RIMS-1980
Hom-versions of the Combinatorial
Grothendieck Conjecture I:
Abelianizations and Graphically Full Actions

By

Reiya TACHIHARA

April 2024

R ESEARCH I NSTITUTE FOR MATHEMATICAL SCIENCES
KYOTO UNIVERSITY, Kyoto, JapanHom-versions of the Combinatorial
Grothendieck Conjecture I: Abelianizationsand Graphically Full Actions
Reiya Tachihara
April 3, 2024
Abstract
Semi-graphs of anabelioids of PSC-type and their PSC-fundamentalgroups (i.e., a combinatorial Galois-category-theoretic abstraction ofpointed stable curves over algebraically closed ﬁelds of characteristic
zero and their fundamental groups) are central objects in the studyof combinatorial anabelian geometry. In the present series of papers,
which consists of two successive works, we investigate combinatorial
anabelian geometry of (not necessarily bijective) continuous homomorphisms between PSC-fundamental groups. This contrasts with previous researches, which focused only on continuous isomorphisms. Morespeciﬁcally, our main results of the present series of papers roughly
state that, if a continuous homomorphism between PSC-fundamentalgroups is compatible with certain outer representations, then it satisﬁes a certain “group-theoretic compatibility property”, i.e., the property that each of the images via the continuous homomorphism ofcertain VCN-subgroups of the domain are included in certain VCNsubgroups of the codomain. Such results may be considered as Homversions of the combinatorial version of the Grothendieck conjecture
established in some previous works. As in the case of previous works(i.e., the Isom-versions), the proof requires diﬀerent techniques depending on the types of outer representations under consideration.
In the present paper, we will treat the case where the outer representations under consideration are assumed to be “l-graphically full”,
i.e., to satisfy a certain condition concerning “weights” consideredwith respect to the “l-adic cyclotomic character”, where l is a certain1

prime number. In addition, to prepare for this purpose, we include
detailed expositions on “reduction techniques”, namely, techniques of
reduction to the maximal pro-Σ quotients and to the abelianizationsof (various open subgroups of) the PSC-fundamental groups underconsideration, where Σ is a certain set of prime numbers. Though
the discussions of these “reduction techniques” are all essentially wellknown to experts, we present the results in a highly uniﬁed/generalizedfashion.

Contents
0 Notations and Conventions

8

1 Preliminaries

9

2 Reduction to the Case of a Smaller Set of Prime Numbers

19

3 Filtration-preservation and Group-theoretic Compatibility

28

4 Hom-version of the Combinatorial Grothendieck Conjecturefor Graphically Full Outer Representations
34

Introduction
Semi-graphs of anabelioids of PSC-type and their PSC-fundamental groups
are central objects in the study of combinatorial anabelian geometry. In thepresent series of papers, which consists of two successive works, we investigate combinatorial anabelian geometry of (not necessarily bijective) continuous homomorphisms between PSC-fundamental groups. This contrasts withprevious researches, which focused only on continuous isomorphisms. Morespeciﬁcally, our main results (cf. Theorem B below, [HmCbGCII], Theorem A, [HmCbGCII], Theorem B, and [HmCbGCII], Theorem C) roughly
state that, if a continuous homomorphism between PSC-fundamental groups
is compatible with certain outer representations, then it satisﬁes a certain“group-theoretic compatibility property”. Such results may be considered
as Hom-versions of the combinatorial version of the Grothendieck conjecture
established in some previous works.

2

To begin with, we roughly explain our basic setting. The references
are [SemiAn] and [CbGC]. We shall refer to as a semi-graph a triple G =(V, E, f ), where V is a set, E is a set each of whose element is itself a set
consisting of precisely two elements, and f : qe∈E e −→ V q {undeﬁned},
where {undeﬁned} is a singleton whose unique element is a formal symbol“undeﬁned”. Here, we think of V (resp. E; an element b of an element e of E;f ) as the set of vertices of G (resp. the set of edges of G; a branch of the edgee; determining the abutment relation). Note that it is allowed that a branchof an edge abuts (not to a vertex but) to “undeﬁned”, i.e., abuts nowhere.
The vertices and the edges of a semi-graph G are collectively referred to as
the components of G. An edge of a semi-graph is said to be closed if both of
the two branches abut to some vertex. An edge of a semi-graph is said to beopen if it is not closed. Then a semi-graph of anabelioids is, by deﬁnition,
a semi-graph equipped with a Galois category on each component, together
with “an exact functor from the vertex to the edge” on each branch that
abuts to a vertex (i.e., that does not abut to “undeﬁned”).
Let X be a pointed stable curve over an algebraically closed ﬁeld of characteristic zero and Σ a non-empty set of prime numbers. Then X and ΣΣ, which we call “a semi-graph ofdetermine a semi-graph of anabelioids GX
anabelioids of pro-Σ PSC-type associated to X”, roughly as follows. The unΣ
is the dual semi-graph of X, i.e., the semi-graphderlying semi-graph of GX
whose vertices (resp. closed edges; open edges) are the irreducible components (resp. the nodes, i.e., the singular points; the cusps, i.e., the marked
points) of X, and the Galois category on each vertex v is the Galois category of ﬁnite étale coverings of the complement of the cusps in the regularlocus of the irreducible component of X corresponding to v whose “Galois
closure” has degree a number the prime factors of which belong to Σ. We
omit the detail of the Galois categories on edges, but we mention that theconstruction is similar to the construction in the case of vertices and that
their fundamental groups are all isomorphic to the maximal pro-Σ quotientẐΣ of Ẑ.
Next, let G be a semi-graph of anabelioids of pro-Σ PSC-type, i.e., a semiΣ
garph of anabelioids which is isomorphic, in the evident sense, to the “GX” forsome X and Σ introduced above. Then we deﬁne the PSC-fundamental groupe op , where Ge −→ G is a ﬁxed “pro-ΣΠG of G as the proﬁnite group Aut(G/G)
universal covering”, which we omit the explanation here (cf. the discussione (resp. Edge(G);epreceding [NodNon], Deﬁnition 1.1). We shall write Vert( G)3e for the set of vertices (resp. edges; components) of G.
e Then we
VCN(G))e For every c̃ ∈ VCN(G),e wehave a natural right action of ΠG on VCN(G).
shall write Πc̃ ⊂ ΠG for the stabilizer subgroup of c̃. If c̃ is a vertex (resp.
a closed edge; an open edge; arbitrary), then Πc̃ is said to be a verticial
subgroup (resp. a nodal subgroup; a cuspidal subgroup; a VCN-subgroup)
e to the set of VCNof ΠG . An important fact is that the map from VCN(G)subgroups of ΠG deﬁned by c̃ 7−→ Πec is bijective (cf. Lemma 1.3, (2)).
Primitively, an anabelian geometer is interested in the following naive
question.
Question: to what extent the geometric information of an object may be recovered from the group-theoretic information of its
fundamental group?
In the context of combinatorial anabelian geometry, one sometimes takes
the “object” in the above question to be a semi-graph of anabelioids of proΣ PSC-type. Among the various results obtained in the previous researchin this realm, the combinatorial version of the Grothendieck conjecture (or“the combinatorial Grothendieck conjecture” for short), which is an analogueof (a certain version of) the original arithmetic Grothendieck conjecture, isof particular interest and importance. Indeed, it is, on the one hand, a certain Galois-category-theoretic uniﬁed abstraction of some techniques in arithmetic anabelian geometry of hyperbolic curves (cf., e.g., the paragraph thatstarts with “The original motivation for...” in [CbGC], Introduction) and,on the other hand, a basic tool in the study of algebro-geometric anabeliangeometry of conﬁguration spaces of hyperbolic curves (cf., e.g., the proof of[NodNon], Theorem 6.1). The combinatorial Grothendieck conjecture statesroughly that, if a continuous isomorphism between PSC-fundamental groups
is compatible with certain outer representations, then the isomorphism underconsideration preserves certain VCN-subgroups. To be more brief, it statesthat certain information of VCN-subgroups of a PSC-fundamental group isrecovered from a certain outer representation on the PSC-fundamental group.
Various versions of the combinatorial Grothendieck conjecture are obtainedin [CbGC], [NodNon], and [CbTpII].
Considering the intriguing nature and signiﬁcance of the combinatorial
Grothendieck conjecture, it is natural to contemplate its new variants andgeneralizations. This is precisely the aim of the present series of papers.
4

More speciﬁcally, in the present series of papers, we will examine the extent towhich one may generalize the combinatorial Grothendieck conjecture to (possibly non-bijective) continuous homomorphisms between PSC-fundamentalgroups, i.e., the extent to which one may establish certain “Hom-versions”
of the combinatorial Grothendieck conjecture. Here, we should mention thegeneral context that the anabelian geometers are interested also in Homversions of the original arithmetic Grothendieck conjecture.
Let G and H be semi-graphs of anabelioids of PSC-type, α : ΠG −→ ΠH ae and T ⊂ Vert(H).
e Then we shallcontinuous homomorphism, S ⊂ Vert(G),say that α is (S, T )-compatible if, for every c̃ ∈ S, either α(Πc̃ ) = 1 or there
exists d˜ ∈ T such that α(Πc̃ ) ⊂ Πd˜. The properties of being (S, T )-compatible
for some S, T are collectively referred to as the group-theoretic compatibility
properties. The Hom-versions of the combinatorial Grothendieck conjecture
established in the present series of papers provide us with various suﬃcientconditions for certain group-theoretic compatibility properties of continuous
homomorphisms between PSC-fundamental groups.
In §1, we review some basic knowledge of semi-graphs of anabelioids of
PSC-type, and we deﬁne several new terminologies some of which play central
roles in the present series of papers. In particular, we deﬁne the notion ofgroup-theoretic compatibility properties introduced above.
In §2, we give a detailed exposition of a well-known “reduction technique”, namely, reduction to the maximal pro-Σ quotients of (various open
subgroups of) the PSC-fundamental groups under consideration, where Σ isa certain set of prime numbers. We also examine the extent to which these
reduction techniques may be applied to our Hom-version situation. Though
the discussions in §2 are all essentially well-known to experts, the results areformulated in a highly uniﬁed fashion.
In §3, we apply the technique established in §2 to verify another “reduction technique” which is also well-known to experts, namely, reduction
to the abelianizations of (various open subgroups of) the PSC-fundamentalgroups under consideration. As a result, we verify the following highly uniﬁed
generalization of this technique to our Hom-version situation.
Theorem A (Theorem 3.2). Let G be a semi-graph of anabelioids of pro-ΣG
PSC-type, H a semi-graph of anabelioids of pro-ΣH PSC-type, α : ΠG −→ ΠH
a continuous homomorphism between their PSC-fundamental groups, S ⊂e and T ⊂ VCN(H).
e Let us consider the following three conditions,
VCN(G),
5where the closures S and T are taken with respect to the natural proﬁnite
e and VCN(H):etopology of VCN(G)(i) α is (S, T )-compatible.
(ii) α is (S, T )-ﬁltration-preserving (cf. Deﬁnition 1.7, (2)). That is tosay, roughly speaking, for every open subgroup U ⊂ ΠH , the image of
the “S-like” submodule of the abelianization α−1 (U )ab via the induced
homomorphism α−1 (U )ab −→ U ab is included in the “T -like” submoduleof the abelianization U ab .
(iii) α is (S, T )-compatible.
Then the implications (i) =⇒ (ii) =⇒ (iii) holds. In particular, if we
suppose further that S = S and T = T , then the conditions above are all
equivalent.
We observe that Theorem A may be considered as a uniﬁed generalized
Hom-version of [CbGC], Theorem 1.6, (ii).
Finally, in §4, we apply the techniques established in §2 and §3 to obtaina Hom-version of the combinatorial version of the Grothendieck conjecture,
where the outer representations under consideration are assumed to be “lgraphically full”, i.e., to satisfy a certain condition concerning “weights”
considered with respect to the “l-adic cyclotomic character”, where l is acertain prime number (cf. [CbGC], Deﬁnition 2.3, (iii)). In general, an outerrepresentation of a proﬁnite group I on another proﬁnite group G is deﬁned
to be a continuous homomorphism I −→ Out(G), where Out(G) denotes thegroup of continuous outer automorphisms of G equipped with the topology
induced by the compact-open topology on Aut(G), the group of continuous
automorphisms of G. It is well-known (and not diﬃcult to see) that, if G is
topologically ﬁnitely generated, then Aut(G), hence also Out(G), is proﬁnite.
If ρ : I −→ Out(G) is an outer representation of a proﬁnite group I on a
proﬁnite topologically ﬁnitely generated group G which is center-free, then,by pulling back the natural exact sequence of proﬁnite groups and continuous
homomorphisms
1/G/ Aut(G)/ Out(G)/ 1,

where the injection G −→ Aut(G) is given by g 7−→ (h 7−→ ghg −1 ) (cf.
the assumption that G is center-free), via the continuous homomorphism6ρ : I −→ Out(G), we obtain an exact sequence of proﬁnite groups and continuous homomorphisms
/G

1

out

/G o I/I/ 1.

In light of these observations, our main result is formulated as follows.
Theorem B (Theorem 4.4). Let I and J be proﬁnite groups, G a semigraph of anabelioids of pro-ΣG PSC-type, H a semi-graph of anabelioids ofpro-ΣH PSC-type, and ρI : I −→ Out(ΠG ) and ρJ : J −→ Out(ΠH ) outerrepresentations such that the images of ρI and ρJ are respectively includedin Aut(G) ⊂ Out(ΠG ) and Aut(H) ⊂ Out(ΠH ), where the usage of the symbol “⊂” is a slight abuse of notation (cf. the discussion preceding [CbGC],
Lemma 2.1). Let us consider the following commutative diagram of proﬁnite
groups and continuous homomorphisms:
1

1

/ ΠG

/ ΠG o I

out

/I

α

α̃

β


/ ΠH



/ ΠH o J

out


/J

/1

/ 1,

where the two horizontal sequences are the respective exact sequences associated to ρI and to ρJ (cf. the discussion preceding the present theorem; also
[CbGC], Remark 1.1.3, for the fact that ΠG and ΠH are topologically ﬁnitelye Suppose
generated and center-free). Let □ ∈ {Vert, Edge} and S ⊂ □(G).
further that there exists l ∈ ΣG ∩ ΣH for which the following four conditions
hold.
(i) For every c̃ ∈ S, there exists an open subgroup U ⊂ ΠH such that theimage of α(Πc̃ ) ∩ U via the natural surjection U −↠ U (l) ∼= Π(H′ )(l)is non-trivial, where the supersprict “(l)” denotes the maximal pro-lquotient.
(ii) I is l-graphically full with respect to the outer representation ρI : I −→Aut(G) (cf. [CbGC], Deﬁnition 2.3, (iii)).
(iii) I is l-graphically full with respect to the outer representation ρJ ◦β : I −→Aut(H) (cf. [CbGC], Deﬁnition 2.3, (iii)).
7

(iv) If, moreover, □ = Edge, then the two cyclotomic characters I −→ Z×lassociated to ρI : I −→ Aut(G) and to ρJ ◦ β : I −→ Aut(H) coincide.
e -compatible.
Then α is S, □(H)
Though the appearance of Theorem B is slightly diﬀerent from that ofTheorem 4.4, it is immediate that the these are essentially the same. Theorem
B, along with its proof, can be considered as a Hom-version of [CbGC],
Corollary 2.7, (ii). Also, it is possible to verify [CbGC], Corollary 2.7, (ii),utilizing Theorem B, albeit not straightforwardly (cf. the assumption (iv) of
Theorem B; Remark 4.4.1).

0

Notations and Conventions

Numbers
N>0 denotes the set or the (multiplicative) monoid of positive rational
integers. Z denotes the set or the (additive) group of rational integers. Q
denotes the set of rational numbers. Ẑ denotes the proﬁnite completion of
the group Z.

Topological Groups
Let K be a topological group and G, H ⊂ K closed subgroups of K. Then
we shall write CG (H) for the commensurator subgroup of H in G, i.e.,
CG (H) = {g ∈ G | H ∩ g −1 Hg is of ﬁnte index in H and g −1 Hg.}.
def

We shall say that the subgroup H is commensurably terminal in K if H =CK (H).
Let Σ be a set of prime numbers, l a prime number, and G a proﬁnite
group. Then we shall write GΣ for the maximal pro-Σ quotient of G and
def
G(l) = G{l} . We shall write Gab for the abelianization of the proﬁnite groupG, i.e., the quotient of G by the closure of the commutator subgroup of G.
Moreover, if α : G −→ H is a continuous homomorphism of proﬁnite groups,
then we shall write αΣ (resp. α(l) ; αab ) for the homomorphism GΣ −→ H Σ(resp. G(l) −→ H (l) ; Gab −→ H ab ) induced by α. We shall write Aut(G)8for the group of automorphisms of G (as a proﬁnite group) and Inn(G) for
the group of inner automorphisms of G. Thus we have a homomorphism ofgroups G −→ Aut(G) determind by conjugation whose image is Inn(G) ⊂ G.
We shall write Out(G) for the quotient of Aut(G) by the normal subgroup
Inn(G) ⊂ Aut(G). In particular, if G is center-free, then the natural homomorphism G −→ Inn(G) is an isomorphism; thus we have a natural exact
sequence of groups
1 −→ G −→ Aut(G) −→ Out(G) −→ 1.
If, moreover, G is topologically ﬁnitely generated, then one veriﬁes easily thatthe topology of G admits a basis of characteristic open subgroups, which thus
induces a proﬁnite topology on the groups Aut(G) and Out(G), with respect
to which the above exact sequence forms an exact sequence of proﬁnite groups
and continuous homomorphisms. In this situation, if, moreover, ρ : J −→out

Out(G) is a continuous homomorphism, then we shall write G o J for the
proﬁnite group obtained by pulling back the above exact sequence of proﬁnite
groups via ρ. Thus we have a natural exact sequence of proﬁnite groups andcontinuous homomorphisms
out

1 −→ G −→ G o J −→ J −→ 1.

Group Actions
Let G be a group, X a set, and φ : X × G −→ X a right action of G
on X. Then we shall simply write “X x G” and omit “φ” to express thissituation. Also, for x ∈ X and g ∈ G, we shall simply write xg instead ofφ(x, g).

1

Preliminaries

In this section, we prepare some deﬁnitions, notational conventions, andbasic lemmas concerning the PSC-fundamental groups of semi-graphs of anabelioids of PSC-type and continuous homomorphisms between them (which
are not necessarily isomorphisms). In particular, we deﬁne the group-theoretic
compatibility properties of such homomorphisms (cf. Deﬁnition 1.7 and Remark 1.7.4). The study of these properties is the central issue of the present
9series of papers.
A basic reference for the theory of semi-graphs of anabelioids of PSCtype is [CbGC]. We shall use the terminologies “semi-graph of anabelioidsof PSC-type”, “PSC-fundamental group of a semi-graph of anabelioids of
PSC-type”, “ﬁnite étale covering of semi-graphs of anabelioids of PSC-type”,“vertex ”, “edge”, “node”, “cusp”, “verticial subgroup”, “edge-like subgroup”,“nodal subgroup”, and “cuspidal subgroup”, as they are deﬁned in [CbGC],
Deﬁnition 1.1. Moreover, if G is a semi-graph of anabelioids of PSC-type,then we shall write ΣG for the (necessarily unique — cf. [CbGC], Remark
1.1.2) set of prime numbers such that G is a semi-graph of anabelioids ofpro-ΣG PSC-type. Also, we shall apply the various notational conventions
established in [NodNon], Deﬁnition 1.1; in particular, if G is a semi-graph ofanabelioids of PSC-type, then the pro-ΣG universal covering Ge −→ G is ﬁxed
throughout the discussion, and the PSC-fundamental group Π G is always
considered to be associated to the ﬁxed pro-ΣG universal covering Ge −→ G.
Thus there is a natural action Ge x ΠG , which induces a natural bijection
the set of open subgroups of ΠGU ⊂ ΠG



∼
=

−→
7−→

the set of the isomorphism classes!
of the connected ﬁnite étale;e
subcoverings of G −→ GGe U −→ G .

If G ′ −→ G is a connected ﬁnite étale subcovering of the pro-ΣG universal
covering Ge −→ G, then we always choose the implicit structure morphism
Ge −→ G ′ as the pro-ΣG ′ universal covering of G ′ . (Here, we note that ΣG =
ΣG ′ .) Under this convention,
the inverseof the natural bijection above isgiven by G ′ −→ G 7−→ ΠG ′ ⊂ ΠG . Finally, we shall refer to the “PSCfundamental group of a semi-graph of anabelioids of PSC-type” simply as
the “fundamental group” (of the semi-graph of anabelioids of PSC-type).
In the following, the letters “G” and “H” always denote semi-graphs ofanabelioids of PSC-type. Note that we do not assume that Σ G = ΣH .
e
Deﬁnition 1.1. We shall refer to any element of VCN(G) (resp. VCN(G))
e Moreover, if c̃ ∈ VCN(G),e then we shall write
as a component of G (resp. G).
Πc̃ for the VCN-subgroup of ΠG associated to c̃ (cf. also Remark 1.3.2 forthe notation “Πc ” for c ∈ VCN(G)).
Let us recall here the following fundamental results, which will be of
frequent use in the present series of papers.
10Lemma 1.2. The following assertions hold.
e Then one of the following two (mutually exclusive)(1) Let c̃, d˜ ∈ Edge(G).
conditions holds.
˜ In particular, the equality Πc̃ = Π ˜ holds.
• It holds that c̃ = d.
d
˜• It holds that c̃ 6= d, and the equality Πc̃ ∩ Π ˜ = 1 holds.
de and d˜ ∈ Vert(G).
e Then one of the following two(2) Let c̃ ∈ Edge(G)(mutually exclusive) conditions holds.
˜ In particular, the inclusion Πc̃ ( Π ˜ holds.
• c̃ abuts to d.
d˜• c̃ does not abut to d, and the equality Πc̃ ∩ Π ˜ = 1 holds.
de Then one of the following three (mutually exclusive)(3) Let c̃, d˜ ∈ Vert(G).
conditions holds.
˜ In particular, the equality Πc̃ = Π ˜ holds.
• It holds that c̃ = d.
d˜e
• It holds that c̃ 6= d, and there exists a unique node ẽ ∈ Node(G)˜ Moreover, the equality Πc̃ ∩Π ˜ = Πẽwhich abuts both to c̃ and to d.
dholds.
˜ and the equality Πc̃ ∩ Π ˜ = 1 holds. In partic• It holds that c̃ 6= d,dular (cf. the assertion (2)), there exists no edge which abuts both˜
to c̃ and to d.
Proof. The assertion (1) follows immediately from [NodNon], Lemma 1.5.
The assertion (2) follows immediately from [NodNon], Lemma 1.7 (cf. also
[CbGC], Remark 1.1.3, for the “6=” of “(”). The assertion (3) follows, in lightof the assertion (1), immediately from [NodNon], Lemma 1.8 and Lemma 1.9.
This completes the proof of Lemma 1.2.
e there exist
Remark 1.2.1. It is also well-known that, for every ẽ ∈ Node(G),precisely two vertices of Ge to which ẽ abuts, i.e., the two branches of ẽ abutto two distinct vertices (cf. [NodNon], Remark 1.2.1, (iii), or [HmCbGCII],
Remark 4.5.1). Combining this fact with Lemma 1.2, (3), we deduce that a
nodal subgroup of ΠG is uniquely presented as the intersection of two verticial
subgroups of ΠG .

11

Lemma 1.3. Recall that we have a natural right action Ge x ΠG which,e x ΠG . For a connectedin particular, determines a natural action VCN(G)′ﬁnite étale Galois subcovering G −→ G of Ge −→ G, deﬁne a right action
the set of VCN-subgroups of ΠG ′ x ΠG by conjugation, i.e., by the formuladef(Πc̃ ∩ ΠG ′ ) · γ = γ −1 Πc̃ γ ∩ ΠG ′ (cf. Remark 1.5.2). Relative to these actions,the following assertions hold.
e the VCN-subgroup Πc̃ ⊂ ΠG coincides with the(1) For any c̃ ∈ VCN(G),estabilizer of c̃ ∈ VCN(G).
(2) For any connected ﬁnite étale Galois subcovering G ′ −→ G of Ge −→ G,
we have a ΠG -equivariant bijection


∼
=
e
VCN(G) −→ the set of VCN-subgroups of ΠG ′ ; c̃ 7−→ Πc̃ ∩ ΠG ′ .
Proof. The assertion (1) follows immediately from the deﬁnitions (cf. [SemiAn],
Remark 2.2.1).
Let us verify the assertion (2). It follows immediately from the deﬁnitions
that the map under consideration is a ΠG -equivariant surjection. Moreover,in light of Remark 1.5.2, it follows from Lemma 1.2, applied to “G” = G ′ ,that the map under consideration is injective. This completes the proof of
the assertion (2), hence also the proof of Lemma 1.3.
Remark 1.3.1. It follows immediately from Lemma 1.3, (1), that, for every
e and γ ∈ ΠG , it holds that Πc̃γ = γ −1 Πc̃ γ.
c̃ ∈ VCN(G)Remark 1.3.2. It also follows from Lemma 1.3, (1), that, for c ∈ VCN(G)e such that c̃(G) = c, the ΠG -conjugacy class of Πc̃ isand c̃ ∈ VCN(G)completely determined by c. We shall write Πc for (some representativeof) this ΠG -conjugacy class. Note that (every representative of) Πc is precisely the image of (some representative of) the natural outer homomorphism
π1 (Gc ) −→ ΠG , where we write Gc for the consituent anabelioid of G at c.
Remark 1.3.3. Unfortunately, the equality Πc̃γ = γ −1 Πc̃ γ pointed out
in Remark 1.3.1 is inconsistent with the equality Π ṽγ = γΠṽ γ −1 (wheree implicitly given in the proof of [NodNon], Lemma 3.6. Thisṽ ∈ Vert(G))
inconsistency is not because of the logical ﬂaw of either loc. cit. or thepresent paper, but just because of the diﬀerence of the deﬁnitions. The author decided to adopt a diﬀerent notation from that of loc. cit. in order toavoid the confusing equality “c̃ab = (c̃b )a ” caused by the notation of loc. cit.
12

e and H a closed subgroup of ΠG . Then theLemma 1.4. Let c̃ ∈ VCN(G)following assertions hold.
(1) If 1 6= H ⊂ Πc̃ , then CΠG (H) ⊂ Πc̃ (cf. the discussion entitled “Topological Groups” in §0).
(2) If U is an open subgroup of H and U ⊂ Πc̃ , then H ⊂ Πc̃ .
Proof. First we verify the assertion (1). Note that any open subgroup of
H is non-trivial, since H is non-trivial, and ΠG is torsion-free (cf. [CbGC],
Remark 1.1.3).
Let γ ∈ CΠG (H). Then we have
1 6= H ∩ γ −1 Hγ ⊂ Πc̃ ∩ γ −1 Πc̃ γ,
where the “6=” follows from the fact that H ∩ γ −1 Hγ ⊂ H is open. In
particular, the two VCN-subgroups Πc̃ , γ −1 Πc̃ γ = Πc̃γ intersect non-trivially.
In light of Lemma 1.2, this non-triviality, in fact, implies either the equalityΠc̃ = γ −1 Πc̃ γ or the equality Πc̃ ∩ γ −1 Πc̃ γ = Πẽ , where in the latter case c̃ is
e is the unique node which abuts bothnecessarily a vertex and ẽ ∈ Node(G)γto c̃ and to c̃ .
Suppose that we are in the former case, i.e., that Πc̃ = γ −1 Πc̃ γ. (Fore Then, by the commensurable
instance, this will be the case if c̃ ∈ Edge(G).)
terminality of Πc̃ in ΠG (cf. [CbGC], Proposition 1.2, (ii)), we obtain thedesired relation γ ∈ CΠG (Πc̃ ) = Πc̃ .
Next, suppose that we are in the latter case, i.e., that Πc̃ ∩ γ −1 Πc̃ γ = Πẽ .
Then we have the relations 1 6= H ∩γ −1 Hγ ⊂ Πc̃ ∩γ −1 Πc̃ γ = Πẽ . By applying
the already veriﬁed former case to “(c̃, H)” = (ẽ, H ∩ γ −1 Hγ), it follows thatCΠG (H ∩ γ −1 Hγ) ⊂ Πẽ . Since Πẽ ⊂ Πc̃ and CΠG (H) = CΠG (H ∩ γ −1 Hγ)(where we note that H ∩γ −1 Hγ is open in H), we obtain the desired inclusionCΠG (H) ⊂ Πc̃ . This completes the proof of the assertion (1).
Next, we verify the assertion (2). If H is trivial, obviously we are done.
Thus we may assume that H is non-trivial. In this case U is also non-trivial,
since ΠG is torsion-free (cf. [CbGC], Remark 1.1.3). Thus the assertion (2)follows immediately from the assertion (1) applied to “H” = U , in light of thefact that H ⊂ CΠG (H) = CΠG (U ). This completes the proof of the assertion(2), hence also the proof of Lemma 1.4.
Now we proceed to give some deﬁnitions which play the most central rolesin the present series of papers.
13

e and G ′ −→ G a connected ﬁnite étale
Deﬁnition 1.5. Let S ⊂ VCN(G)subcovering of Ge −→ G.
e via the natural(1) We shall write S(G ′ ) for the image of S ⊂ VCN(G)e −↠ VCN(G ′ ).
surjection VCN(G)e is said to be S-like if it belongs(2) An element of VCN(G ′ ) (resp. VCN(G))
to S(G ′ ) (resp. S).
(3) A subgroup of ΠG ′ is said to be S-like if it arises as the VCN-subgroupe
associated to an S-like component of G.
(4) We shall write MG ′ for the abelianization Πab
G ′ of ΠG ′ . Moreover, we shallwrite MGS′ for the closed submodule of MG ′ generated by the images of
the S-like subgroups of ΠG ′ via the natural surjection ΠG ′ −↠ MG ′ .
defeVert(G)= MG ′(5) We shall write MGvert
′

def

e
Edge(G)

and MGedge= MG ′′.

edge
Remark 1.5.1. The deﬁnitions of the notations “MGvert” coin′ ” and “MG ′cide with those of [CbGC], Deﬁnition 1.1, (ii).

Remark 1.5.2. Suppose that we are in the situation of Deﬁnition 1.5. Thenit follows immediately from Lemma 1.3, (1), that the “Π c̃ ” of ΠG ′ , i.e., thee = VCN(Ge′ ), is equal to
VCN-subgroup of ΠG ′ corresponding to c̃ ∈ VCN(G)Πc̃ ∩ ΠG ′ . In particular, it holds that a subgroup of ΠG ′ is S-like if and only
if it can be written as the intersection of an S-like subgroup of ΠG with ΠG ′ .
Remark 1.5.3. Suppose that we are in the situation of Deﬁnition 1.5. Thenit holds, as is sketched below, that MG /MGS is free over ẐΣG , hence torsionfree. In particular, MGS ⊂ MG coincides with the “saturation” of (i.e., theinverse image via the natural surjection of the torsion part of the quotient
module of MG by) the image of MGS′ via the natural homomorphism MG ′ −→MG induced by the inclusion homomorphism ΠG ′ −→ ΠG .
One way to verify the freeness of MG /MGS over ẐΣG is as follows. Write
X for “the compact Riemann surface minus some points corresponding to
G”. Then it follows from basic algebraic topology, together with the link
between topological and algebraic fundamental groups, that MG is naturally
isomorphic to H1 (X, ∅)⊗Z ẐΣG , where “H1 ” denotes the ﬁrst relative singularhomology group. Then one constructs an appropriate subset Y ⊂ X suchthat MG /MGS injects (over ẐΣG ) to H1 (X, Y ) ⊗Z ẐΣG . (For example, if S ⊂14

Vert(G), then one may take such a “Y ” to be the disjoint union inside X ofthe various “sub Riemann surfaces” of X determined by various c ∈ S(G).)
Finally, one veriﬁes (by excision isomorphism, for example) that H1 (X, Y ) is
free over Z.
Remark 1.5.4. In the situation of Deﬁnition 1.5,
S for the we shall write′′
projective limit of the projective system S(G ) G ′ −→G , where G −→ G runs
through the connected ﬁnite étale subcoverings of Ge −→ G. Then it ise and ineasily veriﬁed that S is naturally regarded as a subset of VCN(G)e with respect to the proﬁnite
fact coincides with the closure of S in VCN(G)etopology of VCN(G).
e and G ′ −→ G a connected ﬁnite étale
Remark 1.5.5. Let S ⊂ VCN(G)subcovering of Ge −→ G. Then another immediate consequence of the definition of S (cf. Remark 1.5.4) is that S(G ′ ) = S(G ′ ). Put another way,roughly speaking, the diﬀerence between S and S is “invisible” in any ﬁnite approximation level. It also follows formally from this observation that
MGS′ = MGS′ .
e Then it follows
Remark 1.5.6. Let A be a subset of ΠG and S ⊂ VCN(G).
from Lemma 1.3, (1), and Remark 1.5.5, together with the well-known factthat the projective limit of a projective system consisting of non-empty ﬁnite
sets is non-empty, that A is included in an S-like subgroup of ΠG if and only
if there exists a coﬁnal subsystem (Gλ −→ G)λ∈Λ of (the projective system
which gives rises to) the pro-ΣG universal covering Ge −→ G constituted byﬁnite étale Galois subcoverings of Ge −→ G such that, for every λ ∈ Λ, there
exists cλ ∈ S(Gλ ) stabilized by every element of A with respect to the naturalaction VCN(Gλ ) x ΠG ⊃ A.
e
Deﬁnition 1.6. Let S ⊂ VCN(G).
(1) We shall writee | ẽ abuts to a vertex in S.}.
ES = S ∪ {ẽ ∈ Edge(G)def(2) We shall say that S is edge-complete if it holds that S = ES .
e and A a subset of ΠG . Then it follows
Remark 1.6.1. Let S ⊂ VCN(G)immediately from Lemma 1.2, (2), that A is included in an S-like subgroupof ΠG if and only if it is included in an ES -like subgroup of ΠG .
15

Remark 1.6.2. It follows immediately from the deﬁnitions (cf. Remark
e E = ES .
1.5.4) that, for every S ⊂ VCN(G),S
Deﬁnition 1.7. Let α : ΠG −→ ΠH be a continuous homomorphism, S ⊂e and T ⊂ VCN(H).
eVCN(G),(1) We shall say that α is (S, T )-compatible if, for every c̃ ∈ S, α(Πc̃ ) = 1
or there exists d˜ ∈ T such that α(Πc̃ ) ⊂ Πd˜.
(2) We shall say that α is (S, T )-ﬁltration-preserving if, for every cone −→ H, it holds thatnected ﬁnite étale subcovering H′ −→ H of HabST′α (MG ′ ) ⊂ MH′ , where we write G −→ G for the connected ﬁnite
étale subcovering of Ge −→ G corresponding to the open subgroupα−1 (ΠH′ ) of ΠG (i.e., ΠG ′ = α−1 (ΠH′ )) and αab for the homomorphism
MG ′ −→ MH′ naturally induced by α.
Remark 1.7.1. Suppose that we are in the situation of Deﬁnition 1.7. Thenthe part “α(Πc̃ ) = 1 or” in Deﬁnition 1.7, (1), is relevant only when T = ∅,and we often omit the mention of this case. The reason we allowed such αthat α(Πc̃ ) = 1 for every c̃ ∈ S to be (S, ∅)-compatible (even when S 6= ∅) isto simplify things. See Remark 3.2.2 for an example.
Example 1.7.2. Suppose that G is sturdy (cf. [CbGC], Deﬁnition 1.1, (ii),and [CbGC], Remark 1.1.5). Write G cpt for the compactiﬁcation of G (cf.
[CbGC], Remark 1.1.6), which is again a semi-graph of anabelioids of pro-Σ G
PSC-type. Then it is immediate that the natural surjection α : ΠG −↠ ΠG cptcpt ))-compatible, (Node(G),cpt ))-compatible, ande Vert(Gge Node(Ggis (Vert(G),cpt ))-compatible. Note, however, that if assuming the parte Cusp(Gg(Cusp(G),cpt ))e Cusp(Gg“α(Πc̃ ) = 1 or” in Deﬁnition 1.7, (1), is omitted, then (Cusp( G),compatibility of α is not satisﬁed unless Cusp(G) = ∅.
Remark 1.7.3. Suppose that we are in the situation of Deﬁnition 1.7. Thenit follows immediately from Remark 1.5.3 that, in order to verify that α
is (S, T )-ﬁltration-preserving, it suﬃces to verify the following (apparently
weaker) condition: there exists a coﬁnal subsystem (Hλ −→ H)λ∈Λ of (thee −→ Hprojective system which gives rise to) the pro-ΣH universal covering H
e −→ H such that
constituted by connected ﬁnite étale subcoverings of H
αab (MGSλ ) ⊂ MHT λ for every λ ∈ Λ, where we write Gλ −→ G for the connectedﬁnite étale subcovering of Ge −→ G corresponding to the open subgroupα−1 (ΠHλ ) of ΠG (i.e., ΠGλ = α−1 (ΠHλ )).
16Remark 1.7.4. As is noted in Remark 1.7.5 below, the property of being(S, T )-compatible, deﬁned for arbitrary continuous homomorphisms α : ΠG −→ΠH , can be considered as the uniﬁed generalized Hom-version of the properties of being “group-theoretically verticial”, “group-theoretically edge-like”,“group-theoretically nodal”, and “group-theoretically cuspidal”, deﬁned only∼=
for isomorphisms α : ΠG −→ ΠH (cf. [CbGC], Deﬁnition 1.4, (iv), and[NodNon], Deﬁnition 1.12). In light of this observation, we shall collectively refer to the properties of being (S, T )-compatible for some S, T as thegroup-theoretic compatibility properties.
Similarly, we shall refer to the properties of being (S, T )-ﬁltration-preserving
for some S, T as the ﬁltration-preservation properties.
∼=

Remark 1.7.5. Let α : ΠG −→ ΠH be an isomorphism of proﬁnite groups.
Then it holds that, for α to be group-theoretically verticial in the sense ofe Vert(H)e [CbGC], Deﬁnition 1.4, (iv), it is necessary and suﬃcient that α is Vert(G),e Vert(G)e -compatible. Similar remarks also
compatible and α−1 is Vert(H),hold for the terms “group-theoretically edge-like”, “group-theoretically nodal”,“group-theoretically cuspidal”, “verticially ﬁltration-preserving”, and “edgewise ﬁltration-preserving” (cf. [CbGC], Deﬁnition 1.4, (iii), (iv), and [NodNon],Deﬁnition 1.12). Indeed, the equivalences on the ﬁltration-preservation properties follow formally from the deﬁnitions. The equivalences on the grouptheoretic compatibility properties are also easy, though we include the proof
in Lemma 1.9 below.
Remark 1.7.6. Let α : ΠG −→ ΠH be a continuous homomorphism. Then it
follows from [NodNon], Proposition 1.13, that, if α is an isomorphism, thenthe group-theoretic verticiality (cf. [CbGC], Deﬁnition 1.4, (iv)) of α implies
the group-theoretic nodality (cf. [NodNon], Deﬁnition 1.12) of α. However,eeone may verify that the Vert(G), Vert(H) -compatibility of α does not implye Node(H)e -compatibility of α in general. A counter-example
the Node(G),is given in Example 1.7.7 below.
Here, we explain what a problem occurs when one tries to apply to oursituation a similar argument to the proof of [NodNon], Proposition 1.13.
eee and ṽ1 , ṽ2
Suppose that α is Vert(G), Vert(H) -compatible. Let ẽ ∈ Node(G)the two distinct vertices to which ẽ abuts (cf. Remark 1.2.1). By assumption,e such that α(Πṽ ) ⊂ Πw̃ . Byfor each i ∈ {1, 2}, there exists w̃i ∈ Vert(H)iiLemma 1.2, (3), it holds that Πẽ = Πṽ1 ∩ Πṽ2 , hence α(Πẽ ) ⊂ Πw̃1 ∩ Πw̃2 .
However, it is still possible that w̃1 = w̃2 , which implies that Πw̃1 ∩ Πw̃2 is17

not a nodal subgroup (but a verticial subgroup) of Π H . In particular, one
cannot conclude that α(Πẽ ) is included in a nodal subgroup.
Example 1.7.7. Let G be a semi-graph of anabelioids of PSC-type such that
Node(G) 6= ∅, H its generization associated to the full subset Node(G) ⊂∼=Node(G) (cf. [CbTpI], Deﬁnition 2.8), and α : ΠG −→ ΠH the inverse of∼=(some representative of) the specialization outer isomorphism Π H −→ ΠGe Vert(H)e (cf. [CbTpI], Deﬁnition 2.10). Then it is clear that α is Vert(G),e Node(H)e -compatible.
compatible while it is not Node(G),Corollary 1.8. Let α : ΠG −→ ΠH be a continuous homomorphism, G ′ −→ G
(resp. H′ −→ H) a connected ﬁnite étale subcovering of Ge −→ G (resp.
e −→ H), S ⊂ VCN(G),e and T ⊂ VCN(H).
e Suppose that α(ΠG ′ ) ⊂ ΠH′ .
H′
Write α : ΠG ′ −→ ΠH′ for the restriction of α. Then α is (S, T )-compatible
if and only if α′ is (S, T )-compatible.
Proof. This follows easily from Lemma 1.4, (2), and Remark 1.5.2.
∼=

Lemma 1.9. Let α : ΠG −→ ΠH be an isomorphism of proﬁnite groups,
e and T ⊂ □(H).
e□ ∈ {Vert, Edge}, S ⊂ □(G),
Then the following two
conditions are equivalent.
(i) α maps each S-like subgroup of ΠG isomorphically onto a T -like subgroup of ΠH , and, moreover, every T -like subgroup of ΠH arises in thisfashion.
(ii) α is (S, T )-compatible, and α−1 is (T, S)-compatible.
Proof. The implication (i) =⇒ (ii) is clear. To show the converse, supposethat α is (S, T )-compatible and that α−1 is (T, S)-compatible. Let c̃ ∈ S.
The assumptions on α imply that there exists d˜ ∈ T such that α(Πc̃ ) ⊂Πd˜. In a similar vein, we have a component c̃′ ∈ S satisfying the relation
α−1 (Πd˜) ⊂ Πc̃′ , hence the relation Πc̃ ⊂ α−1 (Πd˜) ⊂ Πc̃′ . Then, if □ = Edge
(resp. □ = Vert), then it follows from Lemma 1.2, (1) (resp. Lemma 1.2,(2), (3)), that Πc̃ = α−1 (Πd˜) = Πc̃′ . Thus α(Πc̃ ) is equal to Πd˜, hence a
T -like subgroup of ΠH . By switching the roles played by α and α−1 in theabove argument, it also follows that every T -like subgroup of ΠH arises inthis fashion. This completes the proof of the implication (ii) =⇒ (i), hence
also the proof of Lemma 1.9.
18

Lemma 1.10. Let α : ΠG −→ ΠH be a continuous homomorphism, S ⊂e and T ⊂ VCN(H).
e We shall write S ΠG def
e | c̃ ∈VCN(G),= {c̃γ ∈ VCN(G)e in a similar vein. Then, if α isS, γ ∈ ΠG } and we deﬁne T ΠH ⊂ VCN(H)(S, T )-compatible, then α is (S ΠG , T ΠH )-compatible.
Proof. By Remark 1.3.1, the S ΠG -like subgroups of ΠG are precisely the ΠG conjugates of the S-like subgroups of ΠG ; a similar assertion holds also for
T ΠH -subgroups of ΠH . Lemma 1.10 follows immediately from this observation.

2

Reduction to the Case of a Smaller Set ofPrime Numbers

In this section, we give a detailed exposition on the argument of “reduction to the case of a smaller set of prime numbers”, which is well-known
to experts. As a consequence, we prove that, in order to show a certain
group-theoretic compatibility property of a homomorphism between PSCfundamental groups, it suﬃces to verify certain group-theoretic compatibility properties of homomorphisms between the maximal pro-Σ quotients ofvarious open subgroups, where Σ is a certain “smaller” set of prime numbers(cf. Lemma 2.5 below). Even though only the particular case where the“smaller set of prime numbers” is given by “{l}”, a singleton, is applied in
the present series of papers, the author decided to include the general casebecause the proof is entirely the same.
The notational and terminological conventions established in the discussion preceding Deﬁnition 1.1 remains valid; in particular, the letters “G” and“H” always denote semi-graphs of anabelioids of PSC-type; we do not assume that ΣG = ΣH .
First, let us make it explicit how to consider the maximal pro-Σ quotientΠΣ
G (where ∅ 6= Σ ⊂ ΣG ) itself as a PSC-fundamental group.
Lemma 2.1. Let Σ be a non-empty subset of ΣG and G ′ −→ G a connectedﬁnite étale subcovering of Ge −→ G. Write (G ′ )Σ for the pro-Σ completion of
G ′ , i.e., the semi-graph of anabelioids obtained by replacing the constituent
anabelioids of G ′ by their pro-Σ completions (cf. [SemiAn], Deﬁnition 2.9,(ii)).
19

(1) The semi-graph of anabelioids (G ′ )Σ is of pro-Σ PSC-type. Moreover,′ )Σ −→ (G ′ )Σ from]one can construct the pro-Σ universal covering (Gthe pro-ΣG universal covering Ge −→ G in a natural way.
(2) The fundamental group Π(G ′ )Σ of (G ′ )Σ is naturally isomorphic to the′ Σ]
maximal pro-Σ quotient ΠΣG ′ of ΠG ′ . Here, we choose the “(G ) −→(G ′ )Σ ” constructed naturally from the pro-ΣG universal covering Ge −→G (cf. the assertion (1)) as the pro-Σ universal covering of (G ′ )Σ , andwe think Π(G ′ )Σ as associated to that pro-Σ universal covering.
(3) Suppose that we are in the situation of the assertion (2). Write X forthe set of VCN-subgroups of ΠG ′ ; write Y for the set of VCN-subgroupsof Π(G ′ )Σ . Then we have a natural commutative diagram∼=e
VCN(G)′ )Σ
]VCN (G



/X

∼
=


/ Y,

where the right-hand vertical arrow X −↠ Y maps a VCN-subgroupof ΠG ′ to its image via the natural surjection ΠG ′ −↠ Π(G ′ )Σ (cf. theassertion (2)).
(4) Suppose that we are in the situation of the assertion (3) and moreoverthat the connected ﬁnite étale covering G ′ −→ G is Galois. Recall thate and on X (cf. Lemma
we have a natural right actions of ΠG on VCN(G)′ )Σ and on]
1.3). Then we have natural right actions of ΠG on VCN (GY with respect to which the four arrows in the diagram of the assertion(3) are all ΠG -equivariant.
(5) By forming the quotients of (each vertex of ) the diagram in the assertion (3) by the actions of ΠG ′ ⊂ ΠG (cf. the assertion (4)), one obtains
a commutative diagram of sets with right ΠG actions and ΠG -equivariant
maps∼=/ X/ΠG ′VCN(G ′ )∼=



VCN((G ′ )Σ )
20

∼
=



∼
=

/ Y /ΠG ′ .

Moreover, the left-hand vertical arrow of this diagram coincides with
∼=the “identity” map VCN(G ′ ) −→ VCN (G ′ )Σ ; c 7−→ c.
(6) Suppose that we are in the situation of the assertion (2); in particular,
by conjugation, we have a natural right action of ΠG on ΠG ′ , which∼
thus induces a right action of ΠG on ΠΣG ′ = Π(G ′ )Σ . Then, for every∼=γ ∈ ΠG , the isomorphism Π(G ′ )Σ −→ Π(G ′ )Σ induced by γ is graphic in
the sense of [CbGC], Deﬁnition 1.4, (i). Moreover, the action Y x ΠGin the assertion (4) coincides with the action induced by the action
Π(G ′ )Σ x ΠG .
Proof. First, we verify the assertion (1). The assertion that the semi-graphof anabelioids (G ′ )Σ is of pro-Σ PSC-type follows immediately from thedeﬁnitions. Thus we have only to construct the pro-Σ universal covering′ )Σ −→ (G ′ )Σ naturally from the pro-Σ universal covering G]
e −→ G. Since(GG
the implicit structure morphism Ge −→ G ′ of the subcovering G ′ −→ G of
Ge −→ G gives the pro-ΣG ′ universal covering of G ′ , we may assume thatG ′ = G. However, for the sake of the later use (cf. the proof of the assertion(4)), we continue to use the notation “G ′ ”.
Suppose that the pro-ΣG ′ universal covering Ge −→ G ′ arises from a proobject Pe = (Pλ )λ∈Λ of the Galois category B(G ′ ). Suppose moreover, forsimplicity, that Pλ is a connected object of B(G ′ ) for every λ ∈ Λ and thatevery connected object of B(G ′ ) is isomorphic to Pλ for some λ ∈ Λ. Then wedef
deﬁne M = {λ ∈ Λ | Pλ is Galois of degree ∈ N>0 (Σ) in B(G ′ ).}, where wewrite N>0 (Σ) for the (multiplicative) submonoid of N>0 (freely) generated by
the elements of Σ. In light of Claim 2.1.A below, it follows immediately thatPµ belongs to the full subcategory B((G ′ )Σ ) of B(G ′ ) for every µ ∈ M andthat the pro-object (Pµ )µ∈M of B((G ′ )Σ ) deﬁnes a pro-Σ universal coveringof (G ′ )Σ . Thus it suﬃces to verify Claim 2.1.A below:
Claim 2.1.A: Let A be an object of B(G ′ ). Then A is a connected
Galois object of B(G ′ ) of degree in N>0 (Σ) if and only if A belongs
to the full subcategory B((G ′ )Σ ) and is a connected Galois objectof B((G ′ )Σ ) of degree in N>0 (Σ).
Recall that the ﬁnite direct sums in the Galois category B(G ′ ) (resp. B((G ′ )Σ ))are constructed by taking the ﬁnite direct sums in (the underlying Galois category of) each constituent anabelioid on each vertex and then gluing them21together in an appropriate way; similar constructions are valid also for thequotients by ﬁnite group actions. It follows immediately from this observation that the inclusion functor B((G ′ )Σ ) −→ B(G ′ ) preserves ﬁnite directsums and quotients by ﬁnite group actions. Thus we have only to show that,
if A is a connected Galois object of B(G ′ ) of degree in N>0 (Σ), then A belongs to B((G ′ )Σ ). Let v ∈ Vert(G ′ ). Write Gv′ for (the underlying Galois
category of) the constituent anabelioid of G ′ at v; write φv for the natural
exact functor B(G ′ ) −→ Gv′ . Then, in light of the construction of the quotients by a ﬁnite group actions in B(G ′ ), together with the assumption that
A is Galois in B(G ′ ), it follows that φv (A) is a direct sum of some mutually isomorphic connected Galois objects of Gv′ . If φv (A) is a direct sum ofm pieces of mutually isomorphic connected Galois objects of Gv′ of degree n(where m, n ∈ N>0 ), then mn is in N>0 (Σ) by assumption, hence n is alsoin N>0 (Σ). This completes the proof of Claim 2.1.A, hence also the proof of
the assertion (1).
Next, let us verify the assertion (2). We continue to use the notations
applied in the proof of the assertion (1). By deﬁnition, Π G ′ is the opposite
group of the (proﬁnite) automorphism group of the pro-object Pe = (Pλ )λ∈Λin B(G ′ ); hence, by deﬁnition of M , ΠΣG ′ is the opposite group of the (proﬁnite)
automorphism group of the pro-object (Pµ )µ∈M in B(G ′ ). On the other hand,
as is stated essentially in the assertion (2), we think Π(G ′ )Σ as the opposite
group of the (proﬁnite) automorphism group of the pro-object Pe = (Pλ )λ∈M
in B((G ′ )Σ ) (cf. the proof of the assertion (1)). Since the inclusion functor B((G ′ )Σ ) −→ B(G ′ ) is fully faithful, we obtain a canonical isomorphism∼=
Π(G ′ )Σ −→ ΠΣG ′ . This completes the proof of the assertion (2).
Let us explain the commutative diagram in the assertion (3). The horizontal arrows are given in Lemma 1.3, (2). The left-hand vertical arrowis just the natural projection (cf. the proof of the assertion (4) for moredetail). The right-hand vertical arrow is as in the statement. The only nontrivial point is the well-deﬁnedness of the right-hand vertical arrow; this iseasily veriﬁed and left to the reader. Then the commutativity of the diagram
follows immediately.
Next, let us verify the assertion (4). It suﬃces to give a natural right′ )Σ which is compatible with the natural projection]
action of ΠG on VCN (G′ )Σ . Suppose that the pro-Σ universal covering G]e −↠ VCN (Ge −→VCN(G)
Ge = (Qν )ν∈N of B(G) and that the factorization
G arises from a pro-object Q

22

e −→
Ge −→ G ′ −→ G under consideration corresponds to a factorization Q′
Q −→ (terminal object). Suppose moreover that Qν is a connected object ofB(G) for every ν ∈ N and that every connected object of B(G) is isomorphic
to Qν for some ν ∈ N . Recall that the Galois category B(G ′ ) is naturallye −→ Q′ determines
isomorphic to the slice category of B(G) over Q′ . Then Q′
a pro-object of B(G ) (which corresponds to the pro-ΣG ′ universal coveringGe −→ G ′ ); this is precisely the “Pe” in the proof of the assertion (1). Write′ )Σ for the pro-object of B(G) corresponding to “(G′ )Σ ”, i.e., the pro-object]](Qof B(G) obtained as the image of “(Pµ )µ∈M ” in the proof of the assertion (1)via the forgetful functor B(G ′ ) −→ B(G). Then it follows from the various′ )Σ −→ Q′ .
]e −→ (Qdeﬁnitions involved that we have a natural factorization Qe =Moreover, since Q′ is a connected Galois object of B(G), every γ ∈ Aut(Q)opΠG induces a commutative diagrameQ
natural

γ
∼
=



e
/Q


naturalinduced by γ
/ Q′ ,Q′∼=′ )Σ , a commutative diagram]
hence, by the deﬁnition of (QeQnatural

γ
∼
=

e
/Q
naturalinduced by γ
′ )Σ
′ )Σ
]
]
/ (Q
(Q
∼
=
naturalQ′

induced by γ∼=natural/ Q′ .

Since, for every µ ∈ M in the proof of the assertion (1), the three semigraphs of anabelioids corresponding to Pµ in B((G ′ )Σ ), to Pµ in B(G ′ ), and to
the image of Pµ in B(G), respectively, have mutually canonically isomorphic′ )Σ x Π .
]underlying semi-graphs, we obtain the desired action VCN (GG
By the commutativity of the upper half square of the second diagram above,′ )Σ is Π ]e −↠ VCN (Git also follows that the natural projection VCN(G)G
equivariant with respect to this action. This completes the proof of theassertion (4).
23The assertion (5) follows immediately from the various deﬁnitions involved, in light of the assumption that the covering G ′ −→ G is Galois.
Finally, let us verify the assertion (6). The graphicity of the isomorphism∼=
Π(G ′ )Σ −→ Π(G ′ )Σ induced by γ ∈ ΠG follows from the graphicity of the
conjugation action of ΠG on ΠG ′ , together with the well-deﬁnedness of theright-hand vertical arrow of the diagram in the assertion (3) (the proof ofwhich we left to the reader). To show that the action Y x ΠG in the assertion(4) coincides with the action induced by the action Π(G ′ )Σ x ΠG , it suﬃces
to show that the right-hand vertical arrow X −↠ Y of the diagram in theassertion (3) is ΠG -equivariant with respect to the natural action of X x ΠG
and the action of Y x ΠG induced by the action Π(G ′ )Σ x ΠG . On the other
hand, this is immediate from the deﬁnitions. This completes the proof of
Lemma 2.1.
Lemma 2.2. Let H ⊂ ΠG be a closed subgroup. Then the following assertions
hold.
(1) Suppose that H 6= 1. Then there exist l ∈ ΣG and a connected ﬁnite
étale subcovering G ′ −→ G of Ge −→ G such that the image of H ∩ ΠG ′(l)
via the natural surjection ΠG ′ −→ ΠG ′ is non-trivial.
(2) Let Σ be a non-empty subset of ΣG such that the image of H via thenatural surjection ΠG −→ ΠΣG is non-trivial. Then, for every connected′ﬁnite étale subcovering G −→ G of Ge −→ G, the image of H ∩ ΠG ′ via
the natural surjection ΠG ′ −→ ΠΣG ′ is non-trivial.
Proof. First, let us verify the assertion (1). We may assume that H is procyclic. Write J for the H assumed to be non-trivial and pro-cyclic. Since J is
non-trivial, there exists an open normal subgroup N of ΠG such that J ∩N (J. Write V = JN ; G ′ −→ G for the connected ﬁnite étale subcoveringG ′ −→ G of Ge −→ G corresponding to V (i.e., V = ΠG ′ ). Let l be a prime
number which divides the order of J/(J ∩ N ) = V /N (hence is contained in
ΣG ). Then we obtain a natural commutative diagram
J




J/(J ∩ N )/V/ / V (l)





V /N

24

/ / (V /N )(l)̸=1,where the “6=” follows from the choice of l, together with the cyclicity of
J/J ∩ N . Then the desired non-triviality of J −→ V (l) follows formally fromthe easily veriﬁed non-triviality of the composition J −↠ V /N −↠ (V /N )(l) .
This completes the proof of the assertion (1).
The assertion (2) follows immediately from the natural commutative diagram/ ΠG/ / ΠΣ
HO 
G
O
O

?

H ∩ ΠG ′ ?
/ ΠG ′/ / ΠΣ′ ,G

together with the assumption that the image of H in ΠΣG is non-trivial andΣthe fact that ΠG is torsion-free (cf. Lemma 2.1, (2), and [CbGC], Remark
1.1.3). This completes the proof of the assertion (2), hence also the proof of
Lemma 2.2.
e Then,Deﬁnition 2.3. Let Σ be a non-empty subset of ΣG and S ⊂ VCN(G).
for every connected ﬁnite étale subcovering G ′ −→ G of Ge −→ G, we shall′ )Σ for the inverse image of S(G ′ ) ⊂ VCN(G ′ ) ∼]e ′ , Σ) ⊂ VCN (Gwrite S(G
=



′ Σ
′
Σ
′ )Σ −↠ VCN (G )]VCN (G ) via the natural surjection VCN (G.
e and H ⊂ ΠG
Lemma 2.4. Let Σ be a non-empty subset of ΣG , S ⊂ VCN(G),a closed subgroup. For every connected ﬁnite étale subcovering G ′ −→ Gof Ge −→ G, write Im(H ∩ ΠG ′ ) for the image of H ∩ ΠG ′ via the natural∼surjection ΠG ′ −↠ ΠΣ
G ′ = Π(G ′ )Σ (cf. Lemma 2.1, (2)). Suppose that thefollowing condition holds:
there exists a connected ﬁnite étale subcovering G ′ −→ G of Ge −→G such that Im(H ∩ ΠG ′ ) 6= 1.
Then the following conditions are equivalent.
(i) H is included in an S-like subgroup of ΠG .
(ii) There exists a coﬁnal subsystem (Gλ −→ G)λ∈Λ of (the projective system
which gives rise to) the pro-ΣG universal covering Ge −→ G constituted
by connected ﬁnite étale subcoverings of Ge −→ G such that, for every
e λ , Σ)-like subgroup of Π(G )Σ .
λ ∈ Λ, Im(H ∩ ΠGλ ) is included in an S(Gλ25Proof. The implication (i) =⇒ (ii) is easy.
Let us verify the converse. We may assume that S is edge-complete (cf.
Remark 1.6.1 and Remark 1.6.2).
First, we claim that we may assume that Gλ −→ G is Galois for every
λ ∈ Λ. To verify this claim, it suﬃces to show that, for every connectedﬁnite étale Galois subcovering G ′ −→ G of Ge −→ G, Im(H ∩ ΠG ′ ) ⊂ ΠΣG ′ is′e
included in a S(G , Σ)-like subgroup of Π(G ′ )Σ . (In fact, as is evident fromthe following proof, the Galois assumption on G ′ −→ G is even superﬂuous.)
By the coﬁnality assumption on (Gλ −→ G)λ∈Λ , there exists λ ∈ Λ such thatthe morphism Gλ −→ G factors through G ′ −→ G. Then we form a naturalcommutative diagram
H ∩O ΠG ′ ?

H ∩ ΠGλ / ΠG ′
O/ / ΠΣ′
OG

?
/ ΠG

/ / ΠΣ ,λGλwhere we note that the image of the composite of the upper (resp. lower)Σhorizontal arrows is precisely Im(H ∩ ΠG ′ ) ⊂ ΠΣG ′ (resp. Im(H ∩ ΠGλ ) ⊂ ΠGλ ).
Σ
Now it follows from the assumption that Im(H ∩ ΠGλ ) ⊂ ΠGλ is included
e λ , Σ)-like subgroup of Π(G )Σ , together with the commutativity of
in a S(Gλthe above diagram, that an open subgroup of Im(H ∩ ΠG ′ ) ⊂ ΠΣG ′ , hence
e ′ , Σ)-likealso Im(H ∩ ΠG ′ ) itself (cf. Lemma 1.4, (2)), is included in a S(Gsubgroup of Π(G ′ )Σ . This competes the proof of the reduction to the casewhere Gλ −→ G is Galois for every λ ∈ Λ.
Also, we observe that it follows from Lemma 2.2, (2), applied to “(G, H, Σ)” =′(G , H ∩ ΠG ′ , Σ), that there exists a coﬁnal subsystem (Gλ −→ G)λ∈Λ′ of(Gλ −→ G)λ∈Λ (where Λ′ ⊂ Λ) such that, for every λ ∈ Λ′ , Im(H ∩ ΠGλ ) 6= 1.
By replacing (Gλ −→ G)λ∈Λ by (Gλ −→ G)λ∈Λ′ , we may assume that, forevery λ ∈ Λ, it holds that Im(H ∩ ΠGλ ) 6= 1.
Now we show the condition (i). Here, the readers are recommended torecall the contents of Lemma 2.1, because the group actions, identiﬁcations,and notational conventions given in Lemma 2.1 will be applied without further mention below. It follows from Remark 1.5.6 that it suﬃces to showthat there exists an element of S(Gλ ) stabilized by the action of H ⊂ ΠG .
For this, in light of Lemma 2.1, (3), (4), and (5), it suﬃces to verify thefollowing claim:

26

e λ , Σ) ⊂Claim 2.4.A: For every λ ∈ Λ, there exists ceλ ∈ S(GΣ such that the subgroup Π^
VCN (G⊂ Π(Gλ )Σ is stabilized
λ)cf
λby the action of H ⊂ ΠG .
e λ , Σ) such that Im(H ∩ ΠG ) ⊂To this end, take (by assumption) c̃ ∈ S(Gλ
Π(Gλ )Σ is included in Πc̃ ⊂ Π(Gλ )Σ . Since (it follows immediately from thedeﬁnition of the action of ΠG on Π(Gλ )Σ that) the equality Im(H ∩ ΠGλ )γ =Im(H ∩ΠGλ ) holds for every γ ∈ H, the inclusion Im(H ∩ΠGλ ) ⊂ Πc̃ ∩Πγc̃ holds
for every γ ∈ H. In light of Lemma 1.2, (1) and (3), together with our further
assumption that Im(H ∩ ΠGλ ) 6= 1 (cf. the sentence preceding the sentence“Now we show the condition (i).”), the inclusion Im(H ∩ ΠGλ ) ⊂ Πc̃ ∩ Πγc̃
implies the following claim:
Claim 2.4.B: For every γ ∈ H, either the equalityΠc̃ = Πγc̃or the equalityΠc̃ ∩ Πγc̃ = Πẽholds, where in the latter case c̃ is necessarily a vertex and ẽ ∈Σ is the unique node which abuts both to c̃ and to c̃γ .
^Node (Gλ)
e λ , Σ)If, for every γ ∈ H, the former case of Claim 2.4.B holds, then c̃ ∈ S(Ggives the desired “ceλ ” of Claim 2.4.A. Thus we may assume that there existsΣ^γ ′ ∈ H for which the latter case of Claim 2.4.B holds. Take ẽ ∈ Node (Gλ)′
e λ , Σ) because of the edgesuch that Πc̃ ∩ Πc̃γ = Πẽ . It holds that ẽ ∈ S(Gcompleteness assumption on S (cf. the second sentence of the second paragraph of this proof). Moreover, we observe, as in the argument preceding
Claim 2.4.B, that 1 6= Im(H ∩ ΠGλ ) ⊂ Πẽ ∩ Πγẽ . In light of Lemma 1.2, (1), ite λ , Σ) gives the desired
follows that Πẽ = Πγẽ for every γ ∈ H. Thus ẽ ∈ S(G“ceλ ” of Claim 2.4.A. This completes the proof of Claim 2.4.A, hence also theproof of the implication (ii) =⇒ (i). This completes the proof of Lemma
2.4.
Lemma 2.5. Let α : ΠG −→ ΠH be a continuous homomorphism, S ⊂e T ⊂ VCN(H),e and ∅ 6= Σ′ ⊂ ΣG ∩ ΣH . Moreover, let (Hλ −→VCN(G),27H)λ∈Λ be a coﬁnal subsystem of (the projective system which gives rise to)e −→ H constituted by connected ﬁnite étale
the pro-ΣH universal covering H
e −→ H. For every λ ∈ Λ, write Gλ −→ G for the consubcoverings of H
nected ﬁnte étale subcovering of Ge −→ G corresponding to the open subgroupα−1 (ΠHλ ) ⊂ ΠG ; write αλ : ΠGλ −→ ΠHλ for the continuous homomorphism
induced by α. Suppose that the following condition holds:
for every c̃ ∈ S, there exists a connected ﬁnite étale subcovere −→ H such that the image of α(Πc̃ ) ∩ ΠH′
ing H′ −→ H of H∼in Π(H′ )Σ via the natural surjection ΠH′ −↠ ΠΣ
H′ = Π(H′ )Σ (cf.
Lemma 2.1, (2)) is non-trivial.
Then the following conditions are equivalent.
(i) α is (S, T )-compatible.
(ii) For every λ ∈ Λ, the continuous homomorphism (αλ )Σ : Π(Gλ )Σ −→
e λ , Σ), Te(Hλ , Σ) -compatible.
Π(Hλ )Σ is S(GProof. The implication (i) =⇒ (ii) is immediate. The implication (ii) =⇒(i) follows immediately from Lemma 2.4, (ii) =⇒ (i).
Remark 2.5.1. In Lemma 2.5, the condition (i) is equivalent to saying that,for every c̃ ∈ S, there exists d˜ ∈ T such that α(Πc̃ ) ⊂ Πd˜, because of the nontriviality assumption. That is to say, the part “α(Πc̃ ) = 1 or” in Deﬁnition
1.7, (1), is irrelevant here.

3

Filtration-preservation and Group-theoretic
Compatibility

In this section, we study the relationship between the ﬁltration-preservation
properties and the group-theoretic compatibility properties (cf. Theorem3.2). This result is not only of use in the proof of the main result of thepresent paper, Theorem 4.4, but also of independent interest. The key stepis Proposition 3.1, where, roughly, we extract a characteristic property of thesubgroups of S-like subgroups which can be described in a purely “abelianized” way.
The notational and terminological conventions established in the discussion preceding Deﬁnition 1.1 remains valid; in particular, the letters “G”
28and “H” always denote semi-graphs of anabelioids of PSC-type; we do notassume that ΣG = ΣH .
e and H a closed
Proposition 3.1. Let S be a non-empty subset of VCN(G)esubgroup of ΠG . Recall here that we write S for the closure of S ⊂ VCN(G)e with respect to the proﬁnite topology (cf. Remark 1.5.4). Let us
in VCN(G)consider the following seven conditions:(i) H is included in an S-like subgroup of ΠG .
(ii) An open subgroup of H is included in an S-like subgroup of ΠG .
(iii) Every element of H is contained in an S-like subgroup of ΠG .
(iv) H is included in an S-like subgroup of ΠG .
(v) For any connected ﬁnite étale subcovering G ′ −→ G of Ge −→ G, theimage of the composite of the natural homomorphisms
H ∩ ΠG ′ ,−→ ΠG ′ −↠ MG ′ /MGS′
is trivial.
(vi) There exists a connected ﬁnite étale subcovering G † −→ G of Ge −→G such that, for every connected ﬁnite étale subcovering G ′ −→ G of
Ge −→ G which factors through G † −→ G, the image of the composite of
the natural homomorphisms
H ∩ ΠG ′ ,−→ ΠG ′ −↠ MG ′ /MGS′
is trivial.
(vii) There exists a coﬁnal subsystem (Gλ −→ G)λ∈Λ of (the projective system
which gives rise to) the pro-ΣG universal covering Ge −→ G constituted
by ﬁnite étale Galois subcoverings of Ge −→ G such that, for every
λ ∈ Λ, the image of the composite of the natural homomorphisms
H ∩ ΠGλ ,−→ ΠGλ −↠ MGλ /MGSλis trivial.

29Then the implications(i) ⇐⇒ (ii) ⇐⇒ (iii) =⇒ (iv) ⇐⇒ (v) ⇐⇒ (vi) ⇐⇒ (vii)hold. In particular, if we suppose further that S = S, then the conditions
above are all equivalent.
Proof. The implication (i) =⇒ (ii) is clear, while the converse follows
immediately from Lemma 1.4, (2). Thus the equivalence (i) ⇐⇒ (ii) holds.
Next, let us verify the equivalence (i) ⇐⇒ (iii). The implication (i) =⇒(iii) is clear. Conversely, suppose that the condition (iii) holds. We may
assume that H 6= 1 and S is edge-complete (cf. Remark 1.6.1). If every nontrivial element of H is contained in an edge-like subgroup of ΠG , it follows
from Lemma 1.2, (1) and (2), and the edge-completeness assumption thatevery non-trivial element of H is contained in an S-like edge-like subgroup ofΠG . Then it follows from Claim 1.4.A in the proof of [CbTpII], Proposition
1.4 that H is included in an S-like subgroup. Thus we may assume that there
exists an element γ ∈ H which is not contained in any edge-like subgroup.
In this case, there is a(n) (necessarily unique — cf. Lemma 1.2, (3)) S-likevertex ṽ such that γ ∈ Πṽ . Note that we are now in the situation where• every element of H is contained in a verticial subgroup of ΠG ,• γ ∈ H is not contained in any edge-like subgroup of ΠG , and• ṽ is the unique vertex of Ge such that γ ∈ Πṽ ,to which we can apply Claim 1.5.A in the proof of [CbTpII], Proposition 1.5.
This shows H ⊂ Πṽ , as desired. This completes the proof of the implication(iii) =⇒ (i).
The implications (i) =⇒ (iv) =⇒ (v) =⇒ (vi) =⇒ (vii) are
immediate (cf. also Remark 1.5.5), while the implication (vii) =⇒ (v)follows immediately from Remark 1.5.3. Now we have only to show the
implication (v) =⇒ (iv). For this, we may assume that S = S (cf. Remark
1.5.5). Then, in light of the already veriﬁed equivalence (i) ⇐⇒ (iii), we
may assume that H is pro-cyclic. Moreover, since (we have assumed that)
S 6= ∅, we may also assume that H is non-trivial. We shall write J for the
H assumed to be non-trivial and pro-cyclic.
To show the implication (v) =⇒ (iv) for H = J, ﬁrst we concentrateon the case where ΣG is equal to {l}, a singleton. The general case will be30

treated after that. For U an open subgroup of ΠG , write GU for the connectedﬁnite étale subcovering of Ge −→ G corresponding to U , i.e., ΠGU = U ⊂ ΠG .
Then we claim as follows:
Claim 3.1.A: For any normal open subgroup N of ΠG , there exists
an element of S(GJ·N ) at which the connected ﬁnite étale coveringGN −→ GJ·N is totally ramiﬁed, i.e., there exists an element c ∈S(GJ·N ) that satisﬁes the condition that the composite of thenatural homomorphisms
Πc ,−→ J · N −↠ (J · N )/Nis surjective. (Note that, even though the subgroup “Πc ” ⊂J · N = ΠGJ·N is determined by c only up to J · N -conjugation,
the composite morphism under consideration is completely determined by c since (J · N )/N is abelian.)
We verify Claim 3.1.A as follows. Since J is pro-cyclic, and (J · N )/N is its
quotient, (J · N )/N is a ﬁnite cyclic (hence abelian) l-group; in particular,
we obtain a natural surjection Πab
GJ·N −↠ (J · N )/N . Moreover, again since(J · N )/N is a quotient of J, it follows from the condition (v) that thecomposite of the natural homomorphismsYΠc −→ Πab
GJ·N −↠ (J · N )/Nc∈S(GJ·N )is surjective. Therefore, it follows from the fact that (J · N )/N is a cyclic lgroup and S(GJ·N ) 6= ∅ that there exists c ∈ S(GJ·N ) such that the compositeof the natural homomorphisms Πc ,−→ ΠGJ·N −↠ (J · N )/N is surjective, as
desired. This completes the proof of Claim 3.1.A.
We apply Claim 3.1.A as follows. In light of Remark 1.5.6, it suﬃces togive, for every connected ﬁnite étale Galois subcovering G ′ −→ G of Ge −→ G,
an element c′ ∈ S(G ′ ) stabilized by the action of J ⊂ ΠG on VCN(G ′ ). On
the other hand, if we apply Claim 3.1.A to the open normal subgroup N =ΠG ′ ⊂ ΠG , we immediately obtain the desired “c′ ” as the unique componentof G ′ = GN lying over the “c” in Claim 3.1.A. This completes the proof of
the implication (v) =⇒ (iv) for H = J and ΣG = {l}.
Now we treat the general case of the implication (v) =⇒ (iv) for H = J.
Here, the readers are recommended to recall the contents of Lemma 2.1,31

because the group actions, identiﬁcations, and notational conventions given
in Lemma 2.1 will be applied without further mention below. Suppose thatthe condition (v) holds for H = J. It follows immediately from Lemma 2.2,(1), that there exist l ∈ ΣG and a connected ﬁnite étale subcovering G ′ −→ Gof Ge −→ G such that Im(J ∩ ΠG ′ ) ⊂ Π(G ′ )(l) is non-trivial, where we write(G ′ )(l) for (G ′ ){l} and Im(J ∩ ΠG ′ ) for the image of J ∩ ΠG ′ ⊂ ΠG ′ via thenatural surjection ΠG ′ −↠ Π(G ′ )(l) . We ﬁx such l ∈ ΣG . Then, in light ofLemma 2.4, (ii) =⇒ (i), it suﬃces to show that, for every connected ﬁnite
étale Galois subcovering G ′ −→ G of Ge −→ G, Im(J ∩ ΠG ′ ) ⊂ Π(G ′ )(l) is
e ′ , {l})-like subgroup of Π(G ′ )(l) (cf. Deﬁnition 2.3). On the
included in a S(Gother hand, this follows immediately from the already veriﬁed “Σ G = {l}”case of the implication (v) =⇒ (iv) applied to the following Claim 3.1.B:
Claim 3.1.B: The condition (v) holds for “(G, H = J, S)” =
′ (l)
′
e
(G ) , Im(J ∩ ΠG ′′ ), S(G , {l}) . That is to say, for every con^′ )(l) −→ (G ′ )(l) ,nected ﬁnite étale subcovering K −→ (G ′ )(l) of (Gthe image of the composite of the natural homomorphisms
e ′ ,{l})S(GIm(J ∩ ΠG ′ ) ∩ ΠK ,−→ ΠK −↠ MK /MKis trivial.

Finally, Claim3.1.B, i.e., the triviality of the composite of the lower horizontal
arrows of the following natural commutative diagram, follows immediatelyfrom the triviality of the composite of the upper horizontal arrows of thefollowing natural commutative diagram, where we write p for the natural(l)surjection ΠG ′ −↠ ΠG ′ ∼= Π(G ′ )(l) and L −→ G ′ for the connected ﬁnite étale
subcovering of Ge −→ G ′ corresponding to the inverse image p−1 (ΠK ) ⊂ ΠG ′ ,i.e., ΠL = p−1 (ΠK ):J ∩ ΠL 




/ ΠL

inducedby pIm(J ∩ ΠG ′ ) ∩ ΠKinducedby p/ ΠK/ / ML /M S
L

induced by pe ′/ / MK /M S(G ,{l}) .
K

This completes the proof of the implication (v) =⇒ (iv), hence also theproof of Proposition 3.1.
32

Remark 3.1.1. Proposition 3.1, along with its proof, should be thought ofas a uniﬁed generalization of [CbTpII], Proposition 1.4, [CbTpII], Proposietion 1.5, and [GrphPIPSC], Lemma 1.2. Indeed, if one takes S to be Edge(G)e Node(G)),
e which is manifestly closed in VCN(G),e then the(resp. Vert(G);equivalences (i) ⇐⇒ (ii) ⇐⇒ (iii) ⇐⇒ (viii) in Proposition 3.1 are
precisely the statement of [CbTpII], Proposition 1.4 (resp. [CbTpII], Proposition 1.5; [GrphPIPSC], Lemma 1.2). Note that the assumption S 6= ∅e while, in fact, it should have been assumed
is automatic for S = Vert(G),e 6= ∅ (resp. Node(G)e 6= ∅) in [CbTpII], Proposition 1.4 (resp.
that Edge(G)[GrphPIPSC], Lemma 1.2), as is noted in Remark 3.1.2 below.
Remark 3.1.2. We observe that the assumption “S 6= ∅” in Proposition
3.1 is used only in the veriﬁcation of the implication (v) =⇒ (iv). If one
e to be empty, each of the conditions (i), (ii), (iii), (iv) of
takes S ⊂ VCN(G)Proposition 3.1 can never be true, while (it is easily veriﬁed by an argument
entirely similar to the proof of Lemma 2.2, (1) and (2), that) each of theconditions (v), (vi), (vii) is true if and only if H = 1.
For essentially the same reason, in [CbTpII], Proposition 1.4 (resp. [GrphPIPSC],e 6= ∅ (resp. Node(G)e 6=Lemma 1.2), it should have been assumed that Edge(G)∅). The author believes that these slight errors in [CbTpII], Proposition 1.4,
and [GrphPIPSC], Lemma 1.2, may not cause any serious problems in theirapplications obtained so far, since the trivial case H = 1 is the only one
possible counter-example.
Theorem 3.2. Let α : ΠG −→ ΠH be a continuous homomorphism, S ⊂e and T ⊂ VCN(H).
e Let us consider the following three conditions:
VCN(G),(i) α is (S, T )-compatible.
(ii) α is (S, T )-ﬁltration-preserving.
(iii) α is (S, T )-compatible.
Then the implications (i) =⇒ (ii) =⇒ (iii) holds. In particular, if we
suppose further that S = S and T = T , then the conditions above are all
equivalent.
Proof. The implication (i) =⇒ (ii) follows immediately from the deﬁnitions.
If T 6= ∅, then the implication (ii) =⇒ (iii) follows immediately from
Proposition 3.1, (v) =⇒ (iv), and Remark 1.5.5. If T = ∅, then the33

implication (ii) =⇒ (iii) follows immediately from the observation given in
Remark 3.1.2, where we recall the part “α(Πc̃ )=1 or” of Deﬁnition 1.7, (1).
This completes the proof of Theorem 3.2.
Remark 3.2.1. The last assertion of Theorem 3.2 can be thought of as a
uniﬁed generalized Hom-version of [CbGC], Theorem 1.6, (ii). Indeed, todeduce [CbGC], Theorem 1.6, (ii), from Theorem 3.2, (4), we have only to∼=apply the equivalences given in Remark 1.7.5 to the isomorphism α : ΠG −→ΠH under consideration, and set “(S, T )” in Theorem 3.2, (4), to be, foreeinstance, Vert(G), Vert(H) .
On the other hand, the proof of Theorem 3.2 is substantially diﬀerent fromthe original proof of [CbGC], Theorem 1.6, (ii). Indeed, it does not use thetechnique of reconstruction of various VCN-subgroups involving numerical
data concerning ramiﬁcation, which plays a central role in the original proofof [CbGC], Theorem 1.6, (ii). Rather, Theorem 3.2 is essentially a formal
consequence of Proposition 3.1, and the proof of Proposition 3.1 relies deeplyon the technique established in the proof of [CbTpII], Proposition 1.4 and[CbTpII], Proposition 1.5 (cf. Remark 3.1.1). It seems to the author that
it is diﬃcult to apply the argument involving numerical data to our “Homversion” situation.
Remark 3.2.2. Suppose that we are in the situation of Theorem 3.2. Then
we observe that, if, in Deﬁnition 1.7, (1), we had not allowed such α thatα(Πc̃ ) = 1 for every c̃ ∈ S to be (S, ∅)-compatible when S 6= ∅ (cf. Remark
1.7.1), then the implication (ii) =⇒ (iii) of Theorem 3.2 would be false.
Indeed, under this altered deﬁnition, if S 6= ∅, T = ∅, and α is the trivial
homomorphism, then α satisﬁes the condition (ii) while α would not satisfy
the condition (iii).

4

Hom-version of the Combinatorial Grothendieck
Conjecture for Graphically Full Outer Representations

In this section, we prove that a continuous homomorphism between PSCfundamental groups ΠG −→ ΠH satisfying a certain compatibility with “le Vert(H))e and (Edge(G),e Edge(H))egraphically full outer representations” is (Vert(G),compatible. This result may be considered as a Hom-version of the combi34

natorial Grothendieck conjecture given in [CbGC], Theorem 2.7, (ii). Before
that, we also give a short review of l-graphically full outer representations.
The notational and terminological conventions established in the discussion preceding Deﬁnition 1.1 remains valid; in particular, the letter “G” always denotes a semi-graph of anabelioids of PSC-type.
Before proceeding, let us recall the issue concerning the proﬁnite topology
on Aut(G). Since the fundamental group ΠG of G is topologically ﬁnitely
generated (cf. [CbGC], Remark 1.1.3), the proﬁnite topology of Π G induces(proﬁnite) topologies on Aut(ΠG ) and Out(ΠG ) (cf. the discussion entitled“Topological Groups” in §0). Moreover, if we write Aut(G) for the group of
automorphisms of G, then by the discussion preceding [CbGC], Lemma 2.1,
the natural homomorphism Aut(G) −→ Out(ΠG ) is an injection with closed
image. (Here, we recall that an automorphism of a semi-graph of anabelioids
consists of an automorphism of the underlying semi-graph, together with acompatible system of isomorphisms between the various anabelioids at eachof the vertices and edges of the underlying semi-graph which are compatible
with the various morphisms of anabelioids associated to the branches of theunderlying semi-graph — cf. [SemiAn], Deﬁnition 2.1.) Thus, by equipping
Aut(G) with the topology induced via this homomorphism by the topologyof Out(ΠG ), we may regard Aut(G) as a proﬁnite group.
Let I be a proﬁnite group and ρ : I −→ Aut(G) a continuous homomorphism. Then, as is deﬁned in [NodNon], Deﬁnition 2.1, (i), the pair(G, ρ : I −→ Aut(G)) is referred to as an outer representation of PSC-type.
Moreover, for l ∈ ΣG , if I is l-cyclotomically full (resp. l-graphically full)with respect to ρ : I −→ Aut(G) in the sense of [CbGC], Deﬁnition 2.3 (ii)(resp. (iii)), then we shall say that the outer representation of PSC-type(G, ρ : I −→ Aut(G)) is l-cyclotomically full (resp. l-graphically full ).
In the remainder of the present paper, we are concerned with the exact
sequence of proﬁnite groups and continuous homomorphisms
1 −→ ΠG −→ Πρ −→ I −→ 1,
def

out

(∗)

where we write Πρ = ΠG o I, associated to an outer representation of PSCtype (G, ρ : I −→ Aut(G)). In other words, we are interested in the exact
sequence obtained by pulling back the exact sequence
1 −→ ΠG −→ Aut(ΠG ) −→ Out(ΠG ) −→ 1
35

naturalρ

via the continuous homomorphism I −→ Aut(G) ,−→ Out(ΠG ) (cf. thediscussion entitled “Topological Groups” in §0). Here, we note that ΠG is
topologically ﬁnitely generated and center-free (cf. [CbGC], Remark 1.1.3).
Lemma 4.1. Let (G, ρ : I −→ Aut(G)) be an outer representation of PSCtype. Let us consider the exact sequence (∗) associated′ to G, ρ : I −→Aut(G) . Let Πρ′ be an open subgroup of Πρ . Write I for the image ofΠρ via the surjection Πρ −↠ I and G ′ −→ G for the connected ﬁnite étale
subcovering of Ge −→ G corresponding to the open subgroup Πρ′ ∩ ΠG of ΠG ,i.e., ΠG ′ = Πρ′ ∩ ΠG . Then the following assertions hold.
(1) The exact sequence (∗) naturally induces the following exact sequence:
1 −→ ΠG ′ −→ Πρ′ −→ I ′ −→ 1.
Moreover, this exact sequence arises from a unique outer representationof PSC-type G ′ , ρ′ : I ′ −→ Aut(G ′ ) .
(2) Let l ∈ ΣG . Then, if G, ρ : I −→ Aut(G) is l-cyclotomically full
(resp. l-graphically full) (cf. [CbGC], Deﬁnition 2.3, (ii) (resp.(iii))),then G ′ , ρ′ : I ′ −→ Aut(G ′ ) in (1) is also l-cyclotomically full (resp.
l-graphically full).
Proof. Let us verify the assertion (1). The well-deﬁnedness and the exactness of the sequence are immediate. Thus it suﬃces to show that the homomorphism I ′ −→ Out(ΠG ′ ) factors through the natural closed immersion
Aut(G ′ ) ,−→ Out(ΠG ′ ), or, equivalently, that the conjugation action of any
element of Πρ′ on the closed normal subgroup ΠG ′ ⊂ Πρ′ is graphic in thesense of [CbGC], Deﬁnition 1.4, (i) (cf. the discussion preceding [CbGC],
Lemma 2.1). On the other hand, in light of [CbGC], Proposition 1.5, (ii),this follows immediately from the assumption and Remark 1.5.2. This completes the proof of the assertion (1).
Next, we verify the non-resp’d case of the assertion (2). We apply thenotation of [CbGC], Deﬁnition 2.3. Moreover, we recall the pro-l cyclotomic
cyc.
character Aut(G) −→ Z×
l deﬁned in [CbGC], Lemma 2.1. It suﬃces to showρ
cyc.
that the restriction of the composite Πρ −↠ I −→ Aut(G) −→ Z×l (whose
image is assumed to be open) to the open subgroup Πρ′ ⊂ Πρ coincides
ρ′

cyc.

with the composite Πρ′ −↠ I ′ −→ Aut(G ′ ) −→ Z×
l . It follows from thedeﬁnitions that the composite morphism Πρ −→ Z×l can be computed as36

follows (and entirely similar statement holds also for the composite morphismΠρ′ −→ Z×l ).
(i) Take an arbitrary connected ﬁnite étale subcovering G ′′ −→ G of Ge −→G such that G ′′ is sturdy (cf. [CbGC], Deﬁnition 1.1, (ii), and [CbGC],
Remark 1.1.5) and such that the corresponding open subgroup Π G ′′ ⊂ΠG is stable (as a subgroup) under the conjugation action of Π ρ on ΠG .
Then Πρ acts graphically (cf. [CbGC], Deﬁnition 1.4, (i), and [CbGC],
Proposition 1.5, (ii), for the terminology “graphic”) on Π G ′′ , hence alsoon Π(G ′′ )cpt (cf. Example 1.7.2 for the notation “Π(G ′′ )cpt ”).
(ii) The action of Πρ on Π(G ′′ )cpt deﬁned in (i) determines a continuous
action of Πρ on H 2 (Π(G ′′ )cpt , Zl ), which is a free Zl -module of rank 1,
hence a continuous homomorphism Πρ −→ Z×l . Finally, we take the−1
inverse (i.e., “(–) ”) of this homomorphism.
Here, we emphasize that, in (i) above, we may take an open subgroup which
is not necessarily characteristic in ΠG , despite that only the characteristic
open subgroups of ΠG are considered in (the discussion preceding) [CbGC],
Lemma 2.1. The veriﬁcation of this slight generalization is immediate froman entirely similar argument to the proof of [CbGC], Lemma 2.1. Now the desired coincidence follows immediately from this description of the composite
×
morphisms Πρ −→ Z×l and Πρ′ −→ Zl .
Finally, we verify the resp’d case of the assertion (2). Since we have already shown the non-resp’d case of the assertion (2), we have only to show
that, for every connected ﬁnite étale subcovering G ′′ −→ G ′ of Ge −→ G ′ correedge
sponding to a characteristic open subgroup ΠG ′′ ⊂ ΠG ′ , wl (MGvert′′ /MG ′′ )⊗ẐΣG
def
Zl is included in (0, 2)Q = {q ∈ Q | 0 < q < 2.}, where the “wl ” is takenwith respect to the natural action of Πρ′ . To this end, take a connected ﬁnite étale subcovering G ′′′ −→ G of Ge −→ G corresponding to a characteristic
open subgroup ΠG ′′′ ⊂ ΠG such that it holds that ΠG ′′′ ⊂ ΠG ′′ . Then it follows
edge
vert
from the assumption that it holds that wl (MG ′′′ /MG ′′′ ) ⊗ẐΣG Zl ⊂ (0, 2)Q ,where the “wl ” is taken with respect to the natural action of Πρ . Moreover, by the “coincidence of the cyclotomic character” demonstrated
in theedge
Z
isinvari/M)⊗
proof of the non-resp’d case, the set wl (MGvert
Σ
′′′
l
G ′′′
Ẑ G
ant if one changes the acting group from Πρ to Πρ′ . Finally, since we haveedge
a natural Πρ′ -equivariant open homomorphism (MGvert′′′ /MG ′′′ ) ⊗ẐΣG Zl −→

edge
edge
edge
vert
(MGvert⊂ wl (MGvert′′ /MG ′′ )⊗ẐΣG Zl , we have w l (MG ′′ /MG ′′ )⊗ẐΣG Zl′′′ /MG ′′′ )⊗ẐΣG
37


Zl . This completes the proof of the resp’d case of the assertion (2), hence
also the proof of Lemma 4.1.
Lemma 4.2. Let l ∈ ΣG and (G, ρ : I −→ Aut(G)) an l-graphically fullouter representation of PSC-type. Let us consider the outer representationof PSC-type (G (l) , ρ(l) : I −→ Aut(G (l) )) naturally determined by (G, ρ : I −→Aut(G)). Then the following assertions hold.
(l) (l)(1) The pro-l cyclotomic character I −→ Z×l associated to (G , ρ : I −→(l)
Aut(G )) coincides with the pro-l cyclotomic character I −→ Z×l associated to (G, ρ : I −→ Aut(G)) (cf. [CbGC], Lemma 2.1 for the deﬁnition of pro-l cyclotomic character).

(2) If (G, ρ : I −→ Aut(G)) is l-graphically full, then (G (l) , ρ(l) : I −→Aut(G (l) )) is also l-graphically full.
(3) The exact sequence (∗) in the case where we take “(G, ρ : I −→ Aut(G))”
to be (G (l) , ρ(l) : I −→ Aut(G (l) )) coincides with the exact sequence obtained by taking the quotients of ΠG and Πρ of the exact sequence (∗)(l)
by the kernel of the natural homomorphism ΠG −↠ ΠG ∼= ΠG (l) (cf.
Lemma 2.1, (2)). Here, we note that the kernel of the natural homo(l)morphism ΠG −↠ ΠG ∼= ΠG (l) is characteristic in ΠG and thus normalin ΠρI .
Proof. The assertions (1) and (2) follow from a similar argument to the proof
of Lemma 4.1, (2). The assertion (3) follows immediately from the deﬁnitions.
This completes the proof of Lemma 4.2.
Next, we recall the following result from [CbGC], which plays the most
crucial role in the proof of Theorem 4.4.
Lemma 4.3. Let l ∈ ΣG and (G, ρ : I −→ Aut(G)) an l-graphically full outerrepresentation of PSC-type. Then the following assertions hold.
(1) The quotient MG ⊗ẐΣG Zl −↠ (MG /MGvert )⊗ẐΣG Zl is characterized as the
maximal torsion-free quasi-trivial Zl [I]-quotient module of MG ⊗ẐΣG Zl(cf. [CbGC], Deﬁnition 2.3, (i)).
(2) The submodule MGedge ⊗ẐΣG Zl ⊂ MG ⊗ẐΣG Zl is characterized as the maximal quasi-toral Zl [I]-submodule of MG ⊗ẐΣG Zl (cf. [CbGC], Deﬁnition
2.3, (i)).
38

Proof. This is [CbGC], Proposition 2.4, (viii) and (ix).
This crucial result, combined with various results obtained in the present
paper, naturally leads to the following theorem.
Theorem 4.4. Let (G, ρI : I −→ Aut(G)) and (H, ρJ : J −→ Aut(H)) be
outer representations of PSC-type. Let us consider the following commutative
diagram of proﬁnite groups and continuous homomorphisms:
1

1

/ ΠG

/ Πρ

α

α̃



/ ΠH

I

/I

/1

β



/ Πρ

J


/J

/ 1,

where the two horizontal sequences are the respective exact sequences (∗)
associated to (G, ρI : I −→ Aut(G)) and to (H, ρJ : J −→ Aut(H)). Let □ ∈e Suppose further that there exists l ∈ ΣG ∩ ΣH
{Vert, Edge} and S ⊂ □(G).
for which the following four conditions hold.
(i) For every c̃ ∈ S, there exists a connected ﬁnite étale subcovering H′ −→e −→ H such that the image of α(Πc̃ ) ∩ ΠH′ via the natural
H of H(l)surjection ΠH′ −↠ ΠH′ ∼= Π(H′ )(l) is non-trivial.
(ii) The outer representation of PSC-type (G, ρI : I −→ Aut(G)) is l-graphicallyfull.
(iii) The outer representation of PSC-type (H, ρJ ◦ β : I −→ Aut(H)) isl-graphically full.
(iv) If, moreover, □ = Edge, then the two cyclotomic characters I −→ Z×lassociated to (G, ρI : I −→ Aut(G)) and to (H, ρJ ◦ β : I −→ Aut(H))
coincide.
e -compatible.
Then α is S, □(H)
Proof. We ﬁx l ∈ ΣG ∩ ΣH as in the statement. By pulling back the lower
horizontal exact sequence of the commutative diagram under consideration
via β, we may assume that β is an isomorphism (or even the identity) of
proﬁnite groups.
Let (ΠρJ ,λ ⊂ ΠρJ )λ∈Λ be the family of all open subgroups of ΠρJ . This
family determines a family (ΠHλ = ΠρJ ,λ ∩ ΠH )λ∈Λ of open subgroups of ΠG
39

Tsuch that λ∈Λ ΠHλ = 1, or, equivalently, a coﬁnal subsystem (Hλ −→ H)λ∈Λof (the projective system which gives rise to) the pro-ΣH universal covering
e −→ H. In light of Lemma 2.5, (ii) =⇒ (i), it suﬃces to show that theHcontinuous homomorphism (αλ )(l) : Π(Gλ )(l) −→ Π(Hλ )(l) induced by α, where
we write Gλ −→ G for the connected ﬁnite étale subcovering of Ge −→ G cor^(l)e λ , {l}), □((H
responding to α−1 (ΠH ) ⊂ ΠG , is S(Gλ ) ) -compatible. Here,λwe note that the assumption (i) makes it possible for us to apply Lemma
2.5. By replacing, for every λ ∈ Λ, each proﬁnite group (“ΠρJ ” at the centerof the lower horizontal sequence, for example) in the commutative diagram
under consideration by an appropriate open subgroup (Π ρJ ,λ ⊂ ΠρJ for exam
g(l) ) e {l}), □(Hple), we are reduced to showing that ΠG (l) −→ ΠH(l) is S(G,compatible. Thus, by Lemma 4.2, we may assume that ΣG = ΣH = {l}.
e -compatibility of α in the case where ΣG = ΣH =To verify the S, □(H){l}, again, we take the family (ΠρJ ,λ ⊂ ΠρJ )λ∈Λ of all open subgroups of ΠρJ
and we put ΠHλ = ΠρJ ,λ ∩ ΠH ⊂ ΠH ; ΠGλ = α−1 (ΠHλ ) ⊂ ΠG ; αλ : ΠGλ −→ΠHλ the restriction of α. Then, in light of Theorem 3.2, (ii) =⇒ (iii),
and Remark 1.7.3, it suﬃces to show that αλab (MGSλ ) ⊂ MH◦ λ , where we set◦ = vert (resp. ◦ = edge) if □ = Vert (resp. □ = Edge). On the other hand,e easily from Lemma
this follows, in light of the assumption that S ⊂ □(G),
4.3 (resp. Lemma 4.3 and the assumption (iv)), where the “I” in Lemma
4.3 is taken as the image of ΠρJ ,λ via the natural surjection ΠρJ −↠ J. This
completes the proof of Theorem 4.4.
Remark 4.4.1. Theorem 4.4, along with its proof, can be considered as a
Hom-version of [CbGC], Corollary 2.7, (ii). Indeed, they share the technicalcore of the proof, say, the characterization of MGvert and MGedge (in the casewhere ΣG = {l}) in terms of the l-graphically full actions (cf. Lemma 4.3).
Also, we observe that [CbGC], Corollary 2.7, (ii), follows, at least if we(inappropriately) ignore the necessity of the assumption (iv) of Theorem
4.4, formally from Theorem 4.4 and Lemma 1.9. Thus Theorem 4.4 is, in acertain sense, a generalization of [CbGC] Corollary 2.7, (ii), though not in astraightforward manner.
Regarding the assumption (iv) of Theorem 4.4, which we ignored in theabove discussion, we note that, in the situation of [CbGC], Corollary 2.7,(ii), one may verify (an “Isom-version” counterpart of) the assumption (iv)of Theorem 4.4 by an argument involving numerical data. In particular,
we can indeed prove [CbGC], Corollary 2.7, (ii), using Theorem 4.4 in an
40

essential way.
The author decided to impose the assumption (iv) in Theorem 4.4 precisely due to the diﬃculty of applying such an argument to our “Hom-version”
situation (cf. the proof of [CbGC], Corollary 2.7, (i); also Remark 3.2.1). It
is an interesting remaining work to examine the extent to which assumption(iv) can be relaxed.
Remark 4.4.2. It follows immediately from Lemma 1.10 that one can strengthenethe conclusion of Theorem 4.4 from α being (S, □(H))-compatible
to α being
ΠG
ΠGe(S , □(H))-compatible (cf. Lemma 1.10 for the notation “S ”).
eRemark 4.4.3. In Theorem 4.4, the conclusion “α is (S, □(H))-compatible”
e such thatis equivalent to saying that, for every c̃ ∈ S, there exists d˜ ∈ □(H)α(Πc̃ ) ⊂ Πd˜, i.e., the part “α(Πc̃ ) = 1 or” in Deﬁnition 1.7, (1), is irrelevant,because of the non-triviality assumption (i).

Acknowledgements
I would like to express my deep gratitude to Professor Yuichiro Hoshi,
Professor Shinichi Mochizuki, and Professor Shota Tsujimura for numerous
helpful discussions and warm encouragements. Further I would like to thankProfessor Yuichiro Hoshi for suggesting the theme of the present paper, reading the drafts of the present paper carefully, and giving numerous helpfulcomments. Also I am really grateful to my family for their support.

References[GrphPIPSC] Y. Hoshi, Reconstruction of proﬁnite graphs from proﬁnite
groups of PIPSC-type, Hokkaido Math. J. 49 (2020), pp. 399–430.
[NodNon] Y. Hoshi and S. Mochizuki, On the combinatorial anabeliangeometry of nodally nondegenerate outer representations, Hiroshima Math. J. 41 (2011), pp. 275–342.
[CbTpI]Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial anabelian geometry of hyperbolic curves I: inertia groups andproﬁnite Dehn twists, In: Galois-Teichmüler theory and arithmetic geometry, Adv. Stud. Pure Math., 63, Math. Soc. Japan,
Tokyo, 2012, pp. 659-811.
41

[CbTpII]

Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial
anabelian geometry of hyperbolic curves II: tripods and combinatorial cuspidalization, Lecture Notes in Math., 2299, Springer
Nature Singapore (2022).

[SemiAn]

S. Mochizuki, Semi-graphs of Anabelioids, Publ. Res. Inst. Math.
Sci. 42 (2006), pp. 221-332.

[CbGC]S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math. J. 59 (2007), pp. 455–479.

[HmCbGCII] R. Tachihara,
Hom-versions of the Combinatorial
Grothendieck Conjecture II: Outer Representations of PIPSCand NN-type, RIMS Preprint 1981 (April 2024).

42