# Oriented right-angled Artin pro-$\ell$ groups
and maximal pro-$\ell$ Galois groups

Simone Blumer  ,  Claudio Quadrelli  and  Thomas S. Weigel  Department of
Mathematics and Applications, University of Milano Bicocca, Milan (Italy)
<EMAIL_ADDRESS>Department of Science and High-Tech, University of
Insubria, Como (Italy<EMAIL_ADDRESS>Department of
Mathematics and Applications, University of Milano Bicocca, Milan (Italy)
<EMAIL_ADDRESS>Dedicated to the memory of Avinoam Mann.

###### Abstract.

For a prime number $\ell$ we introduce and study oriented right-angled Artin
pro-$\ell$ groups $G_{\Gamma,\lambda}$(oriented pro-$\ell$ RAAGs for short)
associated to a finite oriented graph $\Gamma$ and a continuous group
homomorphism $\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$. We
show (cf. Thm. 1.1) that an oriented pro-$\ell$ RAAG $G_{\Gamma,\lambda}$ is a
Bloch-Kato pro-$\ell$ group if, and only if,
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is an oriented pro-$\ell$ group
of elementary type generalizing a recent result of I. Snopche and P. Zalesskiĭ
(cf. [SZ:RAAGs]). Here $\theta_{\Gamma,\lambda}\colon
G_{\Gamma,\lambda}\to\mathbb{Z}_{p}^{\times}$ denotes the canonical
$\ell$-orientation on $G_{\Gamma,\lambda}$. We invest some effort in order to
show that oriented right-angled Artin pro-$\ell$ groups share many properties
with right-angled Artin pro-$\ell$-groups or even discrete RAAG’s, e.g., if
$\Gamma$ is a specially oriented chordal graph, then $G_{\Gamma,\lambda}$ is
coherent (cf. Thm. 1.3(ii)) generalizing a result of C. Droms (cf.
[droms:coh]) . Moreover, in this case
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ has the Positselski-Bogomolov
property (cf. Thm. 1.3(i)) generalizing a result of H. Servatius, C. Droms and
B. Servatius for discrete RAAG’s (cf. [serv]). If $\Gamma$ is a specially
oriented chordal graph and $\operatorname{Im}(\lambda)\subseteq
1+4\mathbb{Z}_{2}$ in case that $\ell=2$, then
$H^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})\simeq\Lambda^{\bullet}(\ddot{\Gamma}^{\mathrm{op}})$
(cf. Thm. 1.3(iii)) generalizing a well known result of M. Salvetti (cf.
[salv:coh]).

###### Key words and phrases:

Maximal pro-$\ell$ Galois groups, Galois cohomology, Bloch-Kato pro-$\ell$
groups, oriented pro-$\ell$ groups, the Bogomolov-Positselski property, right-
angled Artin pro-$\ell$ groups

###### 2010 Mathematics Subject Classification:

Primary 12G05; Secondary 20E18, 20J06, 12F10

##  1\. Introduction

In a recent paper (see [SZ:RAAGs]) I. Snopce and P.A. Zalesskiĭ showed that
for pro-$\ell$ completions of right-angled Artin groups, the following four
properties are equivalent: the Bloch-Kato property, 1-cyclotomicity, the
elementary type property, and the realizability as a maximal pro-$\ell$ Galois
group . This theorem can be seen as a pro-$\ell$ analogue of a result of C.
Droms (see [droms:sub]) for discrete right-angled Artin groups, and provides
evidence to the conjecture that a finitely generated cyclotomically oriented
Bloch-Kato pro-$\ell$ group $(G,\theta)$ is necessarily of elementary type
(see [qw:cyclotomic, § 7.5]), and therefore to the Elementary Type Conjecture
formulated by I. Efrat in [efrat:etc1].

In this paper we generalize the concept of right-angled Artin pro-$\ell$ group
to non-necessarily trivially oriented examples.

By an oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ we will understand a
non-empty set of vertices $\mathscr{V}$ together with a set of edges

$\mathscr{E}\subseteq\mathscr{V}\times\mathscr{V}\smallsetminus\Delta(\mathscr{V}),\qquad\text{where
}\Delta(\mathscr{V})=\\{\>(v,v)\>\mid\>v\in\mathscr{V}\>\\}.$

Thus, for an edge $\mathbf{e}=(v,w)\in\mathscr{E}$, its inverse
$\bar{\mathbf{e}}=(w,v)$ might be contained in $\mathscr{E}$ (in which case
$\mathbf{e}$ will be said to be an ordinary edge), or not (in which case
$\mathbf{e}$ will be said to be special). Hence $\mathscr{E}$ is the disjoint
union $\mathscr{E}_{\mathrm{o}}\sqcup\mathscr{E}_{\mathrm{s}}$ of the sets of
ordinary edges and special edges.

In a similar fashion, $\mathscr{V}$ is the disjoint union
$\mathscr{V}_{\mathrm{o}}\sqcup\mathscr{V}_{\mathrm{s}}$ of the sets of
ordinary vertices and special vertices, satisfying the following condition:
the origin (i.e., the first coordinate) of every edge is an ordinary vertex,
i.e., a vertex can be special only if either it is isolated or it is the end
of a special edge (see § 2.2 below).

For an oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ and a continuous
homomorphism $\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ we
call the pro-$\ell$ group $G_{\Gamma,\lambda}$ given by the presentation

(1.1)
$G_{\Gamma,\lambda}=\left\langle\>v\in\mathscr{V}\>\mid\>\forall\>\mathbf{e}=(v,w)\in\mathscr{E},\>wvw^{-1}=\begin{cases}v&\text{if
}\mathbf{e}\in\mathscr{E}_{\mathrm{o}},\\\ v^{\lambda(1)}&\text{if
}\mathbf{e}\in\mathscr{E}_{\mathrm{s}}\end{cases}\>\right\rangle_{\mathrm{pro-}{\ell}}$

the oriented right-angled Artin pro-$\ell$ group (oriented pro-$\ell$ RAAG for
short) associated to $\Gamma$ and $\lambda$. By definition, every oriented
right-angled Artin pro-$\ell$ group $G_{\Gamma,\lambda}$ carries the
orientation $\theta_{\Gamma,\lambda}\colon
G_{\Gamma,\lambda}\to\mathbb{Z}_{\ell}^{\times}$ given by
$\theta_{\Gamma,\lambda}(v)=1$ if $v\in\mathscr{V}_{\mathrm{o}}$, and
$\theta_{\Gamma,\lambda}(v)=\lambda(1)$ if $v\in\mathscr{V}_{\mathrm{s}}$.
Thus, if $\Gamma$ is an oriented graph without special edges, then
$G_{\Gamma,\lambda}$ is just a right-angled Artin pro-$\ell$ group, and
$\theta_{\Gamma,\lambda}=\mathbf{1}$ is the constant $1$-function. In spite of
the rather elementary presentation (1.1), oriented pro-$\ell$ RAAGs yield
surprising variety and flexibility — for example, among oriented pro-$\ell$
RAAGs one finds free pro-$\ell$ groups, $\ell$-adic analytic groups, and even
finite $\ell$-groups.

Our main goal is to prove the following “oriented analogue” of [SZ:RAAGs, Thm.
1.2, Thm. 1.5].

######  Theorem 1.1.

Let $\Gamma$ be an oriented graph, let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a continuous
homomorphism (satisfying $\operatorname{Im}(\lambda)\subseteq
1+4\mathbb{Z}_{2}$ if $\ell=2$), and let
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ be the oriented pro-$\ell$ RAAG
associated to $\Gamma$ and $\lambda$. The following are equivalent.

  * (0)

The oriented graph $\Gamma$ is of elementary type (see Definition 2.9).

  * (i)

The pro-$\ell$ group $G_{\Gamma,\lambda}$ is isomorphic to the maximal
pro-$\ell$ Galois group of a field containing a primitive $\ell$-th root of 1
(and also $\sqrt{-1}$ if $\ell=2$).

  * (ii)

The pro-$\ell$ group $G_{\Gamma,\lambda}$ is a Bloch-Kato pro-$\ell$ group.

  * (iii)

The oriented pro-$\ell$ RAAG $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is
a 1-cyclotomic oriented pro-$\ell$ group.

  * (iv)

The oriented pro-$\ell$ RAAG $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is
an oriented pro-$\ell$ group of elementary type.

  * (v)

Every finitely generated closed subgroup of $G_{\Gamma,\lambda}$ is again
isomorphic to an oriented pro-$\ell$ RAAG.

Note that condition (0) in Theorem 1.1 gives a combinatorial (and rather
immediate) criterion to check whether an oriented pro-$\ell$ RAAG may occur as
a maximal pro-$\ell$ Galois group or not. On one handside, Theorem 1.1
provides a plethora of brand new examples of pro-$\ell$ groups which do not
occur as maximal pro-$\ell$ Galois groups. It is worth mentioning that the
relations involved in the presentation of an oriented pro-$\ell$ RAAG are
elementary — just an elementary commutator times, possibly, the
$\lambda(1)$-th power of a generator —, especially in comparison with other
examples of pro-$\ell$ groups which are known not to occur as maximal
pro-$\ell$ Galois groups (see, e.g., [BLMS, cq:noGal], [cem, § 9], [MT:Gal, §
7]), whose presentations require higher commutators.

On the other hand, considering the richness of the family of oriented
pro-$\ell$ RAAGs, Theorem 1.1 corroborates the Elementary Type Conjecture, and
the Smoothness Conjecture, formulated by Ch. De Clercq and M. Florence (see
[dcf:lift]).

In fact, there are several other conjectures concerning the structure of the
maximal pro-$\ell$ Galois group $G_{\mathbb{K}}(\ell)$ of a field $\mathbb{K}$
containing a primitive $\ell$-th root of 1, and its
$\mathbb{F}_{\ell}$-cohomology, e.g., L. Positselski’s formulation of
Bogomolov Conjecture, which predicts that $G_{\mathbb{K}}(\ell)$ is a free-by-
locally uniform pro-$\ell$ group (see [pos:k] and [qw:bogomolov]); J. Minač
and N.D. Tân’s Massey Vanishing Conjecture, which predicts that
$G_{\mathbb{K}}(\ell)$ satisfies the $n$-Massey product vanishing property for
every $n>2$ (see [MT:conj]); and the Universal Koszulity Conjecture,
formulated by J. Minač et al., which predicts that
$\mathbf{H}^{\bullet}(G_{\mathbb{K}}(\ell),\mathbb{F}_{\ell})$ is a
universally Koszul algebra (see [MPPT]). These three properties are satisfied
by oriented pro-$\ell$ groups of elementary type (see respectively
[qw:bogomolov, Thm. 1.2], [cq:massey, Thm. 1.3], and [MPPT, § 1]). Therefore,
from Theorem 1.1 one concludes the following providing evidence for the three
aforementioned conjectures.

######  Corollary 1.2.

Let $\Gamma$ be an oriented graph and let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a continuous
homomorphism (satisfying $\operatorname{Im}(\lambda)\subseteq
1+4\mathbb{Z}_{2}$ if $\ell=2$). If $G_{\Gamma,\lambda}\simeq
G_{\mathbb{K}}(\ell)$ for some field $\mathbb{K}$ containing a primitive
$\ell$-th root of 1, then

  * (i)

$G_{\Gamma,\lambda}$ has the Bogomolov-Positselski property, i.e., it is the
Frattini pro-$\ell$ cover of a locally uniform pro-$\ell$ group with free
pro-$\ell$ kernel;

  * (ii)

for every $n>2$, $G_{\Gamma,\lambda}$ satisfies the $n$-Massey product
vanishing property, i.e., every non-empty $n$-fold Massey product associated
to an $n$-tuple of elements of
$\mathrm{H}^{1}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$ contains 0;

  * (iii)

the algebra $\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$ is
universally Koszul.

Subsequently, we will focus on oriented pro-$\ell$ RAAGs associated to chordal
oriented graphs.

Recall that an oriented graph $\Gamma$ is said to be chordal if it does not
contain cycles other than triangles as induced subgraphs.

Moreover, an oriented graph will be said to be specially oriented if the
terminus of a special edge is always a special vertex (see Definition 2.5).
Note that an oriented graph of elementary type is always specially oriented
and chordal, but not vice-versa.

For oriented pro-$\ell$ RAAGs associated to chordal specially oriented graphs
we prove the following.

######  Theorem 1.3.

Let $\Gamma$ be a chordal specially oriented graph, let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a continuous
homomorphism (satisfying $\operatorname{Im}(\lambda)\subseteq
1+4\mathbb{Z}_{2}$ if $\ell=2$), and let
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ be the oriented pro-$\ell$ RAAG
associated to $\Gamma$ and $\lambda$. Then:

  * (i)

$G_{\Gamma,\lambda}$ has the Bogomolov-Positselski property;

  * (ii)

every finitely generated closed subgroup of $G_{\Gamma,\lambda}$ is of type
$FP_{\infty}$. In particular, $G$ is coherent;

  * (iii)

the $\mathbb{F}_{\ell}$-cohomology algebra
$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$ is quadratic — in
particular, it is isomorphic to the exterior Stanley-Reisner
$\mathbb{F}_{\ell}$-algebra
$\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})$ associated to
$\ddot{\Gamma}^{\mathrm{op}}$.

Note that Theorem 1.3–(i) can be seen as an oriented pro-$\ell$ analogue of a
theorem of H. Servatius, C. Droms and B. Servatius (see [serv]). One may
speculate whether for a generalized pro-$\ell$ RAAG’s $G_{\Gamma,\lambda}$
based on a specially oriented graph $\Gamma$ the Bogomolov-Positselski
property of $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is equivalent to
the chordality of $\Gamma$. Theorem 1.3–(ii) can be seen as an oriented
pro-$\ell$ analogue of another theorem of C. Droms (see [droms:coh]). As
before one is tempted to believe that the coherence property of
$G_{\Gamma,\lambda}$ for a specially oriented graph $\Gamma$ is equivalent to
the chordality of $\Gamma$. Theorem 1.3–(iii) can be seen as an oriented
analogue of a result of M. Salvetti on discrete RAAG’s (cf. [salv:coh]) which
holds independently of the chordality of the underlying graph. Therefore, one
may conjecture that Theorem 1.3–(iii) holds for all $G_{\Gamma,\lambda}$ based
on specially oriented graphs $\Gamma$ (see 1.4 below).

One reason why it is interesting to investigate right-angled Artin pro-$\ell$
groups $G_{\Gamma}$ (with $\Gamma$ a graph without special vertices and edges)
in a Galois-theoretic context is the fact that
$\mathbf{H}^{\bullet}(G_{\Gamma},\mathbb{F}_{\ell})$ is isomorphic to the
exterior Stanley-Reisner $\mathbb{F}_{\ell}$-algebra
$\mathbf{\Lambda}_{\bullet}(\Gamma^{\mathrm{op}})$ associated to $\Gamma$, and
thus it is quadratic and Koszul (see, e.g., [papa, § 1.2]). Moreover, one has
that the $\mathbb{N}_{0}$-graded $\mathbb{F}_{\ell}$-group algebra
$\mathrm{gr}_{\bullet}(G_{\Gamma})$ associated to the augmentation filtration
is isomorphic to the Cartier-Foita $\mathbb{F}_{\ell}$-algebra
$\mathbf{R}_{\Gamma}$ associated to $\Gamma$, and thus it is quadratic and
Koszul as well. In particular, it is the quadratic dual of
$\mathbf{H}^{\bullet}(G_{\Gamma},\mathbb{F}_{\ell})$ (see [bartholdi, Thm.
1.2]).

The same phenomenon occurs for oriented pro-$\ell$ groups $(G,\theta)$ of
elementary type: if $G$ is such a pro-$\ell$ group, then both algebras
$\mathbf{H}^{\bullet}(G,\mathbb{F}_{\ell})$ and $\mathrm{gr}_{\bullet}(G)$ are
Koszul and quadratic dual to each other (see [MPQT, Thm. A–B]). Moreover, the
same phenomenon is conjectured to hold for finitely generated maximal
pro-$\ell$ Galois groups of fields containing a primitive $\ell$-th root of 1,
see [MPQT, Conj. 1.3 and Ques. 1.5]. We suspect that the same holds also for
oriented pro-$\ell$ RAAGs associated to specially oriented graphs. Altogether
we ask the following.

######  Question 1.4.

Let $\Gamma$ be a specially oriented graph, let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a continuous
homomorphism (satisfying $\operatorname{Im}(\lambda)\subseteq
1+4\mathbb{Z}_{2}$ if $\ell=2$), and let
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ be the oriented pro-$\ell$ RAAG
associated to $\Gamma$ and $\lambda$.

  1. (1)

Is it true that the $\mathbb{N}_{0}$-graded $\mathbb{F}_{\ell}$-group algebra
$\mathrm{gr}_{\bullet}(G_{\Gamma,\lambda})$ is quadratic — and thus isomorphic
to the Cartier-Foita $\mathbb{F}_{\ell}$-algebra associated to $\Gamma$?

  2. (2)

Is it true that the cohomology ring
$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$ is isomorphic to
the exterior Stanley-Reisner $\mathbb{F}_{\ell}$-algebra
$\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})$?

### Acknowledgments

The authors wish to warmly thank Ido Efrat, Ilir Snopce, Matteo Vannacci and
Pavel A. Zalesskiĭ for several helpful discussions.

##  2\. Graphs

Although there is no standard notion of a graph, all notions known to the
authors have one common feature: a graph consists of a pair of sets
$(\mathscr{V},\mathscr{E})$, a set of vertices $\mathscr{V}$, and a set of
edges $\mathscr{E}$. In this paper we make use of two different notions —
naïve graphs and oriented graphs —, which are now discussed in more detail.

###  2.1. Naïve graphs

A naïve graph $\Gamma=(\mathscr{V},\mathscr{E})$ consists of a non-empty set
of vertices $\mathscr{V}$ and a set of edges
$\mathscr{E}\subseteq\mathscr{P}_{2}(\mathscr{V})$, where
$\mathscr{P}_{2}(\mathscr{V})$ denotes the set of subsets of $\mathscr{V}$ of
cardinality $2$. The naïve graph $\Gamma=(\mathscr{V},\mathscr{E})$ satisfying
$\mathscr{E}=\mathscr{P}_{2}(\mathscr{V})$ is said to be a complete graph.
Subgraphs are defined in the obvious way, and a subgraph
$\Lambda=(\mathscr{V}^{\prime},\mathscr{E}^{\prime})$ of the naïve graph
$\Gamma=(\mathscr{V},\mathscr{E})$ is said to be induced, if

(2.1)
$\mathscr{E}^{\prime}=\mathscr{E}\cap\mathscr{P}_{2}(\mathscr{V}^{\prime}).$

Moreover, the subgraph is said to be proper if
$\mathscr{V}^{\prime}\subsetneq\mathscr{V}.$

Henceforth, we will always consider finite naïve graphs, i.e., naïve graphs
with a finite number of vertices.

Recall that a tree is a connected naïve graph
$\mathrm{T}=(\mathscr{V},\mathscr{E})$, $\mathscr{V}\neq\varnothing$, without
circuits as subgraphs (cf. [serre:trees, § 2.2]). Every connected naiv̈e graph
$\Gamma=(\mathscr{V},\mathscr{E})$ contains a maximal subtree, i.e., a tree
$\mathrm{T}=(\mathscr{V}_{\mathrm{T}},\mathscr{E}_{\mathrm{T}})$ which is a
subgraph of $\Gamma$ such that $\mathscr{V}_{\mathrm{T}}=\mathscr{V}$ (cf.
[serre:trees, Prop. 2.11]).

######  Example 2.1.

The naïve graphs $\mathrm{C}_{4}$ and $\mathrm{L}_{3}$ are respectively the
naïve graphs with geometric realization

|
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$\textstyle{v_{1}}$$\textstyle{v_{2}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{4}}$$\textstyle{v_{3}}$ and |
---|---
$\textstyle{v_{1}}$$\textstyle{v_{4}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{2}}$$\textstyle{v_{3}}$

i.e., $\mathrm{C}_{4}$ is a square and $\mathrm{L}_{3}$ is a line of length 3.

A finite complete subgraph $\Xi=(\mathscr{V}^{\prime},\mathscr{E}^{\prime})$
of a naïve graph $\Gamma=(\mathscr{V},\mathscr{E})$ is said to be a
$|\mathscr{V}^{\prime}|$-clique. Note that such a subgraph is always induced.
For $n\geq 1$, an $n$-clique $\Xi$ of $\Gamma$ is said to be maximal if there
are no $(n+1)$-cliques of $\Gamma$ containing $\Xi$ as subgraph.

######  Definition 2.2.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a naïve graph. The clique-graph of
$\Gamma$ is the graph $\Upsilon(\Gamma)=(\mathrm{\bf mx}(\Gamma),\mathrm{\bf
mx}^{2}(\Gamma))$ with

(2.2) $\displaystyle\mathrm{\bf mx}(\Gamma)$
$\displaystyle=\\{\,\Xi\subseteq\Gamma\mid\Xi\ \text{a maximal clique in}\
\Gamma\,\\}$ $\displaystyle\mathrm{\bf mx}^{2}(\Gamma)$
$\displaystyle=\\{\,\\{\Xi,\Xi^{\prime}\\}\mid\Xi,\Xi^{\prime}\in\mathrm{\bf
mx}(\Gamma):\mathscr{V}(\Xi)\cap\mathscr{V}(\Xi^{\prime})\not=\varnothing\,\\}.$

A maximal subtree $\mathrm{T}_{\Upsilon(\Gamma)}$ of $\Upsilon(\Gamma)$ is
said to have the clique-intersection property if, for every pair of distinct
maximal cliques $\Xi,\Xi^{\prime}\in\mathrm{\bf mx}(\Gamma)$, with path

$\begin{split}P_{\Xi,\Xi^{\prime}}&=\left(\mathscr{V}({P_{\Xi,\Xi^{\prime}}}),\mathscr{E}({P_{\Xi,\Xi^{\prime}}})\right)\\\
\mathscr{V}(P_{\Xi,\Xi^{\prime}})&=\\{\>\Xi_{1}=\Xi,\>\Xi_{2},\>\ldots,\>\Xi_{r-1},\Xi_{r}=\Xi^{\prime}\>\\}\subseteq\mathrm{\bf
mx}(\Gamma),\\\
\mathscr{E}(P_{\Xi,\Xi^{\prime}})&=\left\\{\>\\{\Xi_{1},\Xi_{2}\\},\>\ldots,\>\\{\Xi_{r-1},\Xi_{r}\\}\>\right\\}\subseteq\mathscr{E}(\mathrm{T}_{\Upsilon(\Gamma)}),\end{split}$

connecting them in the tree $\mathrm{T}_{\Upsilon(\Gamma)}$, the clique

$\Xi\cap\Xi^{\prime}=(\mathscr{V}(\Xi)\cap\mathscr{V}(\Xi^{\prime}),\mathscr{E}(\Xi)\cap\mathscr{E}(\Xi^{\prime}))$

is a subgraph of $\Xi_{i}$ for every $i=1,\ldots,r$ (cf. [chordalgraphs, §
3.1]).

For naïve graphs one has the following construction. Let
$\Gamma_{1}=(\mathscr{V}_{1},\mathscr{E}_{1})$ and
$\Gamma_{2}=(\mathscr{V}_{2},\mathscr{E}_{2})$ be two naïve graphs, with a
common induced proper subgraph
$\Lambda=(\mathscr{V}^{\prime},\mathscr{E}^{\prime})$. The patching of
$\Gamma_{1},\Gamma_{2}$ along $\Lambda$ is the graph
$\Gamma=(\mathscr{V},\mathscr{E})$ with

$\mathscr{V}=\mathscr{V}_{1}\cup\mathscr{V}_{2},\qquad\mathscr{E}=\mathscr{E}_{1}\cup\mathscr{E}_{2},$

where we identify the vertices lying in
$\mathscr{V}_{1}\cap\mathscr{V}^{\prime}$ and in
$\mathscr{V}_{2}\cap\mathscr{V}^{\prime}$, and the edges lying in
$\mathscr{E}_{1}\cap\mathscr{E}^{\prime}$ and in
$\mathscr{E}_{2}\cap\mathscr{E}^{\prime}$.

###  2.2. Oriented graphs

An oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ consists of a non-empty
set of vertices $\mathscr{V}$, partitioned as a disjoint union
$\mathscr{V}=\mathscr{V}_{\mathrm{s}}\sqcup\mathscr{V}_{\mathrm{o}}$; and a
set of edges

$\mathscr{E}\subseteq\mathscr{V}\times\mathscr{V}\smallsetminus\Delta(\mathscr{V}),$

where $\Delta(\mathscr{V})=\\{\,(v,v)\mid v\in\mathscr{V}\,\\}$ denotes the
diagonal in $\mathscr{V}\times\mathscr{V}$. By definition, every oriented
graph $\Gamma=(\mathscr{V},\mathscr{E})$ comes equipped with two maps, the
origin $o\colon\mathscr{E}\to\mathscr{V}$ given by the projection on the first
coordinate, and the terminus $t\colon\mathscr{E}\to\mathscr{V}$, given by the
projection on the second coordinate. One has a partition
$\mathscr{E}=\mathscr{E}_{\mathrm{s}}\sqcup\mathscr{E}_{\mathrm{o}}$, where

$\mathscr{E}_{\mathrm{s}}=\\{\>\mathbf{e}\in\mathscr{E}\>\mid\>(t(\mathbf{e}),o(\mathbf{e}))\notin\mathscr{E}\>\\}$

is the set of special edges, while $\mathscr{E}_{\mathrm{o}}$ is the set of
ordinary edges. Analogously, the elements of $\mathscr{V}_{\mathrm{s}}$ and
$\mathscr{V}_{\mathrm{o}}$ are called special vertices and ordinary vertices
respectively, and they must satisfy the following condition: if
$\mathbf{e}\in\mathscr{E}$, then $o(\mathbf{e})\in\mathscr{V}_{\mathrm{o}}$.

Every oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ defines a naïve graph
$\ddot{\Gamma}=(\mathscr{V},\ddot{\mathscr{E}})$, where

(2.3)
$\ddot{\mathscr{E}}=\\{\,\\{o(\mathbf{e}),t(\mathbf{e})\\}\mid\mathbf{e}\in\mathscr{E}\,\\}.$

The notions of subgraphs, induced subgraphs, cliques and patching of oriented
graphs are defined in the obvious way. Henceforth, we will consider only
finite oriented graphs.

######  Example 2.3.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph. We realize a
special edge $(v,w)\in\mathscr{E}_{\mathrm{s}}$ as an arrow originating at $v$
and pointing at $w$. If $(v,w),(w,v)\in\mathscr{E}$, then we realize them as a
unique “unoriented” edge joining $v$ and $w$. For example, the two pictures

|
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$\textstyle{v_{1}}$$\textstyle{v_{4}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{2}}$$\textstyle{v_{3}}$ |
---|---
$\textstyle{v_{1}}$$\textstyle{v_{4}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{v_{2}}$$\textstyle{v_{3}}$

realize the same oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$, with

$\mathscr{E}=\underbrace{\\{\>(v_{1},v_{2}),\>(v_{1},v_{4}),\>(v_{2},v_{3})\>\\}}_{\mathscr{E}_{\mathrm{s}}}\sqcup\underbrace{\\{\>(v_{2},v_{4}),\>(v_{4},v_{2}),\>(v_{3},v_{4}),\>(v_{4},v_{2})\>\\}}_{\mathscr{E}_{\mathrm{o}}}$

and $\mathscr{V}_{\mathrm{o}}=\mathscr{V}$. (Henceforth, we will use the
second type of realization for oriented graphs.)

######  Remark 2.4.

Let $\Gamma_{1}=(\mathscr{V}_{1},\mathscr{E}_{1})$ and
$\Gamma_{2}=(\mathscr{V}_{2},\mathscr{E}_{2})$ be two oriented graphs, with
$\mathscr{V}_{1}=\mathscr{V}_{2}=\\{v,w\\}$,
$\mathscr{E}_{1}=\mathscr{E}_{2}=\varnothing$, and such that both $v,w$ are
ordinary vertices of $\Gamma_{1}$, but $v$ is a special vertex of
$\Gamma_{2}$. Then $\Gamma_{1}$ and $\Gamma_{2}$ are not the same oriented
graph, even if their geometric realizations are equal.

###  2.3. Specially oriented graphs

######  Definition 2.5.

An oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ with decompositions
$\mathscr{V}=\mathscr{V}_{\mathrm{o}}\sqcup\mathscr{V}_{\mathrm{s}}$ and
$\mathscr{E}=\mathscr{E}_{\mathrm{o}}\sqcup\mathscr{E}_{\mathrm{s}}$ is said
to be a specially oriented graph (or just a special graph), if the terminus of
every special edge is a special vertex, i.e.,

$\mathscr{V}_{\mathrm{s}}\supseteq\left\\{\>t(\mathbf{e})\>\mid\>\mathbf{e}\in\mathscr{E}_{\mathrm{s}}\>\right\\}.$

In other words, an oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ is
special if, and only if, $\Gamma$ does not contain a subgraph
$\Gamma^{\prime}$ whose geometric realization is either

(2.4)
$\textstyle{y}$$\textstyle{x}$$\textstyle{z}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
or
$\textstyle{y}$$\textstyle{x}$$\textstyle{z}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

— we underline that we do not require $\Gamma^{\prime}$ to be induced, so that
the vertices $y$ and $z$ may be joined by an edge.

######  Remark 2.6.

By definition, a specially oriented graph without special vertices is a
combinatiorial graph in the sense of J-P. Serre (cf. [serre:trees, § 2.1]).

######  Example 2.7.

A complete oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ is specially
oriented if, and only if, there is at most one special vertex and — if there
is one, say $v$ — $(v,w)\in\mathscr{E}_{\mathrm{s}}$ for every
$w\in\mathscr{V}\smallsetminus\\{v\\}$.

In other words, an oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ is
special if, and only if, for every special vertex
$v\in\mathscr{V}_{\mathrm{s}}$ one has $(v,w)\notin\mathscr{E}$ for every
$w\in\mathscr{V}$.

###  2.4. Oriented graphs of elementary type

For oriented graphs we have the following two constructions.

  * (a)

Let $\Gamma_{1}=(\mathscr{V}_{1},\mathscr{E}_{1})$ and
$\Gamma_{2}=(\mathscr{V}_{2},\mathscr{E}_{2})$ be two oriented graphs. The
disjoint union of $\Gamma_{1}$ and $\Gamma_{2}$ is the oriented graph

$\Gamma_{1}\sqcup\Gamma_{2}:=\left(\mathscr{V}_{1}\sqcup\mathscr{V}_{2},\mathscr{E}_{1}\sqcup\mathscr{E}_{2}\right).$

  * (b)

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, with
$\mathscr{V}=\mathscr{V}_{\mathrm{s}}\sqcup\mathscr{V}_{\mathrm{o}}$. The cone
graph with basis $\Gamma$ is the oriented graph

$\nabla(\Gamma):=(\mathscr{V}(\nabla(\Gamma)),\mathscr{E}(\nabla(\Gamma))),$

where $\mathscr{V}(\nabla(\Gamma))=\mathscr{V}\sqcup\\{v\\}$, with $v$ a “new”
ordinary vertex, called the tip of the cone, and

$\mathscr{E}(\nabla(\Gamma))=\mathscr{E}\sqcup\\{(v,w)\mid
w\in\mathscr{V}_{\mathrm{s}}\\}\sqcup\\{(v,w),(w,v)\mid
w\in\mathscr{V}_{\mathrm{o}}\\}.$

Observe that
$\mathscr{V}(\nabla(\Gamma))_{\mathrm{s}}=\mathscr{V}_{\mathrm{s}}$.

######  Remark 2.8.

It is straightforward to see that if $\Gamma_{1}$ and $\Gamma_{2}$ are two
special graphs, then also the disjoint union $\Gamma_{1}\sqcup\Gamma_{2}$ is a
special graph. Similarly, if $\Gamma$ is a special graph, then also the cone
$\nabla(\Gamma)$ is a special graph.

The following definition generalizes the notion of graphs of elementary type
given in [ac:RAAGs, § 3.3].

######  Definition 2.9.

The family of special graphs of elementary type is the smallest family of
oriented graphs containing graphs consisting of a single vertex, and such
that:

  * (a)

if $\Gamma_{1},\Gamma_{2}$ are special graphs of elementary type, then also
their disjoint union $\Gamma_{1}\sqcup\Gamma_{2}$ is a special graph of
elementary type.

  * (b)

if $\Gamma$ is a special graph of elementary type, then also the cone
$\nabla(\Gamma)$ is a special graph of elementary type.

######  Example 2.10.

Any special complete graph is of elementary type, as it may be constructed as
iterated cone — starting from the unique special vertex, if there is one. For
instance, the displayed graph may be obtained by iterating three times the
cone-construction by starting with the single special vertex $v$, i.e.,
$\Gamma=\nabla(\nabla(\nabla(\Gamma^{\prime})))$, where
$\Gamma^{\prime}=(\\{v\\},\varnothing)$ and $v$ is a special vertex.

$\textstyle{v}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{1}}$$\textstyle{v_{2}\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
v_{3}}$

######  Remark 2.11.

  * (a)

Since an oriented graph consisting of a single vertex is specially oriented,
by Remark 2.8 an oriented graph of elementary type is specially oriented.

  * (b)

If a connected oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ is of
elementary type, then either $|\mathscr{V}|=1$ or one has
$\Gamma=\nabla(\Gamma^{\prime})$ for some proper induced subgraph
$\Gamma^{\prime}\subseteq\Gamma$, and therefore there is an ordinary vertex of
$\Gamma$ which is joined to all other vertices.

######  Example 2.12.

  * (a)

By Remark 2.11, if $\Gamma$ is a special graph such that either
$\ddot{\Gamma}=\mathrm{C}_{4}$ or $\ddot{\Gamma}=\mathrm{L}_{3}$, then
$\Gamma$ is not a special graph of elementary type, as $\Gamma$ is connected
but there are no vertices which are joined to each other vertex.

  * (b)

The special graph $\Lambda_{\mathrm{s}}=(\mathscr{V},\mathscr{E})$ with
geometric realization

$\textstyle{v_{2}}$$\textstyle{v_{1}}$$\textstyle{\bullet}$$\textstyle{v_{3}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

is not of elementary type, as it is connected but $v_{2}$ is special, so that
$\Lambda_{\mathrm{s}}$ does not decompose as a cone by Remark 2.11–(b).

Observe that the inductive definition of oriented graphs of elementary type
implies that every induced subgraph of an oriented graph of elementary type is
again an oriented graph of elementary type. Oriented graphs of elementary type
are characterized as follows.

######  Proposition 2.13.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph. Then $\Gamma$ is
of elementary type if, and only if, $\Gamma$ is specially oriented and it does
not contain an induced subgraph $\Gamma^{\prime}$ such that either
$\ddot{\Gamma}^{\prime}=\mathrm{C}_{4}$,
$\ddot{\Gamma}^{\prime}=\mathrm{L}_{3}$, or
$\Gamma^{\prime}=\Lambda_{\mathrm{s}}$ (cf. Examples 2.1, 2.12–(b)).

###### Proof.

If $\Gamma$ is not specially oriented, then it is not of elementary type by
Remark 2.11–(a). If $\Gamma$ contains an induced subgraph $\Gamma^{\prime}$
such that either $\ddot{\Gamma}^{\prime}=\mathrm{C}_{4}$ or
$\ddot{\Gamma}^{\prime}=\mathrm{L}_{3}$, or
$\Gamma^{\prime}=\Lambda_{\mathrm{s}}$, then $\Gamma^{\prime}$ is not of
elementary type by Example 2.12–(b)–(c), and thus $\Gamma$ is not of
elementary type as well.

Conversely, assume that $\Gamma$ contains no such induced subgraphs, and that
it is specially oriented. By Remark 2.11–(b), without loss of generality we
may assume that $\Gamma$ is connected. We proceed by induction on
$|\mathscr{V}|$. If $|\mathscr{V}|\leq 2$, then clearly $\Gamma$ is of
elementary type. Now, the naïve graph
$\ddot{\Gamma}=(\mathscr{V},\ddot{\mathscr{E}})$ contains no induced subgraphs
$\ddot{\Gamma}^{\prime}$ equal to $\mathrm{C}_{4}$ or $\mathrm{L}_{3}$. By
[wolk], $\ddot{\Gamma}$ has a _central vertex_ , i.e., a vertex
$w\in\mathscr{V}$ such that $\\{w,v\\}\in\ddot{\mathscr{E}}$ for every
$v\in\mathscr{V}\smallsetminus\\{w\\}$. We have two cases.

Case (1). If $w\in\mathscr{V}_{\mathrm{o}}$, then
$\Gamma=\nabla(\Gamma^{\prime})$, with $\Gamma^{\prime}$ the induced subgraph
of $\Gamma$ with vertices $\mathscr{V}\smallsetminus\\{w\\}$, which is of
elementary type by induction — and thus also $\Gamma$ is of elementary type.

Case (2). If $w\in\mathscr{V}_{\mathrm{s}}$, then for every
$v\in\mathscr{V}\smallsetminus\\{w\\}$ one has $(v,w)\in\mathscr{E}$,
$(w,v)\notin\mathscr{E}$, and
$\mathscr{V}_{\mathrm{o}}=\mathscr{V}\smallsetminus\\{w\\}$, as $\Gamma$ is
specially oriented (cf. Remark 2.11–(a)). Let $v_{1},v_{2}$ be two distinct
ordinary vertices of $\Gamma$. If
$(v_{1},v_{2}),(v_{2},v_{1})\notin\mathscr{E}$, then the induced subgraph of
$\Gamma$ with vertices $v_{1},v_{2},w$ is isomorphic to
$\Lambda_{\mathrm{s}}$, a contradiction. On the other hand,
$(v_{1},v_{2}),(v_{2},v_{1})\notin\mathscr{E}_{\mathrm{s}}$, as
$v_{1},v_{2}\in\mathscr{V}_{\mathrm{o}}$. Therefore,
$(v_{1},v_{2})\in\mathscr{E}_{\mathrm{o}}$, and hence $\Gamma$ is a complete
specially oriented graph — in particular, $\Gamma$ is of elementary type by
Example 2.12–(a). ∎

###  2.5. Graphs of pro-$\ell$ groups

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a connected oriented graph without
special edges (i.e., a combinatorial graph in the sense of Serre — see
[serre:trees, §2.1]), and let

$\mathcal{G}(\Gamma)=\\{G(v),G(\mathbf{e})\mid
v\in\mathscr{V},\mathbf{e}\in\mathscr{E}\\}$

be a collection of finitely generated pro-$\ell$ groups, endowed with
monomorphisms of pro-$\ell$ groups

$\partial_{\mathbf{e}}\colon G({\mathbf{e}})\longrightarrow G(t(\mathbf{e}))$

for every edge $\mathbf{e}\in\mathscr{E}$. Then $\mathcal{G}(\Gamma)$ is
called a finite graph of (finitely generated) pro-$\ell$ groups based on
$\Gamma$. A finite graph of pro-$\ell$ groups $\mathcal{G}(\Gamma)$ is said to
be reduced if for every edge $\mathbf{e}\in\mathscr{E}$,
$\partial_{\mathbf{e}}\colon G({\mathbf{e}})\to G(t(\mathbf{e}))$ is an
isomorphism.

Suppose that $\Gamma=\mathrm{T}$ is a tree. The fundamental group
$\Pi_{1}(\mathcal{G}(\mathrm{T}))$ of a finite graph of pro-$\ell$ groups
$\mathcal{G}(\mathrm{T})$ based on $\mathrm{T}$ is defined by the following
pro-$\ell$ presentation:

$\Pi_{1}(\mathcal{G}(\mathrm{T}))=\left\langle G(v)\mid
v\in\mathscr{V},\partial_{\mathbf{e}}(g)=\partial_{\bar{\mathbf{e}}}(g)\;\forall\>g\in
G(\mathbf{e}),\mathbf{e}\in\mathscr{E}\right\rangle$

(cf. [zalmel]). We have the following [zalmel, Cor. 4.5].

######  Theorem 2.14.

Let $U$ be an open subgroup of the fundamental pro-$\ell$ group
$G=\Pi_{1}(\mathcal{G},\Gamma)$ of a finite graph of pro-$\ell$ groups
$\mathcal{G}$ based on the graph $\Gamma$, and let $\mathrm{T}$ be the Bass-
Serre tree associated to $\mathcal{G}$. Then $U$ is the fundamental pro-$\ell$
group $U=\Pi_{1}(\mathcal{U},\Delta)$, with
$\Delta=U\backslash\\!\backslash\mathrm{T}$, and vertex and edge groups
$\mathcal{U}(m)$, $m\in\Delta$, are stabilizers $U_{s(m)}$, where
$s\colon\Delta\to\mathrm{T}$ is a connected transversal of $\Delta$ in
$\mathrm{T}$.

##  3\. Pro-$\ell$ groups and orientations

###  3.1. Preliminaries on pro-$\ell$ groups

We work in the world of pro-$\ell$ groups. Henceforth, every subgroup of a
pro-$\ell$ group will be tacitly assumed to be closed, and the generators of a
subgroup will be intended in the topological sense.

In particular, for a pro-$\ell$ group $G$ and a positive integer $n$, $G^{n}$
will denote the closed subgroup of $G$ generated by the $n$-th powers of all
elements of $G$. Moreover, for two elements $g,h\in G$, we set

${}^{g}h=ghg^{-1},\qquad\text{and}\qquad[g,h]={}^{g}h\cdot h^{-1}.$

Given two subgroups $H_{1},H_{2}$ of $G$, $[H_{1},H_{2}]$ will denote the
closed subgroup of $G$ generated by all commutators $[h,g]$ with $h\in H_{1}$
and $g\in H_{2}$. In particular:

  * (a)

$G^{\prime}$ will denote the closure of the commutator subgroup $[G,G]$ of
$G$;

  * (b)

$\Phi(G)=G^{\ell}\cdot G^{\prime}$ will denote the Frattini subgroup of $G$;

  * (c)

$G_{(3)}$ will denote the subgroup of $G$ defined by

$G_{(3)}=\begin{cases}G^{\ell}\cdot[G^{\prime},G]&\text{if }{\ell}\neq 2\\\
G^{4}\cdot(G^{\prime})^{2}\cdot[G^{\prime},G]&\text{if }{\ell}=2.\end{cases}$

For the properties of Galois cohomology of pro-$\ell$ groups we refer to
[serre:gal, Ch. I] and to [nsw:cohn, Ch. I and Ch. III–§ 9].

###  3.2. Oriented pro-$\ell$ groups

Let $G$ be a pro-$\ell$ group. An orientation of $G$ is a continuous
homomorphism $\theta\colon G\to\mathbb{Z}_{\ell}^{\times}$, where
$\mathbb{Z}_{\ell}^{\times}$ denotes the group of units of
$\mathbb{Z}_{\ell}$. Observe that $\operatorname{Im}(\theta)\subseteq
1+{\ell}\mathbb{Z}_{\ell}$, as the pro-$\ell$ Sylow subgroup of
$\mathbb{Z}_{\ell}^{\times}$ is

$1+{\ell}\mathbb{Z}_{\ell}=\\{1+{\ell}\lambda,\>\lambda\in\mathbb{Z}_{\ell}\\}.$

We say that the orientation $\theta$ is torsion-free if $\ell$ is odd, or if
$\ell=2$ and $\operatorname{Im}(\theta)\subseteq 1+4\mathbb{Z}_{2}$.

We call a couple $(G,\theta)$ consisting of a pro-$\ell$ group together with
an orientation $\theta\colon G\to\mathbb{Z}_{\ell}^{\times}$ an oriented
pro-$\ell$ group (in [efrat:small, eq:kummer] an oriented pro-$\ell$ group is
called a cyclotomic pro-$\ell$ pair). A morphism of oriented pro-$\ell$ groups
$(G,\theta)\to(H,\tau)$ is a morphism of pro-$\ell$ groups $\varphi\colon G\to
H$ satisfying $\theta=\tau\circ\varphi$.

An oriented pro-$\ell$ group $(G,\theta)$ comes endowed with a continuous left
$G$-module $\mathbb{Z}_{\ell}(1)$, which is isomorphic to $\mathbb{Z}_{\ell}$
as an abelian pro-$\ell$ group, and with $G$-action

$g.\lambda=\theta(g)\cdot\lambda\qquad\text{ for every }g\in
G,\>\lambda\in\mathbb{Z}_{\ell}(1).$

Observe that the $G$-module $\mathbb{Z}_{\ell}(1)/{\ell}$ is simply the
trivial module $\mathbb{Z}/{\ell}$, as $\theta(g)\equiv 1\bmod{\ell}$ for all
$g\in G$.

In the category of oriented pro-$\ell$ groups one has the following
constructions (cf. e.g., [efrat:small, § 3]).

  * (a)

If $(G,\theta)$ is an oriented pro-$\ell$ group and $N$ is a normal subgroup
of $G$ such that $N\subseteq\operatorname{ker}(\theta)$, then the quotient
$G/N$ yields an oriented pro-$\ell$ group $(G/N,\theta_{/N})$, such that the
canonical projection $G\to G/N$ induces a morphism of oriented pro-$\ell$
groups $(G,\theta)\to(G/N,\theta_{/N})$.

  * (b)

If $(G_{1},\theta_{1})$ and $(G_{2},\theta_{2})$ are oriented pro-$\ell$
groups, then the free pro-$\ell$ product of $G_{1}$ and $G_{2}$ yields an
oriented pro-$\ell$ group

$(G_{1},\theta_{1})\amalg_{\hat{\ell}}(G_{2},\theta_{2}):=(G_{1}\amalg_{\hat{\ell}}G_{2},\theta),$

where $\theta$ is the orientation obtained by the universal property of the
free product.

  * (c)

If $(G,\theta)$ is an oriented pro-$\ell$ group and $A$ is an abelian
pro-$\ell$ group, then one has an oriented pro-$\ell$ group

$A\rtimes(G,\theta):=(A\rtimes G,\tilde{\theta}),$

where $gag^{-1}=a^{\theta(g)}$ for every $a\in A$ and $g\in G$, and such that
the canonical projection $A\rtimes G\to G$ induces a morphism of oriented
pro-$\ell$ groups $(A\rtimes G,\tilde{\theta})\to(G,\theta)$.

The most relevant examples of oriented pro-$\ell$ groups arise from maximal
pro-$\ell$ Galois groups (cf. [eq:kummer, § 4]).

######  Example 3.1.

For a field $\mathbb{K}$, let $\bar{\mathbb{K}}$ denote a separable closure of
$\mathbb{K}$, and let $\mu_{\ell^{\infty}}$ denote the group of roots of 1 of
$\ell$-power order lying in $\bar{\mathbb{K}}$. If $\mathbb{K}$ contains a
root of 1 of order $\ell$, then $\mu_{\ell^{\infty}}$ is contained in the
maximal pro-$\ell$-extension $\mathbb{K}(\ell)$ of $\mathbb{K}$, and the
action of the maximal pro-$\ell$ Galois group
$G_{\mathbb{K}}(\ell)=\mathrm{Gal}(\mathbb{K}(\ell)/\mathbb{K})$ of
$\mathbb{K}$ on $\mu_{\ell^{\infty}}$ fixes the roots of order $\ell$, and
induces a natural orientation, the so-called pro-$\ell$-cyclotomic character

$\hat{\theta}_{\mathbb{K},\ell}\colon
G_{\mathbb{K}}(\ell)\longrightarrow\mathbb{Z}_{\ell}^{\times},$

as $\mu_{\ell^{\infty}}\simeq\mathbb{Z}[\frac{1}{\ell}]/\mathbb{Z}$ and
$\operatorname{Aut}(\mathbb{Z}[\frac{1}{\ell}]/\mathbb{Z})$ is isomorphic to
$\mathbb{Z}_{\ell}^{\times}$. In particular,

$\sigma(\zeta)=\zeta^{\hat{\theta}_{\mathbb{K},\ell}(\sigma)}\qquad\text{for
all }\sigma\in G_{\mathbb{K}}(\ell),\zeta\in\mu_{\ell^{\infty}}.$

Furthermore, one has
$\operatorname{Im}(\hat{\theta}_{\mathbb{K},\ell})=1+\ell^{f}\mathbb{Z}_{\ell}$,
where $f$ is the positive integer satisfying
$|\mu_{\ell^{\infty}}\cap\mathbb{K}^{\times}|=\ell^{f}$ in case
$\mu_{\ell^{\infty}}\cap\mathbb{K}^{\times}$ is finite, and
$\operatorname{Im}(\hat{\theta}_{\mathbb{K},\ell})=\\{1\\}$ if
$\mu_{\ell^{\infty}}\subseteq\mathbb{K}^{\times}$. The continuous
$G_{\mathbb{K}}(\ell)$-module $\mathbb{Z}_{\ell}(1)$ induced by the cyclotomic
character is called the 1st Tate twist of $\mathbb{Z}_{\ell}$ (cf. [nsw:cohn,
Def. 7.3.6]), and for every $n\geq 1$, $\mathbb{Z}_{\ell}(1)/\ell^{n}$ is
isomorphic to the $G_{\mathbb{K}}(\ell)$-module of the $\ell^{n}$-th roots of
1.

###  3.3. $\theta$-abelian oriented pro-$\ell$ groups and locally uniform
pro-$\ell$ groups

An oriented pro-$\ell$ group $(G,\theta)$ with torsion-free orientation is
said to be $\theta$-abelian if $\operatorname{ker}(\theta)$ is a free abelian
pro-$\ell$ group and one has an isomorphism of oriented pro-$\ell$ groups

$(G,\theta)\simeq\operatorname{ker}(\theta)\rtimes(G/\operatorname{ker}(\theta),\theta_{/\operatorname{ker}(\theta)}).$

One has the following group-theoretic “translation” of $\theta$-abelianity.
Recall that a pro-$\ell$ group $G$ is called locally uniform if every finitely
generated subgroup $H$ of $G$ is uniform — i.e., $H$ is torsion free and the
commutator subgroup $H^{\prime}$ is contained in $H^{\ell}$, and also in
$H^{4}$ if ${\ell}=2$ (cf. [cq:bk, § 3]). One has the following (cf. [cq:bk,
Thm. A]).

######  Proposition 3.2.

A pro-$\ell$ group $G$ is locally uniform if, and only if, there exists a
torsion-free orientation $\eth_{G}\colon G\to\mathbb{Z}_{\ell}^{\times}$ such
that $(G,\eth_{G})$ is a $\eth_{G}$-abelian oriented pro-$\ell$ group.

In other words, a pro-$\ell$ group $G$ is locally uniform if, and only if, $G$
has a presentation

$G=\left\langle\>x_{0},x_{i}:i\in
I\>\mid\>{}^{x_{0}}x_{i}=x_{i}^{1+\ell^{f}},\>[x_{i},x_{j}]=1\;\forall\>i,j\in
I\>\>\right\rangle$

for some set $I$ and some $f\in\mathbb{N}\cup\\{\infty\\}$ ($f\geq 2$ if
${\ell}=2$), and in this case $\eth_{G}(x_{0})=1+p^{f}$ and
$\eth_{G}(x_{i})=1$ for all $i\in I$. We call $\eth_{G}$ the canonical
orientation of $G$. Observe that for every subgroup $H\subseteq G$ one has
$\eth_{H}=\eth_{G}|_{H}$.

######  Remark 3.3.

By Lazard’s work [lazard], the $\mathbb{F}_{\ell}$-cohomology algebra
$\mathbf{H}^{\bullet}(G,\mathbb{F}_{\ell})$ of a (finitely generated) uniform
pro-$\ell$ group $G$ is isomorphic to the exterior algebra generated by
$\mathrm{H}^{1}(G,\mathbb{F}_{\ell})$ (cf. e.g., [sw:cohomology, Thm. 5.1.5]).
Hence, a locally uniform pro-$\ell$ group is Bloch-Kato.

###  3.4. Kummerian oriented pro-$\ell$ groups and 1-cyclotomicity

An oriented pro-$\ell$ group $(G,\theta)$ comes endowed with a distinguished
normal subgroup of $G$,

$K_{\theta}(G)=\left\langle\>{}^{g}h\cdot h^{-\theta(g)}\>\mid\>g\in
G,\>h\in\operatorname{ker}(\theta)\>\right\rangle.$

(cf. [eq:kummer]). One has $K_{\theta}(G)\subseteq\Phi(G)$, and moreover
$K_{\theta}(G)\supseteq\operatorname{ker}(\theta)^{\prime}$, so that the
quotient $\operatorname{ker}(\theta)/K_{\theta}(G)$ is an abelian pro-$\ell$
group. Observe that if $\theta$ is trivial (i.e., constantly equal to 1), then
$K_{\theta}(G)=G^{\prime}$.

One has the following (cf., e.g., [qw:bogomolov, Prop. 2.6 and § 3.2]).

######  Proposition 3.4.

Let $(G,\theta)$ be an oriented pro-$\ell$ group with torsion-free
orientation. The following are equivalent:

  * (i)

The map

(3.1)
$\mathrm{H}^{1}(G,\mathbb{Z}_{\ell}(1)/{\ell}^{n})\longrightarrow\mathrm{H}^{1}(G,\mathbb{F}_{\ell}),$

induced by the epimorphism of continuous $G$-modules
$\mathbb{Z}_{\ell}(1)/{\ell}^{n}\to\mathbb{F}_{\ell}$, is surjective for every
$n\geq 1$.

  * (ii)

The quotient $\operatorname{ker}(\theta)/K_{\theta}(G)$ is a free abelian
pro-$\ell$ group.

  * (iii)

There exists an epimorphism of pro-$\ell$ groups $\varphi\colon G\to\bar{G}$
where $\bar{G}$ is locally uniform and
$\operatorname{ker}(\varphi)\subseteq\Phi(G)$, inducing an epimorphism of
oriented pro-$\ell$ groups $(G,\theta)\to(\bar{G},\eth_{\bar{G}})$.

If these conditions hold, then $\operatorname{ker}(\varphi)=K_{\theta}(G)$ and

$(\bar{G},\eth_{\bar{G}})\simeq\operatorname{ker}(\theta)/K_{\theta}(G)\rtimes(G/\operatorname{ker}(\theta),\theta_{/\operatorname{ker}(\theta)}).$

An oriented pro-$\ell$ group $(G,\theta)$ with torsion-free orientation
$\theta$ satisfying the above equivalent conditions is said to be Kummerian
(cf. [eq:kummer, Def. 3.4]). If moreover the oriented pro-$\ell$ group
$(H,\theta|_{H})$ is Kummerian for every subgroup $H$ of $G$, then the
oriented pro-$\ell$ group $(G,\theta)$ is said to be 1-cyclotomic (cf.
[qw:cyclotomic]). (Observe that in [dcf:lift, cq:galfeat, cq:1smoothBK], a
1-cyclotomic oriented pro-$\ell$ group is said to be 1-smooth.)

######  Example 3.5.

  * (a)

If $(G,\theta)$ is a $\theta$-abelian oriented pro-$\ell$ group with torsion-
free orientation, then, by Proposition 3.4–(iii), it is 1-cyclotomic, as
$(H,\theta|_{H})$ is $\theta|_{H}$-abelian for every subgroup $H$.

  * (b)

If $G$ is a free pro-$\ell$ group, then the oriented pro-$\ell$ group
$(G,\theta)$ is 1-cyclotomic for any orientation $\theta\colon
G\to\mathbb{Z}_{\ell}^{\times}$ (cf. [qw:cyclotomic, § 2.2]).

  * (c)

If $(G,\theta)$ is an oriented pro-$\ell$ group with trivial orientation
$\theta\equiv\mathbf{1}$, then $(G,\theta)$ is Kummerian if, and only if, the
abelianization $G/G^{\prime}$ is a free abelian pro-$\ell$ group (cf.
[eq:kummer, Example 3.5–(1)]). Consequently, $(G,\theta)$ is 1-cyclotomic if,
and only if, $H/H^{\prime}$ is a free abelian pro-$\ell$ group for every
subgroup $H$ of $G$ (pro-$\ell$ groups satisfying this property are also
called absolutely torsion-free pro-$\ell$ groups, cf. [wurfel]).

1-cyclotomic oriented pro-$\ell$ groups arise naturally in Galois theory (cf.
[eq:kummer, Thm. 4.2] and [qw:cyclotomic, § 2.3]).

######  Theorem 3.6.

Let $\mathbb{K}$ be a field containing a primitive $\ell$-th root of 1 (and
also $\sqrt{-1}$ if $\ell=2$). Then
$(G_{\mathbb{K}}(\ell),\hat{\theta}_{\mathbb{K},\ell})$ is a torsion-free
1-cyclotomic oriented pro-$\ell$ group.

The pro-$\ell$ version of De Clercq-Florence’s Smoothness Conjecture (cf.
[dcf:lift, Conj. 14.25], see also [cq:galfeat, Conj. 2.10]) states the
following.

######  Conjecture 3.7.

If a pro-$\ell$ group $G$ may be endowed with a 1-cyclotomic orientation
$\theta\colon G\to\mathbb{Z}_{\ell}^{\times}$, then $G$ is weakly Bloch-Kato.

The definition of weakly Bloch-Kato pro-$\ell$ group may be found in
[dcf:lift, Def. 14.23] — for our purposes it is enough to remark that a Bloch-
Kato pro-$\ell$ group is (unsurprisingly) also weakly Bloch-Kato (see also
[cq:galfeat, § 2.3]).

######  Remark 3.8.

Let $(G,\theta)$ be a Kummerian oriented pro-$\ell$ group, with $\theta$ a
torsion-free orientation. Then by Proposition 3.4 one has

$(G/K_{\theta}(G),\theta_{/K_{\theta}(G)})\simeq\operatorname{ker}(\theta)/K_{\theta}(G)\rtimes(G/\operatorname{ker}(\theta),\theta_{/\operatorname{ker}(\theta)})$

with $\operatorname{ker}(\theta)/K_{\theta}(G)$ a free abelian pro-$\ell$
group. Now let $N$ be a normal subgroup of $G$ contained in
$\operatorname{ker}(\theta)$ such that the map

$\mathrm{res}_{G,N}^{1}\colon\mathrm{H}^{1}(G,\mathbb{F}_{\ell})\longrightarrow\mathrm{H}^{1}(N,\mathbb{F}_{\ell})^{G}$

is injective — namely, by duality the map $N/N^{\ell}[N,G]\to G/\Phi(G)$
induced by the inclusion $N\hookrightarrow G$ is injective. Then the group
$\bar{N}=NK_{\theta}(G)/K_{\theta}(G)$ is an isolated subgroup of the free
pro-$\ell$ group $\operatorname{ker}(\theta)/K_{\theta}(G)$. Therefore,

$\frac{\operatorname{ker}(\theta)}{K_{\theta}(G)}\simeq\bar{N}\times
A\qquad\text{and}\qquad\frac{\operatorname{ker}(\theta_{/N})}{K_{\theta_{/N}}(G/N)}\simeq\frac{\operatorname{ker}(\theta)/K_{\theta}(G)}{\bar{N}}\simeq
A,$

where $A$ is a free pro-$\ell$ group. Thus, also the oriented pro-$\ell$ group
$(G/N,\theta_{/N})$ is Kummerian by Proposition 3.4.

###  3.5. Oriented pro-$\ell$ groups of elementary type

The following definition is due to I. Efrat (cf. [efrat:small, § 3], see also
[qw:cyclotomic, § 7.5]).

######  Definition 3.9.

The family of oriented pro-$\ell$ groups of elementary type is the smallest
class of oriented pro-$\ell$ groups such that

  * (a)

the oriented pro-$\ell$ group $(\\{1\\},\mathbf{1})$ consisting of the trivial
group and the trivial orientation is of elementary type, as well as the
oriented pro-$\ell$ group $(\mathbb{Z}_{\ell},\lambda)$ for any linear
orientation $\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$;

  * (b)

any oriented pro-$\ell$ group $(G,\theta)$ with $G$ a Demushkin group and
$\theta\colon G\to\mathbb{Z}_{\ell}^{\times}$ its canonical orientation (cf.
[labute:demushkin, Thm. 4] and [qw:cyclotomic, § 5.3]);

  * (c)

if $(G_{1},\theta_{1})$ and $(G_{2},\theta_{2})$ are two oriented pro-$\ell$
groups of elementary type, then also the free product

$(G_{1},\theta_{1})\amalg^{\hat{\ell}}(G_{2},\theta_{2})$

is an oriented pro-$\ell$ group of elementary type;

  * (d)

if $(G,\theta)$ is an oriented pro-$\ell$ group of elementary type and $A$ is
a free abelian pro-$\ell$ group, then also the oriented pro-$\ell$ group
$A\rtimes(G,\theta)$ is of elementary type.

By [qw:cyclotomic, Thm. 1.3], if $(G,\theta)$ is an oriented pro-$\ell$ group
of elementary type, then $G$ is Bloch-Kato and $(G,\theta)$ is 1-cyclotomic.
Moreover, one has the following fact (cf. [qw:cyclotomic, § 3.3–3.4]).

######  Fact 3.10.

Let $(G,\theta)$ be an oriented pro-$\ell$ group and let $H\subseteq G$ be a
subgroup.

  * (i)

If $(G,\theta)=(G_{1},\theta_{1})\amalg^{\hat{\ell}}(G_{2},\theta_{2})$, then

$(H,\theta|_{H})\simeq{\coprod_{i\in I}}^{\hat{\ell}}(H_{i},\tau_{i}),$

where $H_{i}$ is a subgroup of either $G_{1}$ or $G_{2}$ — and $\tau_{i}$
denotes the restriction of $\theta_{1}$ or $\theta_{2}$ respectively —, or
$H_{i}$ is a free pro-$\ell$ group.

  * (ii)

If $(G,\theta)=A\rtimes(G_{\circ},\theta_{\circ})$ for some oriented
pro-$\ell$ group $(G_{\circ},\theta_{\circ})$ and $A\simeq\mathbb{Z}_{\ell}$,
then

$(H,\theta|_{H})\simeq(A\cap H)\rtimes(H_{\circ},\theta_{\circ}|_{H_{\circ}})$

for some subgroup $H_{\circ}\subseteq G_{\circ}$.

In particular, if $(G,\theta)$ is of elementary type, then for every finitely
generated subgroup $H\subseteq G$ the oriented pro-$\ell$ group
$(H,\theta|_{H})$ is again of elementary type.

Efrat’s Elementary Type conjecture predicts that every finitely generated
pro-$\ell$ group which is isomorphic to a maximal pro-$\ell$ Galois group of a
field containing a primitive $\ell$-th root of unity, endowed with the
cyclotomic character, should be completed into an oriented pro-$\ell$ group of
elementary type (cf. [efrat:etc1], see also [efrat:etc2, Ques. 4.8],
[efrat:etc3, Conj. 1.2] and [marshall:etc, § 10]).

######  Conjecture 3.11.

Let $\mathbb{K}$ be a field containing a primitive $\ell$-th root of 1, and
suppose that $[\mathbb{K}^{\times}:(\mathbb{K}^{\times})^{\ell}]<\infty$
(i.e., the maximal pro-$\ell$ Galois group $G_{\mathbb{K}}({\ell})$ is
finitely generated). Then
$(G_{\mathbb{K}}({\ell}),\hat{\theta}_{\mathbb{K},\ell})$ is an oriented
pro-$\ell$ group of elementary type.

##  4\. Oriented pro-$\ell$ RAAGs

###  4.1. Oriented pro-$\ell$ RAAGs

From now on a non-trivial torsion-free orientation
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ will be called a
linear orientation. Recall from the Introduction the definition of an oriented
pro-$\ell$ RAAG $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ associated to
an oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ and a linear orientation
$\lambda$. The following fact is a direct consequence of the definition of an
oriented pro-$\ell$ RAAG.

######  Fact 4.1.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, and let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a linear
orientation.

  * (i)

If $\Gamma=\Gamma_{1}\sqcup\Gamma_{2}$, then the associated oriented
pro-$\ell$ group $(\mathcal{G}_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$
decomposes as free pro-$\ell$ product of oriented pro-$\ell$ groups

$(G_{\Gamma_{1},\lambda},\theta_{\Gamma_{1},\lambda})\amalg^{\hat{\ell}}(G_{\Gamma_{2},\lambda},\theta_{\Gamma_{2},\lambda}).$

  * (ii)

More generally, if $\Gamma$ is the patching of two induced subgraphs
$\Gamma_{1},\Gamma_{2}$ along a common induced subgraph $\Lambda$, then then
the associated oriented pro-$\ell$ group
$(\mathcal{G}_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ decomposes as
amalgamated free pro-$\ell$ product of oriented pro-$\ell$ groups

$(G_{\Gamma_{1},\lambda},\theta_{\Gamma_{1},\lambda})\amalg_{G_{\Lambda,\lambda}}^{\hat{\ell}}(G_{\Gamma_{2},\lambda},\theta_{\Gamma_{2},\lambda}),$

with amalgam $G_{\Lambda,\lambda}$ (for the definition of amalgamated free
pro-$\ell$ product of oriented pro-$\ell$ groups see [qw:bogomolov, Def.
5.4]).

  * (iii)

If $\Gamma=\nabla(\Gamma^{\prime})$ for some induced subgraph
$\Gamma^{\prime}$, with tip $v\in\mathscr{V}$, then one has

$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})\simeq\langle\>v\>\rangle\rtimes(G_{\Gamma^{\prime},\lambda},\theta_{\Gamma^{\prime},\lambda}).$

###  4.2. Oriented pro-$\ell$ RAAGs and generalized pro-$\ell$ RAAGs

Following [qsv:quadratic, § 5.1], we say that an $\ell$-labelled graph is an
oriented graph $\Gamma=(\mathscr{V},\mathscr{E})$ such that
$\mathscr{E}=\mathscr{E}_{\mathrm{s}}$, endowed with a map
$f=(f_{o},f_{t})\colon\mathscr{E}_{\mathrm{s}}\to\ell\mathbb{Z}_{\ell}\times\ell\mathbb{Z}_{\ell}$,
which we call the $\ell$-labelling of $\Gamma$.

The generalized pro-$\ell$ RAAG associated to a $\ell$-labelled graph
$\Gamma=(\mathscr{V},\mathscr{E})$ with labelling $f$ is the pro-$\ell$ group
with presentation

$\left\langle\>v\in\mathscr{V}\>\mid\>[v,w]v^{f_{o}(\mathbf{e})}w^{f_{t}(\mathbf{e})}=1\text{
for }\mathbf{e}=(v,w)\in\mathscr{E}\>\right\rangle.$

Now let $\lambda$ be a linear orientation, and let
$\Gamma=(\mathscr{V},\mathscr{E})$ be any oriented graph, with
$\mathscr{E}=\mathscr{E}_{\mathrm{s}}\sqcup\mathscr{E}_{\mathrm{o}}$. Then the
pro-$\ell$ group $G_{\Gamma,\lambda}$ is the generalized pro-$\ell$ RAAG
associated to a $\ell$-labelled graph
$\Gamma^{\prime}=(\mathscr{V},\mathscr{E}^{\prime})$ such that
$\ddot{\mathscr{E}}^{\prime}=\ddot{\mathscr{E}}$, with $\ell$-labelling

$f(\mathbf{e})=\begin{cases}(\lambda(1)-1,0)&\text{if
}\mathbf{e}\in\mathscr{E}_{\mathrm{s}}\\\ (0,0)&\text{if
}\mathbf{e}\in\mathscr{E}_{\mathrm{o}}.\end{cases}$

###  4.3. Examples

Let $\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a linear
orientation. One has the following examples of oriented pro-$\ell$ RAAGs.

######  Example 4.2.

If $\Gamma=(\mathscr{V},\mathscr{E})$ is an oriented graph with
$\mathscr{E}=\mathscr{E}_{\mathrm{o}}$ (namely, no edge is an “arrow”), then
$G_{\Gamma,\lambda}$ is the pro-$\ell$ completion of the discrete RAAG
associated to the naïve graph $\ddot{\Gamma}$. In particular, if $\Gamma$ has
no edges, then $G_{\Gamma,\lambda}$ is the free pro-$\ell$ group generated by
$\mathscr{V}$.

######  Example 4.3.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be the specially oriented graph with
geometric realization

$\textstyle{v}$$\textstyle{w}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$

Then

$G=G_{\Gamma,\lambda}=\left\langle\>v,w\>\mid\>{}^{w}v=v^{\lambda(1)}\>\right\rangle,$

and one has $\theta_{\Gamma,\lambda}(v)=1$ and
$\theta_{\Gamma,\lambda}(w)=\lambda(1)$. Observe that
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is
$\theta_{\Gamma,\lambda}$-abelian.

######  Remark 4.4.

A $2$-generated pro-$\ell$ group $G$ is isomorphic to an oriented pro-$\ell$
RAAG associated to an oriented graph (with two vertices) if, and only if,
either of the following occurs: $G$ is a free pro-$\ell$ group (in which case,
the associated oriented graph consists of two disjoint vertices); or $G$ is
locally uniform (in which case, the two vertices of the associated oriented
graph are joined).

######  Example 4.5.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a complete specially oriented graph
and let $\lambda:\mathbb{Z}_{p}\to\mathbb{Z}_{p}^{\times}$ be a linear
orientation. Then $\Gamma$ has at most one special vertex (cf. Example 2.7),
say $w$, and

$G_{\Gamma,\lambda}=\left\langle\>w,\mathscr{V}_{\mathrm{o}}\>\mid\>[v,v^{\prime}]=1,\>{}^{w}v=v^{\lambda(1)}\;\forall\>v,v^{\prime}\in\mathscr{V}_{\mathrm{o}}\>\right\rangle$

(where we set implicitly $w=1$ and we omit the relations
${}^{w}v=v^{\lambda(1)}$, if $\mathscr{V}_{\mathrm{s}}=\varnothing$), which is
locally uniform, with canonical orientation $\eth_{G}=\theta_{\Gamma,\lambda}$
— in particular, $\theta_{\Gamma,\lambda}(w)=\lambda(1)$, if
$\mathscr{V}_{\mathrm{s}}\neq\varnothing$, and $\theta_{\Gamma,\lambda}(v)=1$
for every $v\in\mathscr{V}_{\mathrm{o}}$.

######  Example 4.6.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be the oriented graph with geometric
realization

$\textstyle{v_{1}}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{2}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{4}}$$\textstyle{v_{3}}$

— observe that $\Gamma$ is specially oriented, but not of elementary type by
Remark 2.11–(b), as the induced subgraph $\Gamma^{\prime}$ with vertices
$v_{1},v_{2},v_{4}$ is as the graph $\Lambda_{\mathrm{s}}$ in Example
2.12–(c), and $\Gamma=\nabla(\Gamma^{\prime})$. Moreover, $\Gamma$ is the
patching of the induced vertices $\Gamma_{1},\Gamma_{2}$ — with vertices
respectively $v_{1},v_{2},v_{3}$ and $v_{1},v_{3},v_{4}$ — along the induced
subgraph $\Delta$ with vertices $v_{1},v_{3}$. Then

$\begin{split}G_{\Gamma,\lambda}&=\left\langle\>v_{1},\ldots,v_{4}\>\mid\>{}^{v_{1}}v_{i}=v_{i}^{\lambda(1)},\>[v_{2},v_{3}]=[v_{3},v_{4}]=1,\>i=2,3,4\>\right\rangle\\\
&\simeq
G_{\Gamma_{1},\lambda}\amalg_{G_{\Delta,\lambda}}^{\hat{\ell}}G_{\Gamma_{2},\lambda},\end{split}$

where $G_{\Gamma_{1},\lambda}$, $G_{\Gamma_{2},\lambda}$ and
$G_{\Delta,\lambda}$ are all locally uniform. (The oriented graph $\Gamma$ is
chordal, cf. § 7 below.)

Unlike (pro-$\ell$ completions of) RAAGs, an oriented pro-$\ell$ RAAG may
yield non-trivial torsion, as shown by the following example.

######  Example 4.7.

For $\ell$ odd, set
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ such that
$\lambda(1)=1+{\ell}$, and let $\Gamma=(\mathscr{V},\mathscr{E})$ be the
oriented graph

$\textstyle{v_{1}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{3}}$$\textstyle{v_{2}}$

Then one has

$G_{\Gamma,\lambda}=\left\langle\>v_{1},v_{2},v_{3}\>|\>{}^{v_{2}}v_{1}=v_{1}^{1+\ell},\>{}^{v_{3}}v_{2}=v_{2}^{1+\ell},\>{}^{v_{1}}v_{3}=v_{3}^{1+\ell}\>\right\rangle,$

and this pro-$\ell$ group is a finite $\ell$-group, as shown by J. Mennicke
(cf. [serre:gal, Ch. I, § 4.4, Ex. 2(e)]). Observe that on the one hand the
oriented graph $\Gamma$ is not specially oriented; on the other hand, the
orientation $\theta_{\Gamma,\lambda}$ is constantly equal to 1 as
$\mathscr{V}=\mathscr{V}_{\mathrm{o}}$, and the oriented pro-$\ell$ group
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is not Kummerian, as
$\operatorname{ker}(\theta_{\Gamma,\lambda})/K_{\theta_{\Gamma,\lambda}}(G_{\Gamma,\lambda})=G_{\Gamma,\theta}/G_{\Gamma,\lambda}^{\prime}\simeq(\mathbb{Z}/\ell\mathbb{Z})^{3}$.

###  4.4. Specially oriented graphs and Kummerianity

The goal of this subsection is to prove that Kummerianity characterises
oriented pro-$\ell$ RAAGs associated to specially oriented graphs.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a linear
orientation, and let $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ be the
associated oriented pro-$\ell$ RAAG. First, we need the following lemma.

######  Lemma 4.8.

Let $\theta\colon G_{\Gamma,\lambda}\to\mathbb{Z}_{\ell}^{\times}$ be a
torsion-free orientation such that the oriented pro-$\ell$ group
$(G_{\Gamma,\lambda},\theta)$ is Kummerian, and let $\varphi\colon
G_{\Gamma,\lambda}\to\bar{G}$ be an epimorphism satisfying condition–(iii) of
Proposition 3.4. Moreover, for every edge $\mathbf{e}\in\mathscr{E}$, let
$H_{\mathbf{e}}$ be the subgroup of $G_{\Gamma,\lambda}$ generared by
$o(\mathbf{e}),t(\mathbf{e})$. Then the restriction

$\varphi|_{H_{\mathbf{e}}}\colon H_{\mathbf{e}}\longrightarrow\bar{G}$

is a monomorphism of locally uniform pro-$\ell$ groups.

###### Proof.

Set $G=G_{\Gamma,\lambda}$. By Proposition 3.4,
$\operatorname{ker}{\varphi}=K_{\theta}(G)$, and thus
$\eth_{\bar{G}}=\theta_{/\operatorname{ker}(\theta)}$. For every edge
$e=(v,w)\in\mathscr{E}$, $\varphi(H_{\mathbf{e}})$ is a subgroup of the
locally uniform pro-$\ell$ group $\bar{G}$, and thus it is locally uniform as
well, with associated canonical orientation
$\eth_{\varphi(H_{\mathbf{e}})}=\eth_{\bar{G}}|_{\varphi(H_{\mathbf{e}})}$.
Moreover, $K_{\theta}(G)\subseteq\Phi(G)$ by definition, so that $\varphi(v)$
and $\varphi(w)$ are linearly independent modulo $\Phi(\bar{G})$, and
$H_{\mathbf{e}}$ is 2-generated.

On the other hand, consider the epimorphism of pro-$\ell$ groups

$\varphi|_{H_{\mathbf{e}}}\colon
H_{\mathbf{e}}\longrightarrow\varphi(H_{\mathbf{e}}).$

By Example 4.3–(a), $H_{\mathbf{e}}$ is a quotient of the 2-generated locally
uniform pro-$\ell$ group

$H=\langle\>v,w\>\mid\>R_{\lambda}(\mathbf{e})=1\>\rangle,$

with associated canonical orientation $\eth_{H}$. Altogether, one has a chain
of epimorphisms of pro-$\ell$ groups $H\to
H_{\mathbf{e}}\to\varphi(H_{\mathbf{e}})$, where both $H$ and
$\varphi(H_{\mathbf{e}})$ are 2-generated locally uniform pro-$\ell$ groups.
Therefore, both homomorphism are isomorphisms of 2-generated locally uniform
pro-$\ell$ groups, and one has a chain of isomorphism of oriented pro-$\ell$
groups

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{$\textstyle{(H,\eth_{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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42.72859pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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112.79124pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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as the canonical orientations depend uniquely by the structures of the locally
uniform pro-$\ell$ groups by Proposition 3.2. In particular, one has

(4.1)
$\eth_{H}(v)=\eth_{H_{\mathbf{e}}}(v)=\eth_{\bar{G}}(\varphi(v))=\theta(v)$

— where the last equality follows by the fact that
$\eth_{\bar{G}}\circ\varphi=\theta$ by Proposition 3.4 —, and analogously for
$w$. ∎

######  Theorem 4.9.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a linear
orientation, and let $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ be the
associated oriented pro-$\ell$ RAAG. Then there exists a torsion-free
orientation $\theta\colon G\to\mathbb{Z}_{\ell}^{\times}$ such that the
oriented pro-$\ell$ group $(G_{\Gamma,\theta},\theta)$ is Kummerian if, and
only if, $\Gamma$ is specially oriented and $\theta=\theta_{\Gamma,\lambda}$.

###### Proof.

Set $G=G_{\Gamma,\lambda}$.

Suppose first that $\Gamma$ is specially oriented. Let $F$ be the free
pro-$\ell$ group generated by $\mathscr{V}$, and let $R_{\Gamma,\lambda}$ be
the normal subgroup of $F$ generated (as a normal subgroup) by
$\\{R_{\lambda}(\mathbf{e}),\mathbf{e}\in\mathscr{E}\\}$ — namely, one has a
short exact sequence of pro-$\ell$ groups

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0.0pt\hbox{$\textstyle{\\{1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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34.50002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{R_{\Gamma,\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
79.87016pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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79.87016pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 117.68962pt\raise 0.0pt\hbox{\hbox{\kern
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3.0pt\raise
0.0pt\hbox{$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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3.0pt\raise
0.0pt\hbox{$\textstyle{\\{1\\}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Set $\tilde{\theta}=\theta_{\Gamma,\lambda}\circ\pi$, and consider the
oriented pro-$\ell$ group and $(F,\tilde{\theta})$. By Example 3.5–(b),
$(F,\tilde{\theta})$ is Kummerian, and thus $F/K_{\tilde{\theta}}(F)$ is a
locally uniform pro-$\ell$ group by Proposition 3.4. We claim that
$R_{\Gamma,\lambda}\subseteq K_{\tilde{\theta}}(F)$. Indeed, for every edge
$\mathbf{e}=(x,w)\in\mathscr{E}$, one has the following.

  * (i)

If $\mathbf{e}\in\mathscr{E}_{\mathrm{s}}$, then
$x\in\mathscr{V}_{\mathrm{s}}$, so that $y\in\mathscr{V}_{\mathrm{o}}$ by
condition (ii) in Definition 2.5. Consequently,
$\theta_{\Gamma,\lambda}(x)=\lambda(1)$ and
$y\in\operatorname{ker}(\theta_{\Gamma,\lambda})$, so that

$R_{\lambda}({\mathbf{e}})=[x,y]y^{1-\lambda(1)}=[x,y]y^{1-\theta_{\Gamma,\lambda}(x)}\in
K_{\tilde{\theta}}(F).$

  * (ii)

If $\mathbf{e}\in\mathscr{E}_{\mathrm{o}}$, then
$x,y\in\mathscr{V}_{\mathrm{o}}$ by condition (i) in Definition 2.5.
Consequently, $x,y\in\operatorname{ker}(\theta_{\Gamma,\lambda})$, so that

$R_{\lambda}({\mathbf{e}})=[x,y]\in K_{\tilde{\theta}}(F).$

Therefore,

$\frac{G}{K_{\theta_{\Gamma,\lambda}}(G)}\simeq\frac{F/R_{\Gamma,\lambda}}{K_{\tilde{\theta}}(F)/R_{\Gamma,\lambda}}\simeq\frac{F}{K_{\tilde{\theta}}(F)},$

where the latter is a locally uniform pro-$\ell$ group. Hence,
$(G,\theta_{\Gamma,\lambda})$ is Kummerian by Proposition 3.4.

Suppose now that $\Gamma$ is not specially oriented, and that $\theta\colon
G\to\mathbb{Z}_{\ell}^{\times}$ is a torsion-free orientation such that
$(G,\theta)$ is a Kummerian oriented pro-$\ell$ group. Then $\Gamma$ contains
a (non-necessarily induced) subgraph
$\Gamma^{\prime}=(\mathscr{V}^{\prime},\mathscr{E}^{\prime})$, with
$\mathscr{V}^{\prime}=\\{x,y,z\\}$, as in (2.4). Set
$\mathbf{e}_{1}=(x,y)\in\mathscr{E}$ and
$\mathbf{e}_{2}=(z,x)\in\mathscr{E}_{\mathrm{s}}$. By Lemma 4.8, one has

$H_{\mathbf{e}_{2}}=\left\langle\>x,z\>\mid\>{}^{x}z=z^{\lambda(1)}\>\right\rangle,$

and $\eth_{H_{\mathbf{e}_{2}}}(x)=\theta(x)=\lambda(1)$ by (4.1). Now one has
two cases.

  * (i)

If $\Gamma^{\prime}$ is as the left-side one in (2.4) — i.e.,
$\mathbf{e}_{1}\in\mathscr{E}_{\mathrm{o}}$ —, then $[x,y]=1$ and
$H_{\mathbf{e}_{1}}\simeq\mathbb{Z}_{\ell}^{2}$ by Lemma 4.8. In particular,

$\theta(x)=\eth_{H_{\mathbf{e}_{1}}}(x)=1\neq\lambda(1)$

by (4.1), a contradiction.

  * (ii)

If $\Gamma^{\prime}$ is as the right-side one in (2.4) — i.e.,
$\mathbf{e}_{1}\in\mathscr{E}_{\mathrm{s}}$ —, one has

$H_{\mathbf{e}_{1}}\simeq\left\langle\>x,y\>\mid\>{}^{y}x=x^{\lambda(1)}\>\right\rangle$

by Lemma 4.8. In particular,
$\theta(x)=\eth_{H_{\mathbf{e}_{1}}}(x)=1\neq\lambda(1)$ by (4.1), a
contradiction.

Therefore, if $\Gamma$ is not specially oriented, $(G,\theta)$ cannot be
Kummerian. ∎

From Theorem 3.6 and from Theorem 4.9, one deduces that the oriented
pro-$\ell$ RAAG $(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ associated to
an oriented graph $\Gamma$ and a linear orientation
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ may occur as the
maximal pro-$\ell$ Galois group of a field $\mathbb{K}$ containing a primitive
$\ell$-th root of unity (and also $\sqrt{-1}$ if $\ell=2$) only if $\Gamma$ is
specially oriented, and with
$\theta_{\Gamma,\lambda}=\hat{\theta}_{\mathbb{K}}$. This is a refinement of
[qsv:quadratic, Thm. 5.32].

###  4.5. Cliques and locally uniform oriented pro-$\ell$ RAAGs

From Theorem 4.9 we deduce the following.

######  Corollary 4.10.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, and let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a linear
orientation. The associated oriented pro-$\ell$ RAAG
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is locally uniform if, and only
if, $\Gamma$ is complete and specially oriented.

###### Proof.

Set $G=G_{\Gamma,\lambda}$. Suppose that $\Gamma$ is complete and specially
oriented. If $\mathscr{V}_{\mathrm{s}}=\varnothing$, then $G$ is the
pro-$\ell$ completion of the discrete RAAG associated to the naïve graph
$\ddot{\Gamma}$, and thus it is a free abelian pro-$\ell$ group. Otherwise,
there is only one special vertex $w\in\mathscr{V}_{\mathrm{s}}$ (cf. Example
2.7), and thus

$G=\left\langle\>\mathscr{V}\>\mid\>[v,v^{\prime}]=1,\>{}^{w}v=v^{\lambda(1)}\;\forall\>v,v^{\prime}\in\mathscr{V}_{\mathrm{o}}\>\right\rangle$

is locally uniform.

Conversely, suppose that $G$ is locally uniform, with canonical orientation
$\eth_{G}$. Then $(G,\eth_{G})$ is a Kummerian oriented pro-$\ell$ group, and
therefore by Theorem 4.9, $\Gamma$ is specially oriented and
$\eth_{G}\equiv\theta_{\Gamma,\lambda}$. In particular, given two vertices
$v,v^{\prime}\in\mathscr{V}$, let $H$ be the subgroup of $G$ generated by
$v,v^{\prime}$. Then $H$ is locally uniform with
$\eth_{H}\equiv\theta_{\Gamma,\lambda}|_{H}$ (cf. Lemma 4.8), so that
$v,v^{\prime}$ are joined by an edge. ∎

By the proof of Theorem 4.9, the subgroup of an oriented pro-$\ell$ RAAG,
associated to a specially oriented graph $\Gamma$ and a linear orientation,
generated by two adjacent vertices of $\Gamma$ is isomorphic to the oriented
pro-$\ell$ RAAG associated to the induced subgraph of $\Gamma$ with these two
vertices. This is true also for every clique of $\Gamma$.

######  Proposition 4.11.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a specially oriented graph and
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ a linear
orientation. If $\Delta=(\mathscr{V}(\Delta),\mathscr{E}(\Delta))$ is a clique
of $\Gamma$, then the inclusion
$\mathscr{V}(\Delta)\hookrightarrow\mathscr{V}$ induces a monomorphism of
oriented pro-$\ell$ groups

$(G_{\Delta,\theta},\theta_{\Delta,\lambda})\longrightarrow(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda}).$

###### Proof.

By Corollary 4.10, $(G_{\Delta,\lambda},\theta_{\Delta,\lambda})$ is
$\theta_{\Delta,\lambda}$-abelian — and thus the pro-$\ell$ group
$G_{\Delta,\theta}$ is locally uniform —, as $\Delta$ is complete and
specially oriented.

Set $G=G_{\Gamma,\lambda}$. Since $\Gamma$ is specially oriented,
$(G,\theta_{\Gamma,\lambda})$ is a Kummerian oriented pro-$\ell$ group by
Theorem 4.9, and thus $G/K_{\theta_{\Gamma,\lambda}}(G)$ is a locally uniform
pro-$\ell$ group by Proposition 3.4. Let

(4.2) $\phi_{\Delta}\colon G_{\Delta,\theta}\longrightarrow
G\qquad\text{and}\qquad\varphi\colon G\longrightarrow
G/K(\mathcal{G}_{\Gamma})$

denote respectively the morphism induced by the inclusion
$\mathscr{V}(\Delta)\hookrightarrow\mathscr{V}$ and the canonical projection —
note that both maps $\pi|_{\mathscr{V}}$ and
$(\pi\circ\phi_{\Delta})|_{\mathscr{V}(\Delta)}$ are injective, as
$K_{\theta_{\Gamma,\lambda}}(G)\subseteq\Phi(G)$ — and let $H$ be the subgroup
of $G$ generated by $\mathscr{V}(\Delta)$. Then one has a chain of morphism of
pro-$\ell$ groups

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
11.67569pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G_{\Delta,\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.9236pt\raise 6.11389pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{\phi_{\Delta}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 35.67569pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
35.67569pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
55.00835pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\pi|_{H}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 74.80066pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
74.80066pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\pi(H)}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where both $G_{\Delta,\lambda}$ and $\pi(H)$ are locally uniform pro-$\ell$
groups, minimally generated by $\mathscr{V}(\Delta)$. Therefore, both
morphisms are isomorphism, and $H$ is locally uniform as well. In particular,
$\theta_{\Delta,\lambda}=\eth_{\pi(H)}\circ\pi\circ\phi_{\Delta}$, as the
canonical orientations are uniquely determined by the structure of the locally
uniform pro-$\ell$ groups (cf. Proposition 3.2). ∎

###  4.6. Cohomology of oriented pro-$\ell$ RAAGs

Let $\ddot{\Gamma}=(\ddot{\mathscr{V}},\ddot{\mathscr{E}})$ be a naïve graph,
and let $V$ denote the $\mathbb{F}_{\ell}$ vector space with basis
$\ddot{\mathscr{V}}$. The exterior Stanley-Reisner $\mathbb{F}_{\ell}$-algebra
associated to $\ddot{\Gamma}^{\mathrm{op}}$ is the $\mathbb{F}_{\ell}$-algebra

$\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})=\frac{\mathbf{\Lambda}_{\bullet}(V)}{\left(\>v\wedge
w\>\mid\>\\{v,w\\}\notin\ddot{\mathscr{E}}\>\right)},$

where $\mathbf{\Lambda}_{\bullet}(V)$ denotes the exterior
$\mathbb{F}_{\ell}$-algebra generated by $V$. Clearly,
$\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})$ is a quadratic
algebra.

Now let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, and let
$\lambda$ be a linear orientation. By duality, one has an isomorphism of
$\mathbb{F}_{\ell}$-vector spaces

(4.3)
$V=\mathbf{\Lambda}_{1}(\ddot{\Gamma}^{\mathrm{op}})\overset{\sim}{\longrightarrow}\mathrm{H}^{1}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})=\mathrm{Hom}(G_{\Gamma,\lambda},\mathbb{F}_{\ell}).$

(cf. [serre:gal, Ch. I, § 4.2]). By [qsv:quadratic, Lemma 5.8], the cup-
product extends the isomorphism (4.3) to a homomorphism of graded
$\mathbb{F}_{\ell}$ algebras

(4.4)
$\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})\longrightarrow\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$

which is an isomorphism in degree 2 too. Therefore, if
$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$ is quadratic,
then
$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})\simeq\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})$
(cf. [qsv:quadratic, Thm. E]).

######  Example 4.12.

Let $\lambda$ be a linear orientation, and let
$\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph without triangles as
induced subgraphs. Then
$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$ is isomorphic to
$\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})$ (cf. [qsv:quadratic,
Thm. F]).

######  Remark 4.13.

Put $\ell=2$, and let $\lambda\colon\mathbb{Z}_{2}\to\mathbb{Z}_{2}^{\times}$
be a linear orientation — i.e., $\lambda(1)=1+4\mu$ for some
$\mu\in\mathbb{Z}_{2}\smallsetminus\\{0\\}$. Given an oriented graph
$\Gamma=(\mathscr{V},\mathscr{E})$, set $G=G_{\Gamma,\lambda}$. Then the
quotient $G/G^{4}G^{\prime}$ is isomorphic to $(\mathbb{Z}/4\mathbb{Z})^{d}$,
where $d$ is the number of vertices of $\Gamma$, so that the map

$\mathrm{H}^{1}(G_{\Gamma,\lambda},\mathbb{Z}/4\mathbb{Z})\longrightarrow\mathrm{H}^{1}(G_{\Gamma,\lambda},\mathbb{F}_{2}),$

induced by the canonical projection $\mathbb{Z}/4\mathbb{Z}\to\mathbb{F}_{2}$,
is surjective. This implies that the Bockstein morphism
$\mathfrak{b}_{G}\colon\mathrm{H}^{1}(G,\mathbb{F}_{2})\to\mathrm{H}^{2}(G,\mathbb{F}_{2})$
is trivial (cf., e.g., [cq:1smoothBK, p. 415]). Furthermore, if $H$ is a
subgroup of $G$ such that the restriction map
$\mathrm{res}_{G,H}^{1}\colon\mathrm{H}^{1}(G,\mathbb{F}_{\ell})\to\mathrm{H}^{1}(H,\mathbb{F}_{\ell})$
is surjective, the commutativity of the diagram

$\textstyle{\mathrm{H}^{1}(G,\mathbb{F}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{b}_{G}}$$\scriptstyle{\mathrm{res}_{G,H}^{1}}$$\textstyle{\mathrm{H}^{2}(G,\mathbb{F}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{res}_{G,H}^{1}}$$\textstyle{\mathrm{H}^{1}(H,\mathbb{F}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{b}_{H}}$$\textstyle{\mathrm{H}^{2}(H,\mathbb{F}_{\ell})}$

implies that also the Bockstein morphism
$\mathfrak{b}_{H}\colon\mathrm{H}^{1}(H,\mathbb{F}_{2})\to\mathrm{H}^{2}(H,,\mathbb{F}_{2})$
is trivial.

##  5\. Bloch-Kato Oriented pro-$\ell$ RAAGs

###  5.1. A Tits’ alternative

Recall that by Remark 3.3 a locally uniform pro-$\ell$ group is Bloch-Kato.
For Bloch-Kato pro-$\ell$ groups one has the following Tits’ alternative type
result (cf. [qw:cyclotomic, § 7.1]).

######  Proposition 5.1.

Let $G$ be a Bloch-Kato pro-$\ell$ group. If $\ell=2$ suppose further that the
Bockstein morphism
$\mathfrak{b}_{G}\colon\mathrm{H}^{1}(G,\mathbb{F}_{2})\to\mathrm{H}^{2}(G,\mathbb{F}_{2})$
is trivial. Then either $G$ is locally uniform, or $G$ contains a subgroup
which is a free non-abelian pro-$\ell$ group. In particular, if $G$ is
2-generated, then either $G$ is a free pro-$\ell$ group; or $G$ is locally
uniform.

By Remark 4.4, the same phenomenon occurs for oriented pro-$\ell$ RAAGs.
Moreover, observe that by Remark 4.13, if $\ell=2$ the Bockstein morphism is
always trivial for every subgroup of an oriented pro-$\ell$ RAAG such that the
restriction map of degree 1 is surjective.

###  5.2. Non-specially oriented graphs

######  Theorem 5.2.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, and let $\lambda$
be a linear orientation. If $G_{\Gamma,\lambda}$ is Bloch-Kato, then $\Gamma$
is specially oriented.

###### Proof.

Set $G=G_{\Gamma,\lambda}$, and suppose that $\Gamma$ is not specially
oriented. Then $\Gamma$ contains a (non-necessarily induced) subgraph with
geometric realization as in (2.4) (cf. Definition 2.5). Set
$\mathbf{e}_{1}=(x,y),\mathbf{e}_{2}=(z,x)\in\mathscr{E}$. Let $H$ be the
subgroup of $G$ generated by $x,y,z$, and let $H_{0}$ be the subgroup of $H$
generated by $y,z$. If $H_{0}$ is not free nor locally uniform, then by
Proposition 5.1 $H_{0}$ (and thus also $G$) is not Bloch-Kato. So, let us
assume that $H_{0}$ is locally uniform or free.

Observe that the subgroups $H_{\mathbf{e}_{1}}$ and $H_{\mathbf{e}_{2}}$ of
$G$, generated respectively by $x,y$ and $x,z$, are not free. Therefore, if
$G$ is Bloch-Kato, then they must be locally uniform by Proposition 5.1.

Suppose first that $H_{0}$ is locally uniform: then $[y,z]\in
H_{0}^{\ell}\subseteq H^{\ell}$, and thus $H$ is powerful, as also
$[x,y],[x,z]\in H^{\ell}$. If $H$ is locally uniform, then one has

$\eth_{H}(x)=\eth_{H_{\mathbf{e}_{1}}}(x)=1\qquad\text{and}\qquad\eth_{H}(x)=\eth_{H_{\mathbf{e}_{2}}}(x)=\lambda(1),$

a contradiction. Therefore, $H$ is powerful but not locally uniform, and since
powerful pro-$\ell$ groups contain no non-abelian free subgroups (cf. [ddsms,
Thm. 3.13]), $H$ is not Bloch-Kato by Proposition 5.1.

Suppose now that $H_{0}$ is a free pro-$\ell$ group, and set $t=yz$. Then

(5.1)
${}^{x}t={}^{x}y\cdot{}^{x}z=\begin{cases}y\cdot{}^{x}z=yz^{\lambda(1)}=tz^{\lambda(1)-1},&\text{if
}(x,y)\in\mathscr{E}_{\mathrm{o}},\\\
x^{1-\lambda(1)}yz^{\lambda(1)}=x^{1-\lambda(1)}tz^{\lambda(1)-1},&\text{if
}(x,y)\in\mathscr{E}_{\mathrm{s}}.\end{cases}$

Let $H_{t}$ be the subgroup of $H$ generated by $x,t$. Observe that (5.1)
implies that $z^{\lambda(1)-1}$ lies in $H_{t}$, and therefore $H_{t}$ is not
a free pro-$\ell$ group, as

${}^{x}\left(z^{\lambda(1)-1}\right)=z^{\lambda(1)(\lambda(1)-1)}.$

Moreover, Proposition 5.1 may apply also to $H_{t}$ (cf. § 5.1). If $H_{t}$ is
locally uniform, then also the subgroup
$\langle\>t,z^{\lambda(1)-1}\>\rangle\subseteq H_{t}$ is locally uniform, but
$\langle\>t,z^{\lambda(1)-1}\>\rangle$ is also a subgroup of $H_{0}$ which is
free: therefore,

$\langle\>t,z^{\lambda(1)-1}\>\rangle\simeq\mathbb{Z}_{\ell}\;\Longrightarrow\;t^{\mu_{1}}=z^{\mu_{2}(\lambda(1)-1)}\qquad\text{
for some }\mu_{1},\mu_{2}\in\mathbb{Z}_{\ell}\smallsetminus\\{0\\}.$

This implies that $[z,t^{\mu_{1}}]=1$, while $H_{0}=\langle\>t,z\>\rangle$ is
free, a contradiction. Therefore, $H_{t}$ is not free nor locally uniform, and
thus Proposition 5.1 implies that $H_{0}$ (and hence also $G$) is not Bloch-
Kato. ∎

From the proof of Theorem 5.2 we deduce the following.

######  Corollary 5.3.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, and let $\lambda$
be a linear orientation. If every finitely generated subgroup of
$G_{\Gamma,\lambda}$ is isomorphic to an oriented pro-$\ell$ RAAG, then
$\Gamma$ is specially oriented.

###### Proof.

Set $G=G_{\Gamma,\lambda}$. Suppose that $\Gamma$ is not specially oriented,
and let $x,y,z\in\mathscr{V}$ and $H,H_{0},H_{t}\subseteq G$ be as in the
proof of Theorem 5.2.

If the 2-generated pro-$\ell$ group $H_{0}$ is not a free pro-$\ell$ group nor
locally uniform, then $H_{0}$ does not occur as an oriented pro-$\ell$ RAAG by
Remark 4.4. So, let us suppose that $H_{0}$ is free or locally uniform.

If $H_{0}$ is a free pro-$\ell$ group, then the 2-generated pro-$\ell$ group
$H_{t}$ is not free nor locally uniform, and hence $H_{t}$ does not occur as
an oriented pro-$\ell$ RAAG, again by Remark 4.4.

Finally, suppose that $H_{0}$ is locally uniform. Then $H$ is powerful — but
not locally uniform, by the proof of Theorem 5.2 —, and by [ddsms, Thm. 3.13]
$H$ has no free 2-generated subgroups. Suppose that every 2-generated subgroup
of $H$ is locally uniform. Thus, $H$ is torsion-free. Moreover, for every
couple of elements $u,w\in H$, the subgroup $\langle\>u,w\>\rangle$ is
uniform, and

$[u,w]\in\begin{cases}\langle\>u,w\>\rangle^{\ell}&\text{if }\ell\neq 2,\\\
\langle\>u,w\>\rangle^{4}&\text{if }{\ell}=2.\end{cases}$

Therefore, for every subgroup $V$ of $H$ one has $V^{\prime}\subseteq
V^{\ell}$ ($V^{\prime}\subseteq V^{4}$ if ${\ell}=2$), so that $V$ is
powerful, and also uniform as $H$ is torsion-free. Hence, $H$ is locally
uniform, a contradiction. Thus, there exists a 2-generated subgroup of $H$
which is neither free nor locally uniform, and consequently it does not occur
as an oriented pro-$\ell$ RAAG. ∎

######  Remark 5.4.

From the proof of Corollary 5.3, one deduces that a pro-$\ell$ group $G$ is
locally uniform if, and only if, every 2-generated subgroup of $G$ is uniform.

###  5.3. The oriented graph $\Lambda_{\mathrm{s}}$

Recall that the oriented graph $\Lambda_{\mathrm{s}}$ is specially oriented
but not of elementary type (cf. Example 2.12–(c)).

######  Proposition 5.5.

Let $\lambda$ be a linear orientation. Then the pro-$\ell$ group
$G_{\Lambda_{\mathrm{s}},\lambda}$ is not Bloch-Kato.

###### Proof.

Set $G=G_{\Lambda_{\mathrm{s}},\lambda}$. Up to a change of the generator $x$,
we may assume that

$\theta_{\Lambda_{\mathrm{s}},\lambda}(x)=\lambda(1)=1+p^{f}\qquad\text{for
some }f\geq 1$

(in particular, if ${\ell}=2$ then $f\geq 2$). Put $q=p^{f}$. Then $G$ has a
minimal presentation

$G=\left\langle\>x,z_{1},z_{2}\>\mid\>{}^{x}z_{1}=z_{1}^{1+q},\>{}^{x}z_{2}=z_{2}^{1+q}\>\right\rangle.$

Hence, we can consider $G$ as the fundamental group of the graph of pro-$\ell$
groups

$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Z}$$\textstyle{\bullet}$$\textstyle{G_{1}}$$\textstyle{G_{2}}$

where $Z$ is the pro-$\ell$-cyclic subgroup of $G$ generated by $x$, and for
$i=1,2$ the group $G_{i}$ is the subgroup of $G$ generated by $x,z_{i}$ —
i.e., $G_{i}\simeq\langle z_{i}\rangle\rtimes Z$ —, and the monomorphisms from
$Z$ to $G_{1},G_{2}$ are the canonical embeddings.

Let $\phi_{U}\colon G\to\mathbb{Z}/q\mathbb{Z}$ be the homomorphism defined by
$\phi_{U}(z_{i})=1$ for $i=1,2$, and $\phi_{U}(x)=0$. Set $u_{i}=z_{i}^{q}$,
for $i=1,2$, and $t=z_{1}z_{2}^{-1}$. Then $U$ is the normal subgroup of $G$
generated by $x,u_{1},u_{2},t$. Note that $x,u_{1},t$ are enough to generate
$U$ (as normal subgroup), as

(5.2) ${}^{x}t=z_{1}^{1+q}z_{2}^{-1-q}=u_{1}tu_{2}.$

Let $U_{1},U_{2}$ be the subgroups of $U$ generated by $u_{1},x$ and $u_{2},x$
respectively. By Theorem 2.14, $U$ is the fundamental pro-$\ell$ group of the
graph of pro-$\ell$ groups $\mathcal{U}$ based on the graph
$\Delta=U\backslash\mathrm{T}$, with $q$ edges, where
$\mathrm{T}=(\mathscr{V}(\mathrm{T}),\mathscr{E}(\mathrm{T}))$ is the second-
countable pro-$\ell$ tree associated to the decomposition as pro-$\ell$
amalgam $G\simeq G_{1}\amalg_{Z}^{\hat{\ell}}G_{2}$, namely

$\mathscr{V}(\mathrm{T})=\left\\{\>gG_{1},\>gG_{2}\>\mid\>g\in
G\>\right\\}\qquad\text{and}\qquad\mathscr{E}(\mathrm{T})=\\{\>gZ\>\mid\>g\in
G\>\\}.$

In particular $\mathcal{U}(v_{i})=U_{i}$ for $i=1,2$.

Let $H$ be the subgroup of $U$ generated as a pro-$\ell$ group by
$\\{u_{1},x,t\\}$, and set $y=u_{1}^{-1}x$. Then $u_{2}\in H$, and
$U_{1}=\langle\>u_{1}\>\rangle\rtimes\langle\>y\>\rangle$. Moreover, $H$ is
the fundamental pro-$\ell$ group of the graph of pro-$\ell$ groups

$\mathcal{H}=\qquad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.35901pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
32.35901pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 81.20113pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern-8.35901pt\raise-14.72221pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{U_{1}}$}}}}}}}{\hbox{\kern
32.35901pt\raise-14.72221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
62.35901pt\raise-14.72221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\langle\>u_{2}^{-1}x\>\rangle\simeq\mathbb{Z}_{\ell}}$}}}}}}}\ignorespaces}}}}\ignorespaces$

which is a restriction of $\mathcal{U}$. In particular, the monomorphisms of
pro-$\ell$ groups associated to the bottom edge of $\mathcal{U}$ are the
inclusion $Z\hookrightarrow U_{1}$ and the isomorphism
$Z\to\langle\>u_{2}^{-1}x\>\rangle$, $x\mapsto u_{2}^{-1}x$, while the
monomorphisms of pro-$\ell$ groups associated to the top edge of $\mathcal{U}$
are the monomorphism $\langle\>y\>\rangle\to U$, $y\mapsto x$, and the
isomorphism $\langle\>y\>\rangle\to\langle\>u_{2}^{-1}x\>\rangle$, $y\mapsto
u_{2}^{-1}x$. Hence

$H=\left\langle\>u_{1},y,t,u_{2}^{-1}x\>\mid\>{}^{y}u_{1}=u_{1}^{1+q},\>t^{-1}yt=u_{2}^{-1}x\>\right\rangle,$

and this yields the minimal presentation

(5.3)
$H=\left\langle\>u_{1},y,t\>\mid\>[y,u_{1}]u_{1}^{-q}=\left[u_{1}y,[t^{-1},y]\right]u_{2}^{q}\left({}^{(u_{2}^{-1})}u_{1}^{-1}\right)^{q}=1\>\right\rangle$

— observe that $u_{2}$ may be generated by $u_{1},y,t$, as
$u_{2}=[t^{-1},y]u_{1}^{-1}$ by (5.2).

Now let $F$ be the free pro-$\ell$ group generated by $\\{u_{1},y,t\\}$, and
let $R$ be the normal subgroup of $F$ such that $H=F/R$. Then
$R/R^{\ell}[R,F]\neq RF_{(3)}/F_{(3)}$, because

$[u_{1},y]u_{1}^{-q}\notin
F_{(3)}\qquad\text{and}\qquad\left[u_{1}y,[t^{-1},y]\right]u_{2}^{q}\left({}^{(u_{2}^{-1})}u_{1}^{-1}\right)^{q}\in
F_{(3)}$

(recall that $4\mid q$ if ${\ell}=2$). Hence, by [MPQT, Thm. 7.3], the cup-
product

(5.4)
$\mathrm{H}^{1}(H,\mathbb{F}_{\ell})\times\mathrm{H}^{1}(H,\mathbb{F}_{\ell})\overset{\smallsmile}{\longrightarrow}\mathrm{H}^{2}(H,\mathbb{F}_{\ell})$

is not surjective, and $\mathbf{H}^{\bullet}(H,\mathbb{F}_{\ell})$ is not a
quadratic algebra (see also [qsv:quadratic, Prop. 2.4]). ∎

######  Corollary 5.6.

Let $\lambda$ be a linear orientation. Then the pro-$\ell$ group
$G_{\Lambda_{\mathrm{s}},\lambda}$ contains a finitely generated subgroup
which does not occur as an oriented pro-$\ell$ RAAG.

###### Proof.

Let $G$ and $H$ be as in the proof of Proposition 5.5. Since the cup-product
(5.4) is not surjective, $\mathrm{H}^{2}(H,\mathbb{F}_{\ell})$ is not
isomorphic to $\Lambda_{2}(\ddot{\Gamma}^{\mathrm{op}})$ for any oriented
graph $\Gamma$. Therefore, $H$ can not be isomorphic to the oriented
pro-$\ell$ RAAG associated to an oriented graph $\Gamma$ and a linear
orientation. ∎

##  6\. Oriented pro-$\ell$ RAAGs and maximal pro-$\ell$ Galois groups

###  6.1. Oriented pro-$\ell$ RAAGs of elementary type

The following fact is rather straightforward, and it follows from the
inductive procedure to construct oriented graphs of elementary type and
oriented pro-$\ell$ groups of elementary type (cf. § 3.5).

######  Fact 6.1.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, and let $\lambda$
be a non trivial linear orientation. Then
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is an oriented pro-$\ell$ group
of elementary type if, and only if, $\Gamma$ is of elementary type.

Moreover, from Proposition 3.10 one deduces the following.

######  Fact 6.2.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, and let $\lambda$
be a non trivial linear orientation. If $\Gamma$ is of elementary type, then
for every subgroup $H$ of $G_{\Gamma,\lambda}$ one has

$(H,\theta_{\Gamma,\lambda}|_{H})\simeq(G_{\Gamma^{\prime},\lambda^{\prime}},\theta_{\Gamma^{\prime},\lambda^{\prime}})$

for some oriented graph $\Gamma^{\prime}$ of elementary type and some non
trivial linear orientation $\lambda^{\prime}$.

The following is the group-theoretic analogue of Proposition 2.13.

######  Proposition 6.3.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a specially oriented graph, and let
$\lambda$ be a linear orientation. Then $\Gamma$ is of elementary type if, and
only if, $G_{\Gamma,\lambda}$ has no subgroups isomorphic to
$G_{\Gamma^{\prime},\lambda^{\prime}}$, with
$\Gamma^{\prime}\in\\{\Lambda_{\mathrm{s}},\mathrm{C}_{4},\mathrm{L}_{3}\\}$
and $\lambda^{\prime}$ a linear orientation.

In the statement of Proposition 6.3, $\mathrm{C}_{4}$ and $\mathrm{L}_{3}$
denote — with an abuse of notation — those specially oriented graphs with no
special vertices and edges whose associated naïve graphs are equal to
$\mathrm{C}_{4}$ and $\mathrm{L}_{3}$ (as defined in Example 2.1)
respectively. In order to prove Proposition 6.3, we need the following
technical lemma.

######  Lemma 6.4.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a specially oriented graph and let
$\lambda$ be a linear orientation. If $\Gamma$ has an induced subgraph
$\Gamma^{\prime}$ such that $\ddot{\Gamma}^{\prime}$ has geometric realization

$\textstyle{v_{1}}$$\textstyle{x}$$\textstyle{v_{2}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

then the subgroup $V$ of $G$ generated by $\\{v_{1},v_{2}\\}$ is a free
pro-$\ell$ group.

###### Proof.

Let $H,H_{1},H_{2}$ be the subgroups of $G$ generated respectively by
$\\{x,v_{1},v_{2}\\}$, $\\{x,v_{1}\\}$, and $\\{x,v_{2}\\}$. By Proposition
4.11, $H_{1},H_{2}$ are locally uniform — in particular, for $i=1,2$ one has
$H_{i}=\langle\>v_{i}\>\rangle\rtimes\langle\>x\>\rangle$, with
${}^{x}v_{i}=v_{i}^{\lambda(1)}$, if $x\in\mathscr{V}_{\mathrm{s}}$; while
$H_{i}=\langle\>x\>\rangle\rtimes\langle\>v_{i}\>\rangle$, with
${}^{v_{i}}x=x^{\theta_{\Gamma,\lambda}(v_{i})}$, if
$x\in\mathscr{V}_{\mathrm{o}}$.

Set $Z=\langle\>x\>\rangle$ and $V_{i}=\langle\>v_{i}\>\rangle$, with $i=1,2$,
and let $\mathrm{T}=(\mathscr{V}(\mathrm{T}),\mathscr{E}(\mathrm{T}))$ be the
second-countable pro-$\ell$ tree with

$\mathscr{V}(\mathrm{T})=\\{\>hH_{1},hH_{2}\>\mid\>h\in
H\>\\},\qquad\mathscr{E}(\mathrm{T})=\\{\>hZ\>\mid\>h\in H\>\\}.$

Then $H$ acts naturally on $\mathrm{T}$ by $g\cdot(hH_{i})=(gh)H_{i}$ and
$g\cdot(hZ)=(gh)Z$. The stabilizers in $V$ of and edge $hZ$ and a vertex
$hH_{i}$ of $\mathrm{T}$ are respectively

$\begin{split}\mathrm{Stab}_{V}(hZ)&=V\cap{}^{h}Z=V\cap\left\langle\>{}^{h}x\>\right\rangle,\\\
\mathrm{Stab}_{V}(hH_{i})&=V\cap{}^{h}H_{i}=V\cap\left\langle\>{}^{h}v_{i},{}^{h}x\>\right\rangle.\end{split}$

We have two cases. If $x\in\mathscr{V}_{\mathrm{s}}$, then
${}^{v_{i}}x=v_{i}^{1-\lambda(1)}x$ for both $i=1,2$, and we may write
$h=vx^{\mu}$, for some $v\in V$ and $\mu\in\mathbb{Z}_{\ell}$, as $H=V\rtimes
Z$. Then

${}^{h}x={}^{v}x=v^{\prime}\cdot
x,\qquad{}^{h}v_{i}=vv_{i}^{\lambda(1)^{\mu}}v^{-1},$

for some $v^{\prime}\in\Phi(V)$. If $x\in\mathscr{V}_{\mathrm{o}}$, then
$H=Z\rtimes V$, and we may write $h=x^{\mu}v$ for some $v\in V$ and
$\mu\in\mathbb{Z}_{\ell}$. Then

${}^{h}x=x^{\theta_{\Gamma,\lambda}(v)},\qquad{}^{h}v_{i}=x^{\mu(1-\theta_{\Gamma}(v_{i}))}\cdot{}^{v}v_{i}.$

In both cases, $\mathrm{Stab}_{V}(hZ)=\\{1\\}$ and
$\mathrm{Stab}_{V}(hH_{i})=\langle\>vv_{i}v^{-1}\>\rangle\simeq\mathbb{Z}_{\ell}$.
By [melnikov:freeprod, Thm. 5.6], $V$ is isomorphic to the free pro-$\ell$
product of some $\mathrm{Stab}_{V}(hH_{i})$ and of a free pro-$\ell$ group,
and therefore $V$ is a free pro-$\ell$ group. ∎

###### Proof of Proposition 6.3.

Set $G=G_{\Gamma,\lambda}$. If $\Gamma$ is of elementary type, then
$(G,\theta_{\Gamma,\lambda})$ is of elementary type by Fact 6.1. Hence, for
every finitely generated subgroup $H\subseteq G$ the oriented pro-$\ell$ group
$(H,\theta_{\Gamma,\lambda}|_{H})$ is isomorphic to an oriented pro-$\ell$
RAAG associated to some oriented graph of elementary type by Fact 6.2.

Conversely, suppose that $\Gamma$ is not of elementary type. By Proposition
2.13, $\Gamma$ has an induced subgraph
$\Gamma^{\prime}=(\mathscr{V}^{\prime},\mathscr{E}^{\prime})$ such that either
$\Gamma^{\prime}=\Lambda_{\mathrm{s}}$, or
$\ddot{\Gamma}^{\prime}\in\\{\mathrm{C}_{4},\mathrm{L}_{3}\\}$.

Assume first that $\Gamma^{\prime}=\Lambda_{\mathrm{s}}$, with
$\mathscr{V}^{\prime}=\\{x,v_{1},v_{2}\\}$. By Proposition 4.11, the subgroup
of $G$ generated by $x$ is isomorphic to $\mathbb{Z}_{\ell}$, while by Lemma
6.4, the subgroup of $G$ generated by $\\{v_{1},v_{2}\\}$ is a free pro-$\ell$
group. Hence, the subgroup of $G$ generated by $\mathscr{V}^{\prime}$ is

$\left\langle\>x,v_{1},v_{2}\>\mid\>{}^{x}v_{i}=v_{i}^{\lambda(1)},\>i=1,2\>\right\rangle\simeq
G_{\Lambda_{\mathrm{s}},\lambda}.$

Assume now that $\ddot{\Gamma}^{\prime}=\mathrm{C}_{4}$, with

$\mathscr{V}^{\prime}=\\{\>v_{1},\>v_{2},\>v_{3},\>v_{4}\>\\}\qquad\text{and}\qquad\ddot{\mathscr{E}}^{\prime}=\left\\{\>\\{v_{1},v_{2}\\},\>\ldots,\>\\{v_{4},v_{1}\\}\>\right\\}.$

By Lemma 6.4, the two subgroups $H_{1},H_{2}$ of $G$ generated by
$v_{1},v_{3}$ and by $v_{2},v_{4}$ respectively, are 2-generated free
pro-$\ell$ groups. Now pick
$y_{1},y_{3}\in\operatorname{ker}(\theta_{\Gamma,\lambda}|_{H_{1}})$ and
$y_{2},y_{4}\in\operatorname{ker}(\theta_{\Gamma,\lambda}|_{H_{2}})$ such that
the subgroup of $H_{1}$ generated by $y_{1},y_{3}$ is not isomorphic to
$\mathbb{Z}_{\ell}$, and analogously $y_{2},y_{4}$ — this is possible as both
kernels are non-abelian free pro-$\ell$ groups. Then the subgroup of $G$
generated by $y_{1},y_{2},y_{3},y_{4}$ is

$\left\langle\>y_{1},y_{2},y_{3},y_{4}\>\mid\>[y_{i},y_{j}]=1,\>i=1,3,\>j=2,4\>\right\rangle\simeq
G_{{\mathrm{C}_{4}},\lambda^{\prime}}$

with $\lambda^{\prime}$ an arbitrary linear orientation.

Finally, assume that $\ddot{\Gamma}^{\prime}=\mathrm{L}_{3}$, with

$\mathscr{V}^{\prime}=\\{\>v_{1},\>v_{2},\>v_{3},\>v_{4}\>\\}\qquad\text{and}\qquad\ddot{\mathscr{E}}^{\prime}=\left\\{\>\\{v_{1},v_{2}\\},\>\\{v_{2},v_{3}\\},\>\\{v_{3},v_{4}\\}\>\right\\}.$

If $v_{2}\in\mathscr{V}_{\mathrm{s}}$, then
$(v_{2},v_{1}),(v_{2},v_{3})\in\mathscr{E}_{\mathrm{s}}$, as $\Gamma$ is
specially oriented, and thus the induced subgraph of $\Gamma$ with vertices
$v_{1},v_{2},v_{3}$ is $\Lambda_{\mathrm{s}}$: in this case $G$ contains a
subgroup isomorphic to $G_{\Lambda_{\mathrm{s}},\lambda^{\prime}}$ for some
$\lambda^{\prime}$ by the above argument. Analogously if
$v_{3}\in\mathscr{V}_{\mathrm{s}}$. Otherwise, suppose that
$v_{2},v_{3}\in\mathscr{V}_{\mathrm{o}}$, so that
$(v_{2},v_{3})\in\mathscr{E}_{\mathrm{o}}$. By Lemma 6.4, the two subgroups
$H_{1},H_{2}$ of $G$ generated by $v_{1},v_{3}$ and by $v_{2},v_{4}$
respectively, are 2-generated free pro-$\ell$ groups. Put
$y_{1}=[v_{1},v_{2}]$ and $y_{4}=[v_{3},v_{4}]$ — observe that
$y_{1},y_{4}\in\operatorname{ker}(\theta_{\Gamma,\lambda})$. Then the subgroup
of $G$ generated by $y_{1},v_{2},v_{3},y_{4}$ is

$\left\langle\>y_{1},v_{2},v_{3},y_{4}\>\mid\>[y_{1},v_{2}]=[v_{2},v_{3}]=[v_{3},y_{4}]=1\>\right\rangle\simeq
G_{{\mathrm{L}_{3}},\lambda^{\prime}}$

with $\lambda^{\prime}$ an arbitrary linear orientation. ∎

###  6.2. 1-cyclotomic oriented pro-$\ell$ RAAGs

By Theorem 4.9, an oriented pro-$\ell$ RAAG associated to an oriented graph
$\Gamma$ may be 1-cyclotomic only if $\Gamma$ is specially oriented. We show
that, in fact, 1-cyclotomicity is far more restrictive.

######  Proposition 6.5.

Let $\lambda$ be a linear orientation. Then the oriented pro-$\ell$ RAAG
$(G_{\Lambda_{\mathrm{s}},\lambda},\theta_{\Lambda_{\mathrm{s}},\lambda})$ is
not 1-cyclotomic.

###### Proof.

We keep the same notation as in the proof of Proposition 5.5. Recall that
$G_{\Lambda_{\mathrm{s}},\lambda}$ has a subgroup $H$ with minimal
presentation

$H=\left\langle\>u_{1},y,t\>\mid\>{}^{y}u_{1}=u_{1}^{1+q},\>\left[u_{1}y,[t^{-1},y]\right]=u_{2}^{-q}\left({}^{(u_{2}^{-1})}u_{1}\right)^{q}\>\right\rangle,$

where $u_{1}=z_{1}^{q}$, $y=u_{1}^{-1}x$, $t=z_{1}z_{2}^{-1}$, cf. (5.3).

Let $\phi_{V}\colon H\to\mathbb{Z}/{\ell}\mathbb{Z}$ be the homomorphism
defined by $\phi_{V}(y)=1$, $\phi(u_{1})=\phi(t)=0$, and set $v=y^{\ell}$,
$w=[y,t^{-1}]$. Let $\phi_{V}\colon H\to\mathbb{Z}/{\ell}\mathbb{Z}$ be the
homomorphism defined by $\phi_{V}(y)=1$, $\phi(u_{1})=\phi(t)=0$. Then $V$ is
a normal subgroup of $H$ of index $\ell$, and it is generated, as a normal
subgroup, by $v,u_{1},t^{-1}$. In fact, $V$ is generated as a pro-$\ell$ group
by the set

$\left\\{\>v,\>t^{-1},\>[y,t^{-1}]=w,\>\left[y,[y,t^{-1}]\right],\>\ldots,\>[\underbrace{y,[\ldots,[y}_{p-1\text{
times}},t^{-1}]\ldots]],\>u_{1}\>\right\\},$

as $H/V=\\{V,yV,\ldots,y^{p-1}V\\}$, and ${}^{y}u_{1}=u_{1}^{1+q}$. Observe
that by (5.2), $u_{2}$ lies in the subgroup of $V$ generated by $t,w,u_{1}$,
and thus the relation

${}^{u_{1}}\left[y,[y,t^{-1}]\right][u_{1},w]=\left[u_{1}y,[y,t^{-1}]\right]=u_{1}^{-q}\left({}^{u_{1}^{-1}}u_{2}\right)^{q}\in\langle\>t,w,u_{1}\>\rangle^{q}$

implies that every higher commutator $[y,[\ldots,[y,t^{-1}]\ldots]]$ — and
thus the whole pro-$\ell$ group $V$ — is minimally generated by the set
$\\{v,t^{-1},w,u_{1}\\}$.

Set $\theta=\theta_{\Lambda_{\mathrm{s}},\lambda}|_{V}\colon
V\to\mathbb{Z}_{\ell}^{\times}$. Since $\theta(v)=(1+q)^{\ell}$ and
$\theta(u_{1})=\theta(t)=\theta(w)=1$, $u_{1}$ lies in
$\operatorname{ker}(\theta)$. Now let $N$ be the normal subgroup of $V$
generated by $u_{1}$ as a normal subgroup. Then the map $N/N^{\ell}[N,V]\to
V/\Phi(V)$ is injective, and hence Remark 3.8 implies that if the oriented
pro-$\ell$ group $(V,\theta)$ is Kummerian, then also the oriented pro-$\ell$
group $(V/N,\theta_{/N})$ is Kummerian. We claim that the latter is not
Kummerian. Indeed, $\\{vN,tN,wN\\}$ is a minimal generating set of $V/N$.
Moreover, one has $u_{2}\equiv w\bmod N$, and thus

${}^{y}w={}^{y}[y,t^{-1}]\equiv w^{1+q}\mod N.$

Therefore, one computes

$\begin{split}\left[v,t^{-1}\right]=\left[y^{\ell},t^{-1}\right]&={}^{y^{{\ell}-1}}[y,t^{-1}]\cdot{}^{y^{{\ell}-2}}[y,t^{-1}]\cdots{}^{y}[y,t^{-1}]\cdot[y,t^{-1}]\\\
&\equiv w^{(1+q)^{{\ell}-1}}\cdot w^{(1+q)^{{\ell}-2}}\cdots w^{1+q}\cdot
w\mod N\\\ &\equiv w^{1+(1+q)+\ldots+(1+q)^{{\ell}-2}+(1+q)^{{\ell}-1}}\mod
N.\end{split}$

This yields a relation $[vN,tN]=(wN)^{\mu}$, with
$\mu\in{\ell}\mathbb{Z}_{\ell}$, $\mu\neq 0$. Hence, by [eq:kummer, Thm. 8.1],
the oriented pro-$\ell$ group $(V/N,\theta_{/N})$ is not Kummerian. Thus,
$(G,\theta_{\Lambda_{\mathrm{s}},\lambda})$ is not a 1-cyclotomic oriented
pro-$\ell$ group. ∎

###  6.3. Proof of Theorem 1.1

###### Proof of Theorem 1.1.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, let
$\lambda\colon\mathbb{Z}_{\ell}\to\mathbb{Z}_{\ell}^{\times}$ be a linear
orientation, and set $G=G_{\Gamma,\lambda}$ and
$\theta=\theta_{\Gamma,\lambda}\colon G\to\mathbb{Z}_{\ell}$.

The equivalence between (0) and (iv) is stated in Fact 6.1.

It is well-known that (iv) implies (i), as the realizability as a maximal
pro-$\ell$ groups is closed with respect to free pro-$\ell$ products and semi-
direct products with $\mathbb{Z}_{\ell}$ (cf. …). Moreover, (iv) implies also
(ii) and (iii), cf. [qw:cyclotomic, Thm. 1.4] (see also Proposition 3.10).
Finally, (iv) implies also (v), as stated in Fact 6.2.

The Norm Residue Theorem (cf. [HW:book]) yields the implications (i)
$\Rightarrow$ (ii) (cf. [cq:bk, § 2]) and (i) $\Rightarrow$ (iii) (cf.
[qw:cyclotomic, Thm. 1.1]).

Assume that $G$ is Bloch-Kato. Then $\Gamma$ is specially oriented by
Proposition 5.2. Suppose that $G$ contains a subgroup $H$ isomorphic to
$G_{\Gamma^{\prime},\lambda}$ with either
$\Gamma^{\prime}\in\\{\Lambda_{\mathrm{s}},\mathrm{C}_{4},\mathrm{L}_{3}\\}$.
If $\Gamma^{\prime}=\Lambda_{\mathrm{s}}$, then $H$ is not Bloch-Kato by
Proposition 5.5. If $\Gamma^{\prime}=\mathrm{C}_{4},\mathrm{L}_{3}$, then $H$
is not Bloch-Kato by [SZ:RAAGs, Thm. 1.2]. Therefore, $G$ contains no
subgroups isomorphic to $G_{\Gamma^{\prime},\lambda}$ for such an oriented
graph $\Gamma^{\prime}$, and $(G,\theta_{\Gamma,\lambda})$ is of elementary
type by Proposition 6.3. This proves the implication (iii) $\Rightarrow$ (iv).

Assume $(G,\theta)$ is 1-cyclotomic. Then by Theorem 4.9, $\Gamma$ is
specially oriented, and $\theta=\theta_{\Gamma,\lambda}$. Suppose that $G$
contains a subgroup $H$ isomorphic to $G_{\Gamma^{\prime},\lambda}$ with
either
$\Gamma^{\prime}\in\\{\Lambda_{\mathrm{s}}\mathrm{C}_{4},\mathrm{L}_{3}\\}$.
If $\Gamma^{\prime}=\Lambda_{\mathrm{s}}$, then $H$ can not be completed into
a 1-cyclotomic oriented pro-$\ell$ group by Proposition 6.5. If
$\Gamma^{\prime}=\mathrm{C}_{4},\mathrm{L}_{3}$, then $H$ can not be completed
into a 1-cyclotomic oriented pro-$\ell$ group by [SZ:RAAGs, Thm. 1.5] and
Theorem 4.9. Therefore, $G$ contains no subgroups isomorphic to
$G_{\Gamma^{\prime},\lambda}$ for such an oriented graph $\Gamma^{\prime}$,
and $(G,\theta_{\Gamma,\lambda})$ is of elementary type by Proposition 6.3.
This proves the implication (ii) $\Rightarrow$ (iv).

Finally, assume that every finitely generated subgroup of $G$ is isomorphic to
an oriented pro-$\ell$ RAAG. Then $\Gamma$ is specially oriented by Corollary
5.3. Suppose that $G$ contains a subgroup $H$ isomorphic to
$G_{\Gamma^{\prime},\lambda}$ with either
$\Gamma^{\prime}\in\\{\Lambda_{\mathrm{s}},\mathrm{C}_{4},\mathrm{L}_{3}\\}$.
If $\Gamma^{\prime}=\Lambda_{\mathrm{s}}$, then $H$ contains a subgroup which
can not occur as an oriented pro-$\ell$ RAAG by Corollary 5.6. If
$\Gamma^{\prime}=\mathrm{C}_{4},\mathrm{L}_{3}$, then $H$ contains a subgroup
which can not occur as an oriented pro-$\ell$ RAAG by [SZ:RAAGs, Thm. 1.2].
Therefore, $G$ contains no subgroups isomorphic to
$G_{\Gamma^{\prime},\lambda}$ for such an oriented graph $\Gamma^{\prime}$,
and $(G,\theta_{\Gamma,\lambda})$ is of elementary type by Proposition 6.3.
This proves the implication (iv) $\Rightarrow$ (iv). ∎

##  7\. Chordal oriented graphs

###  7.1. Chordal graphs and patching of graphs

######  Definition 7.1.

A naïve graph $\ddot{\Gamma}=(\ddot{\mathscr{V}},\ddot{\mathscr{E}})$ is said
to be chordal (or triangulated) if there are no induced subgraphs of $\Gamma$
which are circuits of length at least 4. An oriented graph
$\Gamma=(\mathscr{V},\mathscr{E})$ is said to be chordal if $\ddot{\Gamma}$ is
a chordal naïve graph.

Chordal graphs have the following characterization (cf. [graphbook, Prop.
5.5.1] and [chordalgraphs, Thm. 3.2]).

######  Proposition 7.2.

Let $\ddot{\Gamma}=(\ddot{\mathscr{V}},\ddot{\mathscr{E}})$ be a naïve graph.
Then the following are equivalent.

  * (i)

The graph $\ddot{\Gamma}$ is chordal.

  * (ii)

The graph $\ddot{\Gamma}$ decomposes as patching of two induced proper
subgraphs $\ddot{\Gamma}_{1},\ddot{\Gamma}_{2}$ which are chordal, along a
common clique $\ddot{\Delta}\subseteq\ddot{\Gamma}_{1},\ddot{\Gamma}_{2}$.

  * (iii)

The graph $\Upsilon(\ddot{\Gamma})$ has a maximal subtree
$\mathrm{T}_{\Upsilon(\ddot{\Gamma})}$ with the clique-intersection property.

######  Example 7.3.

Consider the naïve graph
$\ddot{\Gamma}=(\ddot{\mathscr{V}},\ddot{\mathscr{E}})$ with geometric
realization

|  |  |  |  |
---|---|---|---|---|---
$\textstyle{v_{1}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ddot{\Delta}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{v_{2}}$$\textstyle{v_{3}}$$\textstyle{v_{4}}$$\textstyle{v_{5}}$

Then $\ddot{\Gamma}$ is chordal, and it is the pasting of the two induced
subgraphs $\ddot{\Gamma}_{1}=(\ddot{\mathscr{V}}_{1},\ddot{\mathscr{E}}_{1})$
and $\ddot{\Gamma}_{2}=(\ddot{\mathscr{V}}_{2},\ddot{\mathscr{E}}_{2})$, with
$\ddot{\mathscr{V}}_{1}=\ddot{\mathscr{V}}\smallsetminus\\{v_{5}\\}$ and
$\ddot{\mathscr{V}}_{2}=\mathscr{V}\smallsetminus\\{v_{2}\\}$, along the
common subgraph $\ddot{\Delta}$, which is the triangle with vertices
$v_{1},v_{3},v_{4}$. Moreover, if $\ddot{\Delta}^{\prime}$ and
$\ddot{\Delta}^{\prime\prime}$ are the triangles with vertices
$v_{1},v_{2},v_{3}$ and $v_{1},v_{4},v_{5}$ respectively, then
$\ddot{\Gamma}_{1}$ may be obtained as the pasting of $\ddot{\Delta}^{\prime}$
and $\ddot{\Delta}$ along the common edge with vertices $v_{1},v_{3}$, and
analogously $\ddot{\Gamma}_{2}$ is the pasting of $\ddot{\Delta}$ and
$\ddot{\Delta}^{\prime\prime}$ along the common edge with vertices
$v_{1},v_{4}$.

###  7.2. Chordal oriented graphs and oriented pro-$\ell$ RAAGs

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a specially oriented graph, and let
$\lambda$ be a linear orientation. Recall that by Proposition 4.11, if
$\Delta=(\mathscr{V}(\Delta),\mathscr{E}(\Delta))$ is a clique of $\Gamma$,
then the inclusion $\mathscr{V}(\Delta)\hookrightarrow\mathscr{V}$ induces a
monomorphism of pro-$\ell$ groups $G_{\Delta,\lambda}\to G_{\Gamma,\lambda}$.
Hence, if $\Gamma$ is chordal, then one may find two proper induced subgraphs
$\Gamma_{1},\Gamma_{2}$ of $\Gamma$, whose intersection is a clique $\Delta$,
such that $\Gamma$ is the patching of $\Gamma_{1},\Gamma_{2}$ along $\Delta$,
so that

(7.1)
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})\simeq(G_{\Gamma_{1},\lambda},\theta_{\Gamma_{1},\lambda})\amalg_{G_{\Delta,\lambda}}^{\hat{\ell}}(G_{\Gamma_{2},\lambda},\theta_{\Gamma_{2},\lambda})$

(see 4.1–(ii)). Moreover, by [qsv:quadratic, Prop. 5.22] this amalgamated free
pro-$\ell$ product is proper — i.e., the two factors are subgroups of the
amalgamated free pro-$\ell$ product —, as $G_{\Delta,\lambda}$ is a locally
uniform pro-$\ell$ group. Therefore, an oriented pro-$\ell$ RAAG associated to
a chordal specially oriented graph may be constructed iterating amalgamated
free pro-$\ell$ products over locally uniform subgroups, starting from
oriented pro-$\ell$ RAAGs associated to complete specially oriented graphs.

In particular, let $\mathrm{T}=\mathrm{T}_{\Upsilon(\Gamma)}$ be a maximal
subtree of the clique-graph $\Upsilon(\Gamma)$ with the clique-intersection
property. Then one has a decomposition as proper amalgamated free pro-$\ell$
product

$G_{\Gamma,\lambda}\simeq\dfrac{\coprod_{\Delta\in\mathrm{\bf
mx}(\Gamma)}^{\hat{\ell}}G_{\Delta,\lambda}}{N},$

where $N$ is the normal subgroup of the free pro-$\ell$ product
$\coprod_{\Delta}G_{\Delta,\lambda}$ generated by the elements

$\iota_{\Xi,\Delta}(v)\cdot\iota_{\Xi,\Delta^{\prime}}(v)^{-1},\qquad
v\in\mathscr{V}(\Delta),\>\Xi=\Delta\cap\Delta^{\prime},\>(\Delta,\Delta^{\prime})\in\mathscr{E}(\mathrm{T}),$

and $\iota_{\Xi,\Delta}\colon G_{\Xi,\lambda}\to G_{\Delta,\lambda}$ is the
monomorphism of locally uniform pro-$\ell$ groups induced by
$\mathscr{V}(\Xi)\hookrightarrow\mathscr{V}(\Delta)$ (cf. [simoblu]).

######  Example 7.4.

Let $\lambda$ be a linear orientation, and let
$\Gamma=(\mathscr{V},\mathscr{E})$ be a specially oriented graph with
associated naïve graph $\ddot{\Gamma}$ as the chordal graph in Example 7.3.
Moreover, let $\Delta$ be the clique of $\Gamma$ with vertices
$v_{1},v_{3},v_{4}$, and analogously $\Delta^{\prime}$ and
$\Delta^{\prime\prime}$. Then the clique-graph $\Upsilon(\Gamma)$ has
geometric realization

$\textstyle{\Delta^{\prime}}$$\textstyle{\Delta}$$\textstyle{\Delta^{\prime\prime}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Xi_{1}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Xi_{2}}$$\textstyle{\bullet}$

where $\Xi_{1}=\Delta^{\prime}\cap\Delta$ and
$\Xi_{2}=\Delta\cap\Delta^{\prime\prime}$ are the 2-cliques with vertices
$v_{1},v_{3}$ and $v_{1},v_{4}$ respectively. Therefore,

(7.2)
$G_{\Gamma,\lambda}\simeq\left(G_{\Delta^{\prime}}\amalg_{G_{\Xi_{1},\lambda}}^{\hat{\ell}}G_{\Delta,\lambda}\right)\amalg_{G_{\Xi_{2},\lambda}}^{\hat{\ell}}G_{\Delta^{\prime\prime},\lambda},$

Observe that
$G_{\Delta^{\prime},\lambda},G_{\Delta,\lambda},G_{\Delta^{\prime\prime},\lambda}$
are 3-generated locally uniform pro-$\ell$ groups, while
$G_{\Xi_{1},\lambda},G_{\Xi_{2},\lambda}$ are 2-generated locally uniform
pro-$\ell$ groups.

###  7.3. Chordal oriented graphs and cohomology

In [qsv:quadratic, Thm. H] it is shown that a generalized pro-$\ell$ RAAG
associated to an oriented graph $\Gamma$ satisfying a particular group-
theoretic condition has $\mathbb{F}_{\ell}$-cohomology isomorphic to the
exterior Stanley-Reisner algebra
$\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})$. Thus, the following
theorem is a refinement of the aforementioned result.

######  Theorem 7.5.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be an oriented graph, and let $\lambda$
be a linear orientation. If $\Gamma$ is chordal and specially oriented, then

$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})\simeq\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}}).$

In order to prove Theorem 7.5, we need a technical lemma, which is a slight
modification of [qsv:quadratic, Prop. 5.21]. Given a pro-$\ell$ group $G$, for
$n\geq 1$ let $\Phi^{n}(G)$ denote the $n$-th term of the Frattini series of
$G$ — namely, $\Phi^{1}(G)=G$ and $\Phi^{n+1}(G)=\Phi(\Phi^{n}(G))$. If $G$ is
locally uniform, then $\Phi^{n}(G)=G^{\ell^{n}}$ (cf. [ddsms, Thm. 3.6]).

Recall that, given a specially oriented graph
$\Gamma=(\mathscr{V},\mathscr{E})$ containing a clique
$\Delta=(\mathscr{V}(\Delta),\mathscr{E}(\Delta))$, the inclusion
$\mathscr{V}(\Delta)\hookrightarrow\mathscr{V}$ induces a monomorphism of
pro-$\ell$ groups $\phi_{\Delta}\colon G_{\Delta,\lambda}\to
G_{\Gamma,\lambda}$ (cf. Proposition 4.11), with $\lambda$ a linear
orientation.

######  Lemma 7.6.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a specially oriented graph, and let
$\lambda$ be a linear orientation. If
$\Delta=(\mathscr{V}(\Delta),\mathscr{E}(\Delta))$ is a clique of $\Gamma$,
then

$\Phi^{n}(G_{\Delta,\lambda})=G_{\Delta,\lambda}\cap\Phi^{n}(G_{\Gamma,\lambda})$

for all $n\geq 1$ (where we consider $G_{\Delta,\lambda}$ as a subgroup of
$G_{\Gamma,\lambda}$ via the monomorphism $\phi_{\Delta}$).

###### Proof.

Set $G=G_{\Gamma,\lambda}$, $A=G_{\Delta,\lambda}$, and
$\bar{G}=G/K_{\theta_{\Gamma,\lambda}}(G)$, and let $\varphi\colon
G\to\bar{G}$ denote the canonical projection. Recall that $\varphi|_{A}$ is
injective. Clearly, one has the inclusion

$\Phi^{n}(A)\subseteq A\cap\Phi^{n}(G).$

Let $F$ be the free pro-$\ell$ group generated by $\mathscr{V}^{\prime}$,
where $\mathscr{V}=\mathscr{V}^{\prime}\sqcup\mathscr{V}(\Delta)$, and put
$\tilde{G}=F\amalg^{\hat{\ell}}A$. Let $\pi\colon\tilde{G}\to G$ be the
epimorphism which sends every vertex $v\in\mathscr{V}^{\prime}$, considered as
an element of $F$, to the same vertex, considered as an element of $G$, and
such that $\pi|_{A}=\phi_{\Delta}$. Altogether, one has a chain of
epimorphisms of pro-$\ell$ groups

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
25.7986pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\tilde{G}=F\amalg^{\hat{\ell}}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
32.80351pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 49.7986pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
49.7986pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
70.3715pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 87.66109pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
87.66109pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\bar{G}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Pick an element $x\in A\cap\Phi^{n}(G)$. Then

$\varphi(x)\in\varphi(A)\cap\Phi^{n}(\bar{G})=\varphi(A)\cap\bar{G}^{\ell^{n}}=\varphi(A)^{\ell^{n}},$

as $\bar{G}$ is locally uniform, generated by
$\varphi(\mathscr{V}^{\prime})\sqcup\varphi(\mathscr{V}(\Delta))$. Since
$\varphi\circ\phi_{\Delta}\colon A\to\varphi(A)$ is an isomorphism, there
exists $y\in A^{\ell^{n}}$ such that $\varphi(\phi_{\Delta}(y))=\varphi(x)$,
and hence $x=\phi_{\Delta}(y)$, namely, $x\in A^{\ell^{n}}$. ∎

###### Proof of Theorem 7.5.

By (4.4) one knows that
$\mathrm{H}^{n}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})\simeq\Lambda_{n}(\ddot{\Gamma}^{\mathrm{op}})$
for $n=0,1,2$. Therefore, it sufficies to show that
$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$ is a quadratic
algebra.

Let $\Gamma_{1},\Gamma_{2}$ be proper induced subgraphs of $\Gamma$ whose
intersection is a clique $\Delta$. We claim that the free amalgamated
pro-$\ell$ product (7.1) is proper. Indeed, for every $n\geq 1$ set
$U_{n}=\Phi^{n}(G_{\Gamma_{1},\lambda})$ and
$V_{n}=\Phi^{n}(G_{\Gamma_{2},\lambda})$. Then $\\{U_{n}\mid n\geq 1\\}$ and
$\\{V_{n}\mid n\geq 1\\}$ are basis of open neighbourhoods of 1 in
$G_{\Gamma_{1},\lambda}$ and $G_{\Gamma_{2},\lambda}$ respectively. By Lemma
7.6,

$U_{n}\cap G_{\Delta,\lambda}=V_{n}\cap
G_{\Delta,\lambda}=G_{\Delta,\lambda}^{\ell^{n}}\qquad\text{for every }n\geq
1,$

and [ribzal:book, Thm. 9.2.4] implies that the amalgam is proper.

Now, if $\Gamma$ is complete, then $G_{\Gamma,\lambda}$ is locally uniform,
and thus
$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})\simeq\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}^{\mathrm{op}})$
by Remark 3.3. Otherwise, by the inductive procedure to construct chordal
graphs we may assume that
$\mathbf{H}^{\bullet}(G_{\Gamma_{i},\lambda},\mathbb{F}_{\ell})\simeq\mathbf{\Lambda}_{\bullet}(\ddot{\Gamma}_{i}^{\mathrm{op}})$
for both $i=1,2$. Moreover, one has

$\mathrm{H}^{n}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})\simeq\operatorname{ker}\left(\operatorname{res}_{G_{\Gamma_{1},\lambda},G_{\Delta,\lambda}}^{n}\right)\oplus\mathrm{H}^{n}(G_{\Delta,\lambda},\mathbb{F}_{\ell})\oplus\operatorname{ker}\left(\operatorname{res}_{G_{\Gamma_{2},\lambda},G_{\Delta,\lambda}}^{n}\right)$

for $n=1,2$. Hence, we may apply [qsv:quadratic, Thm. B], and
$\mathbf{H}^{\bullet}(G_{\Gamma,\lambda},\mathbb{F}_{\ell})$ is a quadratic
algebra. ∎

###  7.4. The Bogomolov-Positselski property

A Kummerian oriented pro-$\ell$ group $(G,\theta)$ with torsion-free
orientation is said to have the Bogomolov-Positselski property if
$K_{\theta}(G)$ is a free pro-$\ell$ group (cf. [qw:bogomolov, § 3]). From
Theorem 7.5 we deduce that an oriented pro-$\ell$ RAAG associated to a chordal
specially oriented graph has the Bogomolov-Positselski property.

######  Theorem 7.7.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a chordal specially oriented graph,
and let $\lambda$ be a linear orientation. Then
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ has the Bogomolov-Positselski
property.

###### Proof.

Since $\Gamma$ is specially oriented,
$(G_{\Gamma,\lambda},\theta_{\Gamma,\lambda})$ is Kummerian by Theorem 4.9.
Let $\Gamma_{1},\Gamma_{2}$ be proper induced subgraphs of $\Gamma$ whose
intersection is a clique $\Delta$. By the proof of Theorem 7.5, the
amalgamated free pro-$\ell$ product (7.1) is proper, and moreover it satisfies
the hypothesis of [qw:bogomolov, Thm. 5.5].

If $\Gamma$ is complete, then it has the Bogomolov-Positselski property as
$K_{\theta_{\Gamma,\lambda}}(G_{\Gamma,\lambda})$ is trivial. Otherwise, by
induction we may assume that
$(G_{\Gamma_{i},\lambda},\theta_{\Gamma_{i},\lambda})$ has the Bogomolov-
Positselski property for both $i=1,2$, and [qw:bogomolov, Thm. 5.5] yields the
claim. ∎

Let $\mathbb{K}$ be a field containing a primitive $\ell$-th root of unity
(and also $\sqrt{-1}$ if ${\ell}=2$). In [pos:k, Conj. 1.2], L. Positselski
conjectures that $(G_{\mathbb{K}}(\ell),\hat{\theta}_{\mathbb{K}})$ has the
Bogomolov-Positselski property. Therefore, Theorem 7.7 implies Corollary
1.2–(iii).

###  7.5. Coherent oriented pro-$\ell$ RAAGs

A finitely generated pro-$\ell$ group $G$ is said to be coherent if every
finitely generated subgroup $H\subseteq G$ is also finitely presented, i.e.,
if $|\mathrm{H}^{1}(H,\mathbb{F}_{\ell})|<\infty$ implies
$|\mathrm{H}^{2}(H,\mathbb{F}_{\ell})|<\infty$. Moreover, $G$ is said to be of
type $FP_{\infty}$ if $\mathrm{H}^{n}(G,\mathbb{F}_{\ell})$ is finite for
every $n\geq 1$. We prove that every finitely generated subgroup of an
oriented pro-$\ell$ RAAG associated to a chordal specially oriented graph —
even if it may not occur as an oriented pro-$\ell$ RAAG — is of type
$FP_{\infty}$.

######  Theorem 7.8.

Let $\Gamma=(\mathscr{V},\mathscr{E})$ be a chordal specially oriented graph,
and let $\lambda$ be a linear orientation. Then every finitely generated
subgroup of $G_{\Gamma,\lambda}$ is of type $FP_{\infty}$. In particular,
$G_{\Gamma,\lambda}$ is coherent.

###### Proof.

Set $G=G_{\Gamma,\lambda}$, and let $H$ be a finitely generated subgroup of
$G$. Moreover, set

$K=K_{\theta_{\Gamma,\lambda}}(G)\cap H\qquad\text{and}\qquad
Q=\frac{HK_{\theta_{\Gamma,\lambda}}(G)}{K_{\theta_{\Gamma,\lambda}}(G)}.$

Then one has a short exact sequence of pro-$\ell$ groups

$\textstyle{\\{1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{1\\}}$

where $K$ is a free pro-$\ell$ group by Theorem 7.7, and $Q$ is locally
powerful as $\Gamma$ is specially oriented (cf. Theorem 4.9). By [king, Thm.
2, § 3], $H$ is of type $FP_{\infty}$ if, and only if,
$H_{n}(K,\mathbb{Z}_{\ell})$ is a finitely generated
$\mathbb{Z}_{\ell}[\\![Q]\\!]$-module for each $n\geq 1$. Since $H$ is
finitely generated, $H_{1}(K,\mathbb{Z}_{\ell})$ is a finitely generated
$\mathbb{Z}_{\ell}[\\![Q]\\!]$-module, while $H_{n}(K,\mathbb{Z}_{\ell})=0$
for each $n\geq 2$ as $K$ is a free pro-$\ell$ group. Hence, $H$ is of type
$FP_{\infty}$. In particular, $\mathrm{H}^{2}(H,\mathbb{F}_{\ell})$ is finite,
and thus $G$ is coherent. ∎

## References