismsof the subquotients G and J, then the automorphism

is the identity automorphismofGout J.

If M and N are topological modules, then we shall refer to a ho-momorphism of topological modules : M N as a split injection if

there exists a homomorphism of topological modules : NM suchthat : MMis the identity automorphism ofM.

Semi-graphs: Let be a connected semi-graph. Then we shall saythat is untangled if every closed edge of abuts to two distinctvertices.

Log stacks: Let Xlog and Ylog be log stacks whose underlying (alge-braic) stacks we denote by X and Y, respectively; MX and MY therespective sheaves of monoids on X and Ydefining the log structuresofXlog and Ylog ; flog : Xlog Ylog a morphism of log stacks. Thenwe shall refer to the quotient ofMX by the image of the morphism

f1MY MX induced by flog as the relative characteristic sheafof


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flog; we shall refer to the relative characteristic sheaf of the morphismXlog X (where, by abuse of notation, we write Xfor the log stack

obtained by equipping Xwith the trivial log structure) induced by thenatural inclusion OX MXas the characteristic sheafofX

log.

Curves: We shall use the terms hyperbolic curve, cusp, stable logcurve, smooth log curve, and tripod as they are defined in [Mzk4],0; [Hsh],0. If (g, r) is a pair of natural numbers such that 2g 2+r >0, then we shall denote by Mg,r the moduli stack of r-pointed stablecurves of genus g over Z whose r marked points are equipped withan ordering, Mg,r Mg,r the open substack of Mg,r parametrizing

smooth curves, and Mlog

g,r the log stack obtained by equipping Mg,rwith the log structure associated to the divisor with normal crossings

Mg,r\ Mg,r Mg,r.Let n be a positive integer and Xlog a stable log curve of type (g, r)

over a log scheme Slog. Then we shall refer to the log scheme obtained

by pulling back the (1-)morphism Mlog

g,r+n Mlog

g,r given by forgetting

the last n points via the classifying (1-)morphism Slog Mlog

g,r ofXlog

as the n-th log configuration spaceofXlog.

1.Some complements concerning semi-graphs of

anabelioids of PSC-type

In this section, we give some complements to the theory of semi-graphs of anabelioids of PSC-typedeveloped in [Mzk4].

A basic reference for the theory ofsemi-graphs of anabelioids of PSC-type is [Mzk4]. We shall use the terms semi-graph of anabelioids ofPSC-type, PSC-fundamental group of a semi-graph of anabelioids ofPSC-type, finite etale covering of semi-graphs of anabelioids of PSC-type, vertex, edge, cusp, node, verticial subgroup, edge-likesubgroup, nodal subgroup, cuspidal subgroup, and sturdy as theyare defined in [Mzk4], Definition 1.1. Also, we shall refer to the PSC-fundamental group of a semi-graph of anabelioids of PSC-type simplyas the fundamental group (of the semi-graph of anabelioids of PSC-type). That is to say, we shall refer to the maximal pro- quotientof the fundamental group of a semi-graph of anabelioids of PSC-type(as a semi-graph of anabelioids!) as the fundamental group of thesemi-graph of anabelioids of PSC-type. In this section, let be anonempty set of prime numbers, Ga semi-graph of anabelioids of pro- PSC-type, and Gthe underlying semi-graph ofG. (In particular, Gis afinitesemi-graph.) Also, let us fix a universal coveringG Gwithunderlying projective system of semi-graphs

G (i.e., the projective sys-

tem consisting of the underlying graphs G of the connected finite etale


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subcoveringsG ofG G) and denote by G the (pro-) fundamentalgroup ofG.

Definition 1.1.

(i) We shall denote by Vert(G) (respectively, Cusp(G); Node(G))the set of the vertices (respectively, cusps; nodes) ofG.

(ii) We shall write

Vert( G) def= lim Vert(G) ;Cusp( G) def= lim Cusp(G) ;Node(G)

def= lim

Node(G)

where the projective limits are over all connected finite etale

subcoveringsG Gof the fixed universal coveringG G.(iii) We shall write

VCN(G)def= Vert(G) Cusp(G) Node(G) ;

Edge(G)def= Cusp(G) Node(G) ;

VCN(G) def= Vert( G) Cusp(G) Node( G) ;Edge(

G)

def= Cusp(

G) Node(

G) .

(iv) Let

V: Edge(G)2Vert(G)

(respectively, C: Vert(G)2Cusp(G);

N: Vert(G)2Node(G);

E: Vert(G)2Edge(G))

be the map obtained by sending e Edge(G) (respectively,v Vert(G); v Vert(G); v Vert(G)) to the set of vertices

(respectively, cusps; nodes; edges) ofG to which e abuts (re-spectively, which abut to v ; which abut to v; which abut to v).Also, we shall write

V: Edge( G)2Vert( eG)(respectively, C: Vert( G)2Cusp( eG);

N: Vert(G)2Node( eG);E: Vert( G)2Edge(eG))

for the map induced by the various Vs (respectively, Cs;Ns;Es) involved.


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(v) Let

z VCN( G). Suppose that G G is a connected finite

etale subcovering ofG G. Then we shall denote byz(G)

VCN(G) the image ofz in VCN(G).(vi) Let v Vert(G),v Vert( G) be such thatv(G) =v. Then it is

easily verified that there exists a uniqueverticial subgroup evof G associated to the vertex v such that for every connected

finite etale subcovering G G ofG G, it holds that thesubgroup ev G G where we write G G for theopen subgroup corresponding to G G is a verticial sub-group of G associated tov(G) Vert(G); thus, the element

v determines a particularverticial subgroup of G associatedto the vertex v. We shall refer to this verticial subgroup of Gdetermined byv as the verticial subgroup ofG associated tovand denote it by ev.

In a similar vein, fore Cusp( G) (respectively,e Node( G);e Edge(G)), by a similar argument to the argument just ap-plied to define the verticial subgroup of G associated tov, theelementedetermines a particularcuspidal (respectively, nodal;edge-like) subgroup of G associated to the cusp (respectively,node; edge)e(G) ofG. We shall refer to this cuspidal (respec-tively, nodal; edge-like) subgroup of G asthe cuspidal(respec-tively, nodal; edge-like) subgroup ofG associated to

e and de-

note it by ee.(vii) Let nbe a natural number, and v, w Vert(G). Then we shall

write (v, w) n if the following conditions are satisfied:

(1) Ifn= 0, then v=w.

(2) Ifn 1, then there exist nnodes e1, , enNode(G) ofGandn 1 verticesv1, , vn1 Vert(G) ofGsuch that,for 1 i n, it holds that V(ei) ={vi1, vi} where we

write v0def= v and vn

def= w.

Moreover, we shall write(v, w) =nif(v, w) nand(v, w)

n1. If(v, w) =n, then we shall say that thedistance betweenv andw is equal to n.

(viii) Letv,w Vert( G). Then we shall write(v,w) def= sup

G{(v(G),w(G))} N {}

whereG ranges over the connected finite etale subcoveringsG GofG G. If(v,w) =n N {}, then we shall saythat the distance between

v and

w is equal to n.


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Remark 1.1.1. Let

z VCN( G), and z def=

z(G) VCN(G). Then

whereasz completely determinesthe subgroup ez, zonly determinesthe G-conjugacy class of the subgroup ez.Definition 1.2. We shall say that the semi-graph of anabelioids ofpro- PSC-type G is untangled if the underlying semi-graph of G isuntangled (cf. the discussion entitled Semi-graphs in0).

Remark 1.2.1.

(i) It follows from a similar argument to the argument in the dis-cussion entitled Curves in [Mzk6],0, that there exists a con-nected finite etale covering G G ofGsuch that G is untan-

gled.

(ii) It is easily verified that ifG isuntangled, then every finite etalecovering G GofG is untangled.

(iii) It follows from (i) and (ii) that for everye Node( G), we haveV(e) = 2.

Definition 1.3.

(i) We shall denote by ab/edgeG the quotient of

abG by the closed

subgroup generated by the images in abG of the edge-like sub-

groups of G.

(ii) Letv Vert(G). Then we shall denote by ab/edgeev the quotient

of the abelianization abev by the closed subgroup generated by

the images in abev of eeev wheree ranges over elements

ofE(v). (Here, we note that it follows from [Mzk4], Proposition1.5, (i), that fore Edge( G), it holds thate E(v) if and onlyif ee ev.)

(iii) Let v Vert(G). Then observe that conjugation by elementsof G determines natural isomorphisms between the various

ab/edge

ev , asv ranges over the elements of Vert( G) such thatv =v(G). We shall denote the resulting profinite group byab/edgev .

Lemma 1.4(Verticial decompositions inside ab/edge-quotients).The natural homomorphism

vVert(G)

ab/edgev ab/edgeG

is asplit injection(cf. the discussion entitled Topological groups in

0) whose image is a free

Z-module of finite rank (cf. [Mzk4], Remark

1.1.4).


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Proof. It follows immediately from the well-known structure of themaximal pro- quotient of the fundamental group of a smooth curve

over an algebraically closed field of characteristic that the quotientby the image of the natural homomorphism in question is a freeZ-module. Therefore, to verify Lemma 1.4, it suffices to verify that thenatural homomorphism in question is injective. Now suppose that wehave been given, for eachv Vert(G), a connected finite etale coveringHv Gv of the anabelioid Gv corresponding to v Vert(G) which

arises from an open subgroup of ab/edgev . Then to verify the desiredinjectivity, it suffices to verify that there exists a connected finite etale

covering F G ofGwhich arises from an open subgroup of ab/edgeGsuch that, for each v Vert(G), any connected component of the re-

striction ofF G to Gv is isomorphic to Hv over Gv. To this end,for v Vert(G), write (ab/edgev ) Av for the Galois group of theconnected finite etale covering Hv Gv,

A=vdef=

wVert(G)\{v}

Aw A def

=

wVert(G)

Aw,

Fv Gv for the (not necessarily connected) finite etale covering ofGvobtained as the disjoint union of copies ofHv indexed by the elementsof A=v, and, for e E(v), Fv|Ge Ge for the finite etale coveringof Ge obtained as the restriction of Fv Gv to the anabelioid Gecorresponding to e N(v). Then the natural action ofA

=v on A

=vand the tautological action ofAv on Hv over Gv naturally determinean action of A on Fv over Gv. Moreover, one verifies immediatelythat this A-action determines a structure ofA-torsoron the coveringFv Gv. Therefore, by gluing the various Fv (for vvert(G)) by A-equivariantisomorphisms between the variousFv|Ge (fore Node(G)),we obtain a finite etale covering F G, any connected component ofwhich satisfies the desired condition. This completes the proof of theinjectivityof the homomorphism in question.

Remark 1.4.1. The following two assertions follow immediately from

Lemma 1.4.

(i) If v G is a verticial subgroup of G, then the natural

homomorphism ab/edgev ab/edgeG is injective.

(ii) Ifv1,v2 Vert(G) aredistinct, then for any verticial subgroupsv1 , v2 Gassociated tov1,v2, the intersection of the images

of v1 and v2 in ab/edgeG istrivial.

Lemma 1.5 (Intersections of edge-like subgroups). Let

e1,

e2

Edge(G). Then the following conditions areequivalent:


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(i)e1 =e2.(ii)

ee1

ee2 ={1}.

In particular, ifee1 ee2 ={1}, thenee1 = ee2.

Proof. The implication(i) =(ii)

is immediate; thus, to verify Lemma 1.5, it suffices to prove the impli-cation

(ii) =(i) .

To this end, let us assume that ee1 ee2 ={1}. Since G is torsion-free (cf. [Mzk4], Remark 1.1.3), by projecting to the maximal pro-l

quotients, for some l , of suitable open subgroups of the variouspro- groups involved, we may assume without loss of generality that ={l}. In particular, since ee1 and ee2 are isomorphic to Zl, we mayassume without loss of generality that ee1 ee2 isopenin ee1 and ee2 .Thus, by replacingGby a connected finite etale covering ofG, we mayassume without loss of generality that ee1 = ee2. Then condition (i)follows from [Mzk4], Proposition 1.2, (i).

Lemma 1.6 (Group-theoretic characterization of subgroups ofedge-like subgroups). LetJG be anontrivial procyclicclosedsubgroup ofG. Then the following conditions areequivalent:

(i) Jis contained in a(n) necessarilyunique (cf. Lemma 1.5)edge-like subgroup.

(ii) There exists a connected finite etale coveringG GofGsuchthat for any connected finite etale coveringG G ofG thatfactors throughG G, the image of the composite

J G G ab/edgeG

istrivial.

Proof. The implication

(i) =(ii)is immediate; thus, to verify Lemma 1.6, it suffices to prove the impli-cation

(ii) =(i) .

To this end, let us assume that condition (ii) holds. Now since edge-likesubgroups are commensurably terminal (cf. [Mzk4], Proposition 1.2,(ii)), it suffices to verify condition (i) under the further assumption thatG = G (cf. theuniqueness portion of condition (i)). Moreover, sinceG is torsion-free(cf. [Mzk4], Remark 1.1.3), to verify condition (i),we may assume without loss of generality (cf. theuniquenessportion

of condition (i)) by projecting to the maximal pro- l quotients, for


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some l , of suitable open subgroups of the various pro- groupsinvolved that ={l}.

IfH G is an open subgroup of G , then let us denote by GHG the connected finite etale covering ofGcorresponding to the opensubgroup HG (i.e., GH=HG). Now we claimthat

() for any normal open subgroup N G of G , thereexists an edge ofGJNat which the connected finite etalecovering GN GJN istotally ramified, i.e., there existsan edgee Edge(GJN) such that the composite of nat-ural homomorphisms

eGJN=J N (J N)/N

is surjective.Indeed, since J is procyclic, it follows that (J N)/N iscyclic; in par-ticular, we obtain a natural surjection abGJN (J N)/N. Moreover,since (JN)/Nis generated by the image ofJ, it follows from condition(ii) that the composite of natural homomorphisms

eEdge(GJN)

e abGJN

(J N)/N

issurjective. Therefore, it follows from the fact that (JN)/Nis acyclicl-group that there exists an edge e ofGJN such that the composite of

the natural homomorphisms e GJN = J N (J N)/N issurjective, as desired. This completes the proof of ().

IfN G is a normal open subgroup, then let us denote by EN Edge(GN) the subset of Edge(GN) consisting of edges which are fixedby the natural action ofJonGN. Then it follows from () that for anynormal open subgroup N G , it holds that EN is nonempty; thus,sinceEN isfinite, the projective limit limNEN whereNranges overthe normal open subgroups of G is nonempty. Note that since

NN where N ranges over the normal open subgroups of G is {1}, it follows that each element of the projective limit limNEN

naturally determines an element of Edge( G). Lete Edge(G) be anelement of Edge( G) determined by an element of limNEN = . Thenit follows from the various definitions involved that J ee. Thiscompletes the proof of the implication

(ii) =(i) .

Remark 1.6.1. When G =Gand Node(G) = , Lemma 1.6 followsimmediately from [Naka], Lemma 2.1.4.


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Lemma 1.7(Intersections of verticial and edge-like subgroups).

Letv Vert( G), ande Edge(G). Then the following conditions areequivalent:(i)e E(v).

(ii) ev ee={1}.

In particular, ifev ee={1}, thenee ev.

Proof. The implication(i) =(ii)

is immediate; thus, to verify Lemma 1.7, it suffices to prove the impli-cation

(ii) =(i) .To this end, let us assume that ev ee = {1}. Since G is torsion-free (cf. [Mzk4], Remark 1.1.3), by projecting to the maximal pro-lquotients, for some l , of suitable open subgroups of the variouspro- groups involved, we may assume without loss of generality that = {l}. In particular, since ee is isomorphic to Zl, we may assumewithout loss of generality that eveeis openin ee. Thus, by replacingGby a connected finite etale covering ofG, we may assume without lossof generality that ee ev. Then condition (i) follows from [Mzk4],Proposition 1.5, (i) (cf. also [Mzk4], Proposition 1.2, (i)).

Lemma 1.8 (Nonexistence of loops). Letv1,v2 Vert( G) be suchthatv1=v2. Then

(N(v1) N(v2)) 1 ;{ v Vert(G) | (v, v1) =(v, v2) = 1 } 1 .

Proof. If the cardinality of either of the sets equipped with a super-script is 2, then the offending edges or vertices give rise to aloop ofG, i.e., a projective system of loops(that map isomorphicallyto one another) in the various semi-graphs that appear in the projec-

tive system

G. On the other hand, since

G is a universal coveringof

G, one verifies immediately that no such projective system of loopsexists. Thus, we obtain a contradiction. This completes the proof ofLemma 1.8.

Remark 1.8.1.

(i) Letv Vert( G). Recall that ife N(v), then the inclusionee ev is strict (i.e., ee = ev). In particular, it followsimmediately that either N(v) = or N(v) 2.

(ii) Let

v1,

v2 Vert(

G). Then, in light of (i), it follows immediately

from Lemma 1.8 that the following assertion holds:


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v1=v2 if and only ifN(v1) =N(v2).(iii) Let e1, e2 Node( G). Then it follows immediately from Lemma1.8 that the following assertion holds:

Ife1=e2, then (V(e1) V(e2)) 1.In particular, it follows from Remark 1.2.1, (iii), that the fol-lowing assertion holds:

e1 =e2 if and only ifV(e1) =V(e2).Lemma 1.9(Graph-theoretic geometry via verticial subgroups).

Fori= 1, 2, let

vi Vert(

G). Then the following hold:

(i) If ev1 ev2 = {1}, then either ev1 = ev2 or ev1 ev2 is anodal subgroup ofG.

(ii) Consider the following three (mutually exclusive) conditions:

(1) (v1, v2) = 0.(2) (v1, v2) = 1.(3) (v1, v2) 2.

Then we haveequivalences

(1)(1) ; (2)(2)(2) ; (3)(3)

with the following four conditions:

(1) ev1 = ev2.

(2) ev1 = ev2; ev1 ev2 ={1}.

(2) ev1 ev2 is a nodal subgroup ofG.

(3) ev1 ev2 ={1}.

Proof. First, we consider assertion (i). Suppose thatH def= ev1 ev2 =

{1}, and ev1 = ev2 (so

v1 =

v2 cf. [Mzk4], Proposition 1.2, (i)).

Note that to verify assertion (i), it suffices to show that H is a nodalsubgroup of G. Also, we observe that since nodal and verticial sub-groups of G are commensurably terminalin G (cf. [Mzk4], Proposi-tion 1.2, (ii)), it follows that we may assume without loss of generality by replacing G by a connected finite etale covering of G thatv1(G)=v2(G).

Let JHbe a nontrivial procyclicclosed subgroup ofH. Then weclaimthatJis contained in an edge-likesubgroup of G . Indeed, sinceJH= ev1 ev2 wherev1=v2 it follows from Remark 1.4.1,(ii), together with our assumption thatv1(G)=v2(G), that the imageof J in ab/edgeG is trivial. Thus, by applying this observation to the

various connected finite etale coverings ofGinvolved, we conclude that


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Jsatisfies condition (ii) in the statement of Lemma 1.6. In particular,it follows from Lemma 1.6 thatJis contained in an edge-like subgroup.

This completes the proof of the above claim. On the other hand, if eeis an edge-like subgroup of G such that J ee, then it follows fromLemma 1.7 that the inclusion Jee implies that ee is in fact nodal,and, moreover, that eeev1 ev2 =H.

By the above discussion, it follows that

H=

eeN(ev1)N(ev2)

ee .

On the other hand, it follows from Lemma 1.8 that the cardinality ofthe intersection N(

v1) N(

v2) is 1. Therefore, it follows thatH is

a nodal subgroup of G. This completes the proof of assertion (i).Next, we consider assertion (ii). The equivalence

(1)(1 )

follows from [Mzk4], Proposition 1.2, (i). In light of this equivalence,the implications

(2) =(2) =(2)

follow from assertion (i), while the implication

(2) =(2)

follows from Lemma 1.7. The equivalence

(3)(3

)then follows from the equivalences

(1)(1) ; (2)(2) .

Remark 1.9.1. It follows immediately from the various definitionsinvolved that for any semi-graph of anabelioids of pro- PSC-type G,there exists, in the terminology of [Mzk6], Definition 1.2, (ii), an IPSC-extension

1

G

I

I

1 .

Therefore, Lemma 1.9 may also be obtained as a consequence of [Mzk6],Proposition 1.3, (iv).

Lemma 1.10 (Conjugates of verticial subgroups). Suppose that

G is untangled. Letv,v Vert(G) be such thatv(G) =v(G). Thenv=v if and only ifev ev ={1}.Proof. The sufficiency of the condition is immediate; thus, to proveLemma 1.10, it suffices to verify thenecessityof the condition. To thisend, let us assume that

v =

v. Then there exists a connected finite

etale subcovering G G ofG G such thatv(G) =v(G). On


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the other hand, since G is untangled, andv(G) =v(G), it follows thatN(

v(G)) N(

v(G)) =. Thus, ev ev G ={1}by Lemma 1.9,

(ii); in particular, since ev is torsion-free (cf. [Mzk4], Remark 1.1.3),we obtain that ev ev ={1}, as desired.

Remark 1.10.1. It follows immediately from Lemma 1.10 that thefollowing assertion holds:

Suppose that Gisuntangled. Letv Vert(G) be a vertexofG, vG a verticial subgroup associated to v , andG\ v. Then v v 1 ={1}.

Definition 1.11. Suppose that G is sturdy. Then by eliminating thecusps (i.e., the open edges) of the semi-graph G, and, for each vertexvofG, replacing the anabelioidGv corresponding to v by the anabelioidGv of finite etale coverings ofGv that restrict to a trivial covering overthe cusps ofG that abut to v, we obtain a semi-graph of anabelioidsof pro- PSC-typeG. (Thus, the pro- fundamental group ofGv maybe naturally identified, up to inner automorphism, with the quotientof v by the subgroup of v topologically normally generated by thee v, for e C(v).) We shall refer to Gas the compactificationofG(cf. [Mzk4], Remark 1.1.6).

Remark 1.11.1. It follows immediately from the definition of thecompactification that the quotient of G by the closed subgroup ofG topologically normally generated by the cuspidalsubgroups of Gis naturally isomorphic, up to inner automorphism, to the fundamen-tal group G ofG. In particular, we have a natural outer surjectionG G.

By analogy to the terms group-theoretically verticial and group-theoretically cuspidalintroduced in [Mzk4] (cf. [Mzk4], Definition 1.4,(iv)), we make the following definition.

Definition 1.12. LetH be a semi-graph of anabelioids of pro- PSC-type, H the (pro-) fundamental group ofH, and : G

H an

isomorphism of profinite groups. Then we shall say that is group-theoretically nodal if, for anye Node( G), the image (ee) H isa nodal subgroup of H, and, moreover, every nodal subgroup of Harises in this fashion.

Proposition 1.13(Group-theoretical verticiality and nodality).LetH be a semi-graph of anabelioids of pro- PSC-type, H the fun-damental group ofH, and : G

H agroup-theoretically ver-

ticial isomorphism. Then is group-theoretically nodal.


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Proof. This follows immediately from Lemma 1.9, (i).

Lemma 1.14 (Graphicity of certain group-theoretically cuspi-dal and verticial isomorphisms). LetH be a semi-graph of anabe-lioids of pro- PSC-type, andH the fundamental group ofH. If anisomorphism : G

H satisfies the following two conditions, then

isgraphic (cf. [Mzk4], Definition 1.4, (i)):

(i) isgroup-theoretically cuspidal.

(ii) For any sturdy connected finite etale covering G G of Gsuch that the corresponding coveringH H ofH (relative tothe isomorphism ) is sturdy, the induced isomorphism (cf.

(i), Remark 1.11.1)G

H

where we write G

(respectively, H) for the semi-graph of

anabelioids of PSC-type obtained as thecompactification(cf.Definition 1.11) ofG (respectively, H) isgroup-theoreticallyverticial.

Proof. Since the isomorphism G

H is group-theoretically ver-ticial (cf. condition (ii)), it follows from Proposition 1.13 that theisomorphism G

H is group-theoretically nodal. Therefore, it fol-

lows immediately from (i) that is graphically filtration-preserving(cf.[Mzk4], Definition 1.4, (iii)). Thus, it follows from [Mzk4], Theorem1.6, (ii), that is graphic, as desired.

Lemma 1.15 (Chains of length two lifting adjacent vertices).

Letv1,v2 Vert( G)be such that if we writevi def=vi(G), then(v1, v2) =1. Then there existw1,u1,w2 Vert(G) which satisfy the followingconditions (which imply that( w1, u1) = 2):

(i) v1 =

w1(G) =

u1(G); v2 =

w2(G).

(ii) ( w1, u1) 2.(iii) ( w2,w1) =(w2, u1) = 1.

Proof. First, we observe that by replacingGby a connected finite etalecovering of G, we may assume without loss of generality that G issturdy (cf. [Mzk4], Remark 1.1.5) and untangled (cf. Remark 1.2.1,(i)). Then it is easily verified that there exists a nontrivial connectedfinite etale covering of the anabelioid Gv2 corresponding to v2 which isunramifiedover the nodes and cusps ofGwhich abut to v2. In lightof the unramified nature of this connected finite etale covering ofGv2 ,by gluing this covering to a splitcovering over the remaining portion

of G, we obtain a connected finite etale covering H G. Then it


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follows immediately from the various definitions involved that the setV1 (respectively, V2) of vertices of H which lie over v1 (respectively,

v2) is of cardinality 2 (respectively, of cardinality 1). Thus, thereexist vertices w1, u1 V1, w2 V2 such that w1 = u1 (which, sinceG is untangled, implies that (w1, u1) 2 cf. condition (ii)), and,moreover,(w2, w1) =(w2, u1) = 1 (cf. condition (iii)). In particular,

it follows immediately that there exist elementsw1,u1,w2 Vert(G)which satisfy the three conditions in the statement of Lemma 1.15.This completes the proof of Lemma 1.15.

2. Nodally nondegenerate outer representations

In this section, we define the notion of an outer representation ofNN-typeand verify various fundamental properties of such outer rep-resentations.

If G is a semi-graph of anabelioids of pro- PSC-type for somenonempty set of prime numbers , then since the fundamental groupG ofG is topologically finitely generated, the profinite topology of Ginduces (profinite) topologies on Aut(G) and Out(G) (cf. the discus-sion entitled Topological groups in0). Moreover, if we write

Aut(G)

for the group of automorphisms ofG, then by the discussion preceding

[Mzk4], Lemma 2.1, the natural homomorphismAut(G)Out(G)

is an injection with closed image. (Here, we recall that an automor-phism of a semi-graph of anabelioids consists of an automorphism ofthe underlying semi-graph, together with a compatible system of iso-morphisms between the various anabelioids at each of the vertices andedges of the underlying semi-graph which are compatible with the var-ious morphisms of anabelioids associated to the branches of the under-lying semi-graph cf. [Mzk3], Definition 2.1; [Mzk3], Remark 2.4.2.)Thus, by equipping Aut(G) with the topology induced via this homo-

morphism by the topology of Out(G), we may regard Aut(G) as beingequipped with the structure of a profinite group.

Definition 2.1.

(i) LetIbe a profinite group, a nonempty set of prime numbers,Ga semi-graph of anabelioids of pro- PSC-type, G the fun-damental group ofG, and : I Aut(G) a homomorphism ofprofinite groups. Then we shall refer to the pair

G, : IAut(G) (Out(G))

as an outer representation of pro- PSC-type. Moreover, we

shall refer to an outer representation of pro- PSC-type for


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some nonempty set of prime numbers as an outer represen-tation of PSC-type. For simplicity, we shall also refer to the

underlying homomorphism of an outer representation ofpro- PSC-type (respectively, of PSC-type) as an outer repre-sentation of pro- PSC-type (respectively, outer representationof PSC-type).

(ii) Let (G, I: IAut(G)), (H, J: JAut(H)) be outer repre-sentations of PSC-type. Then we shall refer to a pair

( :G H, : I

J)

consisting of an isomorphism of semi-graphs of anabelioidsand an isomorphismof profinite groups such that the diagram

I

I

Aut(G)

Aut()J

JAut(H)

where the right-hand vertical arrow is the isomorphism in-duced by commutesas an isomorphism of outer represen-tations of PSC-type.

Remark 2.1.1. It follows immediately that a pro-IPSC-extension

1G II1

(i.e., roughly speaking, an extension that arises from a stable log curveover a log point cf. [Mzk6], Definition 1.2, (ii)) gives rise to an outerrepresentation IOut(G) that factors through Aut(G) Out(G);in particular, we obtain an outer representation of pro- PSC-typeIAut(G).

In the following, let us fix a nonempty set of prime numbers andan outer representation of pro- PSC-type

G, I: IAut(G) (Out(G))

and write I def= G out I (cf. the discussion entitled Topologicalgroups in0); thus, we have an exact sequence

1G II1 .

Definition 2.2.

(i) Let v Vert(G) b e a vertex of G and v G a verticialsubgroup of G associated tov. Then we shall write

Dvdef= NI(v) I (respectively, Iv

def= ZI(v) Dv)

and refer to Dv (respectively, Iv) as a decomposition(respec-

tively, an inertia) subgroup of I associated to the vertex v,


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or, alternatively, the decomposition (respectively, inertia) sub-group of Iassociated to the verticial subgroup v G . If,

moreover, the verticial subgroup v is the verticial subgroupassociated to an elementv Vert( G) (cf. Definition 1.1, (vi)),then we shall write Dev

def= Dv (respectively, Iev

def= Iv) and re-

fer to Dev (respectively, Iev) as the decomposition (respectively,inertia) subgroup ofIassociated tov.

(ii) Let e Cusp(G) be a cusp of G and e G an edge-likesubgroup of G associated toe. Then we shall write

Dedef= NI(e) I (respectively, Ie

def= eDe)

and refer to De (respectively, Ie) as a decomposition(respec-

tively, an inertia) subgroup of Iassociated to the cusp e, or,alternatively, the decomposition (respectively, inertia) subgroupof Iassociated to the edge-like subgroup e G . If, more-over, the edge-like subgroup e is the edge-like subgroup as-sociated to an elemente Cusp( G) (cf. Definition 1.1, (vi)),then we shall writeDee

def= De (respectively,Iee

def= Ie) and refer to

Dee (respectively,Iee) as the decomposition(respectively,inertia)subgroup ofIassociated toe.

(iii) Let e Node(G) be a node of G and e G an edge-likesubgroup of G associated toe. Then we shall write

Dedef= NI(e) I (respectively, Ie

def= ZI(e) De)

and refer to De (respectively, Ie) as a decomposition(respec-tively, an inertia) subgroup of Iassociated to the nodee, or,alternatively, the decomposition (respectively, inertia) subgroupof Iassociated to the edge-like subgroup e G . If, more-over, the edge-like subgroup e is the edge-like subgroup as-

sociated to an elemente Node( G) (cf. Definition 1.1, (vi)),then we shall writeDee

def= De (respectively,Iee

def= Ie) and refer to

Dee (respectively,Iee) as the decomposition(respectively,inertia)

subgroup ofIassociated toe.Lemma 2.3 (Basic properties of inertia subgroups).

(i) Letv Vert( G). Then{1}= Iev G; in particular, the homo-morphism Iev I induced by the surjectionI I is injec-tive.

(ii) Lete Node( G),v V(e). ThenIev Iee.Proof. Assertion (i) follows from the commensurable terminalityof evin G (cf. [Mzk4], Proposition 1.2, (ii)), together with the slimnessof

ev (cf. [Mzk4], Remark 1.1.3). Assertion (ii) follows from the fact that


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eeev, together with the definitions of inertia subgroups of verticesand nodes.

The following definition will play a central rolein the present paper.

Definition 2.4.

(i) We shall say that the outer representation of pro- PSC-type Iisof IPSC-type(where the IPSC stands for inertial pointedstable curve) if I is isomorphic, as an outer representationof PSC-type (cf. Definition 2.1, (ii)), to the outer representa-tion of PSC-type determined by (cf. Remark 2.1.1) an IPSC-extension (i.e., roughly speaking, an extension that arises froma stable log curve over a log point cf. [Mzk6], Definition 1.2,

(ii)).

(ii) We shall say that the outer representation of pro- PSC-type Iof VA-type(where the VA stands for verticially admissible)if the following two conditions are satisfied:

(1) I is isomorphic toZ as an abstract profinite group.(2) For everyv Vert( G), the image of the injection Iev I

(cf. Lemma 2.3, (i)) is openinI.

We shall say that the outer representation of pro- PSC-typeI is of SVA-type(where the SVA stands for strictly ver-ticially admissible) if, in addition to the above condition (1),the following condition is satisfied:

(2) For everyv Vert( G), the injection Iev I isbijective.(iii) We shall say that the outer representation of pro- PSC-type I

isof NN-type(where the NN stands for nodally nondegener-ate) ifIis of VA-type, and, moreover, the following conditionis satisfied:

(3) For everye Node( G), the homomorphism Iev1 Iev2 Iee where we write {

v1,

v2}= V(

e) Vert(

G) induced by

the inclusionsIev1 ,Iev2 Iee(cf. Lemma 2.3, (ii)) is injective,and its image is openinIee.

We shall say that the outer representation of pro- PSC-typeI isof SNN-type(where the SNN stands for strictly nodallynondegenerate) ifIis of SVA-type and of NN-type.

Remark 2.4.1. Note that it isnotthe case that condition (2) of Defini-tion 2.4 is implied by conditions (1) and (3) of Definition 2.4. Indeed, itis easily verified that if Vert(G) ={v}, and Node(G) = (so v = G),then any injection

Z Out(G) satisfies conditions (1) and (3), but

failsto satisfy condition (2). (Moreover, it is also easily verified that


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such an injection exists.) On the other hand, when Node(G) = ,it is not clear to the authors at the time of writing whether or not

condition (2) of Definition 2.4 is implied by conditions (1) and (3) ofDefinition 2.4.

Remark 2.4.2. It follows from [Mzk6], Proposition 1.3, (ii), (iii), thatifI is of IPSC-type, then Iis of SNN-type, i.e.,

IPSC-type = SNN-type = NN-type

SVA-type = VA-type .

Lemma 2.5 (Group structure of inertia subgroups). IfI isofVA-type, then the following hold:

(i) Letv Vert( G). Then as an abstract profinite group, Iev isisomorphic toZ.

(ii) Lete Cusp( G). Then as an abstract profinite group, Iee isisomorphic toZ.

(iii) Let

e Node(

G). Then as an abstract profinite group, Iee is

isomorphic toZ Z.(iv) Lete Node(G). Thenee = Iee G; thus, we have an exact

sequence

1eeIeeIm(Iee I)1

where we write Im(Iee I) for the image of the compositeIeeI I. Moreover, the subgroup Im(Iee I) I isopeninI.

In particular, forv V(e), the image of the homomorphismIev ee Iee induced by the natural inclusionsIev, ee Iee is

open in Iee. If, moreover, forv V(e), the composite Iev I I is surjective (or, equivalently, bijective), then thehomomorphismIev eeIee induced by the natural inclusionsIev, ee Iee isbijective.

Proof. Assertion (i) (respectively, (ii)) follows from conditions (1) and(2) of Definition 2.4 (respectively, from Definition 2.2, (ii)). Asser-tion (iv) follows from the commensurable terminalityof ee in G (cf.[Mzk4], Proposition 1.2, (ii)), together with condition (2) of Defini-tion 2.4. Assertion (iii) follows from the fact that Iee is anextensionof

Z by

Z and abelian(cf. assertion (iv)).


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Lemma 2.6 (Stability of verticial admissibility and nodal non-degeneracy). Suppose thatI isof VA-type (respectively, of NN-

type). Then the following hold:

(i) LetI Ibe anopen subgroup ofI, Gdef= I G, and

I the image of the compositeI I I. Thus, we havean exact sequence

1G I I 1 ;

the open subgroup G G determines a coveringG G ofG; the outer representation I Out(G) determined by Ifactors through I : I

Aut(G). Then I is of VA-type

(respectively, of NN-type).(ii) Suppose that G is sturdy. Then the outer representation of

pro- PSC-typeI: I Aut(G) where we writeG for thecompactification ofG induced by I is of VA-type (re-spectively, of NN-type).

Proof. First, we prove assertion (i). It follows immediately from thevarious definitions involved that I is of VA-type. Moreover, it followsfrom Lemma 2.5, (i), (iv), that the various Iev (respectively, Iee)are torsion-free, and, moreover, that the commensurability classof thesubgroup Iev (respectively, Iee) is unaffected by passing fromI to

I. Thus, condition (3) of Definition 2.4 for I follows from condition(3) of Definition 2.4 for I. This completes the proof of assertion (i).

Next, we verify assertion (ii). First, let us observe that condition (1)of Definition 2.4 forIfollows from condition (1) of Definition 2.4 forI.Next, let us observe that it follows from Lemma 2.3, (i) (respectively,

Lemma 2.5, (iv)), that forv Vert( G) (respectively,e Node( G)),the natural surjection G G induces an open injectionbetweenthe respective subgroups Iev (respectively, Iee). Thus, condition (2)(respectively, (3)) of Definition 2.4 for I follows from condition (2)(respectively, (3)) of Definition 2.4 for I. This completes the proof ofassertion (ii).

Lemma 2.7 (Group structure of decomposition subgroups). IfI isof VA-type, then the following hold:

(i) Letv Vert( G). Thenev =Dev G; thus, we have an exactsequence

1ev DevIm(DevI)1

where we write Im(Dev I) for the image of the compositeDev I I. Moreover, the subgroup Im(Dev I) I is

open inI.


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In particular, the image of the homomorphismIev ev Devinduced by the natural inclusionsIev, ev Dev is open inDev.

If, moreover, the composite Iev I I issurjective (or,equivalently, bijective), then the homomorphismIevevDevisbijective.

(ii) Lete Cusp(G). Thenee =Dee G; thus, we have an exactsequence

1eeDeeIm(Dee I)1

where we write Im(Dee I) for the image of the compositeDee I I. Moreover, the subgroup Im(Dee I) I isopen inI.

In particular, forv V(e), the image of the homomorphismIev eeDee induced by the natural inclusionsIev, ee Dee isopen in Dee. If, moreover, forv V(e), the composite Iev I I is surjective (or, equivalently, bijective), then thehomomorphismIev ee Dee induced by the natural inclusionsIev, ee Dee isbijective.

(iii) Lete Node( G). Thenee =Dee G; thus, we have an exactsequence

1eeDeeIm(Dee I)1

where we write Im(Dee I) for the image of the compositeDee I I. Moreover, the subgroup Im(Dee I) I isopen inI.

In particular, the image of the natural inclusionIee Dee isopen inDee. If, moreover, forv V(e), the composite Iev I issurjective (or, equivalently, bijective), then the naturalinclusionIee Dee isbijective.

Proof. The computation of the intersection with G in assertion (i)(respectively, (ii); (iii)) follows from the commensurable terminalityofev (respectively, ee; ee) in G (cf. [Mzk4], Proposition 1.2, (ii)). Thefact that the images of the respective decomposition subgroups in I

are open follows from condition (2) of Definition 2.4. The final portionof assertion (i) (respectively, (ii); (iii)) then follows immediately fromLemma(s) 2.3, (i) (respectively, 2.3, (i); 2.3, (i), and 2.5, (iv))

Remark 2.7.1. It follows immediately from Lemmas 2.5, 2.7 that thefollowing assertion holds:

Let v Vert( G) (respectively, e Cusp( G); e Node( G)).IfI isof SVA-type, then

Dev =Iev ev =Iev ev

(respectively, Dee = Iev ee =Iev ee , for anyv V(e) ;


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Dee=Iee=Iev ee =Iev ee , for anyv V(e) ).Remark 2.7.2. Lete1,e2 Edge(G). If I is of VA-type, then thefollowing three conditions are equivalent:

(i)e1 =e2.(ii) Iee1 =Iee2.

(iii) Dee1 =Dee2.

Indeed, the implications

(i) =(ii) ; (i) =(iii)

are immediate. On the other hand, if condition (ii) (respectively, (iii))is satisifed, then ee1 = Iee1 G = Iee2 G = ee2 [cf. Definition 2.2,(ii); Lemma 2.5, (iv)] (respectively, ee1 =Dee1 G =Dee2 G = ee2[cf. Lemma 2.7, (ii), (iii)]). Thus, it follows from [Mzk4], Proposition1.2, (i), thate1=e2.Definition 2.8. Suppose thatI isof SVA-type. Then we shall denoteby

G[I]

the connected semi-graph of anabelioids (cf. [Mzk3], Definition 2.1) de-fined as follows: The underlying graph ofG[I] is the underlying graphofG. The anabelioid corresponding to a vertex v Vert(G) (respec-tively, an edge e Edge(G)) is the connected anabelioid determinedby the decomposition subgroup, regarded up to inner automorphism,Dv I (respectively, De I) associated to v (respectively, e); forv V(e), the associated morphism of anabelioids is the morphism de-termined by the natural inclusion De (=Iv e) Dv (=Iv v) (cf.Remark 2.7.1).

Remark 2.8.1.(i) Note that the fundamental group of the anabelioid correspond-

ing to a vertex ofG[I] (i.e., the decomposition subgroup, re-garded up to inner automorphism, associated to the vertex) isnot center-free (cf. Lemma 2.7, (i)). In particular, the semi-graph of anabelioidsG[I] isnotofPSC-type.

(ii) Let G[I] be the pro- fundamental group (i.e., the maximalpro- quotient of the fundamental group) of the connectedsemi-graph of anabelioids G[I] (cf. the discussion following[Mzk3], Definition 2.1). Then it follows from the definition of

G[I] that theinductive system of homomorphismsdetermined


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by the natural outer inclusions Dv I and De I givesrise to a natural outer homomorphism

G[I]I.

Lemma 2.9 (An isomorphism of fundamental groups). SupposethatI isof SVA-type. LetG[I] be the pro- fundamental group ofthe connected semi-graph of anabelioidsG[I]. Then the homomorphismG[I] Idefined in Remark 2.8.1, (ii), is anisomorphism.

Proof. First, we observe (cf. Remark 2.7.1) that the decompositionsubgroup Dz wherez VCN(G) is an extension ofIby z. Nowit is easily verified that theprofiniteGalois covering ofG[I] determinedby the various quotients Dz I(i.e., that arise as composites Dz I I) is isomorphic to G; thus, we obtain an exact sequence

1GG[I]I1 .

On the other hand, it follows from the construction of this profinitecovering G G[I], together with the definition of the homomorphismG[I]I, that the composite G G[I]Icoincides with thenatural inclusion G I. Thus, the bijectivity of the homomorphismG[I] Ifollows from the Five Lemma. This completes the proofof Lemma 2.9.

Definition 2.10. Let (g, r) be a pair of natural numbers such that2g 2 + r >0, k an algebraically closed field of characteristic , s Mg,r(k) ak-valued geometric point ofMg,r (cf. the discussion entitled

Curves in 0), and slog : Spec(k)log Mlog

g,r the strict morphismof log stacks whose underlying morphism of stacks is the morphismcorresponding to s.

(i) We shall denote byXlogs the stable log curve determined byslog.

(ii) We shall denote by Gs the semi-graph of anabelioids of pro-PSC-type determined by the stable log curve Xlogs (cf. [Mzk4],Example 2.5).

(iii) Write Qs for the monoid obtained as the stalk of the charac-teristic sheaf (cf. the discussion entitled Log stacks in 0) of

Mlog

g,r at s, and

Isdef= Hom(Qgps ,

Z(1)) where the (1) denotes a Tate twist. Recall (cf. [Knud],Theorem 2.7) that it follows from the well-known geometry of

the irreducible components of the divisor that defines the log


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structure ofMlog

g,r that we have a natural decomposition

Qs eNode(Gs)

Ne

where we write Ne for a copy ofN indexed bye Node(Gs);thus, we obtain a decomposition

Is

eNode(Gs)

[e]

where we write [e] for a copy ofZ(1) indexed by e Node(Gs).

(iv) It follows from the various definitions involved that, if we write1(X

logs ) for the maximal pro- quotient of the logarithmic fun-

damental group ofXlogs , then we have a natural exact sequenceof profinite group

1Gs 1(X

logs )Is1

which gives rise to an outer representation Is Out(Gs)that factors through Aut(Gs) Out(Gs). Write

s : IsAut(Gs)

for the resulting homomorphism of profinite groups and Isdef=

Gsout

Is. Thus, we have a natural isomorphism of profinitegroups 1(X

logs )

Is.

(v) Let s Mg,r+1(k) be a k-valued geometric point ofMg,r+1that corresponds to a node ofXs. Then it follows immediatelyfrom the various definitions involved that the quotient of Gs where we use the notation obtained by applying (ii) to s by the closed subgroup of Gs topologically normally generatedby the edge-like subgroups of Gs associated to the (r+ 1)-stcusp is naturally isomorphic to Gs; in particular, we have anatural surjection Gs Gs . We shall denote by

Ns/s : Node(Gs)Node(Gs)

the map which as is easily verified is uniquely determinedby the following condition:

Ife Node(Gs), and e Gs is an edge-like sub-group associated to e, then the image of e via theabove surjection Gs Gs is an edge-like subgroupassociated to Ns/s(e) Node(Gs).

Lemma 2.11 (Log fundamental groups in a neighborhood of a

node). In the notation of Definition 2.10, lete Node(Gs) be a node


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ofGs, ands Mg,r+1(k) ak-valued geometric point that correspondsto the node ofXs determined bye. Then the following hold:

(i) The inverse imageN1s/s(e)consists ofprecisely two elements

e1, e2 Node(Gs); the map

Node(Gs) \ {e1, e2} Node(Gs) \ {e}

determined byNs/s isbijective.

(ii) WriteIs for the result of applying Definition 2.10, (iii), to s.

Then the homomorphism Is Is induced on maximal pro- quotients of log fundamental groups by (the strict morphismof log schemes whose underlying morphism of schemes is themorphism corresponding to) s isinjective, and its image isan inertia subgroup Ie ofIs associated to e. Moreover, if,in the notation of (i), we write

M=e1,e2def=

fNode(Gs )\{e1,e2}

[f] Is,

then for i= 1, 2, there exists a vertexvi V(e) Vert(Gs) ofGssuch that the subgroup obtained as the image of the compositeof the injections

[ei] M=e1,e2 Is Is

is an inertia subgroup Ivi

of Is

associated to vi. In this

situation, we shall refer to vi as thevertex associated to ei.

(iii) Let

M=edef=

fNode(Gs)\{e}

[f] Is .

Then the homomorphismIs Is induced byMlog

g,r+1 Mlog

g,r

(i.e., the compositeIs Is Is) coincides with the homo-morphism

[e1] [e2] M=e1,e2 =Is Is = [e] M=e

determined by the homomorphism[e1] [e2] [e]

(a, b) a+b

and the isomorphism

M=e1,e2

M=e

induced by the bijective portion ofNs/s (cf. (i)).

Proof. Assertion (i) follows immediately from the various definitionsinvolved. Assertions (ii) and (iii) follow by computing the log structuresinvolved by means of a chart for the morphism Xlogs Spec (k)

log at

the k-valued point s ofXs.


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Remark 2.11.1. In the notation of Definition 2.10, let

e Node( Gs),

e

def

=e(Gs) Node(Gs), s

Mg,r+1(k) a k-valued geometric pointthat corresponds to the node ofXs determined by e, and {e1, e2} =N1s/s(e) Node(Gs) (cf. Lemma 2.11, (i)). Moreover, for i= 1, 2, let

us denote byvi the (unique!) element of Vert( Gs) such thatvi V(e),and, moreover,vi(Gs) is the vertex associated to ei (cf. Lemma 2.11,(ii)). (Thus, V(e) ={v1, v2}.) Then it follows from Lemma 2.11 thatif we identify M=e with M=e1.e2 via the isomorphism of Lemma 2.11,(iii), then the following assertion holds:

The isomorphismsIevi

[ei] M=e (cf. Lemma 2.11,(ii)), Iee

Is

[e1] [e2] M=e (cf. Lemma 2.11,

(ii), (iii); Definition 2.10, (iii)), and Is [e] M=e(cf. Lemma 2.11, (iii); Definition 2.10, (iii)) fit into thefollowingcommutative diagram

Iev1 Iev2 Iee Is

[e1] M=e

[e2] M=e

[e1] [e2] M=e [e] M=e

(a,m,b,n) (a,b,m+n)

(a,b,m) (a+b, m)

where the upper left-hand horizontal arrow Iev1 Iev2 Iee is the homomorphism induced by the natu-ral inclusions Iev1, Iev2 Iee (cf. Lemma 2.3, (ii)), andthe upper right-hand horizontal arrow Iee Is is thecomposite IeeIs Is.

Lemma 2.12 (The invertibility of a certain homomorphism offree modules). Let A be a commutative ring with unity, M a freeA-module of finite rank, Na freeA-module of rank1, : NN Ma homomorphism of A-modules, 1 : N N the composite of and

the first projectionN M N, and

N1 N2def=

(N M) NMN

(N M) NMN

N0def= (N N M) NMN(N M) NMN

where the definition of N1 (respectively, N2) is to be understoodas the first (respectively, second) module in brackets {}; the nota-tion () NM() denotes the fiber product of modules overN M the diagram obtained via from the diagram

(N M) (N M)N N MN M

(a,m,b,n) (a,b,m+n)


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(a,b,m) (a+b, m) .

Then the following hold:

(i) N1 andN2 are free A-modules of rank 1, andN0 is a free A-module of rank2.

(ii) If is a homomorphism of freeA-modules of rank1, then let usdenote byD() Spec (A) the open subscheme ofSpec (A) onwhich (the homomorphism ofOSpec(A)-modules determined by) is an isomorphism. ThenD(1) = D(det(N1 N2 N0))(cf. (i)).

Proof. Assertion (i) is immediate from the definition of N1, N2, andN0. Thus, to complete the proof of Lemma 2.12, it suffices to verify

assertion (ii). To this end, since the various definitions of modulesand homomorphisms in the statement of Lemma 2.12 are compatiblewith base-change, we may assume without loss of generality that A isa field. On the other hand, ifA is a field, then 1 is either zero or anisomorphism, so it follows immediately from an easy computation thatD(1) and D(det(N1 N2 N0)) coincide. This completes the proofof assertion (ii).

Lemma 2.13 (Injectivity and images of homomorphisms of

Z-modules). In the notation of Definition 2.10, let : I

def=

Z Is

be a homomorphism of profinite groups, and

Jev1 Jev2def=

Iev1IsI

Iev2IsI

Jeedef= Iee IsII

the diagram of homomorphisms of profinite groups obtained viafromthe upper row of the diagram in Remark 2.11.1. Then the followingconditions are equivalent:

(i) The image of the composite

I Is [e] M=e

pr[e]

isopen in[e].

(ii) The first arrowJev1 Jev2 Jee of the above sequence isinjec-tive, and its image isopen inJee.

Proof. It follows immediately from the various definitions involved thatthe implication

(i) =(ii) (respectively, (ii) =(i))

follows from the inclusion D() D(det(N1 N2 N0)) (respec-tively, D(det(N1 N2 N0)) D()) implicit in Lemma 2.12, (ii).Here, we consider the case of D() that arise from an openideal ofthe topological ring

Z, i.e., an ideal generated by a nonzero element

ofZ.


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Proposition 2.14 (Nodal nondegeneracy of certain outer rep-

resentations).In the notation of Definition 2.10, let : I def=Z

Isbe a homomorphism of profinite groups, I: I Aut(Gs) the outerrepresentation of pro- PSC-type obtained as the composite

I Is

sAut(Gs) ,

andIdef= Gs

out I. In the following, forz VCN( Gs), we shall write

Jez for the inertia subgroup Iez ofI(i.e., to avoid confusion with thecorresponding inertia subgroups ofIs). Then the following hold:

(i) I isof SVA-type.

(ii) If

e Node(

Gs), then the following two conditions areequiva-

lent:

(1) The image of the composite

I Is

eNode(Gs)

[e]pree(Gs) [e(Gs)]

isopen in[e(Gs)].(2) IfV(e) ={v1, v2}, then the homomorphismJev1 Jev2 Jee

induced by the inclusionsJev1, Jev2 Jee isinjective, andits image isopen inJee.

In particular, if the image of the composite

I Is

eNode(Gs)

[e]prf

[f]

isopen in[f] for everyfNode(G), thenI isof SNN-type.

Proof. The various assertions of Proposition 2.14 follow immediatelyfrom the various definitions involved, together with Lemma 2.13.

Remark 2.14.1. In the notation of Proposition 2.14, it is not difficultto show by applying various well-known group-theoretic construc-

tions of certain natural isomorphisms between the various copies ofZ(1) involved that the condition on the homomorphism : IIsthat the composites beof IPSC-typeis equivalent to the conditionon that there exists an isomorphism

IHom(Ngp,Z(1))with respect to which is positive definite in the sense that it arises(by applying the functor Hom(,Z(1))) from a homomorphism ofmonoids Qs Nsuch that for any fNode(Gs), the composite

Nf

eNode(Gs)

Ne Qs N


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is nonzero. On the other hand, it follows from Proposition 2.14 thatthe (necessarily strict) nodal nondegeneracyofI is equivalent to the

nondegeneracyof, i.e., the condition that the image of the composite

I Is

eNode(Gs)

[e] prf[f]

be openfor every fNode(Gs). That is to say,

IPSC-type = (S)NN-type

positive definite = nondegenerate.

3. Group-theoretic aspects of the geometry of theunderlying semi-graphs

In this section, we consider the geometry of the underlying semi-graph associated to a semi-graph of anabelioids of PSC-type from agroup-theoretic point of view in the context of outer representations ofNN-type (cf. [Mzk6], Proposition 1.3, for an analogous discussion inthe case of outer representations of IPSC-type).

In this section, let be a nonempty set of prime numbers, Ga semi-graph of anabelioids of pro- PSC-type, G the fundamental group

ofG, I: I Aut(G) an outer representation of NN-type, and Idef

=G

out I.

Lemma 3.1(Contagious conditions). Let(C)be a condition on an

element ofVert( G) which satisfies the following property():() :Letv1,v2 Vert( G) be such that(v1(G), v2(G))1. Thenv1 satisfies the condition(C) if and only ifv2satisfies the condition(C).

Suppose that there exists an element ofVert( G)which satisfies the con-dition(C). Then every element ofVert(

G) satisfies the condition(C).

Proof. This follows immediately from the connectednessof the under-lying semi-graph of a semi-graph of anabelioids of PSC-type.

Lemma 3.2 (Verticial decompositions inside ab/(edge+iner)-

quotients). Let ab/edgeI be the quotient of the abelianizationabI by

the closed subgroup generated by the images in abI of the edge-likesubgroups ofG. Suppose thatI isof SNN-type. Then the followinghold:

(i) For

v Vert(

G), writeMev for the image of the compositeIev

I ab/edgeI . Then the closed subgroup Mev

ab/edgeI is


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independent of the choice of the element

v Vert( G). Denote

this closed subgroup byM. In the following, we shall write

ab/(edge+iner)I

def=

ab/edgeI /M .

(ii) The composite of the injection of Lemma 1.4 with the naturalinclusionG I induces asplit injection (cf. the discus-sion entitled Topological groups in0)

vVert(G)

ab/edgev ab/(edge+iner)I

(cf. Definition 1.3, (i)) whose image is a freeZ-module offinite rank.

Proof. First, we verify assertion (i). If Node(G) = , then assertion(i) is immediate; thus, assume that Node(G) = . Next, let us fix anelementv0 Vert(G). Forv Vert(G), we shall say thatv satisfiesthe condition (triv) if the image ofIev in the quotient

ab/edgeI /Mev0 is

trivial. To verify assertion (i), it is immediate that it suffices to show

that anyv Vert( G) satisfies (triv). Therefore, to verify assertion (i),it follows from Lemma 3.1 that it suffices to show that the condition(triv) satisfies the property () in the statement of Lemma 3.1. To

this end, let

v,

v Vert(G) be such that (

v(G),

v(G)) 1, and

v

satisfies (triv). Let De Ibe a decomposition subgroup associated

toe N(v(G)) N(v(G)). Then since the image ofIev in ab/edgeI /Mev0istrivial, andDeis generated by an edge-like subgroup and a conjugate

ofIev (cf. Remark 2.7.1), it follows that the image ofDein ab/edgeI /Mev0

is trivial. Therefore, since there exists a conjugate ofIev contained in

De, we conclude that the image of Iev in ab/edgeI /Mev0 is trivial; in

particular,v satisfies (triv). This completes the proof of assertion (i).Finally, we observe that assertion (ii) follows from a similar argument

involving coverings this time ofG[I] (cf. Definition 2.8) as opposedtoG to the argument applied in the proof of Lemma 1.4.

Remark 3.2.1. Suppose that I isof SNN-type. Letv1,v2 Vert(G).Then it follows immediately from Remark 1.4.1, (ii); Lemma 3.2 thatthe following assertion holds:

Ifv1(G)=v2(G), then the image of the intersection(Iev1 Dev2) ev1 G

in ab/edgeG is trivial.

Indeed, it follows from Lemma 3.2, (i); Remark 2.7.1, together withthe various definitions involved, that the image of Iev1 (respectively,

Dev2) in ab/(edge+iner)I is trivial (respectively, coincides with the image

of ev2 G). But, by Lemmas 1.4; 3.2, (ii), this implies that the


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image of (Iev1 Dev2) ev1 in ab/edgeG is contained in the intersection of

the images of ev2 and

ev1 in

ab/edge

G . Therefore, the above assertionfollows from Remark 1.4.1, (ii).

Remark 3.2.2. In fact, it is not difficult to verify that both the state-ment and the proof of Lemma 3.2 remain valid even under the weakerassumptionthat I isof SVA-type.

Lemma 3.3 (Submodules of free Zl-modules). Let l be a primenumber, r a positive integer; also, for 1 j r, let cj Zl\ {0}.

For1 i l, 1 j r, setMi,jdef= Zl, M0

def= Zl; write i,j Mi,j,

0 M0 for the generators corresponding to the element 1. Next, letus writeMdiag

i,jMi,j for the submodule obtained as the image ofthe diagonal homomorphismZl

i,jMi,j,

N def=

i,j

Mi,j

/Mdiag

M0

and regardM0 as a submodule ofNvia the inclusionM0 0 M0 N. Then if we denote by H the submodule of N generated by theelements ofNdetermined by the(lr+ 1)-tuples of the form

(0, , 0, cj i,j , 0, , 0, 0)

where(i, j)ranges over pairs of natural numbers such that1 i l,1 j r thenH M0 l M0.

Proof. Suppose that the element h H determined byi,j

di,j(0, , 0, cj i,j, 0, , 0, 0)

= ( , di,jcj i,j, ,i,j

di,j 0)

i,j

Mi,j

M0

where di,j Zl is contained in M0. Now let us observe that thehomomorphism

N=

i,jMi,j

/Mdiag

M0

(i,j)=(1,1)Mi,j

M0

([(i,j)i,j], 0) ((i,j 1,1)(i,j)=(1,1), 0)

where we write [?] for the image of ? in the module {},

and () is an element of M() is an isomorphism. Thus, byapplying to h HM0, we conclude that di,jcj d1,1c1 = 0, for1 i l, 1 j r; in particular, it follows that di,jcj isindependentof the pair (i, j), hence that (di,j di,j)cj = 0. But, since cj = 0, thisimplies that di,j is independentof i, hence since i ranges over the

integers from 1 to l that i,jdi,j l Zl, as desired.


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Lemma 3.4 (Existence of certain coverings). Suppose that thefollowing two conditions are satisfied:

(a) I isof SNN-type.

(b) G issturdy anduntangled (cf. Definition 1.2).

If, by abuse of notation, we writeG for the underlying semi-graph ofG[I] (cf. Definition 2.8), then, for a vertex v (respectively, an edgee) ofG, let us writeDv (respectively,De) for the connected anabelioidcorresponding to v (respectively, e), and Dv (respectively, De) forthe fundamental group of the connected anabelioidDv (respectively,De)[so it follows from the definition ofG[I] thatDv, De are naturallyisomorphic, up to inner automorphism, to Dv, De, respectively]. Fix

a vertex v0 Vert(G). Then there exists a connected covering ofsemi-graphs of anabelioids (cf. [Mzk3], Definition 2.2, (i))

H G[I]

ofG[I] such that if we denote the underlying semi-graph ofH byHand use analogous notation forH to the notation introduced above forG[I], then the following conditions are satisfied:

(1) The set of vertices ofH which lie overv0 consists ofpreciselyone elementw0, and the image of the outer injectionDw0 Dv0 Dv0 induced by the morphismDw0 Dv0 does not

contain the normal subgroup Iv0 Dv0, i.e., Iv0 Dw0.(2) For any v1 Vert(G) such that (v0, v1) = 1, the set of ver-

tices ofH which lie overv1 consists ofprecisely one elementw1, and the image of the outer injectionDw1 Dv1 Dv1induced by the morphismDw1 Dv1 contains the normal sub-group Iv1 Dv1, i.e., Iv1 Dw1.

(3) For anyv Vert(G) such that (v0, v) 2, and any vertexwofH which lies overv, the morphismDw Dv is anisomor-phism.

(4) For any e Node(G) such that v0 V(e), and any closededge f ofH which lies over e, the morphismDf De is anisomorphism.

(5) For anye Cusp(G), and any open edgef ofHwhich lies overe, the image of the outer injection Df De De inducedby the morphismDf De containsthe normal subgroupeDe, i.e., e Df.

Proof. To verify Lemma 3.4, by replacing Gby the compactification ofG(cf. Definition 1.11), we may assume without loss of generality thatCusp(G) = , and hence that condition (5) is satisfied automatically.

Moreover, by projecting to the maximal pro-lquotients, for somel ,


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of the various pro- groups involved, to verify Lemma 3.4, we mayassume without loss of generality that = {l}.

WriteV1 Vert(G)

for the set of vertices v ofGsuch that (v0, v) 1,

V=1 V1

for the set of vertices v ofG such that (v0, v) = 1 (i.e., V=1 = V1\{v0}),

N0 Node(G)

for the set of nodes ofGwhich abut to v0 (i.e., N0def= N(v0)), and

N1 Node(G)

for the set of nodes e ofG such that V(e) V=1 = and v0 V(e).Then we claim that there exists a connected finite etale covering ofsemi-graphs of anabelioids F G[I] ofG[I] such that if we denotethe underlying semi-graph ofF by F and use analogous notation forFto the notation introduced in the statement of Lemma 3.4, then thefollowing conditions are satisfied:

(i) The connected finite etale covering of semi-graphs of anabelioidsF G[I] ofG[I] isGalois, and its Galois group is isomorphic

to Z/lZ.(ii) For anyv V1, the set of vertices ofF which lie overv consists

ofprecisely oneelement u, and the image of the outer injectionDu Dv Dv induced by the morphism Du Dv containsthe normal subgroup IvDv, i.e., Iv Du.

(iii) For any v Vert(G) such that(v0, v) 2, and any vertex u ofFwhich lies over v , the morphismDu Dv is anisomorphism.

(iv) For any e Node(G), and any edge h ofF which lies over e,the morphismDh De is an isomorphism.

Indeed, since G is sturdy(cf. condition (b)), it follows that ab/edgev ={1} (cf. Definition 1.3, (iii)) for anyv Vert(G). Thus, the aboveclaimfollows immediately from the existence of the nat

nodally nondegenerate outer representations

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