# On the connectivity of the Sylow graph of a finite group

Francesco G. Russo  University of Palermo
Department of Mathematics
via Archirafi 34, 90123, Palermo, Italy<EMAIL_ADDRESS>
###### Abstract.

The Sylow graph $\Gamma(G)$ of a finite group $G$ originated from recent
investigations on the so–called $\mathbf{N}$–closed classes of groups. The
connectivity of $\Gamma(G)$ was proved only few years ago, involving the
classification of finite simple groups, and the structure of $G$ may be
strongly restricted, once information on $\Gamma(G)$ are given. The first
result of the present paper deals with a condition on $\mathbf{N}$–closed
classes of groups. The second result deals with a computational criterion,
related to the connectivity of $\Gamma(G)$.

###### Key words and phrases:

Sylow graph of a finite group; normalizers; almost–simple groups; formations;
symmetric coverings

###### 2010 Mathematics Subject Classification:

20E32, 20D20, 20F17.

##  1\. Sylow graph and formations of groups

All the groups, which are considered in the present paper, are finite. If $G$
is a group in which every Sylow subgroup is self–normalizing, then [12,
Theorem 1] shows that $G$ is a $p$–group for some prime $p$. [12, Theorem 1]
is a classical result and belongs to a long standing line of research which
investigates the structural properties of a group, once restrictions on its
normalizers are given (see [10, Chapter 5]). As noted in [3], we may introduce
the group class operator $\mathbf{N}$:

(1.1) $\mathbf{N}(\mathfrak{X})=(G:N_{G}(G_{p})\in\mathfrak{X},\forall
p\in\pi(G)),$

where $\mathfrak{X}$ is a class of groups, $\pi(G)$ is the set of prime
divisors of the order of $G$ and $G_{p}$ denotes the Sylow $p$– subgroup of
$G$, briefly $G_{p}\in\mathrm{Syl}_{p}(G)$. $\mathbf{N}$ was largely studied
in [1, 2, 3, 5, 6, 7, 8, 9, 13]. Originally, [12, Corollary 2] shows that the
class $\mathfrak{E}_{p}$ of all $p$–groups is $\mathbf{N}$–closed. More
generally, [3, Theorem 2] shows that the class $\mathfrak{N}$ of all nilpotent
groups is $\mathbf{N}$–closed. A further improvement is [1, Theorem 2], where
it is considered the class of all $p$–nilpotent groups. In the class
$\mathfrak{S}$ of all solvable groups, relations among $\mathbf{N}$ and
classes of groups which are closed with respect to forming subgroups (briefly,
$\mathbf{S}$–closed), were investigated in [5, 6]. These researches continued
in [7, 8, 9], introducing some technical notions, which refer to ideas and
techniques in the theory of formations of groups due to Gaschütz, Lubesender
and Shemetkov. The terminology is standard in literature and can be found in
[10]. A significant notion is the following.

######  Definition 1.1 (See [6]).

Assume that $\pi$ is a set of primes. A covering
$\mathcal{R}=(\pi(p))_{p\in\mathbb{P}}$ is called $symmetric$ in $\pi$, if the
following properties hold:

  * (i)

$\pi={\underset{p\in\pi}{\bigcup}}\pi(p),$

  * (ii)

$p\in\pi(p)$, for each $p\in\pi$,

  * (iii)

if $q\in\pi(p)$, then $p\in\pi(q)$.

Let $f_{\mathcal{R}}$ be the formation function, defined by

(1.2)
$f_{\mathcal{R}}(p)=\left\\{\begin{array}[]{lcl}\mathfrak{E}_{\pi(p)},\,\,\mathrm{if}\,\,p\in\pi\\\
\emptyset,\,\,\,\,\,\,\,\,\,\,\mathrm{if}\,\,p\not\in\pi.\end{array}\right.$

The saturated formation $LF(f_{\mathcal{R}})=\mathfrak{E}_{\mathcal{R}},$
locally defined by $f_{\mathcal{R}}$, is said to be a $covering$–$formation$,
associated to $\mathcal{R}$. The intersection
$\mathfrak{E}_{\mathcal{R}}\cap\mathfrak{S}$ is said to be a
covering–formation of solvable groups, associated to $\mathcal{R}$. If
$\mathcal{R}$ is a partition of $\pi$, $\mathfrak{E}_{\mathcal{R}}$ is called
$lattice$–$formation$.

In [1, 2, 3, 5, 6, 7, 8, 9] the covering–formations and the lattice–formations
originated the following graph, which appears in [9] for the first time.

######  Definition 1.2 (See [9]).

Let $G$ be a group and $p,q$ two distinct primes in $\pi(G)$. Then there is a
sequence $p_{1}=p,p_{2},\ldots,p_{n}=q$ in $\pi(G)$ such that either $p_{i}$
divides $|N_{G}(G_{p_{i+1}}):C_{G}(G_{p_{i+1}})|$ or $p_{i+1}$ divides
$|N_{G}(G_{p_{i}}):C_{G}(G_{p_{i}})|,$ for each $i=1,2,\ldots,n$. When this
happen, we write briefly $p\approx q$. $\Delta(G)$ denotes the graph with set
of vertices $\pi(G)$ and edges given by the relation $\approx$.

The connectivity of $\Delta(G)$ was hard to investigate in the form of
Definition 1.2. In fact it was reformulated under a different prospective by
Kazarin and others in [13]. These authors study the following graph, known as
the $Sylow$ $graph$ $of$ $G$.

######  Definition 1.3 (See [13]).

In a group $G$ define the automiser $A_{p}(G)$ to be the group
$N_{G}(G_{p})/G_{p}C_{G}(G_{p})$. The Sylow graph $\Gamma(G)$ of $G$, with set
of vertices $\pi(G)$, is given by the following rules: Two vertices
$p,q\in\pi(G)$ form an edge of $\Gamma_{A}(G)$ if either $q\in\pi(A_{p}(G))$
or $p\in\pi(A_{q}(G))$. Briefly, we write $p\rightarrow q$ if
$q\in\pi(A_{p}(G))$.

It is easy to see that $p\rightarrow q$ implies $p\approx q$, since
$|N_{G}(G_{p}):G_{p}N_{G}(G_{p})|\leq|N_{G}(G_{p}):C_{G}(G_{p})|$. In
particular, if $G_{p}$ is abelian, then $G_{p}\subseteq C_{G}(G_{p})$ and we
conclude that $\Delta(G)=\Gamma(G)$. Therefore the two graphs are related and
we can summarize all in the following observation.

######  Remark 1.4.

While $\Delta(G)$ admits loops, $\Gamma(G)$ does not admit loops. This fact
does not affect the connectivity of $\Delta(G)$ and $\Gamma(G)$ so that
$\Delta(G)$ is connected if, and only if, $\Gamma(G)$ is connected.

The connectivity of $\Gamma(G)$ was proved in [13, Main Theorem] and
influenced heavily the first versions of the present paper. An immediate
consequence, conjectured in [9], is that the lattice–formations are
$\mathbf{N}$–closed. Therefore $\Gamma(G)$ allows us to decide whether a class
of groups is $\mathbf{N}$–closed or not. In Section 2 we describe some
abstract conditions, which are useful for the same scope, without using the
methods of the graph theory. Since the properties of $\Gamma(G)$ are related
to those of the Sylow normalizers of $G$, it is used the classification of
finite simple groups in [13]. In Section 3 we illustrate a criterion for the
connectivity of $\Gamma(G)$, dealing with a computational method. In this way,
we avoid the classification of finite simple groups.

##  2\. Some abstract conditions

We begin with some examples to become confident with $\mathbf{N}$–closed
classes.

######  Example 2.1 (See [13], Examples 2,3 and Remark in Section 5).

Consider $\pi=\\{2,3,5\\},$ $\pi(2)=\\{2,3,5\\},$ $\pi(3)=\\{2,3\\},$
$\pi(5)=\\{2,5\\}$ and $\mathcal{R}=\pi(2)\cup\pi(3)\cup\pi(5)$. Definition
1.1 is satisfied and $\mathcal{R}$ is symmetric in $\pi$. Note that the
alternating groups on 5 elements belong to
$\mathbf{N}(\mathfrak{E}_{\mathcal{R}})\setminus\mathfrak{E}_{\mathcal{R}}$.
This suggests that there is not a characterization of covering–formations in
the universe $\mathfrak{E}$ of all finite groups in terms of $\mathbf{N}$.

Example 2.1 shows the following fact. If $E$ is a simple group, belonging to a
covering–formation, then $E$ is a $\pi(p)$–group, for each $p\in\pi(E)$,
because $E$ is clearly the only chief factor of itself.

######  Remark 2.2.

Still from Definition 1.1, the lattice–formation, related to the minimal
partition of $\pi$, is the formation $\mathfrak{N}_{\pi}$ of all nilpotent
$\pi$–groups. In particular, $\mathfrak{N}_{\mathbb{P}}=\mathfrak{N}$. The
lattice–formation, related to the maximal partition of $\pi$, is
$\mathfrak{E}_{\pi}$. In particular, $\mathfrak{E}_{\mathbb{P}}=\mathfrak{E}$.
From [2], the lattice–formation, related to a partition $(\pi_{i})_{i\in I}$
of $\pi\subseteq\mathbb{P}$, is the class of the groups which are direct
product of $\pi_{i}$–groups.

######  Example 2.3.

Still from Definition 1.1, the class $\mathfrak{D}$ of all groups which have
odd order is both a lattice–formation and a covering–formation of solvable
groups, thanks to a well–known result of J. Thompson and W. Feit. More
precisely, $\mathfrak{D}=LF(f_{\mathcal{R}})$, where
$f_{\mathcal{R}}(2)=\emptyset$ and
$f_{\mathcal{R}}(p)=\mathfrak{S}_{2^{\prime}}$ if $p\not=2.$ Note that
$\mathfrak{D}=\mathfrak{S}_{2^{\prime}}$ and the characteristic of the
formation $\mathfrak{D}$ (see [10] for the terminology) is equal to
$2^{\prime}=\mathbb{P}\setminus\\{2\\}$.

The above considerations show that saturated formations, which are
$\mathbf{S}$–closed and $\mathbf{N}$–closed, could not be covering–formations.
This was already noted in [8, Section 3]. The following condition is more easy
to check in a computational way.

######  Hypothesis.

Assume that in a group $G$, if $p\in\pi(G)$ and $p$ is odd, then there exists
a prime $q$ such that $q<p$ and $q$ divides $|N_{G}(G_{p}):C_{G}(G_{p})|$ for
each $G_{p}\in\mathrm{Syl}_{p}(G)$.

If Hypothesis is true, then $\Delta(G)$ is connected and, consequently,
$\Gamma(G)$ is connected. Unfortunately, the converse is false, as shown
below.

######  Example 2.4.

Let $H=PSL_{2}(3^{3^{k}})$ for any integer $k\geq 1$ and consider
$G=H\langle\sigma\rangle$, where

(2.1) $\sigma:\left(\begin{array}[]{ccccccc}a&b\\\ c&d\\\
\end{array}\right)\in
PSL_{2}(3^{3^{k}})\longmapsto\left(\begin{array}[]{ccccccc}a^{3}&b^{3}\\\
c^{3}&d^{3}\\\ \end{array}\right)\in PSL_{2}(3^{3^{k}}).$

$G$ is an almost–simple group, i.e.: $\mathrm{Inn}(S)\leq
G\leq\mathrm{Aut}(S)$ for some non–abelian simple group $S$, and
$\\{2,3\\}\subseteq\pi(G)$. Now $G$ does not satisfies Hypothesis, because $2$
does not divides $|N_{G}(G_{3}):C_{G}(G_{3})|$. At the same time, one can see
that there is a prime $l\in\pi(G)\setminus\\{2,3\\}$ such that $2$ divides
$|N_{G}(G_{l}):C_{G}(G_{l})|$ and $3$ divides $|N_{G}(G_{l}):C_{G}(G_{l})|$ so
that $2\approx 3$.

Saturated formations, which are both $\mathbf{S}$–closed and
$\mathbf{N}$–closed, are characterized in [8, Theorem 2] and yields to
interesting constructions as in [8, Section 3] or in [13, Section 5]. We
recall just one case.

######  Example 2.5.

Consider a set of primes
$\pi=\\{p\\}\cup(\pi\setminus\\{p\\})=\\{p\\}\cup\omega$ such that there exist
a non–abelian simple $\omega$–group. Then the formation function $f$, defined
by $f(q)=\emptyset$, if $q\not\in\pi$; $f(q)=\mathfrak{S}_{\pi}$, if $q=p$;
$f(q)=\mathfrak{E}_{\pi}$, if $q\in\omega$, determines the saturated formation
$\mathfrak{F}=LF(f)$, locally defined by $f$, which is in many situations both
$\mathbf{S}$–closed and $\mathbf{N}$–closed but not a covering–formation. See
[8, Section 3] for details.

We may extend Example 2.5 in a more appropriate way. Recall that $G^{\infty}$
denotes the largest perfect subgroup of a group $G$. Recall also that
$\mathfrak{F}$ in Example 2.5 is the class of all finite groups $G$ such that
whenever $H/K$ is a chief–factor of $G$ and $q$ is a prime dividing $|H/K|$,
then $G/C_{G}(H/K)\in f(q)$.

######  Lemma 2.6.

$G\in\mathfrak{F}$ if and only if $G$ is a $\pi$–group and $G^{\infty}$ is a
$p^{\prime}$–group.

###### Proof.

Suppose $G\in\mathfrak{F}$. If $q$ is a prime divisor of $|G|$ then there
exists a chief factor $H/K$ of $G$ such that $q$ divides $|H/K|$. Then
$G/C_{G}(H/K)\in f(q)$. Thus $f(q)$ is not empty and so $q\in\pi$. So $G$ is a
$\pi$–group. Suppose for a contradiction that $G^{\infty}$ is not a
$p^{\prime}$–group. Then there exists a chief factor $H/K$ of $G$ such that
$H\leq G^{\infty}$, $p$ divides $|H/K|$ and $G^{\infty}/H$ is a
$p^{\prime}$–group. It follows that $G/C_{G}(H/K)\in f(p)$ and so
$G/C_{G}(H/K)$ is solvable. Thus $G^{\infty}\leq C_{G}(H/K)$ and so $H/K\leq
Z(G^{\infty}/K)$. Since $p$ divides $|H/K|$, $H/K$ is a $p$–elementary abelian
group. Since $G^{\infty}/K$ is $p^{\prime}$–group, we conclude that
$G^{\infty}/K=H/K\times L/K$ for some subgroup $L/K$ of $G^{\infty}/K$. This
is a contradiction, since $G^{\infty}/K$ is perfect and $H/K$ is abelian.

Suppose next that $G$ is a $\pi$–group and $G^{\infty}$ is a
$p^{\prime}$–group. Let $H/K$ be a chief factor of $G$ and $q$ a prime
dividing $|H/K|$. Since $G$ is a $\pi$–group, $q\in\pi$ and $G/C_{G}(H/K)$ is
a $\pi$–group. So if $q\neq p$ we have $G/C_{G}(H/K)\in f(q)$. Suppose $q=p$.
Since $[H/K,G^{\prime}]=[H,G^{\prime}]K/K$ is a $G$–invariant
$p^{\prime}$–subgroup of $H/K$, we conclude that $G^{\prime}\leq C_{G}(H/K)$
and so $G/C_{G}(H/K)$ is solvable. Hence again $G/C_{G}(H/K)\in f(q)$. So
$G\in\mathfrak{F}$. ∎

######  Corollary 2.7.

$\mathfrak{F}$ is $\mathbf{S}$–closed.

###### Proof.

This follows immediately from Lemma 2.6. ∎

In the next results we will use the $generalized$ $Fitting$ $subgroup$
$F^{*}(G)$ of a group $G$ and the $components$ $of$ $G$. These are well–known
notions, which can be found in [10, p.580].

######  Lemma 2.8.

Let $H/K$ be a non–abelian chief–factor of a group $G$, $L$ a component of
$H/K$, $\overline{U}=N_{G/K}(L)/C_{G/K}(L)$, $q$ a prime,
$T\in\mathrm{Syl}_{q}(\overline{U})$ and $S\in\mathrm{Syl}_{q}(G)$. Then
$F^{*}(\overline{U})=\overline{L}\cong L$ and
$N_{\overline{L}}(T)/\overline{L}\cap T$ is isomorphic to a subgroup of
$N_{H}(S)/(S\cap H)K$.

###### Proof.

The first statement is obvious. To simplify the notation we may replace $G$ by
$G/K$ and assume that $K=1$. Put $R=N_{S}(L)$. We also may assume that
$T=\overline{R}$. Then $N_{\overline{L}}(T)/\overline{L}\cap T\cong
N_{L}(R)/L\cap R$. Let $s_{1},\ldots s_{k}$ be a transversal to $R$ in $S$
with $s_{1}=1$ and put $L_{i}=L^{s_{i}}$. Then $M=\langle
L^{S}\rangle=L_{1}\times L_{2}\times\ldots L_{k}$ and it is now easy to see
that $N_{M}(S)/S\cap M\cong N_{L}(R)/L\cap R$.∎

######  Corollary 2.9.

Let $G$, $L$ and $\overline{U}$ be as in Lemma 2.8. If
$G\in\mathbf{N}(\mathfrak{F})$, then
$\overline{U}\in\mathbf{N}(\mathfrak{F})$.

###### Proof.

Since $\mathfrak{F}$ is closed under sections, this follows from Lemma 2.8. ∎

Note that a group $U$ is almost–simple if and only if $F^{*}(U)$ is simple.

######  Theorem 2.10.

$\mathfrak{F}=\mathbf{N}(\mathfrak{F})$ if and only the following statements
are true:

  * (a)

If $U$ is a non–solvable almost–simple group in $\mathbf{N}(\mathfrak{F})$,
then $F^{*}(U)$ is a $p^{\prime}$–group.

  * (b)

If $L$ is a non–abelian finite simple $(\pi\setminus\\{p\\})$–group and
$S\in\mathrm{Syl}_{p}(\mathrm{Aut}(L))$, then $C_{L}(S)$ is non–solvable.

###### Proof.

Suppose first that $\mathfrak{F}=\mathbf{N}(\mathfrak{F})$. Let $U$ be an
almost–simple group in $\mathbf{N}(\mathfrak{F})$. Since
$\mathfrak{F}=\mathbf{N}(\mathfrak{F})$, we conclude that $L\in\mathfrak{F}$
and so $U^{\infty}$ is a $p^{\prime}$–group. Since $U$ is non–solvable,
$F^{*}(U)$ is perfect and so $F^{*}(U)\leq U^{\infty}$ and $F^{*}(U)$ is a
$p^{\prime}$–group. Let $L$ be a finite simple $(\pi\setminus\\{p\\})$–group
and $S\in\mathrm{Syl}_{p}(\mathrm{Aut}(L))$. Let $\mathbb{F}$ be a finite
field of characteristic $p$ such that $q$ divides $|\mathbb{F}^{\sharp}|$ for
all $q\in\pi$ with $q\neq p$ ($\mathbb{F}^{\sharp}$ denotes the multiplicative
group of $\mathbb{F}$). Let $D$ be the $(\pi\setminus\\{p\\})$–Hall subgroup
of $\mathbb{F}^{\sharp}$. Let $V$ be a faithful $\mathbb{F}LS$–module. Then
$V$ is also a module for $D\times LS$ and we can consider the semidirect
product $H=V(D\times LS)$. Note that $H$ is a $\pi$–group. Also $[V,L]\leq
H^{\infty}$ and so $H^{\infty}$ is not a $p^{\prime}$–group. So
$H\not\in\mathfrak{F}$. Since $\mathfrak{F}=\mathbf{N}(\mathfrak{F})$, this
gives $H\not\in\mathbf{N}(\mathfrak{F})$. Hence there exists $q\in\pi$ and
$T\in\mathrm{Syl}_{q}(H)$ such that $N_{H}(T)\not\in\mathfrak{F}$. Since
$N_{H}(R)$ is a $\pi$–group, this means that $N_{H}(T)^{\infty}$ is not a
$p^{\prime}$–group. Since $N_{H}(T)^{\infty}\leq H^{\infty}\leq VL$ and $L$ is
a $p^{\prime}$–group we conclude that $N_{H}(T)\cap V\neq 1$. Assume that
$q=p$. Then $T\cap V=1$ and $1\neq N_{H}(T)\cap V=C_{V}(T)=C_{V}(VT)$. Since
$T$ is a Sylow $q$–subgroup of $H$ and $q$ divides $|D|$, we get $VT\cap D\neq
1$. But each non–trivial element of $D$ acts fixed point freely on $V$ and so
$C_{V}(VT\cap D)=1$, a contradiction to $C_{V}(VT)\neq 1$. Thus $q=p$ and we
may assume that $T=VS$. Then $N_{H}(T)=VDN_{L}(S)$ and, since $N_{H}(T)$ is
non–solvable, we conclude that $N_{L}(S)$ is non–solvable. Suppose next that
(a) and (b) hold. We will first show that

(2.2) $\mathrm{Any}\ \mathrm{non}-\mathrm{abelian}\ \mathrm{composition}\
\mathrm{factor}\ L\ \mathrm{of}\ G\in\mathbf{N}(\mathfrak{F})\ \mathrm{is}\
\mathrm{a}\ p^{\prime}-\mathrm{group}.$

Indeed by Lemma 2.8 and Corollary 2.9, $L\cong F^{*}(\overline{U})$ for the
simple group $\overline{U}\in\mathbf{N}(\mathfrak{F})$. Thus by (a), $L$ is a
$p^{\prime}$–group. So (2.2) holds.

Suppose for a contradiction that there exists
$G\in\mathbf{N}(\mathfrak{F})\setminus\mathfrak{F}$. Then there exist a
chief–factor $H/K$ and a prime $q$ dividing $|H/K|$ such that
$G/C_{G}(H/K)\not\in f(q)$. Since $G$ is a $\pi$–group, this implies that
$q=p$ and $G/C_{G}(H/K)$ is non–solvable. Thus $G^{\infty}\nleq C_{G}(H/K)$
and there exists a normal subgroup $M$ of $G$ maximal with $C_{G}(H/K)\leq
M<G$. Observe that $G^{\infty}/M$ is a non–abelian chief factor of $G$. Let
$L$ be a composition factor of $G^{\infty}/M$. By (2.2), $L$ is a
$p^{\prime}$–group. From (b) and Lemma 2.8 we conclude that
$N_{G^{\infty}}(S)M/M$ is non–solvable and so

(2.3) $N_{G}(S)^{\infty}\nleq C_{G}(H/K).$

Since $G\in\mathbf{N}(\mathfrak{F})$, $N_{G}(S)^{\infty}$ is a
$p^{\prime}$–group. Since $[N_{G}(S)^{\infty},S]\leq N_{G}(S)^{\infty}\cap S,$
we conclude that $N_{G}(S^{\infty})\leq C_{G}(S)$. By (2.2), $H/K$ is a
$p$–group. Thus $H=(S\cap H)K$ and so

(2.4) $[N_{G}(S)^{\infty},H]\leq[N_{G}(S^{\infty}),S]K=K,$

which is in contradiction with (2.3). ∎

######  Remark 2.11.

(a) and (b) in Theorem 2.10 follow from: (c) If $L$ is a finite simple
$\pi$–group, then $\mathrm{Aut}(L)$ is a $p^{\prime}$–group.

##  3\. A computational method

In the present section we illustrate an algorithm in GAP (see [14]), which
allows us to decide whether $\Delta(G)$ is connected or not in an empiric way.
The times of answer of the oracle are quite long, but, in case we use
Hypothesis, they are significatively reduced. The present algorithm has been
called ROSN1, achronymus of ”Restrictions On Sylow Normalizers”. In Appendix
the reader may find the description of $\Delta(G)$ for all the sporadic simple
groups ($\textrm{Div}(k)$ denotes the set of divisors of the integer $k\geq
1$). These results are deduced by looking at [4] and confirm the empiric
results which we get on ROSN1.

TestGroup:=function(G)

local pi,p,S,x,N,C,Q,R,i,res;
pi:=AsSet(FactorsInt(Size(G)));
S:=List(pi,p->SylowSubgroup(G,p));
N:=List(S,x->Normalizer(G,x));
C:=List(S,x->Centralizer(G,x));
Q:=List([1..Size(pi)],i->Size(N[i])/Size(C[i]));
R:=List([1..Size(pi)],i->ForAny(FactorsInt(Q[i]),p->p<pi[i]));
if pi[1]=2 then
res:=ForAll(R{[2..Size(pi)]},x->x);
else res:=ForAll(R,x->x);
fi;
return res;
end;

## acknowledgement

The present work was supported by GNSAGA (Indam, Florence, Italy) in the years
2008–09. This allowed me to visit Michigan State University (USA), TU of
Darmstadt (Germany), Free University of Berlin (Germany) and University
Technology of Malaysia (Malaysia), finding the inspiration for the present
paper.

## References

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##  4\. Appendix

  * (1)

For $G=M_{11}$, $\pi(M_{11})=\\{2,3,5,11\\}$.
$2\in\textrm{Div}(|N_{M_{11}}(G_{3}):C_{M_{11}}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)$. These data are enough.

  * (2)

For $G=M_{12}$, $\pi(M_{12})=\\{2,3,5,11\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$ These data are enough.

  * (3)

For $G=J_{1}$, $\pi(J_{1})=\\{2,3,5,7,11,19\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)$. These data are enough.

  * (4)

For $G=M_{22}$, $\pi(M_{22})=\\{2,3,5,7,11\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.

  * (5)

For $G=J_{2}$, $\pi(J_{2})=\\{2,3,5,7\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|).$
These data are enough.

  * (6)

For $G=M_{23}$, $\pi(M_{23})=\\{2,3,5,7,11,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|)$. These data are enough.

  * (7)

For $G=H_{5}$, $\pi(H_{5})=\\{2,3,5,7,11\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.

  * (8)

For $G=J_{3}$, $\pi(J_{3})=\\{2,3,5,17,19\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$3\in\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|).$ These data are enough.

  * (9)

For $G=M_{24}$, $\pi(M_{24})=\\{2,3,5,17,19\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$3\in\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|).$ These data are enough.

  * (10)

For $G=McL$,
$\pi(McL)=\\{2,3,5,7,11\\}$.$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.

  * (11)

For $G=He$, $\pi(He)=\\{2,3,5,7,17\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|).$ These data are enough.

  * (12)

For $G=Ru$, $\pi(Ru)=\\{2,3,5,7,13,29\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$7\in\textrm{Div}(|N_{G}(G_{29}):C_{G}(G_{29})|).$ These data are enough.

  * (13)

For $G=Suz$, $\pi(Suz)=\\{2,3,5,7,11,13\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.

  * (14)

For $G=O^{\prime}N$, $\pi(O^{\prime}N)=\\{2,3,5,7,11,19,31\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|).$
These data are enough.

  * (15)

For $G=Co3$, $\pi(Co3)=\\{2,3,5,7,11,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|).$ These data are enough.

  * (16)

For $G=Co2$, $\pi(Co2)=\\{2,3,5,7,11,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|).$ These data are enough.

  * (17)

For $G=Co1$, $\pi(Co1)=\\{2,3,5,7,11,13,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|).$ These data are enough.

  * (18)

For $G=Fi_{22}$, $\pi(Fi_{22})=\\{2,3,5,7,11,13\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.

  * (19)

For$G=Fi_{24}$, $\pi(Fi_{24})=\\{2,3,5,7,11,13,17,23,29\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$7\in\textrm{Div}(|N_{G}(G_{29}):C_{G}(G_{29})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|),$ These data are enough.

  * (20)

For $G=HN$, $\pi(HN)=\\{2,3,5,7,11,19\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|).$ These data are enough.

  * (21)

For $G=Ly$, $\pi(Ly)=\\{2,3,5,7,11,31,37,67\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|)l,$
$11\in\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|).$ These data are enough.

  * (22)

For $G=Fi_{23}$, $\pi(Fi_{24})=\\{2,3,5,7,11,13,17,23\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|),$
$11\in\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|).$ These data are enough.

  * (23)

For $G=Th$, $\pi(Th)=\\{2,3,5,7,11,13,19,31\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|)\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|),$
$5\in\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|).$ These data are enough.

  * (24)

For $G=J_{4}$, $\pi(J_{4})=\\{2,3,5,7,11,23,29,31,37,43\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|)\cap\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|),$
$7\in\textrm{Div}(|N_{G}(G_{29}):C_{G}(G_{29})|)\cap\textrm{Div}(|N_{G}(G_{43}):C_{G}(G_{43})|).$
These data are enough.

  * (25)

For $G=B$, $\pi(B)=\\{2,3,5,7,11,13,17,19,23,31,47\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|)$$\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|)\cap\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|)\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|),$
$23\in\textrm{Div}(|N_{G}(G_{47}):C_{G}(G_{47})|).$ These data are enough.

  * (26)

For $G=M$, $\pi(M)=\\{2,3,5,7,11,13,17,19,23,29,31,41,47,59,71\\}$.
$2\in\textrm{Div}(|N_{G}(G_{3}):C_{G}(G_{3})|)\cap\textrm{Div}(|N_{G}(G_{5}):C_{G}(G_{5})|)\cap\textrm{Div}(|N_{G}(G_{17}):C_{G}(G_{17})|),$
$3\in\textrm{Div}(|N_{G}(G_{7}):C_{G}(G_{7})|)\cap\textrm{Div}(|N_{G}(G_{13}):C_{G}(G_{13})|)$$\cap\textrm{Div}(|N_{G}(G_{19}):C_{G}(G_{19})|)\cap\textrm{Div}(|N_{G}(G_{37}):C_{G}(G_{37})|),$
$5\in\textrm{Div}(|N_{G}(G_{11}):C_{G}(G_{11})|)\cap\textrm{Div}(|N_{G}(G_{23}):C_{G}(G_{23})|)$$\cap\textrm{Div}(|N_{G}(G_{31}):C_{G}(G_{31})|)\cap\textrm{Div}(|N_{G}(G_{41}):C_{G}(G_{41})|),$$7\in\textrm{Div}(|N_{G}(G_{71}):C_{G}(G_{71})|),$
$23\in\textrm{Div}(|N_{G}(G_{47}):C_{G}(G_{47})|),$
$29\in\textrm{Div}(|N_{G}(G_{59}):C_{G}(G_{59})|).$ These data are enough.


$W_{\varrho,p}(\alpha_{n},\alpha)\to 0$ and then apply Lemma 3.22. The
$1$-homogeneity of $\varrho^{p}$ yields

$\displaystyle\int_{\mathfrak{C}[\mathsf{X}]}\varrho^{p}(\mathfrak{y},\mathfrak{o})\,\mathrm{d}\alpha_{n}(\mathfrak{y})$
$\displaystyle=\int_{\mathsf{X}}m_{n}^{-1}\varrho^{p}([x,m_{n}],\mathfrak{o})\,\mathrm{d}\mu_{n}(x)$
$\displaystyle=\int_{\mathsf{X}}\varrho^{p}([x,1],\mathfrak{o})\,\mathrm{d}\mu_{n}(x)\stackrel{{\scriptstyle
n\to\infty}}{{\to}}\int_{\mathsf{X}}\varrho^{p}([x,1],\mathfrak{o})\,\mathrm{d}\mu(x)$
$\displaystyle=\int_{\mathsf{X}}m^{-1}\varrho^{p}([x,m],\mathfrak{o})\,\mathrm{d}\mu(x)=\int_{\mathfrak{C}[\mathsf{X}]}\varrho^{p}(\mathfrak{y},\mathfrak{o})\,\mathrm{d}\alpha(\mathfrak{y}),$

and we get that $W_{\varrho,p}(\alpha_{n},\alpha)\to 0$ applying [AGS08,
Proposition 7.1.5]. ∎

##  Appendix B The case of a cost function finite on an open cone

In this appendix we repeat the constructions of Section 4 under different
assumptions on $\mathsf{H}$. As in Section 4, we assume that $\mathsf{X}_{1}$
and $\mathsf{X}_{2}$ are compact and metrizable spaces. Given non negative
numbers $q_{i}$, $i=1,2$, we define the open cone

$U_{q_{1}q_{2}}:=\left\\{([x_{1},r_{1}],[x_{2},r_{2}])\in\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\mid
r_{2}>r_{1}q_{1},\,r_{1}>r_{2}q_{2}\right\\}.$

Using this notation, we assume that

$\begin{split}\mathsf{H}:&\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]\text{
is continuous, radially $1$-homogeneous and convex and }\\\ &\text{that there
exists non-negative numbers $q_{1},q_{2}$ such that
}{\mathrm{D}}(\mathsf{H})=U_{q_{1}q_{2}}\text{ and }\\\ &\lim_{r_{1}\downarrow
q_{2}}\inf_{(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}}\mathsf{H}([x_{1},r_{1}],[x_{2},1])=\lim_{r_{2}\downarrow
q_{1}}\inf_{(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}}\mathsf{H}([x_{1},1],[x_{2},r_{2}])=+\infty.\end{split}$
(B.1)

Notice that, if $q_{1}=q_{2}=0$, we are simply assuming that $\mathsf{H}$ is
finite on the whole open cone.

######  Proposition B.1.

Assume that $\mathsf{H}$ is as in (B.1) and that
$\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ are such that

$\mathsf{supp}\left(\mu_{i}\right)=\mathsf{X}_{i},\,i=1,2,\quad
q_{1}<\frac{\mu_{2}(\mathsf{X}_{2})}{\mu_{1}(\mathsf{X}_{1})},\quad
q_{2}<\frac{\mu_{1}(\mathsf{X}_{1})}{\mu_{2}(\mathsf{X}_{2})},$

where $q_{i}$ are as in (B.1). Then there exists a constant $C>0$ such that,
for every $(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}$ with

$\int_{\mathsf{X}_{1}}\varphi_{1}\,\mathrm{d}\mu_{1}+\int_{\mathsf{X}_{2}}\varphi_{2}\,\mathrm{d}\mu_{2}\geq
0,$

it holds

$\varphi_{1}(x_{1})\leq C,\quad\varphi_{2}(x_{2})\leq C\quad\text{ for every
}(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$

and there exists
$(\bar{x}_{1},\bar{x}_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$ such that
$\varphi_{1}(\bar{x}_{1})\geq-C$, $\varphi_{2}(\bar{x}_{2})\geq-C$.

###### Proof.

We start from the last claim for $\varphi_{1}$. Assume by contradiction that
there exists a sequence
$(\varphi_{1}^{j},\varphi_{2}^{j})_{j}\subset\Phi_{\mathsf{H}}$ with
$\int_{\mathsf{X}_{1}}\varphi_{1}^{j}\,\mathrm{d}\mu_{1}+\int_{\mathsf{X}_{2}}\varphi_{2}^{j}\,\mathrm{d}\mu_{2}\geq
0$ such that $\max_{x_{1}\in\mathsf{X}_{1}}\varphi_{1}^{j}(x_{1})\to-\infty$.
Let $(x_{1}^{j})_{j}\subset\mathsf{X}_{1}$ be the sequence of points where the
maxima are attained. We thus have

$\varphi_{1}^{j}(x_{1}^{j})\mu_{1}(\mathsf{X}_{1})+\int_{\mathsf{X}_{2}}\varphi_{2}^{j}\,\mathrm{d}\mu_{2}\geq
0\quad\text{ for every }j\in\mathbb{N}$

so that we can find $(x_{2}^{j})_{j}\subset\mathsf{X}_{2}$ such that

$\varphi_{2}^{j}(x_{2}^{j})\geq-\frac{\varphi_{1}^{j}(x_{1}^{j})\mu_{1}(\mathsf{X}_{1})}{\mu_{2}(\mathsf{X}_{2})}\quad\text{
for every }j\in\mathbb{N}.$

Since $(\varphi_{1}^{j},\varphi_{2}^{j})\in\Phi_{\mathsf{H}}$, we have

$\varphi_{1}^{j}(x_{1}^{j})\left(r_{1}-\frac{\mu_{1}(\mathsf{X}_{1})}{\mu_{2}(\mathsf{X}_{2})}r_{2}\right)\leq\varphi_{1}^{j}(x_{1}^{j})r_{1}+\varphi_{2}^{j}(x_{2}^{j})\leq\mathsf{H}([x_{1}^{j},r_{1}],[x_{2}^{j},r_{2}])\quad\text{
for every }r_{1},r_{2}\geq 0.$

We can assume, up to passing to a subsequence, that
$(x_{1}^{j},x_{2}^{j})\to(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$.
Thanks to (B.1) and the assumptions on $\mu_{i}$, we can find
$\bar{r}_{1},\bar{r}_{2}>0$ such that

$\mathsf{H}([x_{1},\bar{r}_{1}],[x_{2},\bar{r}_{2}])<+\infty,\quad\bar{r}_{1}-\frac{\mu_{1}(\mathsf{X}_{1})}{\mu_{2}(\mathsf{X}_{2})}\bar{r}_{2}<0.$

We thus have that

$+\infty\leq\mathsf{H}([x_{1},\bar{r}_{1}],[x_{2},\bar{r}_{2}],$

a contradiction with (B.1). The proof for $\varphi_{2}$ is the same; we have
thus proven that there exists a constant $D>0$ independent of
$(\varphi_{1},\varphi_{2})$ and a point
$(\bar{x}_{1},\bar{x}_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$ such that

$\varphi_{1}(\bar{x}_{1})\geq-D,\quad\varphi_{2}(\bar{x}_{2})\geq-D.$

Thus, if we set

$C:=D+\max_{(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}}\mathsf{H}([x_{1},\bar{r}_{1}],[x_{2},\bar{r}_{2}]),$

where $\bar{r}_{1}$ and $\bar{r}_{2}$ are as above, we get that

$\varphi_{1}(x_{1})\leq\frac{1}{\bar{r}_{1}}\left(\mathsf{H}([x_{1},\bar{r}_{1}],[\bar{x}_{2},\bar{r}_{1}])-\bar{r}_{2}\varphi_{2}(\bar{x}_{2})\right)\leq
C\quad\text{ for every }x_{1}\in\mathsf{X}_{1}$

and the corresponding statement for $\varphi_{2}$. ∎

In the following result, which is the analogue of Proposition 4.6, we use the
notation of Definition 4.5.

######  Proposition B.2.

Assume that $\mathsf{H}$ is as in (B.1) and that
$\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ are such that

$\mathsf{supp}\left(\mu_{i}\right)=\mathsf{X}_{i},\,i=1,2,\quad
q_{1}<\frac{\mu_{2}(\mathsf{X}_{2})}{\mu_{1}(\mathsf{X}_{1})},\quad
q_{2}<\frac{\mu_{1}(\mathsf{X}_{1})}{\mu_{2}(\mathsf{X}_{2})},$

where $q_{i}$ are as in (B.1). Then there exist constants
$a_{i},a_{s},b_{i},b_{s},M>0$ such that, if
$(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}$ are such that

$\int_{\mathsf{X}_{1}}\varphi_{1}\,\mathrm{d}\mu_{1}+\int_{\mathsf{X}_{2}}\varphi_{2}\,\mathrm{d}\mu_{2}\geq
0,$

then
$\|\varphi_{1}^{\mathsf{H}}\|_{\infty},\|\varphi_{1}^{\mathsf{H}\mathsf{H}}\|_{\infty}\leq
M$ and

$\displaystyle\varphi_{1}^{\mathsf{H}}(x_{2})$
$\displaystyle=\inf_{x_{1}\in\mathsf{X}_{1}}\inf_{a_{i}\leq\alpha\leq
a_{s}}\biggl{\\{}\mathsf{H}([x_{1},\alpha],[x_{2},1])-\alpha\varphi_{1}(x_{1})\biggr{\\}},\quad
x_{2}\in\mathsf{X}_{2},$ (B.2)
$\displaystyle\varphi_{1}^{\mathsf{H}\mathsf{H}}(x_{1})$
$\displaystyle=\inf_{x_{2}\in\mathsf{X}_{2}}\inf_{b_{i}\leq\alpha\leq
b_{s}}\biggl{\\{}\mathsf{H}([x_{1},1],[x_{2},\alpha])-\alpha\varphi_{1}^{\mathsf{H}}(x_{2})\biggr{\\}},\quad
x_{1}\in\mathsf{X}_{1}.$ (B.3)

Moreover the sets

$\displaystyle\mathfrak{C}_{0}^{1}$
$\displaystyle:=\\{(\mathfrak{y}_{1},\mathfrak{y}_{2})\in\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\mid
a_{i}\leq\mathsf{r}(\mathfrak{y}_{1})\leq
a_{s},\,\mathsf{r}(\mathfrak{y}_{2})=1\\},$
$\displaystyle\quad\mathfrak{C}_{0}^{2}$
$\displaystyle:=\\{(\mathfrak{y}_{1},\mathfrak{y}_{2})\in\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\mid
b_{i}\leq\mathsf{r}(\mathfrak{y}_{2})\leq
b_{s},\,\mathsf{r}(\mathfrak{y}_{1})=1\\}$

are compact subset of $U_{q_{1}q_{2}}$ and
$(\varphi_{1}^{\mathsf{H}\mathsf{H}},\varphi_{1}^{\mathsf{H}})\in\Phi_{\mathsf{H}}$,
$\varphi_{1}^{\mathsf{H}\mathsf{H}}\geq\varphi_{1}$,
$\varphi_{1}^{\mathsf{H}}\geq\varphi_{2}$. Finally, if $\mathsf{d}_{i}$ are
distances metrizing $\mathsf{X}_{i}$, $i=1,2$, then
$\varphi_{1}^{\mathsf{H}\mathsf{H}}$ is $\mathsf{d}_{1}$-uniformly continuous
with the same (uniform)
$\mathsf{d}_{1}\otimes_{\mathfrak{C}}\mathsf{d}_{2}$-modulus of continuity of
$\mathsf{H}$ on $\mathfrak{C}_{0}^{2}$ and $\varphi_{1}^{\mathsf{H}}$ is
$\mathsf{d}_{2}$-uniformly continuous with the same (uniform)
$\mathsf{d}_{1}\otimes_{\mathfrak{C}}\mathsf{d}_{2}$-modulus of continuity of
$\mathsf{H}$ on $\mathfrak{C}_{0}^{1}$.

###### Proof.

Let $(\varphi_{1},\varphi_{2})$ be as in the statement. Let us set

$\delta:=\frac{1}{2}\frac{1-q_{1}q_{2}}{1+q_{1}+q_{2}}$

so that, for every $0<\varepsilon\leq\delta$, we have
$q_{2}+\varepsilon<(q_{1}+\varepsilon)^{-1}$. Let us fix a point
$\bar{\alpha}\in(q_{2}+\delta,\frac{1}{q_{1}+\delta})$ and let us define

$m:=\max_{(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}}\mathsf{H}([x_{1},\bar{\alpha}],[x_{2},1])<+\infty,$

since $([x_{1},\bar{\alpha}],[x_{2},1])\in U_{q_{1}q_{2}}$ for every
$(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$.
By (B.1), we know that for every $L>0$, there exists $\varepsilon_{L}>0$ such
that

$\displaystyle\mathsf{H}([x_{1},r_{1}],[x_{2},1]$ $\displaystyle\geq
L\quad\text{ for every
}(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2},\,0\leq
r_{1}<q_{2}+\varepsilon_{L},$
$\displaystyle\mathsf{H}([x_{1},1],[x_{2},r_{2}]$ $\displaystyle\geq
L\quad\text{ for every
}(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2},\,0\leq
r_{2}<q_{1}+\varepsilon_{L}.$

Let

$L:=\max\left\\{m+C(\bar{\alpha}+q_{2}+\delta),(q_{1}+\delta)(m+\bar{\alpha}C)+C\right\\},$

where $C$ comes from Proposition B.1, and let us take any $a_{i},a_{s}>0$ such
that

$q_{2}<a_{i}<q_{2}+\varepsilon_{L}\wedge\delta,\quad\frac{1}{q_{1}+\varepsilon_{L}\wedge\delta}<a_{s}<\frac{1}{q_{1}},$

so that $0<a_{i}<a_{s}$, $\mathfrak{C}_{0}^{1}\subset U_{q_{1}q_{2}}$ and
$a_{i}\leq\bar{\alpha}\leq a_{s}$.
If $\alpha>a_{s}$, then, for every
$(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$, we have

$\displaystyle\mathsf{H}([x_{1},\alpha],[x_{2},1])-\alpha\varphi_{1}(x_{1})$
$\displaystyle=\alpha\left(\mathsf{H}([x_{1},1],[x_{2},1/\alpha])-\varphi_{1}(x_{1})\right)$
$\displaystyle\geq\alpha(L-C)$ $\displaystyle\geq a_{s}(L-C)$
$\displaystyle\geq m+\bar{\alpha}C$
$\displaystyle\geq\mathsf{H}([\bar{x}_{1},\bar{\alpha}]),[x_{2},1]-\bar{\alpha}\varphi_{1}(\bar{x}_{1}),$

where $\bar{x}_{1}$ comes from Proposition B.1. If $\alpha<a_{i}$, then, for
every $(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$, we have

$\displaystyle\mathsf{H}([x_{1},\alpha],[x_{2},1])-\alpha\varphi_{1}(x_{1})$
$\displaystyle\geq L-\alpha C$ $\displaystyle\geq L-Ca_{i}$ $\displaystyle\geq
m+\bar{\alpha}C$
$\displaystyle\geq\mathsf{H}([\bar{x}_{1},\bar{\alpha}],[x_{2},1])-\bar{\alpha}\varphi_{1}(\bar{x}_{1}).$

Thus, for every $x_{2}\in\mathsf{X}_{2}$, we get

$\displaystyle\inf_{x_{1}\in\mathsf{X}_{1}}\inf_{0\leq\alpha<a_{i}\vee\alpha>a_{s}}\left\\{\mathsf{H}([x_{1},\alpha],[x_{2},1])-\alpha\varphi_{1}(x_{1})\right\\}$
$\displaystyle\geq\mathsf{H}([\bar{x}_{1},\bar{\alpha}],[x_{2},1])-\bar{\alpha}\varphi_{1}(\bar{x}_{1})$
$\displaystyle>\inf_{x_{1}\in\mathsf{X}_{1}}\inf_{a_{i}\leq\alpha\leq
a_{s}}\left\\{\mathsf{H}([x_{1},\alpha],[x_{2},1])-\alpha\varphi_{1}(x_{1})\right\\}$

and this proves (B.2). The proof of (B.3) is analogous.
The remaining part of the proof is identical to the one of Proposition 4.6. ∎

By Proposition B.2 we obtain the analogue of Theorem 4.7 also in this setting
with exactly the same proof.

######  Theorem B.3.

Assume that $\mathsf{H}$ is as in (B.1) and that
$\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ are such that

$\mathsf{supp}\left(\mu_{i}\right)=\mathsf{X}_{i},\,i=1,2,\quad
q_{1}<\frac{\mu_{2}(\mathsf{X}_{2})}{\mu_{1}(\mathsf{X}_{1})},\quad
q_{2}<\frac{\mu_{1}(\mathsf{X}_{1})}{\mu_{2}(\mathsf{X}_{2})},$

where $q_{i}$ are as in (B.1). Then there exists
$(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}$ such that

$\int_{\mathsf{X}_{1}}\varphi_{1}\,\mathrm{d}\mu_{1}+\int_{\mathsf{X}_{2}}\varphi_{2}\,\mathrm{d}\mu_{2}=\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2}).$

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