# Three arithmetic sites

Lieven Le Bruyn  Department of Mathematics, University of Antwerp
Middelheimlaan 1, B-2020 Antwerp (Belgium)
<EMAIL_ADDRESS>
###### Abstract.

Two new arithmetic sites are introduced, based on dynamical Belyi maps and
Conway’s big picture, respectively. We relate these to arboreal Galois
representations, Bost-Connes data, and the original arithmetic site due to
Connes and Consani.

Renewed interest in the presheaf topos of a monoid category (the one object
category with given endomorphism monoid) was spurred by the discovery of
Connes and Consani, see [4], that in the special case of
$A=\mathbb{N}^{\times}_{+}$, the multiplicative monoid of strict positive
numbers, the points of the so called arithmetic site, that is, of the
corresponding presheaf topos

$\mathbf{points}(\widehat{\mathbf{A}})=\mathbb{Q}^{\times}_{+}\backslash\mathbb{A}^{f}/\widehat{\mathbb{Z}}^{*}$

are the finite adèle classes, a horrific topological space, better studied
with the tools coming from non-commutative geometry.

In this note we will introduce two more such sites and investigate their
interplay with the arithmetic site. The first one, Belyi’s site
$\widehat{\mathbf{B}}$, is constructed from the monoid of dynamical Belyi
maps, as in [1], and is relevant to the study of the absolute Galois group
$Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, whereas the arithmetic site is
relevant to the abelianisation $Gal(\overline{\mathbb{Q}}/\mathbb{Q})^{ab}$.
The second one, Conway’s site $\widehat{\mathbf{C}}$, is constructed from a
monoid structure on a subset of $\Gamma\backslash PGL_{2}^{+}(\mathbb{Q})$,
classifying projective classes of commensurable integral lattices of rank two,
and is relevant to the description of groups appearing in moonshine, see for
example [5] or [11].

We will study the points of these sites by constructing suitable localic
covers of them. We will construct sheaves on these covers, relevant in the
study of arboreal Galois representations, as in [3] (in the
$\widehat{\mathbf{B}}$ case), and of Bost-Connes data, as introduced in [13]
(in the $\widehat{\mathbf{C}}$ case). Moreover, these three arithmetic sites
are related via a triangle of geometric morphisms

$\textstyle{\widehat{\mathbf{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbf{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbf{A}}}$

##  1\. The arithmetic site and its localic cover

In order to illustrate the strategy we will use later on in studying the Belyi
and Conway sites, let us recall our approach to the original arithmetic site
$\widehat{\mathbf{A}}$, see [9] for more details.

With $\mathbf{A}_{L}$ we denote the poset category of the poset structure on
$\mathbb{N}_{+}^{\times}$ determined by reverse division, that is,
$\mathbf{A}_{L}$ has an object $[k]$ for every $k\in\mathbb{N}_{+}^{\times}$
and a unique morphism
$\textstyle{[k]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[l]}$
whenever $l|k$. The covariant functor $\pi:\mathbf{A}_{L}\rTo\mathbf{A}$
sending the morphism
$\textstyle{[k]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[l]}$
to the endomorphism $\tfrac{k}{l}$ induces a geometric morphism between the
presheaf toposes

$\pi~{}:~{}\widehat{\mathbf{A}_{L}}\rTo\widehat{\mathbf{A}}$

and, as $\mathbf{A}_{L}$ is a poset-category, $\widehat{\mathbf{A}_{L}}$ is a
locale which we call the localic cover of $\widehat{\mathbf{A}}$. In [9] it is
shown that the points of $\widehat{\mathbf{A}_{L}}$ are in one-to-one
correspondence with subsets $S\subseteq\mathbb{N}_{+}^{\times}$ closed under
division and taking least common multiples. The supernatural numbers
$\mathbb{S}$ is the multiplicative semigroup consisting of formal products
$s=\prod_{p\in\mathbb{P}}p^{s_{p}}$ where $p$ runs over all prime numbers
$\mathbb{P}$ and $s_{p}\in\mathbb{N}\cup\\{\infty\\}$, so we have a one-to-one
correspondence

$\mathbf{points}(\widehat{\mathbf{A}_{L}})=\mathbb{S}\qquad S\mapsto
s=\prod_{p\in\mathbb{P}}p^{max(d~{}:~{}p^{d}\in S)}$

We equip $\mathbb{S}$ with the localic topology having as its sets of opens
the subsets

$\mathbb{X}_{l}(\cup_{i}n_{i}\mathbb{N}^{\times}_{+})=\\{s\in\mathbb{S}~{}:~{}\exists
i~{}:~{}n_{i}|s\\}$

By [10, Theorem 1] we know that the locale $\widehat{\mathbf{A}_{L}}$ is
equivalent to the topos of sheaves of sets $\mathbf{Sh}(\mathbb{S},loc)$ of
the topological space $\mathbb{S}$, equipped with the localic topology.

The upshot being that we can associate to every
$S\in\widehat{\mathbf{A}_{L}}$, that is to every contravariant functor
$S~{}:~{}\mathbf{A}_{L}\rTo\mathbf{Sets}$ a sheaf
$\mathcal{S}\in\mathbf{Sh}(\mathbb{S},loc)$ of which the stalk
$\mathcal{S}_{s}$ in an infinite supernatural number
$s\in\mathbb{S}-\mathbb{N}^{\times}_{+}$ extends the functor as
$S([n])=\mathcal{S}_{n}$ for $n\in\mathbb{N}^{\times~{}}_{+}$. Here a couple
of examples, see also [6, Remark 2.10 and §4.5]:

Finite fields:  Fix a prime number $p$ and let $S$ be the functor

$S~{}:~{}\mathbf{A}_{L}\rTo\mathbf{Sets}\qquad\begin{cases}[n]\mapsto\mathbb{F}_{p^{n}}\\\
\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.77896pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{[n]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.77896pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.77896pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{[m]}$}}}}}}}\ignorespaces}}}}\ignorespaces\mapsto\mathbb{F}_{p^{m}}\rInto\mathbb{F}_{p^{n}}\end{cases}$

for the canonical embedding of fields obtained from taking invariants under
the Frobenius. The stalk of the coresponding sheaf $\mathcal{S}$ at a
supernatural number $s\in\mathbb{S}-\mathbb{N}^{\times}_{+}$ is then the
(infinite) algebraic extension

$\mathcal{S}_{s}=\cup_{n|s}\mathbb{F}_{p^{n}}$

It is well known that algebraic extensions of $\mathbb{F}_{p}$ correspond up
to isomorphism to supernatural numbers in this way.

UHF-algebras:  Let $S$ be the functor

$S~{}:~{}\mathbf{A}_{L}\rTo\mathbf{Sets}\qquad\begin{cases}[n]\mapsto
M_{n}(\mathbb{C})\\\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.77896pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{[n]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.77896pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.77896pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{[m]}$}}}}}}}\ignorespaces}}}}\ignorespaces\mapsto
M_{m}(\mathbb{C})\rInto M_{n}(\mathbb{C})\end{cases}$

UHF-algebras are closures of chains of matrix algebras

$M_{n_{1}}(\mathbb{C})\rInto M_{n_{2}}(\mathbb{C})\rInto\ldots$

with $n_{1}|n_{2}|\ldots$, and are known to be classified up to isomorphism by
supernatural numbers, which in our set-up corresponds to taking the stalk
$\mathcal{S}_{s}$ of the corresponding sheaf
$\mathcal{S}\in\mathbf{Sh}(\mathbb{S},loc)$.

The geometric morphism $\pi:\widehat{\mathbf{A}_{L}}\rTo\widehat{\mathbf{A}}$
induces a surjection on the level of points

$\mathbb{S}=\mathbf{points}(\widehat{\mathbf{A}_{L}})\rOnto\mathbf{points}(\widehat{\mathbf{A}})=[\mathbb{S}]$

by modding out all finite supernaturals, that is, $[\mathbb{S}]$ is the set of
classes for the equivalence relation

$s\sim s^{\prime}\quad\Leftrightarrow\quad\exists
n,m\in\mathbb{N}^{\times~{}}_{+}~{}:~{}n.s=m.s^{\prime}$

and it is easy to verify that indeed $[\mathbb{S}]$ coincides with the set of
finite adèle classes
$\mathbb{Q}^{\times}_{+}\backslash\mathbb{A}^{f}/\widehat{\mathbb{Z}}^{*}$.
This equivalence relation implies that the points of the arithmetic site
correspond to stable isomorphism classes for the objects classified by a
functor $S\in\widehat{\mathbf{A}_{L}}$. For example, two algebraic extentions
of $\mathbb{F}_{p}$ (resp. two UHF-algebras) $\mathcal{S}_{s}$ and
$\mathcal{S}_{s^{\prime}}$ are stably isomorphic if there exist
$m,n\in\mathbb{N}^{\times}_{+}$ such that

$\mathcal{S}_{s}\otimes\mathbb{F}_{p^{n}}\simeq\mathcal{S}_{s^{\prime}}\otimes\mathbb{F}_{p^{m}}\qquad(resp.\quad\mathcal{S}_{s}\otimes
M_{n}(\mathbb{C})\simeq\mathcal{S}_{s^{\prime}}\otimes M_{m}(\mathbb{C})~{}).$

It is known that stable isomorphism classes of UHF-algebras are its Morita
equivalence classes. As noncommutative spaces are Morita equivalence classes
of $C^{*}$-algebras, we can view (the points of) the arithmetic site as the
moduli space of the noncommutative spaces corresponding to UHF-algebras.

##  2\. Conway’s site and its localic cover

In [5] John H. Conway introduced his big picture which is a graph with
vertices the left coset classes $\Gamma\backslash PGL_{2}^{+}(\mathbb{Q})$,
where $PGL_{2}^{+}(\mathbb{Q})$ is the group of all elements in
$PGL_{2}(\mathbb{Q})$ with strictly positive determinant, and $\Gamma$ is the
modular group $PSL_{2}(\mathbb{Z})$. We can identify the set $\Gamma\backslash
PGL_{2}^{+}(\mathbb{Q})$ with $\mathbb{Q}_{+}\times\mathbb{Q}/\mathbb{Z}$ via
the assignment

$\mathbb{Q}_{+}\times\mathbb{Q}/\mathbb{Z}\rTo\Gamma\backslash
PGL_{2}^{+}(\mathbb{Q})\qquad
X=(M,\frac{g}{h})\mapsto\Gamma.\begin{bmatrix}M&\frac{g}{h}\\\
0&1\end{bmatrix}=\Gamma.\alpha_{X}$

For two elements $X,Y\in\mathbb{Q}_{+}\times\mathbb{Q}/\mathbb{Z}$ let
$d_{XY}$ be the smallest positive rational number such that
$d_{XY}.\alpha_{X}\alpha_{Y}^{-1}\in GL_{2}(\mathbb{Z})$. The hyper-distance
between the two elements is then

$\delta(X,Y)=det(d_{XY}.\alpha_{X}\alpha_{Y}^{-1})\in\mathbb{N}_{+}$

Conway showed in [5, p. 329] that this definition is symmetric and that the
$log$ of the hyper-distance is a proper distance function on $\Gamma\backslash
PGL_{2}^{+}(\mathbb{Q})$.

The poset-category $\mathbf{C}_{L}$ has as its objects $X\in\Gamma\backslash
PGL_{2}^{+}(\mathbb{Q})$ and a unique arrow $Y\rTo X$ if and only if
$\delta(1,X)\leq\delta(1,Y)$ and $\delta(1,Y)=\delta(1,X).\delta(X,Y)$ where
$1$ is the class of the identity matrix. Conway’s big picture is the graph
derived from $\mathbf{C}_{L}$ by only drawing an edge for each arrow $Y\rTo X$
such that $\delta(X,Y)$ is a prime number. An object $M=(M,0)$ for
$M\in\mathbb{N}^{\times}_{+}$ is called a number-class.

Conway showed that each $X=(M,\frac{g}{h})$ has exactly $p+1$ neighbours in
the big picture at hyper-distance $p$ for any prime number $p$. These are the
classes

$X_{k}=(\frac{M}{p},\frac{g}{hp}+\frac{k}{p}~{}\text{mod}~{}1)~{}\quad\text{for
$0\leq k<p$}~{}\quad~{}\text{and}~{}\quad
X_{p}=(pM,\frac{pg}{h}~{}\text{mod}~{}1)$

It is convenient to consider these $p$-neighbours as the classes corresponding
to the matrices obtained by multiplying $\alpha_{X}$ on the left with the
matrices

$P_{k}=\begin{bmatrix}\frac{1}{p}&\frac{k}{p}\\\
0&1\end{bmatrix}~{}\quad~{}\text{for $0\leq k<p$,
and}~{}\quad~{}P_{p}=\begin{bmatrix}p&0\\\ 0&1\end{bmatrix}$

The classes at hyper-distance a $p$-power from $1$ form a $p+1$-valent tree in
the big picture, and the big picture itself ’factorizes’ as a product of these
$p$-adic trees, see [5, p. 332] or [11, Lemma 3]. That is, the class (or
object) $X$ at hyper-distance $p^{k}$ from $1$ corresponds to a unique product
$P_{i_{b}}.\ldots.P_{i_{2}}.P_{i_{1}}P_{p}^{a}$ of the matrices $P_{i}$ with
$0\leq i<p$ and $P_{p}$ introduced before with $a+b=k$. Here, $p^{a}$ is the
maximal number-class on the unique path in the big picture from $X$ to $1$.
The matrices $P_{i}$ with $0\leq i<p$ generate a free monoid and satisfy the
relation $P_{p}.P_{i}=id$. Matrices $P_{i}$ and $Q_{j}$ corresponding to
different prime numbers $p$ and $q$ satisfy a meta-commutation relation.

######  Lemma 1.

For distinct primes $p$ and $q$ and for all $0\leq i<p$ and $0\leq j<q$ there
exist unique $0\leq k<p$ and $0\leq l<q$ such that

$P_{i}.Q_{j}=Q_{l}.P_{k}\quad\text{and further we have}\quad
P_{p}.Q_{j}=Q_{a}.P_{p}~{}\text{when $a=pj~{}\text{mod}~{}q$}$

###### Proof.

$P_{i}.Q_{j}=\begin{bmatrix}\frac{1}{pq}&\frac{iq+j}{pq}\\\ 0&1\end{bmatrix}$

and as $0\leq iq+j<pq$ we have unique $0\leq k<p$ and $0\leq l<q$ such that
$iq+j=lp+k$. ∎

######  Lemma 2.

An object $X=(M,\frac{g}{h})$ in $\mathbf{C}_{L}$ at hyperdistance
$N=p^{k}\times q^{l}\times\ldots\times r^{s}=\delta(1,X)$ for prime numbers
$p<q<\ldots<r$ can be identified uniquely with a product

$P_{i_{1}}P_{i_{2}}\ldots P_{i_{b}}Q_{j_{1}}Q_{j_{2}}\ldots Q_{j_{d}}\ldots
R_{z_{1}}R_{z_{2}}\ldots R_{z_{v}}P_{p}^{a}Q_{q}^{c}\ldots
R_{r}^{u}=L_{X}.K_{X}$

for unique $0\leq i_{a}<p$, $0\leq j_{b}<q,\ldots,0\leq z_{u}<r$ and with
$a+b=k,c+d=l,\ldots,u+v=s$.

###### Proof.

A path in the big picture from $1$ to $X=(M,\frac{g}{h})$ of minimal length
can be viewed as a product (left multiplication) $X_{l}X_{l-1}\ldots
X_{2}X_{1}$ with each $X_{i}$ one of the matrices $P_{a},Q_{b},\ldots,R_{u}$.
Now use the meta-commutation relations of the previous lemma. ∎

Here, $K_{X}=P_{p}^{a}Q_{q}^{b}\ldots R_{r}^{u}$ is the unique number-class at
minimal hyper-distance from $X=(M,\frac{g}{h})$. For a number-like class
$X=(M,\frac{g}{h})$, that is with $M\in\mathbb{N}^{\times}_{+}$, we have
$K_{X}=Mh$ and $N=Mh^{2}$.

######  Proposition 1.

The points of $\widehat{\mathbf{C}_{L}}$ are in one-to-one correspondence with
equivalence classes of finite or infinite sequences $X_{1}\geq
X_{2}\geq\ldots$ of classes in $\Gamma\backslash PGL_{2}^{+}(\mathbb{Q})$
under the equivalence relation

$X_{1}\geq X_{2}\geq\ldots\sim Y_{1}\geq
Y_{2}\geq\ldots\Leftrightarrow\forall~{}i,\exists~{}j,k~{}:~{}X_{i}\geq
Y_{j},~{}Y_{i}\geq X_{k}$

In particular, the finite points correspond to the objects in
$\mathbf{C}_{L}$.

###### Proof.

The points of the presheaf topos of a poset-category are the upwards closed
subsets of the poset satisfying the condition that it contains for any two of
its elements an element smaller or equal to both. To any sequence $X_{1}\geq
X_{2}\geq\ldots$ we associate the subset $\\{X\in\Gamma\backslash
PGL_{2}^{+}(\mathbb{Q})~{}|~{}\exists i~{}:~{}X\geq X_{i}\\}$. The claims
follow from this. ∎

In [13] M. Marcolli and G. Tabuada developed a broad generalisation of the
classical Bost-Connes system, where roots of unity are replaced by a Bost-
Connes datum $(\Sigma,\sigma_{n})$ consisting of

  1. (1)

an Abelian group $\Sigma$, possibly equipped with a continuous action of a
Galois group $G$, and

  2. (2)

a family of commuting $G$-equivariant group endomorphisms
$\sigma_{n}~{}:~{}\Sigma\rTo\Sigma$ such that
$\sigma_{n}\circ\sigma_{m}=\sigma_{n\times m}$.

An illustrative example is taking $\Sigma=\overline{\mathbb{Q}}^{*}$, the
algebraic numbers with the canonical Galois action, and with $\sigma_{n}$ the
power map $x\mapsto x^{n}$, see [13] for more instances. In many interesting
examples the Bost-Connes datum $(\Sigma,\sigma_{n})$ will satisfy the
following additional properties

  1. (3)

the kernel of $\sigma_{n}$ is the cyclic group of order $n$, and

  2. (4)

there is a family of set-maps $s_{n}~{}:~{}\Sigma\rTo\Sigma$ commuting among
themselves and with the $\sigma_{m}$ such that their restriction to
$\Sigma_{n}=Im(\sigma_{n})$ is a set-theoretic section of the image map

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{n}}$$\textstyle{\Sigma_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{n}}$$\textstyle{0}$

and $\Sigma$ is the disjoint union of $C_{n}$-translates of the subset
$s_{n}(\Sigma_{n})$.

Let us denote the group operation on $\Sigma$ additively, and consider for
each prime number $p$ maps on $\Sigma$ (identified as vectors via
$x\mapsto\begin{bmatrix}x\\\ 1\end{bmatrix}$) by left-multiplication by the
’matrices’ for $0\leq i<p$

$\mathcal{P}_{i}=\begin{bmatrix}s_{p}&x_{i,p}\\\
0&1\end{bmatrix}\qquad\text{and}\qquad\mathcal{P}_{p}=\begin{bmatrix}\sigma_{p}&0\\\
0&1\end{bmatrix}$

where $C_{p}=Ker(\sigma_{p})=\\{0=x_{0,p},x_{1,p},\ldots,x_{p-1,p}\\}$. That
is, we have for all $x\in\Sigma$

$\mathcal{P}_{i}.x=s_{p}(x)+x_{i,p}\quad\text{and}\quad\mathcal{P}_{p}.x=\sigma_{p}(x)$

As in [13], let $\rho_{n}:\Sigma_{n}\rTo P(\Sigma)$ be the map sending an
element $x\in\Sigma_{n}$ to the set of its pre-images under $\sigma_{n}$, then
we have
$\rho_{p}(x)=\\{\mathcal{P}_{0}.x,\mathcal{P}_{1}.x,\ldots,\mathcal{P}_{p-1}.x\\}$.
If we want these operators to satisfy the same meta-commutation relations as
the $P_{i}$ and $Q_{j}$ we have to impose

  1. (5)

for any two prime numbers $p$ and $q$, for all $0\leq i<p$ and $0\leq j<q$ we
have for the unique $0\leq k<p$ and $0\leq l<q$ such that $i.q+j=l.p+k$ that

$s_{p}(x_{j,q})+x_{i,p}=s_{q}(x_{k,p})+x_{i,q}\quad\text{and}\quad\sigma_{p}(x_{j,q})=x_{a,q}$

for $a\equiv p.j~{}mod~{}q$.

######  Proposition 2.

Let $(\Sigma,\sigma_{p},s_{p})$ be a Bost-Connes datum satisfying conditions
$(1)-(5)$, then we have a presheaf $S_{\Sigma}\in\widehat{\mathbf{C}_{L}}$,
defined by assigning to every object $X$ with unique identifier

$P_{i_{1}}P_{i_{2}}\ldots P_{i_{b}}Q_{j_{1}}Q_{j_{2}}\ldots Q_{j_{d}}\ldots
R_{z_{1}}R_{z_{2}}\ldots R_{z_{v}}P_{p}^{a}Q_{q}^{c}\ldots
R_{r}^{u}=L_{X}.K_{X}$

the set

$S_{\Sigma}(X)=\mathcal{P}_{i_{1}}\circ\ldots\circ\mathcal{P}_{i_{b}}\circ\mathcal{Q}_{j_{1}}\circ\ldots\circ\mathcal{Q}_{j_{d}}\circ\ldots\circ\mathcal{R}_{z_{1}}\circ\ldots\circ\mathcal{R}_{z_{v}}(\Sigma_{K_{X}})$

and this defines a sheaf $\mathcal{S}_{\Sigma}$ on
$\mathbf{points}(\widehat{\mathbf{C}_{L}})$ equipped with the localic
topology.

There is a covariant functor $\delta:\mathbf{C}_{L}\rTo\mathbf{A}_{L}$ sending
$X$ to $\delta(1,X)$.

######  Proposition 3.

The fiber $\delta^{-1}(n)$ consists of $\psi(n)=n\prod_{p|n}(1+\frac{1}{p})$
elements where $\psi$ is Dedekind’s function, and there is a canonical
bijection

$\delta^{-1}(n)\qquad\leftrightarrow\qquad\mathbb{P}^{1}_{\mathbb{Z}/n\mathbb{Z}}$

###### Proof.

The stabilizer subgroup of $1$ is $\Gamma$ and any class $X$ at hyperdistance
$n$ maps under $\Gamma$ to another at hyperdistance $n$. Further, the joint
stabilizer of $1=(1,0)$ and $n=(n,0)$ is the mdular group $\Gamma_{0}(n)$. The
canonical bijection then comes from

$\Gamma/\Gamma_{0}(n)\simeq
GL_{2}(\mathbb{Z})/\widetilde{\Gamma}_{0}(n)\simeq\mathbb{P}^{1}_{\mathbb{Z}/n\mathbb{Z}}$

where the right-most bijection comes from considering the action of
$GL_{2}(\mathbb{Z})$ on $\mathbb{A}^{2}_{\mathbb{Z}/n\mathbb{Z}}$ and noticing
that the stabilizer subgroup of the line through $(1,0)$ is
$\widetilde{\Gamma}_{0}(n)$. ∎

The mental image of the geometric morphism
$\delta:\widehat{\mathbf{C}_{L}}\rTo\widehat{\mathbf{A}_{L}}$ might therefore
be that of the structural morphism

$\pi:\mathbb{P}^{1}_{\mathbb{Z}}\rOnto\mathbf{Spec}(\mathbb{Z})$

with the inclusion $\mathbb{N}^{\times}_{+}\rInto C$ given by $n\mapsto(n,0)$
corresponding to the zero section.

We define the Conway monoid $C$, having $p$ generators $P_{0},\ldots,P_{p-1}$
for each prime $p$, with $\\{P_{0},\ldots,P_{p-1}\\}$ generating a free monoid
of rank $p$. For different prime numbers $p$ and $q$ we have the meta-
commutation relations, given by lemma 1 for all $0\leq i<p$ and $0\leq j<q$

$P_{i}\ast Q_{j}=Q_{l}\ast P_{k},\quad\text{if}\quad iq+j=lp+k$

It follows from lemma 2 that the elements of $C$ are in one-to-one
correspondence with the subcategory $\mathbf{C}_{L}^{\prime}$ of
$\mathbf{C}_{L}$ consisting of the objects $X\in\mathbf{C}_{L}$ such that
$1=(1,0)$ is the number-class of minimal distance from $X$, and that the
poset-structure on $\mathbf{C}_{L}^{\prime}$ is related to the monoid product
via

$Y\leq X\quad\Leftrightarrow\quad\exists(!)Z~{}:~{}Y=Z\ast X$

(uniqueness follows because $C$ is left cancellative).

Note that the multiplication is concatenation of minimal paths from $1$. That
is, if $X$ has minimal path $L_{X}$ from $1$ and $Z$ miniimal path $L_{Z}$,
then $Z\ast X$ is the object of $\mathbf{C}_{L}^{\prime}$ one reaches after
walking along path $L_{Z}$ from $X$. If however $X$ and $Z$ are objects in
$\mathbf{C}_{L}$ with number classes $K_{X}$ at minimal distance from $X$ and
$L_{X}$ a minimal path from $K_{X}$ to $X$, and $K_{Z}$ the number class at
minimal distance of $Z$ and $L_{Z}$ a minimal path from $K_{Z}$ to $Z$, one
might consider the product

$Z\ast X=L_{Z}.L_{X}.K_{Z}.K_{X}$

but this product does not always produces an object at hyperdistance
$\delta(1,X)\delta(1,Z)$ from $1$ as $L_{X}$ might start with a term $P_{0}$
for some prime number dividing $K_{Z}$. For example, the only consistent
choice of $P_{0}\ast P_{p}$ would be $1$, which is not at the expected
hyperdistance $p^{2}$. I thank Jens Hemelaer for pointing out this problem.

With $\mathbf{C}$ we denote the corresponding monoid-category, whose presheaf-
topos $\widehat{\mathbf{C}}$ will be Conway’s site. Because the Conway monoid
$C$ is left cancellative, we have a covariant functor
$\pi:\mathbf{C}_{L}^{\prime}\rTo\mathbf{C}$ which assigns to the unique
morphism $Y\rTo X$ between two objects of $\mathbf{C}_{L}^{\prime}$ such that
$Y\leq X$ the unique element $Z\in C$ such that $Y=Z\ast X$.

We will call the points of $\mathbf{C}_{L}^{\prime}$ the non-commutative
supernatural numbers and denote them with $\mathbb{S}_{nc}$. That is,
$\mathbb{S}_{nc}$ is the set of equivalence classes of finite or infinite
products $\ldots\ast A_{2}\ast A_{1}$ in $C$ under the equivalence relation

$\ldots A_{2}\ast A_{1}\sim\ldots B_{2}\ast B_{1}\Leftrightarrow\forall
i,\exists j,k,U,V~{}:~{}\begin{cases}B_{j}\ast\ldots\ast B_{1}=U\ast
A_{i}\ast\ldots\ast A_{1}\\\ A_{k}\ast\ldots\ast A_{1}=V\ast
B_{i}\ast\ldots\ast B_{1}\end{cases}$

This functor induces a surjection on the points of the corresponding presheaf
toposes, giving us that

$\mathbf{points}(\widehat{\mathbf{C}})=[\mathbb{S}_{nc}]$

where the equivalence relations on noncommutative supernatural numbers is
given by restricting to their tails, that is

$[\ldots\ast A_{2}\ast A_{1}]\sim[\ldots\ast B_{2}\ast
B_{1}]\quad\Leftrightarrow\quad\exists i,j~{}:~{}[\ldots\ast A_{i+1}\ast
A_{i}]=[\ldots\ast B_{j+1}\ast B_{j}]$

The hyper-distance $\delta(1,X)$ from $X$ to $1$ gives a morphism of monoids
$\delta:C\rTo\mathbb{N}^{\times}_{+}$ and hence a geometric morphism to the
arithmetic site $\widehat{\mathbf{C}}\rTo\widehat{\mathbf{A}}$.

Connes and Consani equip the arithmetic site $\widehat{\mathbf{A}}$ with a
structure sheaf $\mathbb{Z}_{max}=(\mathbb{Z}\cup\\{-\infty\\},max,+)$ (the
fundamental semi-ring in characteristic $1$, with addition $x\oplus
y=max(x,y)$ and multiplication $x\otimes y=x+y$) in order to restrict the
large group of auto-equivalences of the topos $\widehat{\mathbf{A}}$, coming
from permutations among prime numbers in the monoid $\mathbb{N}^{\times}_{+}$.

The geometric morphism $\widehat{\mathbf{C}}\rTo\widehat{\mathbf{A}}$ can be
seen as an alternative to this rigidification problem. The only monoid
morphism of $\mathbb{N}^{\times~{}}_{+}$ which lifts to a monoid morphism of
the Conway monoid $C$, compatible with the morphism $\delta$, is the identity
map because a class at hyperdistance $p$ must be mapped to another one at
hyperdistance $p$.

##  3\. Belyi’s site and some localic subcovers

A (dynamical) Belyi map is a rational map
$B:\mathbb{P}^{1}_{\overline{\mathbb{Q}}}\rTo\mathbb{P}^{1}_{\overline{\mathbb{Q}}}$
defined over a number field with exactly three ramification points which we
assume to be $\\{0,1,\infty\\}$ (and such that $B(0)=0,B(1)=1$ and
$B(\infty)=\infty$), see for example [1] or [15]. If
$B^{\prime}:\mathbb{P}^{1}_{\overline{\mathbb{Q}}}\rTo\mathbb{P}^{1}_{\overline{\mathbb{Q}}}$
is another dynamical Belyi map, then so is their composition $B^{\prime}\circ
B$, that is, dynamical Belyi maps form a monoid. We will restrict here to the
submonoid of dynamical Belyi polynomials and call it the Belyi monoid
$B_{\overline{\mathbb{Q}}}$ or $B$ if we restrict to polynomials defined over
$\mathbb{Q}$.

To a (dynamical) Belyi map $B$ we associate its (framed) dessin (d’enfant)
which is a bicolored graph $D(B)$ drawn on $\mathbb{P}^{1}_{\mathbb{C}}=S^{2}$
with

  * •

as its black vertices the inverse images $B^{-1}(0)$, (with the vertex $0$
labeled),

  * •

as its white vertices the inverse images $B^{-1}(1)$, (with the vertex $1$
labeled),

  * •

as its edges the inverse images $B^{-1}(0,1)$ of the open interval $(0,1)$.

If $B$ is a (dynamical) Belyi polynomial, then $D(B)$ is a (framed) tree
dessin, that is, a bicolored planar tree.

Two bicolored planar trees $T$ and $T^{\prime}$ are said to be combinatorial
equivalent is there is a color-preserving tree isomorphism

$\gamma:V(T)\rTo V(T^{\prime})\quad\text{and}\quad\gamma:E(T)\rTo
E(T^{\prime})$

preserving the orientation around each vertex. That is, for any vertex $v$ of
$T$ if $c_{v}=(e_{1},\ldots,e_{k})$ is the cyclic order of edges encountered
when walking counter-clockwise around $v$, then the cyclic order of edges
encountered when walking counter-clockwise around $\gamma(v)$ must be
$c_{\gamma(v)}=(\gamma(e_{1}),\ldots,\gamma(e_{k}))$. A framed tree dessin is
a bicolored planar tree such that a black vertex is labeled $0$ and a white
vertex is labeled $1$. Two framed tree dessins $T$ and $T^{\prime}$ are
isomorphic if there is a combinatorial equivalence $\gamma:V(T)\rTo
V(T^{\prime})$ such that $\gamma(0)=0$ and $\gamma(1)=1$.

Let $Aut(\overline{\mathbb{Q}})=\\{L=\alpha x+\beta:\alpha\not=0\\}$ be the
group of non-constant linear polynomials under composition. We say that two
Belyi polynomials $B$ and $B^{\prime}$ are equivalent if $B^{\prime}=B\circ L$
for some $L\in Aut(\overline{\mathbb{Q}})$. G. Shabat proved that
$B\leftrightarrow D(B)$ gives a one-to-one correspondence between equivalence
classes of Belyi polynomials and combinatorial equivalence classes of tree
dessins, see [8, Thm. 2.2.9].

######  Proposition 4.

Every framed tree dessin $T$ is isomorphic to the framed tree dessin $D(B)$ of
a unique (!) dynamical Belyi polynomial $B$.

###### Proof.

Forgetting the framing of $T$ for a moment, there is Belyi polynomial
$P\in\overline{\mathbb{Q}}[x]$ such that $T$ and the dessin $D(P)$ are
combinatorially equivalent by Shabat’s result. Let $\alpha$ and $\beta$ be the
vertices of $D(P)$ corresponding to the labeled vertices $0$ and $1$ of $T$
under the equivalence. Consider

$L=(\beta-\alpha)x+\alpha\in Aut(\overline{\mathbb{Q}})$

then $B=P\circ L$ is a Belyi polynomial such that $B(0)=P(\alpha)=0$ and
$B(1)=P(\beta)=1$, whence $B$ is a dynamical Belyi polynomial with framed tree
dessin isomorphic to $T$. To prove unicity, assume $B^{\prime}$ is another
dynamical Belyi polynomial such that $D(B^{\prime})$ is isomorphic to $D(B)$.
Then, by Shabat’s result there exists an $L=\alpha x+\beta\in
Aut(\overline{\mathbb{Q}})$ such that $B^{\prime}=B\circ L$. But then, the
combinatorial equivalence on vertices is given by

$L~{}:~{}V(D(B^{\prime}))\rTo V(D(B))$

and in particular $L(0)=0$ and $L(1)=1$ whence $\beta=0$ and $\alpha=1$ whence
$B^{\prime}=B$. ∎

######  Example 1.

The bicolored planar tree

can be realised as the tree dessin of the Belyi polynomial (even dynamical
Belyi polynomial) $B=x^{2}(3-2x)$, as $D(B)$ is

$1$$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{0}$

with leftmost white vertex $-\frac{1}{2}$ and rightmost black vertex
$\frac{3}{2}$. There are $3$ more $(b,w)$-couples:
$a=(0,-\frac{1}{2}),b=(\frac{3}{2},1)$ and $c=(\frac{3}{2},-\frac{1}{2})$. The
corresponding linear maps are $L_{a}=-\frac{1}{2}x$,
$L_{b}=-\frac{1}{2}x+\frac{3}{2}$ and $L_{c}=-2x+\frac{3}{2}$, resulting in
the dynamical Belyi polynomials with corresponding framed tree dessins

$\leavevmode\hbox to73.65pt{\vbox
to17.97pt{\pgfpicture\makeatletter\hbox{\hskip 4.91386pt\lower-8.98589pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{4.71387pt}{0.0pt}\pgfsys@curveto{4.71387pt}{2.60342pt}{2.60342pt}{4.71387pt}{0.0pt}{4.71387pt}\pgfsys@curveto{-2.60342pt}{4.71387pt}{-4.71387pt}{2.60342pt}{-4.71387pt}{0.0pt}\pgfsys@curveto{-4.71387pt}{-2.60342pt}{-2.60342pt}{-4.71387pt}{0.0pt}{-4.71387pt}\pgfsys@curveto{2.60342pt}{-4.71387pt}{4.71387pt}{-2.60342pt}{4.71387pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{48.61957pt}{0.0pt}\pgfsys@curveto{48.61957pt}{4.85237pt}{44.68605pt}{8.78589pt}{39.83368pt}{8.78589pt}\pgfsys@curveto{34.98131pt}{8.78589pt}{31.04779pt}{4.85237pt}{31.04779pt}{0.0pt}\pgfsys@curveto{31.04779pt}{-4.85237pt}{34.98131pt}{-8.78589pt}{39.83368pt}{-8.78589pt}\pgfsys@curveto{44.68605pt}{-8.78589pt}{48.61957pt}{-4.85237pt}{48.61957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.33368pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{0}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{24.6307pt}{0.0pt}\pgfsys@curveto{24.6307pt}{2.60342pt}{22.52026pt}{4.71387pt}{19.91684pt}{4.71387pt}\pgfsys@curveto{17.31342pt}{4.71387pt}{15.20297pt}{2.60342pt}{15.20297pt}{0.0pt}\pgfsys@curveto{15.20297pt}{-2.60342pt}{17.31342pt}{-4.71387pt}{19.91684pt}{-4.71387pt}\pgfsys@curveto{22.52026pt}{-4.71387pt}{24.6307pt}{-2.60342pt}{24.6307pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.91684pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{68.5364pt}{0.0pt}\pgfsys@curveto{68.5364pt}{4.85237pt}{64.60289pt}{8.78589pt}{59.75052pt}{8.78589pt}\pgfsys@curveto{54.89815pt}{8.78589pt}{50.96463pt}{4.85237pt}{50.96463pt}{0.0pt}\pgfsys@curveto{50.96463pt}{-4.85237pt}{54.89815pt}{-8.78589pt}{59.75052pt}{-8.78589pt}\pgfsys@curveto{64.60289pt}{-8.78589pt}{68.5364pt}{-4.85237pt}{68.5364pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{59.75052pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{57.25052pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{4.91383pt}{0.0pt}\pgfsys@lineto{15.00291pt}{0.0pt}\pgfsys@moveto{30.84769pt}{0.0pt}\pgfsys@lineto{24.83061pt}{0.0pt}\pgfsys@moveto{50.76448pt}{0.0pt}\pgfsys@lineto{48.81941pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\leftrightarrow\qquad\frac{1}{4}x^{2}(x+3)$

$\leavevmode\hbox to73.65pt{\vbox
to17.97pt{\pgfpicture\makeatletter\hbox{\hskip 8.98589pt\lower-8.98589pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{8.78589pt}{0.0pt}\pgfsys@curveto{8.78589pt}{4.85237pt}{4.85237pt}{8.78589pt}{0.0pt}{8.78589pt}\pgfsys@curveto{-4.85237pt}{8.78589pt}{-8.78589pt}{4.85237pt}{-8.78589pt}{0.0pt}\pgfsys@curveto{-8.78589pt}{-4.85237pt}{-4.85237pt}{-8.78589pt}{0.0pt}{-8.78589pt}\pgfsys@curveto{4.85237pt}{-8.78589pt}{8.78589pt}{-4.85237pt}{8.78589pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{0}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{44.54755pt}{0.0pt}\pgfsys@curveto{44.54755pt}{2.60342pt}{42.4371pt}{4.71387pt}{39.83368pt}{4.71387pt}\pgfsys@curveto{37.23026pt}{4.71387pt}{35.11981pt}{2.60342pt}{35.11981pt}{0.0pt}\pgfsys@curveto{35.11981pt}{-2.60342pt}{37.23026pt}{-4.71387pt}{39.83368pt}{-4.71387pt}\pgfsys@curveto{42.4371pt}{-4.71387pt}{44.54755pt}{-2.60342pt}{44.54755pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{39.83368pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{28.70273pt}{0.0pt}\pgfsys@curveto{28.70273pt}{4.85237pt}{24.76921pt}{8.78589pt}{19.91684pt}{8.78589pt}\pgfsys@curveto{15.06447pt}{8.78589pt}{11.13095pt}{4.85237pt}{11.13095pt}{0.0pt}\pgfsys@curveto{11.13095pt}{-4.85237pt}{15.06447pt}{-8.78589pt}{19.91684pt}{-8.78589pt}\pgfsys@curveto{24.76921pt}{-8.78589pt}{28.70273pt}{-4.85237pt}{28.70273pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.41684pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{64.46439pt}{0.0pt}\pgfsys@curveto{64.46439pt}{2.60342pt}{62.35394pt}{4.71387pt}{59.75052pt}{4.71387pt}\pgfsys@curveto{57.1471pt}{4.71387pt}{55.03665pt}{2.60342pt}{55.03665pt}{0.0pt}\pgfsys@curveto{55.03665pt}{-2.60342pt}{57.1471pt}{-4.71387pt}{59.75052pt}{-4.71387pt}\pgfsys@curveto{62.35394pt}{-4.71387pt}{64.46439pt}{-2.60342pt}{64.46439pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{59.75052pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{59.75052pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{8.98584pt}{0.0pt}\pgfsys@lineto{10.93091pt}{0.0pt}\pgfsys@moveto{34.9197pt}{0.0pt}\pgfsys@lineto{28.90263pt}{0.0pt}\pgfsys@moveto{54.83649pt}{0.0pt}\pgfsys@lineto{44.7474pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\leftrightarrow\qquad\frac{1}{4}x(x-3)^{2}$

$\leavevmode\hbox to77.72pt{\vbox
to17.97pt{\pgfpicture\makeatletter\hbox{\hskip 8.98589pt\lower-8.98589pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{8.78589pt}{0.0pt}\pgfsys@curveto{8.78589pt}{4.85237pt}{4.85237pt}{8.78589pt}{0.0pt}{8.78589pt}\pgfsys@curveto{-4.85237pt}{8.78589pt}{-8.78589pt}{4.85237pt}{-8.78589pt}{0.0pt}\pgfsys@curveto{-8.78589pt}{-4.85237pt}{-4.85237pt}{-8.78589pt}{0.0pt}{-8.78589pt}\pgfsys@curveto{4.85237pt}{-8.78589pt}{8.78589pt}{-4.85237pt}{8.78589pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{0}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{44.54755pt}{0.0pt}\pgfsys@curveto{44.54755pt}{2.60342pt}{42.4371pt}{4.71387pt}{39.83368pt}{4.71387pt}\pgfsys@curveto{37.23026pt}{4.71387pt}{35.11981pt}{2.60342pt}{35.11981pt}{0.0pt}\pgfsys@curveto{35.11981pt}{-2.60342pt}{37.23026pt}{-4.71387pt}{39.83368pt}{-4.71387pt}\pgfsys@curveto{42.4371pt}{-4.71387pt}{44.54755pt}{-2.60342pt}{44.54755pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{39.83368pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{24.6307pt}{0.0pt}\pgfsys@curveto{24.6307pt}{2.60342pt}{22.52026pt}{4.71387pt}{19.91684pt}{4.71387pt}\pgfsys@curveto{17.31342pt}{4.71387pt}{15.20297pt}{2.60342pt}{15.20297pt}{0.0pt}\pgfsys@curveto{15.20297pt}{-2.60342pt}{17.31342pt}{-4.71387pt}{19.91684pt}{-4.71387pt}\pgfsys@curveto{22.52026pt}{-4.71387pt}{24.6307pt}{-2.60342pt}{24.6307pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.91684pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{68.5364pt}{0.0pt}\pgfsys@curveto{68.5364pt}{4.85237pt}{64.60289pt}{8.78589pt}{59.75052pt}{8.78589pt}\pgfsys@curveto{54.89815pt}{8.78589pt}{50.96463pt}{4.85237pt}{50.96463pt}{0.0pt}\pgfsys@curveto{50.96463pt}{-4.85237pt}{54.89815pt}{-8.78589pt}{59.75052pt}{-8.78589pt}\pgfsys@curveto{64.60289pt}{-8.78589pt}{68.5364pt}{-4.85237pt}{68.5364pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{59.75052pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{57.25052pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{8.98584pt}{0.0pt}\pgfsys@lineto{15.00291pt}{0.0pt}\pgfsys@moveto{34.9197pt}{0.0pt}\pgfsys@lineto{24.83061pt}{0.0pt}\pgfsys@moveto{50.76448pt}{0.0pt}\pgfsys@lineto{44.7474pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\leftrightarrow\qquad
x(4x-3)^{2}$

Note that the bicolored planar trees

and

are combinatorial equivalent as they differ by a rotation.

Following [14, §6] we will identify the following parts of a framed tree
dessin $T$:

  1. (1)

The spine $s(T)$ of $T$ is the unique path in $T$ from $0$ to $1$,

  2. (2)

The body $b(T)$ of $T$ is the subtree consisting of the spine and all branches
attached to a vertex in the spine, different from $0$ and $1$,

  3. (3)

The head $h(T)$ of $T$ consists of all branches attached to $0$, except for
the spine ,

  4. (4)

The tail $t(T)$ of $T$ consists of all branches attached to $1$, except for
the spine.

Given framed tree dessins $T$ and $T^{\prime}$ one constructs a new framed
tree dessin $T\circ T^{\prime}$ as follows (see [14, §6]):

  1. (1)

To each black vertex $b$ in $T^{\prime}$ attach a number of heads of $T$ equal
to the valency of $b$, and separate these copies by the edges in $T^{\prime}$
at $b$ .

  2. (2)

To each white vertex $w$ in $T^{\prime}$ attach a number of tails of $T$ equal
to the valency of $w$, and separate these copies by the edges in $T^{\prime}$
at $w$.

  3. (3)

Every edge $e$ in $T^{\prime}$ will be oriented from the black vertex to the
white vertex. Replace $e$ by a copy of the body of $T$ respecting the
orientation, that is, such that the $0$-vertex in $T$ glues to the black
vertex and the $1$-vertex to the white vertex of $e$.

  4. (4)

The framing of $T\circ T^{\prime}$ is inherited from that of $T^{\prime}$.

If $T\simeq D(B)$ and $T^{\prime}\simeq D(B^{\prime})$, then $T\circ
T^{\prime}\simeq D(B\circ B^{\prime})$, which allows us to study the Belyi
monoids $B_{\overline{\mathbb{Q}}}$ and $B$ purely combinatorially by
investigating the monoid structure on framed tree dessins. The absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on (dynamical) Belyi
polynomials and hence on (framed) tree dessins. We recall some Galois
invariants and how they behave under composition.

For a (framed) tree dessin $T$ the passport $p(T)$ is the couple
$(p_{w},p_{b})$ of partitions of $d$ which are the valencies $p_{w}$ (resp.
$p_{b}$) of the white (resp. black) vertices of $T$. The automorphism group
$Aut(T)$ is the group of combinatorial self-equivalences of $T$. This group is
always cyclic and can be viewed as rotations around a central vertex
preserving colours and valencies. The monodromy group $\mu(T)$ is the
(conjugacy class) of the subgroup of $S_{d}$ generated by the permutation
$\alpha$ having as its cycles the cyclic ordering of edges $c_{v}$ for all
black vertices $v$, and $\beta$ with cycles $c_{v}$ for all white vertices of
$T$.

If the passports of the head (minus $0$) of $T$ be $(h_{b},h_{w})$, of the
tail (minus $1$) of $T$ be $(t_{b},t_{w})$, of the body (minus $0$ and $1$) of
$T$ be $(b_{b},b_{w})$. Let the valency at $0$ in $T$ be $i$ and the valency
at $1$ in $T$ be $j$. Then, if the passport of $T^{\prime}$ is
$(p_{b},p_{w})$, and the degree of $B^{\prime}$ is $d$, the passport of
$T\circ T^{\prime}$ is

$p(T\circ T^{\prime})=(h_{b}^{d}\times b_{b}^{d}\times
t_{b}^{d}\times(i.p_{b}),h_{w}^{d}\times b_{w}^{d}\times
t_{w}^{d}\times(j.p_{w}))$

If $(\alpha_{T},\beta_{T})$ is the pair of permutations of $E=E(T)$ giving the
cyclic order of edges around the back (resp. white) vertices of $T$, and let
$(\alpha_{T^{\prime}},\beta_{T^{\prime}})$ be the permutations on the edges
$F=E(T^{\prime})$ of $T^{\prime}$. Let $e_{0}\in E$ be the edge of the spine
of $T$ connected to $0$ and $e_{1}$ the edge of the spine of $T$ connected to
$1$. If we identify $E\times F$ to the edges $E(T\circ T^{\prime})$, then the
permutations giving the cyclic order of edges of $T\circ T^{\prime}$ around
the black (resp. white) vertices are

$\alpha_{T\circ T^{\prime}}(e,f)=\begin{cases}(\alpha_{T}(e),f)~{}\text{if
$e\not=e_{0}$,}\\\ (\alpha_{T}(e),\alpha_{T^{\prime}}(f))~{}\text{if
$e=e_{0}$}\end{cases}$ $\beta_{T\circ
T^{\prime}}(e,f)=\begin{cases}(\beta_{T}(e),f)~{}\text{if $e\not=e_{1}$,}\\\
(\beta_{T}(e),\beta_{T^{\prime}}(f))~{}\text{if $e=e_{1}$}\end{cases}$

######  Proposition 5.

The Belyi monoids $B_{\overline{\mathbb{Q}}}$ and $B$ are both left- and
right-cancellative, that is, for all dynamical Belyi polynomials
$B,B^{\prime},B"$

$B\circ B^{\prime}=B\circ B"\Rightarrow B^{\prime}=B"\quad\text{and}\quad
B^{\prime}\circ B=B"\circ B\Rightarrow B^{\prime}=B"$

In fact, we have the stronger result that if $deg(B)=deg(B_{1})$ and if

$B^{\prime}\circ B=B"\circ B_{1}\Rightarrow
B^{\prime}=B"~{}\text{and}~{}B=B_{1}$

###### Proof.

If $deg(B)=deg(B_{1})$ and $B^{\prime}\circ B=B"\circ B_{1}$, then by [8,
Lemma 2.4.18] there exists $L=\alpha x+\beta\in Aut(\overline{\mathbb{Q}})$
such that $B_{1}=L\circ B=\alpha B+\beta$. But then, from $B(0)=0=B_{1}(0)$
and $B(1)=1=B_{1}(1)$ it follows that $\alpha=1$ and $\beta=0$ and therefore
$B=B_{1}$. For every $c\in\mathbb{C}$ take a $d\in B^{-1}(c)$, then if
$B^{\prime}\circ B=B"\circ B$ we have that $B^{\prime}(c)=B^{\prime}\circ
B(d)=B"\circ B(d)=B"(c)$, whence $B^{\prime}=B"$. ∎

In [1, Prop. 3.1] the following family of dynamical Belyi polynomials defined
over $\mathbb{Q}$ was introduced: for each degree $d$ and all $0\leq k<d$
consider the polynomial

$B_{d,k}=cx^{d-k}(a_{0}x^{k}+\ldots+a_{k-1}x+a_{k})\in\mathbb{Q}[x]$

with

$a_{i}=\frac{(-1)^{k-i}}{d-i}\binom{k}{i}\quad\text{and}\quad
c=\frac{1}{k!}\prod_{j=0}^{k}(d-j)$

Note that $B_{d,0}=x^{d}$ and $B_{d,d-1}=1-(1-x)^{d}$. The corresponding
framed tree dessin $D(B_{d,k})=E_{d,k}$ has vertex $0$ of valency $d-k$,
vertex $1$ of valency $k+1$ and all remaining vertices are leaf vertices. For
example

$E_{8,3}~{}=~{}\leavevmode\hbox to85.69pt{\vbox
to49.66pt{\pgfpicture\makeatletter\hbox{\hskip 21.027pt\lower-24.8307pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} ;
{}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{8.78589pt}{0.0pt}\pgfsys@curveto{8.78589pt}{4.85237pt}{4.85237pt}{8.78589pt}{0.0pt}{8.78589pt}\pgfsys@curveto{-4.85237pt}{8.78589pt}{-8.78589pt}{4.85237pt}{-8.78589pt}{0.0pt}\pgfsys@curveto{-8.78589pt}{-4.85237pt}{-4.85237pt}{-8.78589pt}{0.0pt}{-8.78589pt}\pgfsys@curveto{4.85237pt}{-8.78589pt}{8.78589pt}{-4.85237pt}{8.78589pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{0}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{64.46439pt}{0.0pt}\pgfsys@curveto{64.46439pt}{2.60342pt}{62.35394pt}{4.71387pt}{59.75052pt}{4.71387pt}\pgfsys@curveto{57.1471pt}{4.71387pt}{55.03665pt}{2.60342pt}{55.03665pt}{0.0pt}\pgfsys@curveto{55.03665pt}{-2.60342pt}{57.1471pt}{-4.71387pt}{59.75052pt}{-4.71387pt}\pgfsys@curveto{62.35394pt}{-4.71387pt}{64.46439pt}{-2.60342pt}{64.46439pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{59.75052pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{59.75052pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{44.54755pt}{19.91684pt}\pgfsys@curveto{44.54755pt}{22.52026pt}{42.4371pt}{24.6307pt}{39.83368pt}{24.6307pt}\pgfsys@curveto{37.23026pt}{24.6307pt}{35.11981pt}{22.52026pt}{35.11981pt}{19.91684pt}\pgfsys@curveto{35.11981pt}{17.31342pt}{37.23026pt}{15.20297pt}{39.83368pt}{15.20297pt}\pgfsys@curveto{42.4371pt}{15.20297pt}{44.54755pt}{17.31342pt}{44.54755pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{39.83368pt}{19.91684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{44.54755pt}{-19.91684pt}\pgfsys@curveto{44.54755pt}{-17.31342pt}{42.4371pt}{-15.20297pt}{39.83368pt}{-15.20297pt}\pgfsys@curveto{37.23026pt}{-15.20297pt}{35.11981pt}{-17.31342pt}{35.11981pt}{-19.91684pt}\pgfsys@curveto{35.11981pt}{-22.52026pt}{37.23026pt}{-24.6307pt}{39.83368pt}{-24.6307pt}\pgfsys@curveto{42.4371pt}{-24.6307pt}{44.54755pt}{-22.52026pt}{44.54755pt}{-19.91684pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{-19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{39.83368pt}{-19.91684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} ;
{}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{48.61957pt}{0.0pt}\pgfsys@curveto{48.61957pt}{4.85237pt}{44.68605pt}{8.78589pt}{39.83368pt}{8.78589pt}\pgfsys@curveto{34.98131pt}{8.78589pt}{31.04779pt}{4.85237pt}{31.04779pt}{0.0pt}\pgfsys@curveto{31.04779pt}{-4.85237pt}{34.98131pt}{-8.78589pt}{39.83368pt}{-8.78589pt}\pgfsys@curveto{44.68605pt}{-8.78589pt}{48.61957pt}{-4.85237pt}{48.61957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.33368pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{10.86858pt}{18.9422pt}\pgfsys@curveto{10.86858pt}{21.54562pt}{8.75813pt}{23.65607pt}{6.15471pt}{23.65607pt}\pgfsys@curveto{3.55128pt}{23.65607pt}{1.44084pt}{21.54562pt}{1.44084pt}{18.9422pt}\pgfsys@curveto{1.44084pt}{16.33878pt}{3.55128pt}{14.22833pt}{6.15471pt}{14.22833pt}\pgfsys@curveto{8.75813pt}{14.22833pt}{10.86858pt}{16.33878pt}{10.86858pt}{18.9422pt}\pgfsys@closepath\pgfsys@moveto{6.15471pt}{18.9422pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.15471pt}{18.9422pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{-11.39926pt}{11.70679pt}\pgfsys@curveto{-11.39926pt}{14.31021pt}{-13.5097pt}{16.42065pt}{-16.11313pt}{16.42065pt}\pgfsys@curveto{-18.71655pt}{16.42065pt}{-20.827pt}{14.31021pt}{-20.827pt}{11.70679pt}\pgfsys@curveto{-20.827pt}{9.10336pt}{-18.71655pt}{6.99292pt}{-16.11313pt}{6.99292pt}\pgfsys@curveto{-13.5097pt}{6.99292pt}{-11.39926pt}{9.10336pt}{-11.39926pt}{11.70679pt}\pgfsys@closepath\pgfsys@moveto{-16.11313pt}{11.70679pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.11313pt}{11.70679pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{-11.39926pt}{-11.70679pt}\pgfsys@curveto{-11.39926pt}{-9.10336pt}{-13.5097pt}{-6.99292pt}{-16.11313pt}{-6.99292pt}\pgfsys@curveto{-18.71655pt}{-6.99292pt}{-20.827pt}{-9.10336pt}{-20.827pt}{-11.70679pt}\pgfsys@curveto{-20.827pt}{-14.31021pt}{-18.71655pt}{-16.42065pt}{-16.11313pt}{-16.42065pt}\pgfsys@curveto{-13.5097pt}{-16.42065pt}{-11.39926pt}{-14.31021pt}{-11.39926pt}{-11.70679pt}\pgfsys@closepath\pgfsys@moveto{-16.11313pt}{-11.70679pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.11313pt}{-11.70679pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{10.86858pt}{-18.9422pt}\pgfsys@curveto{10.86858pt}{-16.33878pt}{8.75813pt}{-14.22833pt}{6.15471pt}{-14.22833pt}\pgfsys@curveto{3.55128pt}{-14.22833pt}{1.44084pt}{-16.33878pt}{1.44084pt}{-18.9422pt}\pgfsys@curveto{1.44084pt}{-21.54562pt}{3.55128pt}{-23.65607pt}{6.15471pt}{-23.65607pt}\pgfsys@curveto{8.75813pt}{-23.65607pt}{10.86858pt}{-21.54562pt}{10.86858pt}{-18.9422pt}\pgfsys@closepath\pgfsys@moveto{6.15471pt}{-18.9422pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.15471pt}{-18.9422pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{8.98584pt}{0.0pt}\pgfsys@lineto{30.84769pt}{0.0pt}\pgfsys@moveto{2.77681pt}{8.54611pt}\pgfsys@lineto{4.6362pt}{14.26874pt}\pgfsys@moveto{-7.26973pt}{5.28172pt}\pgfsys@lineto{-12.13766pt}{8.81847pt}\pgfsys@moveto{-7.26973pt}{-5.28172pt}\pgfsys@lineto{-12.13766pt}{-8.81847pt}\pgfsys@moveto{2.77681pt}{-8.54611pt}\pgfsys@lineto{4.6362pt}{-14.26874pt}\pgfsys@moveto{48.81941pt}{0.0pt}\pgfsys@lineto{54.83649pt}{0.0pt}\pgfsys@moveto{39.83344pt}{8.98584pt}\pgfsys@lineto{39.8335pt}{15.00291pt}\pgfsys@moveto{39.83344pt}{-8.98584pt}\pgfsys@lineto{39.8335pt}{-15.00291pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

$E_{d,k}$ has as passport of its head is $(1^{d-k-1},0)$, of the body $(0,0)$
and of the tail $(0,1^{k})$. But then, the passport of $E_{d,k}\circ T$ for a
framed tree dessin $T$ with passport $(p_{b},p_{w})$ and $n$ edges

$p(E_{d,k}\circ T)=(1^{nk}\times(d-k).p_{b},1^{n(d-k-1)}\times(k+1).p_{w})$

######  Proposition 6.

The Belyi monoids $B_{\overline{\mathbb{Q}}}$ and $B$ contain free sub-
monoids:

  1. (1)

If $\\{B_{1},\ldots,B_{k}\\}$ is a set of dynamical Belyi polynomials of fixed
degree $d\geq 2$, then they generate a free sub-monoid of
$B_{\overline{\mathbb{Q}}}$.

  2. (2)

The dynamical Belyi polynomials $\\{B_{d,k}~{}|~{}1\leq k\leq d-2,d\geq 2\\}$
generate a free sub-monoid of $B$.

###### Proof.

$(1)$ follows by induction on the length of words from proposition 5. For
$(2)$ suppose that $B$ and $B^{\prime}$ are dynamical Belyi polynomials of
degrees $n$ and $n^{\prime}$ such that $B_{d,k}\circ
B=B_{d^{\prime},k^{\prime}}\circ B^{\prime}$ for $1\leq k\leq d-2$ and $1\leq
k^{\prime}\leq d^{\prime}-2$. From the passport of $D(B_{d,k}\circ B)$ it
follows that $D(B_{d,k}\circ B)$ has $n\times k$ black leaf-vertices and
$n\times(d-k-1)$ white leaf vertices. This gives us the equalities

$n\times k=n^{\prime}\times
k^{\prime},~{}n\times(d-k-1)=n^{\prime}\times(d^{\prime}-k^{\prime}-1),~{}\text{and}~{}n\times
d=n^{\prime}\times d^{\prime}$

from which $n=n^{\prime}$ follows, but then proposition 5 finishes the proof.
∎

Note that the Belyi monoids have an involution sending $B$ to $(1-x)\circ
B\circ(1-x)$. At the level of the framed tree dessins the involution reverses
the colours of the vertices, swaps the $0$ and $1$ vertices and rotates the
dessin by 180 degrees.

Let $\mathbf{B}_{L}$ (resp. $\mathbf{B}_{\overline{\mathbb{Q}}L}$) be the
poset-category with objects the elements of the Belyi monoid $B$ (resp.
$B_{\overline{\mathbb{Q}}}$) and a unique arrow $B\rTo B^{\prime}$ if and only
if $B\leq B^{\prime}$, that is $B=B^{\prime}\circ B"$ for some dynamical Belyi
polynomial $B"$.

######  Proposition 7.

For a dynamical Belyi polynomial $B\in\overline{\mathbb{Q}}[x]$ let
$\overline{\mathbb{Q}}(B-t)$ be the finite Galois extension of
$\overline{\mathbb{Q}}(t)$ generated by the roots of the polynomial
$B-t\in\overline{\mathbb{Q}}(t)[x]$, then

$S_{geo}~{}:~{}\mathbf{B}_{\overline{\mathbb{Q}}L}\rTo\mathbf{Sets}\qquad
B\mapsto\overline{\mathbb{Q}}(B-t)$

is a presheaf, that is a contravariant functor. For a dynamical Belyi
polynomial $B\in\mathbb{Q}[x]$ let $\mathbb{Q}(B-t)$ be the finite Galois
extension of $\mathbb{Q}(t)$ generated by the roots of the polynomial
$B-t\in\mathbb{Q}(t)[x]$, then

$S_{arith}~{}:~{}\mathbf{B}_{L}\rTo\mathbf{Sets}\qquad
B\mapsto\mathbb{Q}(B-t)$

is a presheaf.

###### Proof.

For $B\rightarrow B^{\prime}$ with $B=B^{\prime}\circ B"$ let $\alpha$ be a
root of $B^{\prime}-t$. Then there is a root $\beta$ of $B-t$ such that
$B"(\beta)=\alpha$. But then

$\overline{\mathbb{Q}}(t)(\alpha)\subseteq\overline{\mathbb{Q}}(t)(\beta)\subseteq\overline{\mathbb{Q}}(B-t)$

As we can repeat this for all roots of $B^{\prime}-t$, we have that
$\overline{\mathbb{Q}}(B^{\prime}-t)$ is a subfield of
$\overline{\mathbb{Q}}(B-t)$. For $B=B^{\prime}\circ B"$ in $\mathbf{B}_{L}$
we can repeat this argument giving us that

$\mathbb{Q}(t)(\alpha)\subseteq\mathbb{Q}(t)(\beta)\subseteq\mathbb{Q}(B-t)$

and therefore $\mathbb{Q}(B^{\prime}-t)$ is a subfield of $\mathbb{Q}(B-t)$. ∎

Recall that the monodromy group of the framed dessin $D(B)$ is isomorphic to
the Galois group $Gal(\overline{\mathbb{Q}}(B-t)/\overline{\mathbb{Q}}(t))$.

Let $D$ be a sub-monoid of either $B$ or $B_{\overline{\mathbb{Q}}}$ and let
$\mathbf{D}_{L}$ be the corresponding poset-category. As in the previous
section, we know that points of $\widehat{\mathbf{D}_{L}}$ correspond to
specific upwards closed subsets of $\mathbf{D}_{L}$. Repeating the argument
here we get that the points correspond to equivalence classes of finite or
infinite compositions $B_{1}\circ B_{2}\circ\ldots$, with all $B_{i}\in D$,
under the relation

$B_{1}\circ B_{2}\circ\ldots\sim B^{\prime}_{1}\circ
B^{\prime}_{2}\circ\ldots\Leftrightarrow\forall i,\exists
j,k~{}:~{}\begin{cases}B_{1}\circ\ldots\circ B_{i}\geq
B^{\prime}_{1}\circ\ldots\circ B^{\prime}_{j}\\\
B^{\prime}_{1}\circ\ldots\circ B^{\prime}_{i}\geq B_{1}\circ\ldots\circ
B_{k}\end{cases}$

Let $P_{D}=\mathbf{points}(\widehat{\mathbf{D}_{L}})$, then the localic
topology of $P_{D}$ has as basis of open sets

$\mathbb{X}_{l}(B)=\\{p=[B_{1}\circ B_{2}\circ\ldots]\in P_{D}~{}|~{}\exists
n~{}:~{}B_{1}\circ\ldots\circ B_{n}\leq B\\}$

and as there are enough points to separate opens we have
$\widehat{\mathbf{D}_{L}}\simeq\mathbf{Sh}(P_{D},loc)$. The stalk of the sheaf
$\mathcal{S}_{geo}$ or $\mathcal{S}_{arith}$ corresponding to the induced
presheaf $S_{geo}$ or $S_{arith}$ on $\mathbf{D}_{L}$ at a point
$p=[B_{1}\circ B_{2}\circ\ldots]$ is

$(\mathcal{S}_{geo})_{p}=\cup_{n}\overline{\mathbb{Q}}(B_{1}\circ\ldots\circ
B_{n}-t)\quad\text{and}\quad(\mathcal{S}_{arith})_{p}=\cup_{n}\mathbb{Q}(B_{1}\circ\ldots\circ
B_{n}-t)$

If $D$ is a free sub-monoid on $n$ generators (see e.g. proposition 6), then
the points of $\widehat{\mathbf{D}_{L}}$ are in one-to-one correspondence with
the finite or infinite paths from the root vertex in the complete rooted
$n$-tree. The localic topology in this case has a basis of opens $U_{\gamma}$
consisting of all paths containing a fixed finite path $\gamma$.

We will relate the above to the theory of arboreal Galois representations, see
for example [7]. For fixed $d$ let $T_{n}$ be the complete $d$-ary rooted tree
of level $n$. Let $Aut(T_{n})$ be the automorphism group of $T_{n}$ which is a
subgroup of $S_{d^{n}}$ after choice of labelling of the leaf-vertices.
Clearly, $Aut(T_{1})\simeq S_{d}$ and by induction

$Aut(T_{n})\simeq Aut(T_{n-1})\wr Aut(T_{1})$

whence $Aut(T_{n})$ is a subgroup of the $n$-fold iterated wreath product of
$S_{d}$ by itself.

Let $P\in\mathbb{Q}[x]$ of degree $d$, and $\alpha\in\mathbb{Q}$ be such that
for all $n$ we have that

$P^{(n)}-\alpha=\underbrace{P\circ\ldots\circ P\circ P}_{n}-\alpha$

has $d^{n}$ distinct roots in $\overline{\mathbb{Q}}$. Label the vertices of
$T_{n}$ as follows: the root vertex will be $\alpha$, the vertices in the
first layer will be the $d$ roots of $P-\alpha$, those in the second layer the
roots of $P^{(2)}-\alpha$ grouped together so that their images under $P$ map
to the same root of $P-\alpha$, and so on until the leaf-vertices which are
the roots of $P^{(n)}-\alpha$ grouped together such that their images under
$P$ map to the same root of $P^{(n-1)}-\alpha$. Let $K_{n}$ be the Galois
extension of $\mathbb{Q}$ generated by the roots of $P^{(n)}-\alpha$, then the
action of $Gal(K_{n}/\mathbb{Q})$ gives a morphism

$\phi_{n}~{}:~{}Gal(K_{n}/\mathbb{Q})\rInto Aut(T_{n})$

If $T_{\infty}$ is the infinite complete $d$-ary rooted tree, this procedure
results in a continuous morphism

$\phi_{P}~{}:~{}Gal(\overline{\mathbb{Q}}/\mathbb{Q})\rOnto
G_{\infty,P,\alpha}=\underset{\infty\leftarrow
n}{lim}~{}Gal(K_{n}/\mathbb{Q})\rTo Aut(T_{\infty})$

which is called the arboreal Galois representation determined by $P$ and
$\alpha$.

By a result of Jones and Pink, [7, Thm. 3.1], the index
$[T_{\infty}:G_{\infty,P,\alpha}]$ is infinite for post-critically finite
polynomials $P$, hence also for all dynamical Belyi polynomials
$B\in\mathbb{Q}[x]$. In [2] the first explicit description of the arboreal
representation of a cubic polynomial was given for $P=-2x^{3}+3x^{2}=B_{3,2}$
and this was extended in [3] to all rational Belyi polynomials $B_{d,k}$
described above.

Let $D$ be the (free) sub-monoid generated by all dynamical Belyi polynomials
$B_{i}\in\mathbb{Q}[x]$ of degree $d$ (among which are the $B_{d,k}$ for
$0\leq k<d$). Let $\alpha\in\mathbb{Q}\cap(0,1)$ be generic for the finite
number of $B_{i}$, that is $B_{i}(\alpha)\not=0,1$. Let $\mathcal{S}_{\alpha}$
be the sheaf on $(P_{D},loc)$ corresponding to the contravariant functor

$S_{\alpha}~{}:~{}\mathbf{D}_{L}\rTo\mathbf{Sets}\qquad
B\mapsto\mathbb{Q}(B-\alpha)$

######  Proposition 8.

The stalk of the sheaf $\mathcal{S}_{\alpha}$ in the point $p=B_{i_{1}}\circ
B_{i_{2}}\circ\ldots$ is an infinite Galois extension $K_{p}$ of $\mathbb{Q}$
giving rise to a generalised arboreal Galois representation of the absolute
Galois group

$\phi_{p}~{}:~{}Gal(\overline{\mathbb{Q}}/\mathbb{Q})\rOnto
Gal(K_{p}/\mathbb{Q})\rTo Aut(T_{\infty})$

where the $n$-th layer of $T_{\infty}$ is labeled by the $d^{n}$ distinct
roots of $B_{i_{1}}\circ\ldots\circ B_{i_{n}}-\alpha$, grouped together in
$d$-tuples having as their image under $B_{i_{n}}$ the same root of
$B_{i_{1}}\circ\ldots\circ B_{i_{n-1}}-\alpha$. In particular, for
$p=B_{i}\circ B_{i}\circ\ldots=B_{i}^{(\infty)}$ we recover the arboreal
Galois representation corresponding to the dynamical Belyi polynomial $B_{i}$.

Because the Belyi monoids $B$ and $B_{\overline{\mathbb{Q}}}$ are right-
cancellative we have a covariant functor
$\pi:\mathbf{B}_{(\overline{\mathbb{Q}})L}\rTo\mathbf{B}_{(\overline{\mathbb{Q}})}$
which assigns to the unique morphism $B\rTo B^{\prime}$ between two objects of
$\mathbf{B}_{(\overline{\mathbb{Q}})L}$ such that $B\leq B^{\prime}$ the
unique element $B"\in B_{(\overline{\mathbb{Q}})}$ such that
$B=B^{\prime}\circ B"$. This functor induces a surjection on the points of the
corresponding presheaf toposes giving us that

$\mathbf{points}(\widehat{B_{(\overline{\mathbb{Q}})}})=[P_{B_{(\overline{\mathbb{Q}})}}]$

where the equivalence relation on points is given by restricting to their
tails.

The degree function $d$ of a dynamical Belyi polynomial gives a morphism of
monoids $d:B\rTo\mathbb{N}^{\times}_{+}$ and
$d:B_{\overline{\mathbb{Q}}}\rTo\mathbb{N}^{\times}_{+}$, and therefore
geometric morphisms to the arithmetic site

$\widehat{\mathbf{B}}\rTo\widehat{\mathbf{A}}\quad\text{and}\quad\widehat{\mathbf{B}_{\overline{\mathbb{Q}}}}\rTo\widehat{\mathbf{A}}$

Remains to link the Belyi-site(s) to the Conway-site. The approach below is
inspired by [12, Prop. 2.25]. Let $B$ and $B^{\prime}$ be two dynamical Belyi
polynomials, then we have seen that the number of black vertices of $D(B\circ
B^{\prime})$, that is,

$\\#(B\circ
B^{\prime})^{-1}(0)=deg(B^{\prime})(\\#B^{-1}(0)-1)+\\#(B^{\prime})^{-1}(0).$

But then, the map

$\beta~{}:~{}B_{(\overline{\mathbb{Q}})}\rTo C\qquad
B\mapsto\begin{bmatrix}\frac{1}{deg(B)}&\frac{\\#B^{-1}(0)-1}{deg(B)}\\\
0&1\end{bmatrix}$

is a morphism of monoids, with image the sub-monoid of $C$ generated by the
$P_{i}$ with $0\leq i<p$ for all prime numbers $p$ (look at the images of the
$B_{d,k}$). Further, the hyperdistance $\delta(\beta(B),1)=deg(B)$, so the
monoid map $\beta$ induces a geometric morphism between the sites giving rise
to the commuting triangle

$\textstyle{\widehat{\mathbf{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{\beta}$$\textstyle{\widehat{\mathbf{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\widehat{\mathbf{A}}}$

## References

  * [1] Jacqueline Anderson, Irene Bouw, Ozlem Ejder, Neslihan Girgin, Valentijn Karemaker and Michelle Manes, Dynamical Belyi maps, arXiv:1703.08563 (2017)
  * [2] Robert L. Benedetto, Xander Faber, Benjamin Hutz, Jamie Juul and Yu Yasufuku, A large arboreal Galois representation for a cubic postcritically finite polynomial, arXiv:1612.03358 (2016)
  * [3] Irene I. Bouw, Özlem Ejder and Valentijn Karemaker, Dynamical Belyi maps and arboreal Galois groups, arXiv:1811.10086 (2018)
  * [4] Alain Connes and Caterina Consani, The Arithmetic Site, Le Site Arithmétique, arXiv:1405.4527 (2014)
  * [5] John H. Conway, Understanding groups like $\Gamma_{0}(N)$, In ”Groups, difference sets, and the Monster”, Walter de Gruyter (1996)
  * [6] Jens Hemelaer, Azumaya toposes, arXiv:1707.03814 (2017)
  * [7] Rafe Jones, Galois representations from pre-image trees: an arboreal survey, Publ. Math. Besançon Algèbre Théorie (2013)
  * [8] Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of mathematical sciences, Vol. 141, Springer (2004)
  * [9] Lieven Le Bruyn, The sieve topology on the arithmetic site, Journal of Algebra and Applications (2016)
  * [10] Lieven Le Bruyn, Covers of the arithmetic site, arXiv:1602.01627v2 (2017)
  * [11] Lieven Le Bruyn, The monstrous moonshine picture, arXiv:1804.04127 (2018)
  * [12] Yuri I. Manin and Matilde Marcolli, Quantum statistical mechanics of the absolute Galois group, arXiv:1907.13545v1 (2019)
  * [13] Matilde Marcolli and Goncalo Tabuada, Bost-Connes systems, categorification, quantum statistical mechanics, and Weil numbers, arXiv:1411.3223 (2014)
  * [14] George Shabat and Alexander Zvonkin, Plane trees and algebraic numbers, AMS Contemporary Mathematics 178 (1994) 233-275
  * [15] Melanie Wood, Belyi-extending maps and the Galois action on dessins d’enfant, arXiv:0304489v2 (2005)