Let $\Gamma$ be a finitely generated self-simulable group and suppose
$N\leqslant\Gamma$ is a finite index subgroup. Then $N$ is self-simulable.

###### Proof.

Let $N\curvearrowright X$ be an effectively closed action. As $N$ has finite
index in $\Gamma$, we can find a representative of the left cosets
$L=\\{\ell_{0}=1_{\Gamma},\ell_{1},\dots,\ell_{k}\\}$ of $N$ in $\Gamma$.
Every $g\in\Gamma$ can be written uniquely as a product $g=\ell h$ with
$\ell\in L$ and $h\in N$.

Consider the space $X[L]$ of formal sums
$\ell_{0}x_{0}+\ell_{1}x_{1}+\dots+\ell_{k}x_{k}$ with $x_{i}\in X$ for $0\leq
i\leq k$. Notice that if $X\subseteq A^{\mathbb{N}}$, then we can formally
identify $X[L]$ with a subset of $(A^{L})^{\mathbb{N}}$, we shall use this
space to obtain the classical action of $\Gamma$ induced from a subgroup of
finite index. Indeed, we define $\Gamma\curvearrowright X[L]$ through

$g(\ell_{0}x_{0}+\ell_{1}x_{1}+\dots+\ell_{k}x_{k})=\ell^{\prime}_{0}(n_{0}x_{0})+\ell^{\prime}_{1}(n_{1}x_{1})+\dots+\ell^{\prime}_{k}(n_{k}x_{k}).$

Where for every $0\leq i\leq k$, $\ell^{\prime}_{i}\in L$ and $n_{i}\in N$ is
the unique pair of elements such that $\ell^{\prime}_{i}n_{i}=g\ell_{i}$.
Notice that the map
$(\ell_{0},\dots,\ell_{k})\overset{g}{\mapsto}(\ell^{\prime}_{0},\dots,\ell^{\prime}_{k})$
is a permutation, and thus the formal sum in the right is well defined. It is
clear that $\Gamma\curvearrowright X[L]$ is indeed an action. Let us argue
that it is effectively closed. Notice that if $S$ is a set of generators for
$N$, then $S\cup L$ is a finite generating set for $\Gamma$. For each $s\in
S\cup L$, we can find pairs $(\ell^{\prime}_{i,s},w_{i,s})_{0\leq i\leq k}$
where $\ell^{\prime}_{i,s}\in L$ and $w_{i,s}\in S^{*}$ are such that
$s\ell_{i}=\ell^{\prime}_{i,s}\underline{w_{i,s}}$ and then use the Turing
machine which computes $N\curvearrowright X$ to compute
$\underline{w_{i,s}}x_{i}$. As there are finitely many generators, the pairs
$(\ell^{\prime}_{i,s},w_{i,s})_{0\leq i\leq k}$ can be hard-coded in a Turing
machine and thus we conclude that $\Gamma\curvearrowright X[L]$ is effectively
closed.

As $\Gamma$ is self-simulable, there exists a $\Gamma$-subshift of finite type
$Z$ which factors onto $\Gamma\curvearrowright X[L]$. Observe that the
$N$-subaction of a $\Gamma$-subshift of finite type is topologically
conjugated to an $N$-subshift of finite type whenever $N$ has finite index in
$\Gamma$ (this is usually called the higher-block shift. The proof of this
last claim can be found in [17, Proposition 7]). It follows that there is a
$N$-subshift of finite type which factors onto the $N$-subaction of
$\Gamma\curvearrowright X[L]$. Composing this factor map with the projection
of $X[L]$ to the $\ell_{0}$-coordinate we obtain that $N\curvearrowright X$ is
a factor of this $N$-subshift of finite type, and thus $N$ is self-simulable.
∎

######  Lemma 7.2.

Let $\Gamma$ be a finitely generated group and suppose
$N\trianglelefteq\Gamma$ is a finite index normal subgroup which is self-
simulable. Then $\Gamma$ is self-simulable.

Notice that if $\Gamma$ is recursively presented, this follows immediately
from Corollary 6.12 without the finite index assumption.

###### Proof.

Let $\Gamma\curvearrowright X$ be an effectively closed action. As $N$ is a
finitely generated subgroup of $\Gamma$, it follows that the $N$-subaction
$N\curvearrowright X$ of $\Gamma\curvearrowright X$ is also effectively
closed. As $N$ is self-simulable, there exists an $N$-subshift of finite type
$Y$ which factors onto $N\curvearrowright X$ through a map $\phi\colon Y\to
X$.

Let $L\ni 1_{\Gamma}$ be a finite set of representatives of left cosets of $N$
such that $\Gamma=LN$. Let $B$ be the alphabet of $Y$ and consider the
$\Gamma$-subshift $Z$ consisting on the configurations
$z\in(B\cup\\{\varnothing\\})^{\Gamma}$ which satisfy the following two
constraints:

  1. (1)

If $z(g)\in B$, then $(z(gn))_{n\in N}\in Y$.

  2. (2)

For every $g\in\Gamma$, there is a unique $\ell^{-1}\in L^{-1}$ such that
$z(g\ell^{-1})\neq\varnothing$.

As $Y$ is a subshift of finite type and $L^{-1}$ is finite, it follows that
$Z$ is a subshift of finite type. Moreover, for every $y\in Y$ we can
construct $z\in Z$ with $z|_{N}=y$ by letting

$z(g)=\begin{cases}y(g)&\mbox{ if }g\in N\\\ \varnothing&\mbox{ if }g\notin
N.\end{cases}$

By definition, there is a well-defined map $\psi\colon Z\to L$ such that
$z|_{\psi(z)N}\in Y$. We define $\widetilde{\phi}\colon Z\to X$ by

$\widetilde{\phi}(z)=\psi(z)\phi((\psi(z)^{-1}z)|_{N})\mbox{ for every }z\in
Z.$

Where the inner product is the shift map, and the outer product is the action
$\Gamma\curvearrowright X$. We claim $\widetilde{\phi}$ is a factor map. It is
clearly continuous and surjective. To see that $\widetilde{\phi}$ is
$\Gamma$-equivariant, suppose first that $n\in N$. Then we have that
$\psi(nz)=\psi(z)$ and therefore we obtain

$\displaystyle\widetilde{\phi}(nz)$
$\displaystyle=\psi(nz)\phi((\psi(nz)^{-1}nz)|_{N})$
$\displaystyle=\psi(z)\phi((\psi(z)^{-1}n\psi(z)\psi(z)^{-1}z)|_{N})$
$\displaystyle=\psi(z)(\psi(z)^{-1}n\psi(z))\phi((\psi(z)^{-1}z)|_{N})$
$\displaystyle=n\widetilde{\phi}(z).$

Now let $\ell\in L$ and notice that there is $n\in N$ such that $\psi(\ell
z)=\ell\psi(z)n$. We can therefore write

$\displaystyle\widetilde{\phi}(\ell z)$ $\displaystyle=\psi(\ell
z)\phi((\psi(\ell z)^{-1}\ell z)|_{N})$
$\displaystyle=\ell\psi(z)n\phi((n^{-1}\psi(z)^{-1}\ell^{-1}\ell z)|_{N})$
$\displaystyle=\ell\psi(z)\phi((\psi(z)^{-1}z)|_{N})$
$\displaystyle=\ell\widetilde{\phi}(z).$

As $L$ and $N$ generate $\Gamma$, we conclude that $\widetilde{\phi}$ is
$\Gamma$-equivariant and thus a factor map. Hence $\Gamma$ is self-simulable.
∎

Now we are ready to prove Theorem 1.4, namely, that if $\Gamma_{1}$ and
$\Gamma_{2}$ are finitely generated commensurable groups. Then $\Gamma_{1}$ is
self-simulable if and only if $\Gamma_{2}$ is self-simulable.

###### Proof of Theorem 1.4.

Suppose that $\Gamma_{1}$ is self-simulable. As $\Gamma_{1}$ and $\Gamma_{2}$
are commensurable, there are finite index subgroups $H_{1}\leqslant\Gamma_{1}$
and $H_{2}\leqslant\Gamma_{2}$ which are isomorphic. As $H_{1}$ has finite
index in $\Gamma_{1}$, we can find a finite index subgroup $N_{1}\leqslant
H_{1}$ such that $N_{1}$ is finitely generated and normal in $\Gamma_{1}$.
Then by Lemma 7.1 we get that $N_{1}$ is self-simulable. As $N_{1}$ is normal
in $H_{1}$, by Lemma 7.2 we get that $H_{1}$ is self-simulable. As $H_{1}$ and
$H_{2}$ are isomorphic, we obtain that $H_{2}$ is self-simulable. Similarly,
we extract a finite index subgroup $N_{2}\leqslant H_{2}$ which is normal in
$\Gamma_{2}$. By Lemma 7.1 we get that $N_{2}$ is self-simulable and by Lemma
7.2 we get that $\Gamma_{2}$ is self-simulable. ∎

Let us recall that a function $f\colon X\to Y$ between metric spaces
$(X,d_{X})$ and $(Y,d_{Y})$ is a quasi-isometry if there exist real constants
$A\geq 1$ and $B,C\geq 0$ such that

  1. (1)

$f$ is a quasi-isometric embedding: for every $x,y\in X$

$\frac{1}{A}d_{X}(x,y)-B\leq d_{Y}(f(x),f(y))\leq Ad_{X}(x,y)+B.$

  2. (2)

$f(X)$ is relatively dense: for every $z\in Y$ there is $x\in X$ such that
$d(z,f(x))\leq C$.

Let $\Gamma$ be a finitely generated group generated by $S\Subset\Gamma$. For
$g\in\Gamma$, let $|g|_{S}$ be the minimum length of a word $w\in S^{*}$ such
that $g=\underline{w}$. We have that $d_{S}(g,h)=|g^{-1}h|_{S}$ is a metric on
$\Gamma_{1}$ called the word metric. Notice that if two finitely generated
groups are quasi-isometric with respect to the word metrics induced by a
specific choice of generating sets, then they are quasi-isometric with respect
every choice of generating sets. In what follows, when referring to quasi-
isometries of finitely generated groups we implicitly consider the group as a
metric space with respect to some word metric.

The rest of the section is dedicated to the proof of Theorem 1.5, namely, that
for finitely presented groups the property of being self-simulable is stable
under quasi-isometries. By Proposition 3.2, we already know that Theorem 1.5
holds trivially for amenable groups and thus in our proof we may assume that
both groups involved are non-amenable. This will greatly simplify our proof
due to a result of Whyte [50, Theorem 4.1] which implies that every quasi-
isometry between non-amenable finitely generated groups is at bounded distance
from a bilipschitz map.

For the remainder, fix two non-amenable finitely presented groups $\Gamma_{1}$
and $\Gamma_{2}$, an effectively closed action $\Gamma_{1}\curvearrowright X$
and a bilipschitz map $f\colon\Gamma_{2}\to\Gamma_{1}$ such that
$f(1_{\Gamma_{2}})=1_{\Gamma_{1}}$. We will assume that $\Gamma_{2}$ is self-
simulable and prove the existence of a subshift of finite type which factors
onto $\Gamma_{1}\curvearrowright X$.

######  Lemma 7.3.

There exist finite symmetric generating sets $S\Subset\Gamma_{1}$ and
$T,U\Subset\Gamma_{2}$ such that $1_{\Gamma_{1}}\in S$, $1_{\Gamma_{2}}\in
T\cap U$ and

  1. (1)

$\Gamma_{1}$ can be presented by $S$ with relations of length at most $3$.

  2. (2)

$\Gamma_{2}$ can be presented by $T$ with relations of length at most $3$.

  3. (3)

$f(hT)\subseteq f(h)S$ for every $h\in\Gamma_{2}$.

  4. (4)

$f(h)S\subseteq f(hU)$ for every $h\in\Gamma_{2}$.

###### Proof.

Let us fix symmetric generating sets $S_{1}$ and $S_{2}$ of $\Gamma_{1}$ and
$\Gamma_{2}$ respectively. As $f$ is a bilipschitz map, it is a bijection and
there is $D\geq 1$ such that

$\frac{1}{D}d_{S_{2}}(h_{1},h_{2})\leq d_{S_{1}}(f(h_{1}),f(h_{2}))\leq
Dd_{S_{2}}(h_{1},h_{2}),$

As $\Gamma_{2}$ is finitely presented, there is a finite set of words
$W_{2}\subseteq S_{2}^{*}$ which presents $\Gamma_{2}$. Let $K_{2}\geq 1$ be
the largest length of a word in $W_{2}$ and take $T$ as the symmetric closure
of the set of all words on $S_{2}^{*}$ of length at most $K_{2}$. Take $R_{2}$
as the set of all words on $T$ such that

  * •

if $w\in W_{2}$, then $w\in R_{2}$.

  * •

for every $s\in S_{2}$ and $t_{1},t_{2}\in T$ such that ${t_{1}s}={t_{2}}$
then $t_{1}st_{2}^{-1}\in R_{2}$.

It is clear that $R_{2}$ presents $\Gamma_{2}$ with the generating set $T$ and
thus we have (2).

Now let $h\in\Gamma_{2}$ and $t\in T$. notice that
$d_{S_{2}}(h,ht)=|t|_{S_{2}}$ and thus $d_{S_{1}}(f(h),f(ht))\leq
D|t|_{S_{2}}$. This means that we have $f(ht)\in f(h)S_{1}^{\lfloor
DK_{2}\rfloor}$ for every $t\in T$ and thus $f(hT)\subseteq f(h)S_{1}^{\lfloor
DK_{2}\rfloor}$.

As $\Gamma_{1}$ is finitely presented, there is a finite set of words
$W_{1}\subseteq S_{1}^{*}$ which presents $\Gamma_{1}$. Let $K_{1}\geq 1$ be
the largest length of a word in $W_{1}\cup S_{1}^{\lfloor DK_{2}\rfloor}$.
Letting $S$ be the symmetric closure of the set of all words on $S_{1}$ of
length at most $K_{1}$ we conclude, in the same way as with $T$ and
$\Gamma_{2}$, that $\Gamma_{1}$ is presented by words of length at most $3$ in
$S$. By the choice of $K_{1}$, it also follows that for every
$h\in\Gamma_{2}$, $f(hT)\subseteq f(h)S_{1}^{\lfloor DK_{2}\rfloor}\subseteq
f(h)S$ and so (1) and (3) hold.

Finally, let $s\in S$ and $h\in\Gamma_{2}$ such that $f(h)s=f(hu)$ for some
$u\in\Gamma_{2}$. It follows that $d_{S_{1}}(f(h),f(hu))=|s|_{S_{1}}\leq
K_{1}$. Then we have that

$d_{S_{2}}(h,hu)\leq DK_{1}$

Therefore $|u|_{S_{2}}\leq DK_{1}$ Letting $U$ be the symmetric closure of the
union of $T$ with all the words of length at most $DK_{1}$ we obtain (4).∎

Let $S,T,U$ be sets satisfying the conditions of Lemma 7.3 and let
$Y\subseteq(S^{T})^{\Gamma_{2}}$ be the set of configurations $y$ which
satisfy the following cocycle-like conditions:

  1. (1)

For every $h\in\Gamma_{2}$ and $t\in T$ we have

$y(h)(1_{\Gamma_{2}})=1_{\Gamma_{1}}\mbox{ and
}y(h)(t^{-1})=(y(ht^{-1})(t))^{-1}.$

  2. (2)

For every $h\in\Gamma_{2}$ and $t_{1},t_{2},t_{3}\in T$ such that
${t_{3}t_{2}t_{1}}=1_{\Gamma_{2}}$ we have

$\mbox{ if }\left[y(ht_{1}^{-1}t_{2}^{-1})(t_{3}^{-1})=s_{3}\mbox{ and
}y(ht_{1}^{-1})(t_{2}^{-1})=s_{2}\mbox{ and
}y(h)(t_{1}^{-1})=s_{1}\right]\mbox{ then }{s_{1}s_{2}s_{3}}=1_{\Gamma_{1}}.$

From the definition above, it follows that $Y$ equipped with the left shift
action $\Gamma_{2}\curvearrowright Y$ is a $\Gamma_{2}$-subshift of finite
type. It is nonempty as a valid configuration may be defined by letting
$y(h)(t)=f(h)^{-1}f(ht)$ for every $h\in\Gamma_{2}$ and $t\in T$.

Consider the map $\gamma\colon T\times Y\times X\to Y\times X$ given by

$\gamma(t,(y,x))=(ty,(y(1_{\Gamma_{2}})(t^{-1}))^{-1}x)\mbox{ for every }t\in
T\mbox{ and }({y},x)\in{Y}\times X.$

The term $ty$ in the expression above is given by the left shift action, and
the second term is given by the action $\Gamma_{1}\curvearrowright X$.

######  Lemma 7.4.

The map $\gamma$ induces a left action
$\Gamma_{2}\overset{\gamma}{\curvearrowright}Y\times X$ which is topologically
conjugate to an effectively closed left action of $\Gamma_{2}$.

###### Proof.

Let us first argue that $\gamma$ induces an action of $\Gamma_{2}$. As $T$ is
a set of generators of $\Gamma_{2}$, it suffices to show that every relation
in a presentation by $T$ (as an abstract monoid) yields the trivial map. As
$\Gamma_{2}$ can be presented by relators of length at most $3$, the
conditions on configurations of $Y$ ensure that this condition holds,
therefore $\gamma$ induces a left action of $\Gamma_{2}$ on $Y\times X$.

The left shift action $\Gamma_{2}\curvearrowright Y$ is a subshift of finite
type and thus an effectively closed subshift (by patterns). As $\Gamma_{2}$ is
finitely presented, by Proposition 2.18, $\Gamma_{2}\curvearrowright Y$ is
topologically conjugated to an effectively closed action
$\Gamma_{2}\curvearrowright\widetilde{Y}\subseteq(S^{T})^{\mathbb{N}}$ through
a map $\phi\colon\widetilde{Y}\to Y$. Let us note that, by the construction
given in the proof of Proposition 2.18, the map $\phi$ can be chosen (by
letting $\rho(0)$ be the empty word in $T^{*}$) such that
$\widetilde{y}_{0}=\phi(\widetilde{y})(1_{\Gamma_{2}})$ for any
$\widetilde{y}\in\widetilde{Y}$. It follows that the action
$\Gamma_{2}\overset{\gamma}{\curvearrowright}Y\times X$ is topologically
conjugated to the action
$\Gamma_{2}\overset{\gamma}{\curvearrowright}\widetilde{Y}\times X$ given by

$t(\widetilde{y},x)=(t\widetilde{y},\widetilde{y}_{0}(t^{-1})^{-1}x)\mbox{ for
every }t\in\Gamma_{2}\mbox{ and }(\widetilde{y},x)\in\widetilde{Y}\times X.$

Finally, this action is effectively closed because both
$\Gamma_{2}\curvearrowright\widetilde{Y}$ and $\Gamma_{1}\curvearrowright X$
are effectively closed and $\widetilde{y}_{0}(t^{-1})^{-1}$ can be effectively
computed from $\widetilde{y}$ and $t$. ∎

Let us now suppose that $\Gamma_{2}$ is self-simulable. Then there exists a
finite alphabet $A$, a $\Gamma_{2}$-subshift of finite type $Z\subseteq
A^{\Gamma_{2}}$ and a topological factor map $\phi_{2}\colon Z\to Y\times X$
such that $\Gamma_{2}\overset{\gamma}{\curvearrowright}Y\times X$ is the
factor of the left shift action $\Gamma_{2}\curvearrowright Z$ under
$\phi_{2}$.

Without loss of generality (by recoding $Z$) we may assume that $Z$ is
described by a nearest neighbor set of forbidden patterns. In particular,
there is $\mathcal{L}\subseteq A^{T}$ such that $z\in Z$ if and only if
$(hz)|_{T}\in\mathcal{L}$ for every $h\in\Gamma_{2}$.

Furthermore, as $\phi_{2}$ is continuous, if $\phi_{2}(z)=(y,x)$ then the
symbol $y(1_{\Gamma_{2}})$ depends only on finitely many coordinates of $Z$.
We may further recode $Z$ so that this information is contained in the
coordinate $z(1_{\Gamma_{2}})$, that is, such that there is a function
$\Psi\colon A\to S^{T}$ with the property that if $\phi_{2}(z)=(y,x)$ then
$y(h)=\Psi(z(h))$ for every $h\in\Gamma_{2}$. Note that $\Psi$ extends to a
$1$-block factor map $\psi\colon Z\to Y$.

In what follows, we shall describe a $\Gamma_{1}$-subshift of finite type $W$
which encodes configurations of $Z$ on $\Gamma_{1}$ using configurations that
encode the space of bilipschitz maps which resemble $f$. Our description is
essentially based on a construction of Cohen [20] which describes a process to
encode the structure of a finitely presented group into another quasi-
isometric group through a subshift of finite type.

###  7.1. Construction of $W$

Consider the alphabet $A$ which defines $Z$ and recall there is a map
$\Psi\colon A\to S^{T}$. The subshift of finite type $W$ is the set of
configurations $w\in A^{\Gamma_{1}}$ which satisfy the following constraints.

Constraint 1:  For every $g\in\Gamma_{1}$ and $t,t_{1},t_{2},t_{3}\in T$ such
that ${t_{3}t_{2}t_{1}}=1_{\Gamma_{2}}$, we have

  1. (1)

$\Psi(w(g))(1_{\Gamma_{2}})=1_{\Gamma_{1}}$.

  2. (2)

$\Psi(w(g))(t)\Psi(g\Psi(w(g))(t))(t^{-1})=1_{\Gamma_{1}}$.

  3. (3)

$\Psi(w(g)(t_{3}))\Psi(g\Psi(w(g)(t_{3})))(t_{2})\Psi(g\Psi(g\Psi(w(g)(t_{3})))(t_{2}))(t_{1})=1_{\Gamma}$.

While from this definition it is clear that this condition can be implemented
using finitely many forbidden patterns, its meaning is not immediately clear
and it is quite painful to parse. Given $w\in A^{\Gamma_{1}}$, we may define a
map $*_{w}\colon\Gamma_{1}\times T\to\Gamma_{1}$ given by
$g*_{w}t=g\Psi(w(g))(t)$. In terms of this action the three conditions above
translate in following more intuitive presentation.

  1. (1)

$*_{w}1_{\Gamma_{2}}=\operatorname{id}_{\Gamma_{1}}$.

  2. (2)

$*_{w}{t}\circ*_{w}{t^{-1}}=\operatorname{id}_{\Gamma_{1}}$.

  3. (3)

$*_{w}{t_{3}}\circ*_{w}{t_{2}}\circ*_{w}{t_{1}}=\operatorname{id}_{\Gamma_{1}}$.

Therefore this first constraint encodes the fact that this map extends to a
well-defined right action
$\Gamma_{1}\overset{*_{w}}{\curvearrowleft}\Gamma_{2}$ for every $w\in W$. For
$h\in\Gamma_{2}$, we will write $*_{w}h$ to denote this action.

Constraint 2:  For every $g\in\Gamma_{1}$ and every $s\in S$ there exists
$u\in U$ such that

$g*_{w}u=gs.$

This constraint encodes the fact that the actions
$\Gamma_{1}\overset{*_{w}}{\curvearrowleft}\Gamma_{2}$ are transitive. Indeed,
for $g_{0},g\in\Gamma_{1}$ there are $s_{1}\dots s_{n}\in S^{*}$ such that
$g=g_{0}s_{1}\dots s_{n}$. Let $g_{i}=g_{0}s_{1}\dots s_{i}$. By constraint 4,
there is $u_{i}\in U$ such that $g_{i-1}*_{w}u_{i}=g_{i-1}s_{i}=g_{i}$ Letting
$h={u_{0}\dots u_{n-1}}$ we obtain that $g_{0}*_{w}h=g_{n}=g$.

Given $g\in\Gamma_{1}$ we can extract from $w$ a configuration $\chi(w,g)\in
A^{\Gamma_{2}}$ given by $\chi(w,g)(h)=w(g*_{w}h)$. This configuration
satisfies the following two useful equations. For every
$g,g^{\prime}\in\Gamma_{1}$ and $h\in\Gamma_{2}$, we have

$h^{-1}\chi(w,g)=\chi(w,g*_{w}h)\mbox{ and
}\chi(gw,g^{\prime})=\chi(w,g^{-1}g^{\prime}).$

Recall that there is a finite set of patterns $\mathcal{L}\subseteq A^{T}$
such that $z\in Z$ if and only if for every
$h\in\Gamma_{2},(gz)|_{T}\in\mathcal{L}$.

Constraint 3:  For every $g\in\Gamma_{1}$, we have that
$\chi(w,g)|_{T}\in\mathcal{L}$.

This constraint encodes the fact that $\chi(w,g)\in Z$. Indeed, for every
$h\in\Gamma_{2}$ we have
$h^{-1}\chi(w,g)|_{T}=\chi(w,g*_{w}h)|_{T}\in\mathcal{L}$ and thus
$\chi(w,g)\in Z$.

From the considerations above, it is clear that $W$ is a $\Gamma_{1}$-subshift
of finite type.

###  7.2. Construction of the factor map

Let us recall we have a factor map $\phi_{2}\colon Z\to Y\times X$. Let
$\zeta$ denote the projection of $\phi_{2}$ onto the second coordinate, that
is, if $\phi_{2}(z)=(y,x)$ then $\zeta(z)=x$.

Recall we have constructed a $\Gamma_{1}$-subshift of finite type $W$ whose
configurations satisfy constraints (1)-(3). We define $\phi_{1}\colon W\to X$
by

$\phi_{1}(w)=\zeta(\chi(w,1_{\Gamma_{1}})).$

We claim that $\phi_{1}$ is a topological factor map. This will be proven in
the following three claims.

######  Claim 7.5.

The map $\phi_{1}$ is continuous.

###### Proof.

If $h\in\Gamma_{2}$, then $\chi(w,1_{\Gamma_{1}})(h)=w(1_{\Gamma_{1}}*_{w}h)$
and thus it depends upon the values of $w$ on the set $S^{|h|_{T}}$. We
conclude that $\phi_{1}$ is continuous. Furthermore, $\phi_{2}$ is continuous,
we have that $\zeta$ is continuous and thus $\phi_{1}$ is continuous. ∎

######  Claim 7.6.

The map $\phi_{1}$ is $\Gamma_{1}$-equivariant.

###### Proof.

As $\Gamma_{1}$ is generated by $S$, it suffices to show that for every $s\in
S$ and $w\in W$, we have $s\phi_{1}(w)=\phi_{1}(sw)$.

By condition 2, that there is $u\in U$ such that
$1_{\Gamma_{1}}*_{w}u=s^{-1}$. From here, we obtain that

$\chi(sw,1_{\Gamma_{1}})=\chi(w,s^{-1})=\chi(w,1_{\Gamma_{1}}*_{w}u)=u^{-1}\chi(w,1_{\Gamma}).$

Using that $\phi_{2}$ is $\Gamma_{2}$-equivariant, we get

$\phi_{2}(\chi(sw,1_{\Gamma}))=u^{-1}\phi_{2}(\chi(w,1_{\Gamma_{1}})).$

Let us introduce a bit of notation. Let $t_{1}\dots t_{n}\in T^{*}$ such that
$u=\underline{t_{1}\dots t_{n}}$. Let $\ell_{0}=1_{\Gamma_{1}}$,
$\ell_{n}=s^{-1}$ and for $1\leq j<n$ define $\ell_{j}$ as the element such
that $\ell_{0}*_{w}{t_{1}\dots t_{j}}=\ell_{j}$.

Let us write $(y,x)=\phi_{2}(\chi(w,1_{\Gamma_{1}}))$. By the previous
equation we have that $\phi_{2}(\chi(sw,1_{\Gamma_{1}}))=u^{-1}(y,x)$. By
definition of the action $\Gamma_{2}\curvearrowright Y\times X$, we have that

$u^{-1}(y,x)=(u^{-1}y,{\left(y(t_{1}t_{2}\dots
t_{n-1})(t_{n})\right)^{-1}\dots\left(y(t_{1})(t_{2})\right)^{-1}\left(y(1_{\Gamma_{1}})(t_{1})\right)^{-1}}x).$

Recall that $\Psi\colon A\to S^{T}$ extends to a $1$-block factor map
$\psi\colon Z\to Y$ and thus $y=\psi(\chi(w,1_{\Gamma_{1}}))$. Therefore we
obtain that for every $h\in\Gamma_{2}$ we have

$y(h)=h^{-1}y(1_{\Gamma_{2}})=\psi(h^{-1}\chi(w,1_{\Gamma_{1}}))(1_{\Gamma_{2}})=\Psi(\chi(w,1_{\Gamma_{1}}*_{w}h)(1_{\Gamma_{2}})).$

In particular, for every $1\leq j\leq n$, then

$\displaystyle y(t_{1}t_{2}\dots t_{j-1})(t_{j})$
$\displaystyle=\Psi(\chi(w,(1_{\Gamma_{1}}*_{w}t_{1}t_{2}\dots
t_{j-1})(1_{\Gamma_{2}}))(t_{j})$
$\displaystyle=\Psi(\chi(w,\ell_{j-1})(1_{\Gamma_{2}}))(t_{j})$
$\displaystyle=\ell_{j-1}^{-1}\ell_{j}.$

From this equation, we obtain that

$u^{-1}(y,x)=(u^{-1}y,(\ell_{n-1}^{-1}\ell_{n})^{-1}(\ell_{n-2}^{-1}\ell_{n-1})^{-1}\dots(\ell_{0}^{-1}\ell_{1})^{-1}x)=(uy,\ell_{n}^{-1}\ell_{0}x)=(u^{-1}y,sx).$

Therefore

$\phi_{1}(sw)=\zeta(u^{-1}\chi(w,1_{\Gamma_{1}}))=s\zeta(\chi(w,1_{\Gamma_{1}}))=s\phi_{1}(w).$

From where we obtain that $\phi_{1}$ is indeed $\Gamma_{1}$-equivariant. ∎

Notice that up to this point, we have not used $f$ in any meaningful way other
that to determine the constant $D$. As far as the definition goes, the actions
induced by elements of $w$ might not be free. The next claim proving
surjectivity of $\phi_{1}$ relies strongly on the existence of this particular
bilipschitz map $f$.

######  Claim 7.7.

The map $\phi_{1}$ is surjective.

###### Proof.

Let $x\in X$ be arbitrary. Let us first define $y\in S^{T}$ by

$y(h)(t)=f(h)^{-1}f(ht).$

By Lemma 7.3 part (3) we have that $y\in S^{T}$. It is clear by definition
that $y(h)(1_{\Gamma_{2}})=1_{\Gamma_{1}}$. Furthermore, for every $t$ we have

$(y(ht^{-1})(t))^{-1}=(f(ht^{-1})^{-1}f(ht^{-1}t))^{-1}=f(h)^{-1}f(ht^{-1})=y(h)(t^{-1}).$

Finally, for $t_{1},t_{2},t_{3}\in T$ such that
${t_{3}t_{2}t_{1}}=1_{\Gamma_{2}}$ we have

$\displaystyle y(h)(t_{1}^{-1})$
$\displaystyle=\underbracket{f(h)^{-1}f(ht_{1}^{-1})}_{s_{1}},$ $\displaystyle
y(ht_{1}^{-1})(t_{2}^{-1})$
$\displaystyle=\underbracket{f(ht_{1}^{-1})^{-1}f(ht_{1}^{-1}t_{2}^{-1})}_{s_{2}}$
$\displaystyle y(ht_{1}^{-1}t_{2}^{-1})(t_{3}^{-1})$
$\displaystyle=\underbracket{f(ht_{1}^{-1}t_{2}^{-1})^{-1}f(ht_{1}^{-1}t_{2}^{-1}t_{3}^{-1})}_{s_{3}}.$

As ${t_{1}^{-1}t_{2}^{-1}t_{3}^{-1}}=1_{\Gamma_{2}}$ we verify that

${s_{1}s_{2}s_{3}}=f(h)^{-1}f(ht_{1}^{-1}t_{2}^{-1}t_{3}^{-1})=f(h)^{-1}f(h)=1_{\Gamma_{1}}.$

Therefore we conclude that $y\in Y$, and thus $(y,x)\in Y\times X$. Let $z\in
Z$ such that $\phi_{2}(z)=(y,x)$, it suffices to construct $w\in W$ such that
$\chi(w,1_{\Gamma_{1}})=z$, as then we would have
$\phi_{1}(w)=\zeta(\chi(w,1_{\Gamma_{1}}))=\zeta(z)=x$.

Let $w\in A^{\Gamma_{1}}$ be given by $w(g)=z(f^{-1}(g))$ for every
$g\in\Gamma_{1}$. Notice that for every $t\in T$ and $g\in\Gamma_{1}$, we have

$g*_{w}t=g\Psi(w(g))(t)=g\Psi(z(f^{-1}(g)))(t)=g(f(f^{-1}(g))^{-1}f(f^{-1}(g)t)=f(f^{-1}(g)t).$

As $T$ is a generating set, we deduce that for any $h\in\Gamma_{2}$ then
$g*_{w}h=f(f^{-1}(g)h)$.

From this we deduce that constraint $1$ holds on $w$. Indeed, let
$g\in\Gamma_{1}$ be arbitrary. For the first condition we have

$g*_{w}1_{\Gamma_{1}}=f(f^{-1}(g)1_{\Gamma_{1}})=g.$

For the second condition, let $t\in T$. We have

$\displaystyle g*_{w}t*_{w}t^{-1}$ $\displaystyle=f(f^{-1}(g)t)*_{w}t^{-1}$
$\displaystyle=f(f^{-1}(g)tt^{-1})=g.$

For the third condition, let $t_{1},t_{2},t_{3}\in T$ such that
$t_{3}t_{2}t_{1}=1_{\Gamma_{2}}$ we have

$\displaystyle g*_{w}t_{3}*_{w}t_{2}*_{w}t_{1}$
$\displaystyle=f(f^{-1}(g)t_{3})*_{w}t_{2}*_{w}t_{1}$
$\displaystyle=f(f^{-1}(g)t_{3}t_{2})*_{w}t_{1}$
$\displaystyle=f(f^{-1}(g)t_{3}t_{2}t_{1})=g.$

By Lemma 7.3 part (4), we get that for every $h\in\Gamma_{2}$ we have
$f(h)S\subseteq f(hU)$ therefore for every $s\in S$ and $g\in\Gamma_{1}$ we
obtain that $gs$ can be written as $f(f^{-1}(g)u)=g*_{w}u$ for some $u\in U$
and thus constraint 2 holds.

Finally, recall that we have asked that $f(1_{\Gamma_{2}})=1_{\Gamma_{1}}$. By
construction we have that for every $h\in\Gamma_{2}$,

$\chi(w,1_{\Gamma_{1}})(h)=w(1_{\Gamma}*h)=w(f(f^{-1}(1_{\Gamma_{1}})h))=w(f(h))=z(h).$

We conclude that $z=\chi(w,1_{\Gamma_{1}})$, and in particular that constraint
$3$ holds and thus $w\in W$. This shows that $\phi_{1}$ is surjective.∎

We have shown that $W$ is a $\Gamma_{1}$-subshift of finite type and in Claims
7.5, 7.6 and 7.7 that $\phi_{1}\colon W\to X$ is a topological factor map. As
$\Gamma_{1}\curvearrowright X$ was an arbitrary effectively closed action, we
obtain that $\Gamma_{1}$ is self-simulable. This completes the proof of
Theorem 1.5.

##  8\. Examples and applications

In this section we provide several handy versions of the theorems in the last
sections which enables us to show that several well known non-amenable groups
in the literature are self-simulable. We show that self-simulable groups with
decidable word problem admit nonempty strongly aperiodic subshifts of finite
type and that their sunny-side up subshift is sofic.

###  8.1. Branch groups

An immediate consequence of Theorem 1.4 is that whenever a group is
commensurable to a product of two finitely generated non-amenable groups, then
it must be self-simulable. An interesting class of groups for which a slighter
weaker property holds is the class of branch groups. An extensive survey about
these groups can be found in [11].

We will not provide the definition of these groups here and just refer the
reader to [11, Definition 1.1]. The sole property of these groups we will use,
and which can be deduced directly from the definition, is that any branch
group $\Gamma$ admits a finite index subgroup $H$ which is isomorphic to the
direct product of at least two isomorphic copies of a group $L$.

######  Corollary 8.1.

Let $\Gamma$ be a finitely generated branch group. If $\Gamma$ is non-amenable
then $\Gamma$ is self-simulable. The converse holds whenever $\Gamma$ is
recursively presented.

###### Proof.

Suppose $\Gamma$ is non-amenable, then the finite index subgroup $H$ is non-
amenable (and finitely generated). From here it follows that $L$ must be non-
amenable and finitely generated, and thus $H$ is isomorphic to the direct
product of at least two finitely generated non-amenable groups. By Theorem 1.2
it follows that $H$ is self-simulable. As $\Gamma$ is commensurable with $H$
we obtain that $\Gamma$ is self-simulable by Theorem 1.4.

The converse is a direct consequence of Proposition 3.2. ∎

###  8.2. Burger-Mozes groups

A construction of Burger and Mozes [15] provides a class of finitely presented
simple groups which act geometrically on the direct product of two infinite
simplicial trees with constant bounded degree $\geq 3$. By the Švarc-Milnor
lemma, these groups are quasi-isometric to the direct product $F_{2}\times
F_{2}$, which is self-simulable by Theorem 1.2. Putting this fact together
with Theorem 1.5 yields the following corollary.

######  Corollary 8.2.

The groups of Burger and Mozes are self-simulable. In particular, there exist
finitely presented non-amenable simple groups which are self-simulable.

A reader interested in the groups by Burger and Mozes, will find plenty of
information in the following chapter of Bartholdi [10].

###  8.3. Thompson’s groups

Next, we provide a handy tool for showing that a group is self-simulable, by
simply producing a sufficiently rich faithful action on a zero-dimensional
space.

######  Lemma 8.3.

Let $\Gamma$ be a finitely generated and recursively presented group that acts
faithfully on a zero-dimensional space $X$ by homeomorphisms, such that for
each nonempty open set $U\subseteq X$, the subgroup of elements of $\Gamma$
which act trivially on $X\setminus U$ is non-amenable. Then $\Gamma$ is self-
simulable.

###### Proof.

For an open set $U\subseteq X$, write $\Gamma_{U}$ for the subgroup of
elements which act trivially on $X\setminus U$. As $X$ is zero-dimensional, we
can find two nonempty disjoint clopen sets $A,B$ such that $X=A\cup B$. As
$\Gamma\curvearrowright X$ is faithful, it follows that
$\Gamma_{A}\cap\Gamma_{B}=\\{1_{\Gamma}\\}$ and thus they commute and form a
direct product inside $\Gamma$, As both $\Gamma_{A}$ and $\Gamma_{B}$ are non-
amenable, they contain finitely generated non-amenable groups $\Delta_{A}$ and
$\Delta_{B}$, and we thus have that the finitely generated group
$\Delta_{A}\times\Delta_{B}$ embeds into $\Gamma$. Let $T$ be a finite
generating set for $\Gamma$ and for each $t\in T$ pick some arbitrary point
$x_{t}\in A$. We have $tx_{t}\in C_{t}$ where $C_{t}\in\\{A,B\\}$. As $\Gamma$
acts by homeomorphisms, there’s a clopen set $U_{t}$ which contains $x_{t}$
such that $U_{t}\subseteq A$ and $tU_{t}\subseteq C_{t}$.

By our hypothesis, $\Gamma_{tU_{t}}$ is non-amenable and thus we can pick a
finitely generated non-amenable group $\Sigma_{t}$ inside $\Gamma_{tU_{t}}$.
Thus we have that $\Sigma_{t}\leqslant\Gamma_{tU_{t}}\leqslant\Gamma_{C_{t}}$
and $(\Sigma_{t})^{t}\leqslant\Gamma_{U_{t}}\leqslant\Gamma_{A}$. Let
$\widetilde{\Delta}_{A}$ be the group generated by $\Delta_{A}$, the groups
$\Sigma_{t}$ for $t\in T$ and the groups $(\Sigma_{t})^{t}$ for those $t$ such
that $C_{t}=A$. Let $\widetilde{\Delta}_{B}$ be the group generated by
$\Delta_{B}$ and the groups $(\Sigma_{t})^{t}$ for those $t$ such that
$C_{t}=B$.

By definition, the groups $\widetilde{\Delta}_{A}$ and
$\widetilde{\Delta}_{B}$ are finitely generated. As they contain respectively
$\Delta_{A}$ and $\Delta_{B}$, they are non-amenable and hence by Theorem 1.2,
the group $\widetilde{\Delta}_{A}\times\widetilde{\Delta}_{B}$ is self-
simulable. Furthermore, we have that
$\widetilde{\Delta}_{A}\times\widetilde{\Delta}_{B}\leqslant\Gamma_{A}\times\Gamma_{B}\leqslant\Gamma$.
As $\widetilde{\Delta}_{A}\times\widetilde{\Delta}_{B}$ contains both
$\Sigma_{t}$ and $(\Sigma_{t})^{t}$ for every $t\in T$, we conclude that it is
mediated and thus $\Gamma$ is self-simulable by Theorem 1.3. ∎

The Thompson’s groups are three finitely presented groups, commonly denoted by
$F$, $T$ and $V$, which were introduced by R. Thompson as potential
counterexamples for the now disproved Von Neumann-Day conjecture. The group
$V$ was, to the best of our knowledge, the first example of an infinite and
finitely presented simple group. The canonical reference for these groups are
the notes [16].

The group $V$ can be described as a subgroup of homeomorphisms of the Cantor
space $C=\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$. An element $\phi$ of $V$
can be described by fixing two finite partitions
$C=[w_{1}]\cup[w_{2}]\cup\dots\cup[w_{n}]$ and
$C=[u_{1}]\cup[u_{w}]\cup\dots\cup[u_{n}]$, where
$w_{i},u_{i}\in\\{\mathtt{0},\mathtt{1}\\}^{*}$ for every
$i\in\\{1,\dots,n\\}$. Then $\phi$ is the homeomorphism on $C$ which replaces
the prefixes $w_{i}$ by $u_{i}$, that is, $\phi(w_{i}z)=u_{i}z$ for every
$z\in\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$. This is usually described in
terms of “tree pairs” made of carets and a finite permutation, see for
instance [22]. It is well-known that $V$ is non-amenable, a short proof being
that the natural action of $V$ on $\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$
does not admit any Borel $V$-invariant probability measure, as every cylinder
would necessarily have the same measure (and thus if $V$ were amenable it
would contradict the Krylov-Bogolyubov theorem, see for instance [33, Theorem
4.4]).

Brin-Thomson’s group $nV$ was constructed in [14] as a higher dimensional
analogue of Thompson’s $V$ group which consists of homeomorphisms on $C^{n}$
where $C=\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$ is the Cantor set. We will
not provide a precise definition of $nV$, but only use the following two
simple facts: (1) $nV$ is a finitely generated and recursively presented group
(in fact, it is finitely presented and has a decidable word problem), (2) $nV$
is non-amenable, and (3) if we fix any cylinder set
$D=[w_{1}]\times[w_{2}]\times\cdots[w_{n}]$ for non-trivial
$w_{i}\in\\{\mathtt{0},\mathtt{1}\\}^{*}$ then the subgroup of $nV$ which acts
trivially on $C^{n}\setminus D$ is isomorphic to $nV$.

######  Corollary 8.4.

Thompson’s $V$, and more generally the higher Brin-Thompson groups $nV$ for
$n\geq 1$, are self-simulable.

###### Proof.

Consider the natural faithful action of $nV\curvearrowright C^{n}$ by
homeomorphisms, where $C=\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$ is the
Cantor set. By property (3) above, for every nonempty open set $U$ the
restriction of $nV$ to elements which act trivially on $C^{n}\setminus U$
contains a copy of $nV$, thus is non-amenable by property (2). Since $nV$ is
recursively presented by property (1), Lemma 8.3 applies. ∎

The natural action $V\curvearrowright\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$
is expansive. Therefore the previous result has the somewhat funny corollary
that the natural action of $V$ on $\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$
is “sofic”, in the sense that it is topologically conjugate to a sofic
$V$-subshift. Of course, this is (presumably) unrelated to the open question
of whether $V$ itself is a sofic group in the sense of Gromov [26].

The groups $F$ and $T$ are more commonly described as subgroups of
homeomorphisms of $[0,1]$ and $\mathbb{R}/\mathbb{Z}$ respectively. More
precisely, $F$ (resp. $T$) is the group of homeomorphisms of $[0,1]$ (resp.
$\mathbb{R}/\mathbb{Z}$) which consists of piecewise linear maps whose slopes
are powers of $2$ and the endpoints of each linear part are dyadic. Of course,
these two groups can equivalently be described as homeomorphisms of
$C=\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$. In the case of $F$, it is the
subgroup of $V$ where the homeomorphisms are defined by partitions
$C=[w_{1}]\cup[w_{2}]\cup\dots\cup[w_{n}]$ and
$C=[u_{1}]\cup[u_{w}]\cup\dots\cup[u_{n}]$ where the sequences $(w_{i})_{1\leq
i\leq n}$ and $(u_{i})_{1\leq i\leq n}$ are both given in lexicographical
order. $T$ is defined similarly but the sequence $(u_{i})_{1\leq i\leq n}$ is
allowed to be in lexicographical order up to a cyclic permutation.

It is well known that $T$ is non-amenable (the same proof of non-existence of
a Borel $V$-invariant measure works for $T$), but the status of the
amenability of $F$ is a famous open problem. In what follows, we shall
establish a link between the amenability of $F$ and self-simulability of $F$
and $T$.

######  Corollary 8.5.

Thompson’s $F$ is self-simulable if and only if $F$ is non-amenable. If $F$ is
non-amenable then Thompson’s $T$ is self-simulable.

###### Proof.

Suppose that $F$ is amenable, as $F$ is finitely presented it follows by
Proposition 3.2 that there exists an effectively closed action of $F$ which is
not the factor of any subshift. Thus $F$ is not self-simulable.

Conversely, suppose $F$ is non-amenable, let $\Gamma$ denote either $F$ or $T$
and consider the natural faithful action of $\Gamma$ on
$\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$ by homeomorphisms. For every non-
trivial word $w\in\\{\mathtt{0},\mathtt{1}\\}^{*}$ the subgroup of $\Gamma$
which fixes $\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}\setminus[w]$ is
isomorphic to $F$ (this subgroup consists of the elements of $\Gamma$ which
act as the identity on any element of $[0,1]$ (resp. $\mathbb{R}/\mathbb{Z}$)
whose binary expansion does not begin with $w$). By Lemma 8.3, we conclude
that $\Gamma$ is self-simulable. ∎

###  8.4. General linear groups

We will now show that the general linear groups
$\operatorname{GL}_{n}(\mathbb{Z})$ are self-simulable in high enough
dimension. Let us first introduce some helpful notation.

Let $\Gamma$ be a finitely generated group which acts faithfully on a
Cartesian product $A_{1}\times A_{2}\times\cdots\times A_{n}$. For
$I\subseteq\\{1,2,\dots,n\\}$, denote by $\Gamma_{I}$ the subgroup of all
elements $g\in\Gamma$ such that for some bijection $h\colon\prod_{i\in
I}A_{i}\to\prod_{i\in I}A_{i}$, we have
$g(a_{1},a_{2},\dots,a_{n})=(b_{1},b_{2},\dots,b_{n})$ where $b_{j}=a_{j}$ if
$j\notin I$, and $b_{j}=h((a_{i})_{i\in I})_{j}$ otherwise.

######  Lemma 8.6.

Let $\Gamma$ be a finitely generated and recursively presented group which
acts faithfully on a Cartesian product $A_{1}\times A_{2}\times\cdots\times
A_{n}$. Suppose $n\geq 5$ and $\Gamma$ is generated by the subgroups
$\Gamma_{I}$ for $I\subseteq\\{1,2,...,n\\}$ with $|I|=2$, which are all non-
amenable and finitely-generated. Then $\Gamma$ is self-simulable.

###### Proof.

Split $N=\\{1,2,\dots,n\\}$ into two disjoint sets $J$, $J^{\prime}$ both with
cardinality at least two. As the action is faithful, we have
$\Gamma_{J}\cap\Gamma_{J^{\prime}}=\\{1_{\Gamma}\\}$ and that $\Gamma_{J}$
commutes with $\Gamma_{J^{\prime}}$. Therefore the subgroup
$\Delta\leqslant\Gamma$ generated by both $\Gamma_{J}$ and
$\Gamma_{J^{\prime}}$ is isomorphic to $\Gamma_{J}\times\Gamma_{J^{\prime}}$,
which is a product of two finitely generated non-amenable groups and thus
self-simulable by Theorem 1.2. In what follows we shall show that $\Delta$ is
mediated.

By hypothesis, we can take the union of a generating set for each $\Gamma_{I}$
with $|I|=2$ to get a generating set for $\Gamma$. Fix $I\subseteq N$. Then
either (1) $I\subseteq J$, (2) $I\subseteq J^{\prime}$ or (3) $I$ intersects
both $J$ and $J^{\prime}$. In cases (1) and (2) we clearly have that
$\Gamma_{I}\leqslant\Gamma_{J}$ and $\Gamma_{I}\leqslant\Gamma_{J^{\prime}}$
respectively. In case (3), as $n\geq 5$, we can find
$I^{\prime}=\\{j,j^{\prime}\\}$ with $j\neq j^{\prime}$ such that
$I^{\prime}\cap I=\varnothing$ and $I^{\prime}\subseteq J$ or
$I^{\prime}\subseteq J^{\prime}$. We conclude that
$\Gamma_{I^{\prime}}\leqslant\Gamma_{J}\times\Gamma_{J^{\prime}}$ commutes
with $\Gamma_{I}$ and thus can be taken as the group $\Sigma$ in Remark 6.10.
∎

######  Corollary 8.7.

The general linear group $\operatorname{GL}_{n}(\mathbb{Z})$ and the special
linear group $\operatorname{SL}_{n}(\mathbb{Z})$ are self-simulable for $n\geq
5$.

###### Proof.

Let $\Gamma=\operatorname{SL}_{n}(\mathbb{Z})$ for $n\geq 5$ and take
$A_{i}=\mathbb{Z}$ for every $1\leq i\leq 5$. Then clearly for every
$I\subseteq\\{1,\dots,n\\}$ with $|I|=2$ we have that $\Gamma_{I}$ is
isomorphic to $\operatorname{SL}_{2}(\mathbb{Z})$, which is finitely generated
and non-amenable. It is well known that for any Euclidean domain $R$,
$\mathrm{SL}_{n}(R)$ is generated by the transvection matrices which add a
multiple of a row (resp./ column) to another row (resp./ column), and
therefore $\mathrm{SL}_{n}(\mathbb{Z})$ is indeed generated by the
$\Gamma_{I}$ for $|I|=2$. We conclude by Lemma 8.6 that
$\operatorname{SL}_{n}(\mathbb{Z})$ is self-simulable. As
$\operatorname{SL}_{n}(\mathbb{Z})$ is a normal subgroup of
$\operatorname{GL}_{n}(\mathbb{Z})$, we get that
$\operatorname{GL}_{n}(\mathbb{Z})$ is self-simulable by Corollary 6.12. ∎

In Lemma 8.6 there is nothing special about using subgroup generators with
$|I|=2$, and indeed, we will now present a generalized method which extends
this lemma and will allows us to prove that several new groups are also self-
simulable.

###  8.5. Groups generated by subgroups

In this section, we shall present an abstract generalization of Lemma 8.6 that
will enable us to cover (outer) automorphism groups of free groups (with
enough generators), braid groups (with enough braids), and some classes of
right-angled Artin groups. We also give an alternative proof for the Brin-
Thompson’s group $2V$ using this method.

Let $\mathcal{S}$ a set with two relations defined on it: an orthogonality
relation $\bot\subseteq\mathcal{S}^{2}$ which is symmetric, and a containment
relation ${\leq}\subseteq\mathcal{S}^{2}$ which is the order of a join-
semilattice operation $\vee$. Also require that whenever $a\leq b$ and $b\bot
c$ then $a\bot c$.

######  Definition 8.8.

Let $\mathcal{S}$ be as above. We say $A\subseteq\mathcal{S}$ is orthogonal if
for every pair of distinct $a,b\in A$ we have $a\bot b$.

For $A,B,\mathcal{N}\subseteq\mathcal{S}$, we say that the set $B$ is weakly
$\mathcal{N}$-suborthogonal to $A$ if for every $b\in B$ there is
$c\in\mathcal{N}$ and $a\in A$ such that $b\bot c$ and $c\leq a$.

In the definition above, we should think of the set $\mathcal{S}$ as
generalizing the power set of the interval $\\{1,\dots,n\\}$ in Lemma 8.6.
More precisely, let us say that a finitely-generated group $\Gamma$ is
compatible with $(\mathcal{S},\bot,\leq)$ if for each $s\in\mathcal{S}$ we
have a subgroup $\Gamma_{s}\leqslant\Gamma$, and the collection
$(\Gamma_{s})_{s\in\mathcal{S}}$ respects the operations in the sense that
whenever $s\bot t$, then both $\Gamma_{s}\cap\Gamma_{t}=\\{1_{\Gamma}\\}$ and
$[\Gamma_{s},\Gamma_{t}]=1$, and whenever $s\leq t$ then
$\Gamma_{s}\leqslant\Gamma_{t}$. Say $B\subseteq\mathcal{S}$ is generating if
$\Gamma$ is generated by the subgroups $(\Gamma_{b})_{b\in B}$.

######  Lemma 8.9.

Let $(\mathcal{S},\bot,\leq)$ be as above and $\Gamma$ be a finitely
generated, recursively presented group which is compatible with
$(\mathcal{S},\bot,\leq)$. Let $(\Gamma_{s})_{s\in\mathcal{S}}$ be as above,
and let $\mathcal{N},A,B\subseteq\mathcal{S}$ such that

  1. (1)

$\Gamma_{s}$ is finitely generated and non-amenable for every
$s\in\mathcal{N}$,

  2. (2)

$A\subseteq\mathcal{N}$ is finite, orthogonal and $|A|\geq 2$,

  3. (3)

$B$ is generating and weakly $\mathcal{N}$-suborthogonal to $A$.

Then $\Gamma$ is self-simulable.

###### Proof.

The group $\Delta=\langle\bigcup_{a\in A}\Gamma_{a}\rangle$ generated by the
subgroups $\Gamma_{a}$ is isomorphic to the direct product $\prod_{a\in
A}\Gamma_{a}$. As $A$ is finite and $|A|\geq 2$, then $\Delta$ is self-
simulable by Theorem 1.2. We shall show that it is mediated. Observe that
because $\Gamma$ is finitely-generated and $B$ is generating, we can find a
finite generating set $T$ for $\Gamma$ with $T\subseteq\bigcup_{b\in
B}\Gamma_{b}$. If $t\in T\cap\Gamma_{b}$ with $b\in B$, by weak
$\mathcal{N}$-suborthogonality there is $\Sigma=\Gamma_{c}$ with
$c\in\mathcal{N}$, such that $c\bot b$ ($[\Gamma_{c},\Gamma_{b}]=1$) and
$c\leq a$ ($\Gamma_{c}\leqslant\Gamma_{a}$) for some $a\in A$. Now, as
$c\in\mathcal{N}$ implies $\Sigma$ is non-amenable, the condition $c\bot b$
implies that $\Sigma^{t}=\Sigma$ (even $g^{t}=g$ for all $g\in\Sigma$), and
$c\leq a$ implies $\Sigma\leqslant\Gamma_{a}\leqslant\Delta$. This proves that
$\Delta$ is mediated and thus $\Gamma$ is self-simulable. ∎

Notice that we can always take $|A|=2$ by taking the join of all but one
element of $A$, and this approach will be taken in most examples.

######  Corollary 8.10.

For all $n\geq 5$, the groups $\operatorname{Aut}(F_{n})$ and
$\operatorname{Out}(F_{n})$ are self-simulable.

###### Proof.

Consider a finite set $N=\\{1,2,\dots,n\\}$ that freely generates a free group
$F_{N}$, and let $\Gamma$ be its automorphism group or outer automorphism
group (these cases are almost identical). Let $\mathcal{S}$ be the power set
$\mathcal{P}(N)$ of $N$ and define $\bot$ as disjointness and $\leq$ as
containment. To $I\subseteq N$ associate the group $\Gamma_{I}$ which, in the
case of $\operatorname{Aut}(F_{N})$, is the natural copy of
$\operatorname{Aut}(F_{I})$ that fixes generators outside of $I$ (in the case
of $\operatorname{Out}(F_{N})$ it is the quotient of
$\operatorname{Aut}(F_{I})$ inside $\Gamma$). From this definition it is clear
that whenever $I\bot I^{\prime}$ then
$\Gamma_{I}\cap\Gamma_{I^{\prime}}=\\{1_{F_{N}}\\}$ and
$[\Gamma_{I},\Gamma_{I^{\prime}}]=1$.

Let $\mathcal{N}=\\{I\subseteq N:|I|\geq 2\\}$ and fix $I\in\mathcal{N}$. The
group $\Gamma_{I}$ is non-amenable in both cases. Observe that
$\operatorname{Out}(F_{2})$ is isomorphic to
$\operatorname{GL}_{2}(\mathbb{Z})$ which is non-amenable, and thus
$\operatorname{Aut}(F_{2})$ is non-amenable as it admits a non-amenable
quotient. Furthermore, the groups $\Gamma_{I}$ are finitely generated because
for every $k\in\mathbb{N}$, the group $\operatorname{Aut}(F_{k})$ is generated
by Nielsen transformations [39], and thus a fortiori the quotient
$\operatorname{Out}(F_{k})$ is generated by them as well. For the same reason,
the set $B=\\{I\subseteq N:|I|=2\\}$ is generating (notice that the
permutations of generators in the Nielsen transformations are generated by
transpositions). Let $A=\\{\\{1,2\\},\\{3,\dots,n\\}\\}$. It is clear that $A$
is orthogonal, therefore to satisfy the conditions of Lemma 8.9 it suffices to
check that $B$ is weakly $\mathcal{N}$-suborthogonal to $A$. The argument for
this is the same as in the proof of Lemma 8.6, namely, either $b\in B$ is
contained in $\\{1,2\\}$ or $\\{3,\dots,n\\}$ in which case we take $c=a$ as
the set which does not intersect $b$, or $b$ intersects both of those sets in
which case we take $a=\\{3,\dots,n\\}$ and $c=\\{3,\dots,n\\}\setminus b$, and
note that $|c|\geq 2$ and thus $c\in\mathcal{N}$. ∎

Braid groups are a class of finitely presented groups whose elements can be
represented visually as a set of intertwining strands. We will not give a
proper definition of these groups here, readers interested in them can find a
survey on braid groups in [36] or [32]. We will only use the following facts
about braid groups. The braid groups are all nested in the sense that for
$m\leq n$ the group $B_{m}$ occurs as the subgroup of $B_{n}$ where exactly
$m$ fixed contiguous strands can be intertwined, (2) The braid group $B_{n}$
is generated by the braids which pass the $i+1$-th braid over the $i$-th braid
for $i=1,\dots,n-1$, and (3) $B_{n}$ is non-amenable for $n\geq 3$. These
facts are all elementary and can found in the previous two references.

######  Corollary 8.11.

Braid groups $B_{n}$ are self-simulable for $n\geq 7$.

###### Proof.

Consider a finite set $N=\\{1,2,\dots,n\\}$, let $\mathcal{S}$ be the set of
discrete sub-intervals of $N$, and to each $I\in\mathcal{S}$ associate the
braid group $B_{I}$ on these strands. We have the natural inclusion $I\leq
J\implies B_{I}\leq B_{J}$. Take $\mathcal{N}=\\{I\in\mathcal{S}:|I|\geq
3\\}$. By (1) and (3) it follows that every $B_{I}$ is a non-amenable and
finitely generated group. The set $B=\\{I:|I|=2\\}$ is generating by (2), and
the partition $A=\\{\\{1,2,3\\},\\{4,\dots,n\\}\\}$ satisfies the assumptions
of Lemma 8.9, the proof being analogous as those of Corollaries 8.6 and 8.10.
∎

######  Remark 8.12.

With this technique is also possible to obtain again that Brin-Thompson $2V$
is self-simulable. Indeed, consider the natural faithful action of Brin-
Thompson $2V$, which we denote by $\Gamma$, on $C\times C$ where
$C=\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{N}}$ is the Cantor set. Let
$\mathcal{S}$ be the set of all clopen sets in $C\times C$. Let $\bot$ be
disjointness and $\leq$ be containment. For $I\in\mathcal{S}$ let $\Gamma_{I}$
be the subgroup of elements which act trivially on $(C\times C)\setminus I$.
It follows that whenever $I\leq J$ then $\Gamma_{I}\leqslant\Gamma_{J}$ and,
as the action is faithful, that whenever $I\bot J$, then $\Gamma_{I}$ and
$\Gamma_{J}$ commute and have trivial intersection.

Take $\mathcal{N}=\\{[w]\times C:w\in\\{\mathtt{0},\mathtt{1}\\}^{*}\\}$, and
observe that by (2) $\Gamma_{I}\cong\Gamma$ for all $I\in\mathcal{N}$, so
these groups are indeed finitely-generated and non-amenable. Proposition 3.2
of [14] shows that for $\varepsilon=1/4$ the set $B=\\{I\times
C:\mu(I)<\varepsilon\\}$ is generating, where $\mu$ is the uniform Bernoulli
measure on $C$. Taking $A=\\{[0]\times C,[1]\times C\\}$, we have that
$A\subseteq\mathcal{N}$ is a finite orthogonal set, and $B$ is weakly
$\mathcal{N}$-suborthogonal to $A$ because for any $I\times C\in B$,
$([0]\times C)\setminus(I\times C)$ is a positive measure clopen set, thus
nonempty, thus for some $w\in\\{\mathtt{0},\mathtt{1}\\}^{*}$ we have
$([w]\times C)\subseteq([0]\times C)$ with $([w]\times C)\bot B$, concluding
the proof of weak suborthogonality.

###  8.6. Right-angled Artin groups

Let $G=(V,E)$ be a finite simple graph (undirected edges and no loops). The
right-angled Artin group (RAAG) defined by $G$ is group $\Gamma[G]$ generated
by the vertices $V$ which satisfies that $u,v\in V$ commute whenever
$\\{u,v\\}\in E$.

$\Gamma[G]=\langle V\ \mid\ [u,v]=1\mbox{ for }\\{u,v\\}\in E\rangle.$

RAAGs provide a formalism for mapping finite simple graphs into finitely
generated groups, and thus some properties of graphs can be translated into
properties of the groups they generate. A recent survey on RAAGs can be found
in [19]. In what follows we shall provide conditions on a graph which imply
that their corresponding RAAG is self-simulable.

Let us fix some notation beforehand. Let $G=(V,E)$ be a graph. We denote by
$\overline{G}$ the complement of $G$, that is, the graph with vertices $V$ and
where $\\{u,v\\}$ is an edge of $\overline{G}$ if and only if $\\{u,v\\}$ is
not an edge of $G$. We say that an edge $\\{u,v\\}\in E$ is adjacent to a node
$w$ if either $\\{u,w\\}\in E$ or $\\{v,w\\}\in E$. Let us also denote, for
$A,B\subseteq V$ by $E(A,B)$ the set of edges $\\{u,v\\}$ for which $u\in A$
and $v\in B$.

Below, we make our statements in terms of the complement of a graph, as they
are easier to describe.

######  Corollary 8.13.

Suppose $G=(V,E)$ is a finite connected graph which has two edges with the
property that no $v\in V$ is adjacent to both of them. Then
$\Gamma[\overline{G}]$ is self-simulable.

###### Proof.

We shall employ Lemma 8.9. Take $\mathcal{S}=\mathcal{P}(V)$ the set of
subsets of vertices, let $a\bot b$ mean that $E(a,b)=\varnothing$ and
interpret $\leq$ as inclusion. For $a\subseteq V$, let $\Gamma_{a}$ be the
subgroup of $\Gamma[\overline{G}]$ generated by the vertices of $a$.

Notice that $(\Gamma_{a})_{a\subseteq V}$ is compatible with
$(\mathcal{S},\bot,\leq)$, as $a\bot b$ means that $[\Gamma_{a},\Gamma_{b}]=1$
and that they have trivial intersection, and clearly $a\leq b$ implies that
$\Gamma_{a}\leqslant\Gamma_{b}$.

Take as $B$ the set of singleton subsets of $V$. Then $B$ is generating by
definition. Take $\mathcal{N}=E$ and notice that they all give induced copies
of the free group on two generators, therefore they are finitely generated and
non-amenable. Finally, by assumption there are two edges $e=\\{u_{1},u_{2}\\}$
and $e^{\prime}=\\{v_{1},v_{2}\\}$ such that no node is adjacent to both $e$
and $e^{\prime}$. Take $A=\\{e,e^{\prime}\\}$. Notice that this assumption
implies that $A$ is orthogonal.

Finally, we just need to check that $B$ is weakly $\mathcal{N}$-suborthogonal
to $A$. if $b=\\{v\\}\in B$, then there is
$e^{\prime\prime}\in\\{e,e^{\prime}\\}$ which is not adjacent to $v$. Choosing
$c=a=e^{\prime\prime}$ gives that $b\bot c\leq a$. ∎

######  Example 8.14.

Let $G$ be a finite connected simple graph with diameter at least five. Then
by taking a shortest path of length $5$ between two vertices, it follows that
the extremal edges satisfy the condition of Corollary 8.13. It follows that
$\Gamma[\overline{G}]$ is self-simulable. $\fullmoon$

######  Example 8.15.

Let $G$ be a cycle of length $n\geq 7$. Then $\Gamma[\overline{G}]$ is self-
simulable. Indeed, if $n\geq 8$ it follows that $G$ satisfies the condition of
Corollary 8.13. If $n=7$, Let $(\mathcal{S},\bot,\leq)$ and $B$ be the set of
singleton subsets of $V=\\{1,\dots,7\\}$ as in the proof of Corollary 8.13,
take $\mathcal{N}$ as the set of vertices which induce connected subgraphs
with at least two nodes (which are again, finitely generated and non-
amenable), and pick $A\\{\\{1,2\\},\\{4,5,6\\}\\}$. Again, it is clear that
$A$ is orthogonal.

Now let $b\in B$, if $b=\\{3\\}$ pick $c=\\{5,6\\}$ and $a=\\{4,5,6\\}$. If
$b=\\{7\\}$ pick $c=\\{4,5\\}$ and $a=\\{4,5,6\\}$. If $b\subseteq\\{1,2\\}$
pick $a=c=\\{4,5,6\\}$ and finally, if $b\subseteq\\{4,5,6\\}$ pick
$a=c=\\{1,2\\}$. The result then follows from Lemma 8.9. $\fullmoon$

###  8.7. Existence of strongly aperiodic subshifts of finite type

######  Definition 8.16.

We say a $\Gamma$-subshift is strongly aperiodic if $\Gamma$ acts freely on
it, that is, if $gx=x$ for some $x\in X$ then $g=1_{\Gamma}$.

We shall use the following result from [4] which ensures that every finitely
generated group with decidable word problem admits a nonempty effectively
closed subshift which is strongly aperiodic. The proof relies on a
probabilistic argument of [1] which shows that every graph of bounded degree
can be colored with finitely many colors in such a way that the sequence of
colors of every non self-intersecting path does not contain two subsequent
occurrences of the same sequence of colors.

######  Theorem 8.17 (Theorem 2.6 of [4]).

For every finitely generated group with decidable word problem there exists a
nonempty strongly aperiodic and effectively closed subshift.

A clear consequence of Theorem 8.17 is the following.

######  Corollary 8.18.

Every finitely generated self-simulable group with decidable word problem
admits a nonempty strongly aperiodic subshift of finite type.

###### Proof.

Let $\Gamma$ be a finitely generated self-simulable group with decidable word
problem. By Theorem 8.17 there exists a strongly aperiodic and effectively
closed $\Gamma$-subshift $Y$. As $\Gamma$ is self-simulable (and recursively
presented), there exists a nonempty $\Gamma$-subshift of finite type $X$ and a
factor map $\phi\colon X\to Y$. Let $g\in\Gamma$ and $x\in X$ so that $gx=x$.
Then we have that $g\phi(x)=\phi(gx)=\phi(x)$. As $Y$ is strongly aperiodic,
we conclude that $g=1_{\Gamma}$ and thus $X$ is strongly aperiodic. ∎

In particular, there exist nonempty strongly aperiodic subshifts of finite
type on every group with decidable word problem which was shown to be self-
simulable in this section.

###  8.8. Soficity of the Sunny-side up subshift

######  Definition 8.19.

Let $\Gamma$ be a group. The sunny-side up subshift is given by

$X_{\leq 1}=\\{x\in\\{\mathtt{0},\mathtt{1}\\}^{\Gamma}:\mbox{ if
}x(g)=x(h)=1\mbox{ then }g=h\\}.$

In words, the sunny-side up subshift consists of all configurations on the
alphabet $\\{\mathtt{0},\mathtt{1}\\}$ which have at most one position marked
by the symbol $\mathtt{1}$. This object is the symbolic representation of the
translation action of $\Gamma$ on its one-point compactification
$\Gamma\cup\\{\infty\\}$. It is well known that if $\Gamma$ is an infinite and
finitely generated group, then $X_{\leq 1}$ is not a subshift of finite type
(see [3, Proposition 2.10]). An interesting problem is figuring out for which
groups $\Gamma$ the sunny-side up subshift is sofic. It has been shown by
Dahmani and Yaman in [23] that (1) if $\Gamma$ splits in a short exact
sequence $1\to N\to\Gamma\to H\to 1$ and $X_{\leq 1}$ is sofic for both $N$
and $H$ then it is also sofic for $\Gamma$, (2) soficity of the sunny-side up
subshift is an invariant of commensurability and that (3) the sunny-side up
subshift is sofic for finitely generated free groups, finitely generated
abelian groups, hyperbolic groups and poly-hyperbolic groups.

The first examples of finitely generated groups where the sunny-side up
subshift is not sofic were given in [3]. Namely [3, Proposition 2.11] states
that if $\Gamma$ is a recursively presented group then $X_{\leq 1}$ is
effectively closed if and only if the word problem of $\Gamma$ is decidable.
It follows that if $\Gamma$ is a recursively presented group with undecidable
word problem then $X_{\leq 1}$ is not effectively closed and therefore is also
not sofic. To the knowledge of the authors, it is still unknown whether there
exists a finitely generated group $\Gamma$ with decidable word problem for
which the sunny-side up subshift is not sofic.

An obvious consequence of a group being self-simulable, is that if the group
has decidable word problem, then the sunny-side up subshift is sofic, as all
effectively closed subshifts are automatically sofic.

######  Corollary 8.20.

Let $\Gamma$ be a finitely generated, self-simulable group with decidable word
problem. The sunny-side up subshift on $\Gamma$ is sofic.

##  9\. Perspectives and questions

In this final section we present several topics which we either did not
explore, or were unable to answer in this study. We hope some of these
questions might motivate further studies about the class of self-simulable
groups.

###  9.1. Properties of the extension

Our main theorem provides a subshift of finite type extension for any
effectively closed action, but says very little about the properties of said
extension. More precisely, one might wonder what kind of Borel invariant
probability measures the extension can admit, or whether the extension can be
made transitive or minimal provided the effectively closed action is
transitive or minimal.

######  Question 9.1.

Let $\Gamma$ be a self-simulable group, and suppose the effectively closed
action $\Gamma\curvearrowright X$ is minimal (resp. transitive). Is there a
$\Gamma$-subshift of finite type extension which is minimal (resp.
transitive)?

######  Question 9.2.

Let $\Gamma$ be a self-simulable group and $\Gamma\curvearrowright X$ be an
effectively closed action which admits a $\Gamma$-invariant Borel probability
measure $\nu$. Is there a $\Gamma$-subshift of finite type extension $Z$ which
admits a $\Gamma$-invariant Borel probability measure $\mu$? Can we have that
$\Gamma\curvearrowright(X,\nu)$ is a measure-theoretic factor of
$\Gamma\curvearrowright(Z,\mu)$ such that $\nu$ is the push-forward of $\mu$?

Regarding Question 9.1, a first step in a tentative adaptation of Theorem 1.2
would be to produce a minimal analogue of the paradoxical subshift. This can
at least be done in the case when $\Gamma=F_{2}$ (or more generally, any
finitely generated free group on at least two elements). Indeed, consider the
alphabet $A$ given by the four tiles shown below.

We can define a subshift of finite type $\mathbf{M}$ on $F_{2}$ by imposing
that along both generators of $F_{2}$ an arrowhead must be matched with an
arrow tail. It is not hard to produce a collection of paradoxical path covers
from any configuration in $\mathbf{M}$, and it can be shown that the shift
action on $\mathbf{M}$ is minimal. However, adapting the rest of the proof
seems hard. One important issue is that bi-infinite paths on any paradoxical
subshift are unavoidable (by compactness) and therefore there is no evident
way to restrict the contents of those paths in the computation layer.
Furthermore, a second issue arises from the fact that the return times to a
cylinder on the minimal system $F_{2}\curvearrowright X$ we are simulating
might coordinate with the return times of a pattern in
$F_{2}\curvearrowright\mathbf{M}$, thus introducing patterns in the subshift
of finite type cover of $F_{2}\curvearrowright X$ which do not occur with
bounded gaps.

Concerning Question 9.2, our methods are far from providing a useful approach,
due to the fact that they strongly rely on constructions based on paradoxical
decompositions, which thus admit no invariant measures.

###  9.2. The class of self-simulable groups

The rest of our questions are about the class of self-simulable groups. We are
still quite far from having an idea of how to algebraically characterize this
class of groups.

The first candidate for a self-simulable group, which was studied by the first
two named authors in collaboration with N. Aubrun, were the surface groups,
that is, fundamental groups of closed orientable surfaces. A surface group is
either virtually abelian or non-amenable, and all the non-amenable ones are
commensurable. The non-amenable ones are strongly related to tilings of the
hyperbolic plane, and a restricted variant of self-simulation has been proved
in this setting [6]. The methods of the present paper are completely
different, and we are not aware of any method to apply them to surface groups.

######  Question 9.3.

Let $\Gamma$ be the fundamental group of a closed orientable surface of genus
at least two. Is $\Gamma$ self-simulable? More generally, is every one-ended
word-hyperbolic group self-simulable?

Another problem we tried to tackle and were unable to resolve is the
following. By Corollary 6.12, we have that if $\Gamma$ and $\Delta$ are
finitely generated and recursively presented groups, and $\Gamma$ is self-
simulable, then $\Gamma\times\Delta$ is self-simulable. We do not know whether
the following restricted version of the converse holds.

######  Question 9.4.

Suppose $\Gamma\times\Delta$ is self-simulable and $\Delta$ is amenable, is
$\Gamma$ self-simulable?

In fact, we do not know the answer to Question 9.4 even in the case when
$\Delta=\mathbb{Z}$.

Another question is related to how much can Theorem 1.3 be extended. We know
that solely containing a self-simulable subgroup is not enough to be self-
simulable (see Example 6.1). However, our examples all rely on the few
obstructions we constructed in Section 3. Therefore we are unable to answer
the following question.

######  Question 9.5.

Let $\Gamma$ be a one-ended group which contains a self-simulable group. Is
$\Gamma$ self-simulable?

Most of the theorems we proved in Sections 6 and 7 used the hypothesis that
$\Gamma$ was recursively presented. More importantly, the proof of quasi-
isometric rigidity uses in a fundamental way that the groups involved are
finitely presented. We do not know if this condition can be relaxed.

######  Question 9.6.

Is the property of being self-simulable quasi-isometrically rigid for finitely
generated recursively presented groups?

We showed that $\operatorname{GL}_{n}(\mathbb{Z})$ is self-simulable when
$n\geq 5$, and it is clearly not self-simulable when $n=1$ (because it is
finite) or when $n=2$ (because it is virtually $\mathbb{Z}_{2}*\mathbb{Z}_{3}$
and thus multi-ended).

######  Question 9.7.

Are the groups $\operatorname{GL}_{3}(\mathbb{Z})$ and
$\operatorname{GL}_{4}(\mathbb{Z})$ self-simulable?

Similar questions can be posed for $\operatorname{Aut}(F_{n})$ and
$\operatorname{Out}(F_{n})$ for $n\leq 4$ and $B_{n}$ for $n\leq 6$. In the
same spirit, we also wonder what is exactly the class of RAAGs which is self-
simulable.

######  Question 9.8.

For which finite simple graphs $G=(V,E)$ is the right-angled Artin group
$\Gamma[G]$ self-simulable?

We showed that Thompson’s $V$ is self-simulable. In light of Corollary 8.5,
the following question is interesting.

######  Question 9.9.

Are Thompson’s $F$ and $T$ self-simulable?

Notice that if one is able to show that either $F$ or $T$ are not self-
simulable, it would imply that $F$ is amenable.

S. Barbieri, DMCC, Universidad de Santiago de Chile.

E-mail address<EMAIL_ADDRESS>
M. Sablik, Université Paul Sabatier.

E-mail address<EMAIL_ADDRESS>
V. Salo, University of Turku.

E-mail address<EMAIL_ADDRESS>
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Separation of scales in quantum field theories is essential when studying the low-energy phenomenology of a given UV model. To this end, it is necessary to construct an effective field theory containing only the light degrees of freedom and matching it to the full theory, ensuring that both describe the same low-energy dynamics. Performing this matching beyond the leading order is crucial, as a great number of observables, like FCNC, only appear at the loop level in the Standard Model and in many new physics scenarios. One possibility to obtain the low-energy theory is by integrating out the heavy particles from the full theory using path integral techniques. We review this functional matching procedure at the one-loop level and discuss common challenges involved in determining the matching conditions for effective theories. Due to the great diversity of beyond the Standard Model theories and the complexity of the matching computations, an automation of this procedure is desirable. On this matter, we present the ongoing effort to develop the  package Matchete facilitating the fully automatic matching for a broad range of theories. When completed, this will significantly simplify the analysis of the low-energy phenomenology of beyond the Standard Model physics.
These proceedings are based on refs. [1] and [2].

§ INTRODUCTION

Effective Field Theory (EFT) is a powerful tool to study the low-energy dynamics of quantum field theories involving particles of different mass scales. Consider a theory containing heavy particles $\eta_H$ with masses $m_H$ and light particles $\eta_L$ with masses $m_L$ with a large scale separation $m_L \ll m_H$. If an experiment is performed at low energies $\smash{E \sim \sqrt{q^2} \sim m_L}$, where $q$ is the typical momentum scale of the light particles in that process, the heavy degrees of freedom do not contribute as on-shell states, and thus their effects can be considered as sub-leading corrections to the measurements. In this scenario, due to the resummation of large logarithms, it is required to construct an EFT containing only the light particles, where the effects of the heavy particles are encoded in a tower of higher-dimensional operators built out of $\eta_L$ fields, that are suppressed by appropriate powers of $1/m_H$. An example for such a model is Fermi's theory, where the heavy particle removed from the theory is the $W$ boson, and the interactions at low energies are described by four-fermion contact interactions that are suppressed by $1/m_W^{2}$.

There are two main challenges for the construction of EFTs: $i)$ Finding all higher-dimensional operators that have to be considered such that the EFT can describe the same low-energy phenomenology as the corresponding UV theory; $ii)$ Determining the values of the coefficients of all effective operators in terms of the UV Lagrangian parameters. The latter is often referred to as matching the EFT to the UV. Traditionally, this is done by computing all Feynman diagrams that contribute to the effective action $\Gamma[\eta_L]$, as a function of the light fields only, in both theories and equating these $\Gamma_\mathrm{EFT}=\Gamma_\mathrm{UV}$, where $\Gamma_\mathrm{UV}$ has to be expanded as a power series in $1/m_H$ (see e.g. [3]). This method is called diagrammatic matching. In the following, however, we will consider a different approach to EFT matching based on the path integral. When we construct an EFT it is often important to do this at the loop level, meaning we have to consider effective operators whose coefficients get a non-vanishing value when the matching is performed at the loop level. The reason is that many interesting phenomena, such as FCNCs, do not appear at tree level in the Standard Model (SM) and many New Physics (NP) scenarios.

Effective theories also play a crucial role in the analysis of Beyond-the-SM (BSM) theories. The absence of direct signals of NP at the experiments at the LHC indicate that the mass scale of the potential new particles is far above the electroweak scale $v \ll m_H$, although light NP is not fully excluded. In the case of heavy NP, the effects of the BSM particles can be described at lower energies by the so-called Standard Model Effective Field Theory (SMEFT) [5, 6], which allows to study the NP effects at the electroweak scale $v$ in a model independent way.
To analyze the effects of a UV theory at even lower energies, we can first match it to the SMEFT, then use the renormalization group equations (RGE) of the SMEFT [7] to evolve the coefficients to lower scales, and then match the SMEFT to another EFT, often called LEFT [8], where the heaviest particles of the SM (i.e. the top quark, $W/Z$ boson, and the Higgs) are removed. This EFT is then appropriate to compare the theory to the low-energy experiments.

These multi-step matching scenarios can be computationally very challenging, as all matching steps have to be performed at the loop level, and the RGE of all EFTs need to be determined. Therefore, an automation of this procedure is important to allow for analyses of the vast variety of different BSM theories. In these proceeding, which are based on refs. [1] and [2], we review the functional method for EFT matching developed, e.g., in refs. [10, 11, 12, 13, 14, 9], which both derives the effective operators and their matching conditions. The method is particularly suited for an automation and we present the ongoing effort in the development of the  code  [2] – matching effective theories efficiently – which will, when released, fully automate the EFT matching at one loop, thus significantly simplifying the analysis of BSM theories.

§ FUNCTIONAL METHODS FOR EFT MATCHING

In this section we review the functional matching procedure based on the path integral formalism and the background field method, for which we will mostly follow refs. [1, 9].

§.§ Functional formalism and tree-level matching

Consider a general UV theory described by the Lagrangian $\mathcal{L}_\mathrm{UV}$ containing both heavy $\eta_H$ and light $\eta_L$ particles with a mass gap $m_H \gg m_L$. The corresponding fields can be grouped in a multiplet ${\eta=(\eta_H,\,\eta_L)}$. Using the background field method we split all fields by $\eta\to\hat\eta+\eta$, where $\hat\eta$ is the background field configuration satisfying the classical equations of motion (EOM), and $\eta$ are the quantum fluctuations. In Feynman diagrams tree-level lines thus arise from the background fields $\hat\eta$, whereas lines in loops arise from the quantum fluctuations $\eta$. Therefore, to capture all effects up to one loop, it is sufficient to expand the Lagrangian up to quadratic order in the fluctuations $\eta$

\begin{eqnarray}
\label{eq:Lagrangian}
\mathcal{L}_\mathrm{UV}[\hat\eta+\eta]
&= \mathcal{L}_\mathrm{UV}[\hat\eta] + \frac{1}{2} \bar{\eta}_i \left. \frac{\delta^2 \mathcal{L}_\mathrm{UV}}{\delta \bar\eta_i \, \delta\eta_j} \right|_{\eta=\hat\eta} \eta_j + \mathcal{O}(\eta^3) \,,
\end{eqnarray}

where the functional derivative with respect to the quantum fluctuations is denoted $\delta/\delta\eta_i$. The linear terms in the expansion above vanish due to the EOM, and higher-order terms only contribute at higher loop orders.

The effective action of the theory can be expressed as a path integral by

\begin{eqnarray}
\label{eq:effective_action}
\exp\left( i \Gamma_\mathrm{UV}[\hat\eta] \right)
&= \int \mathcal{D}\eta \, \exp\left( i\int\mathrm{d}^dx\,\mathcal{L}_\mathrm{UV}[\hat\eta+\eta] \right)\,.
\end{eqnarray}

Combining eqs. (<ref>) and (<ref>) we find the tree-level effective action $\smash{\Gamma_\mathrm{UV}^{(0)}[\hat\eta] = \int \mathrm{d}^d x \, \mathcal{L}_\mathrm{UV}[\hat\eta]}$.
At energies far below the heavy mass, i.e. for $q^2\sim m_L^2 \ll m_H^2$, we can solve the EOM for $\hat{\eta}_H$ by expanding in inverse powers of $m_H$, thus obtaining a solution of the form $\hat\eta_H=\hat\eta_H(\hat\eta_L)$ as a perturbative series in $1/m_H$. This yields the tree-level EFT Lagrangian

\begin{eqnarray}
\mathcal{L}_\mathrm{EFT}^{(0)}[\hat\eta_L] &= \mathcal{L}_\mathrm{UV}[\hat\eta_L,\hat\eta_H(\hat\eta_L)] \,,
\end{eqnarray}

which only depends on light background fields.

§.§ One-loop matching

Similarly the one-loop effective action is given by

\begin{eqnarray}
\exp\left( i \Gamma_\mathrm{UV}^{(1)} \right)
&= \int\mathcal{D}\eta \, \exp\left( \frac{i}{2} \int\mathrm{d}^d x \, \bar{\eta}_i \mathcal{O}_{ij} \eta_j \right) \,, \quad \mathrm{where} \quad \mathcal{O}_{ij} = \left .\frac{\delta^2 \mathcal{L}_\mathrm{UV}}{\delta \bar{\eta}_i \, \delta{\eta}_j} \right|_{\eta=\hat\eta} \,.
\end{eqnarray}

The operator $\mathcal{O}_{ij}$ is dubbed the fluctuation operator. The one-loop effective action is thus determined by a Gaussian path integral which can be readily solved

\begin{eqnarray}
\Gamma_\mathrm{UV}^{(1)}
&= \frac{i}{2} \log \mathrm{SDet} \, \mathcal{O}
= \frac{i}{2} \mathrm{STr} \log \mathcal{O} \,.
\end{eqnarray}

Here, we introduced the superdeterminant SDet and the supertrace STr, which are generalizations of the determinant and the trace, respectively, to matrices with both Grassmann and ordinary elements, where, e.g., STr carries opposite sign for fermionic and bosonic entries.
The supertrace is a trace over all internal degrees of freedom including momenta, and thus also contains a loop integral

\begin{eqnarray}
\label{eq:loop-integral}
\mathrm{STr} \log \mathcal{O}
\pm \int \frac{\mathrm{d}^d k}{(2\pi)^d} \langle k | \mathrm{tr}\log\mathcal{O} | k \rangle \,,
\end{eqnarray}

where to regulate divergences we use dimensional regularization with the $\overline{\mathrm{MS}}$-scheme.

We split the fluctuation operator into a propagator $\Delta_i$ and an interaction part $X_{ij}$

\begin{align}
\label{eq:fluctuation-operator}
\mathcal{O}_{ij} &= \delta_{ij} \Delta_i^{-1} - X_{ij}
= \Delta_{ij}^{-1} \left( \delta_{ij} - \Delta_i X_{ij} \right) \, ,
\Delta_{i}^{-1} &= \left\{\begin{matrix}
\\
i \slashed{D} - m_i
\\
\end{matrix}
\right.
\end{align}

for scalars, fermions and vector bosons, respectively.
For simplicity, we work in the Feynman gauge for the propagators of the quantum fluctuations. Notice that this does not imply a particular gauge choice for the light background fields which remain in a general $R_\xi$ gauge (see ref. [10] for more details).

The one-loop effective action thus reads

\begin{eqnarray}
\Gamma_\mathrm{UV}^{(1)}
&= \frac{i}{2} \mathrm{STr} \log \Delta^{-1} + \frac{i}{2} \mathrm{STr} \log \left(1-\Delta X\right) \,.
\label{eq:on-loop_effective_action}
\end{eqnarray}

To further simplify $\Gamma_\mathrm{UV}^{(1)}$ we apply the method of expansion by regions [15, 16]. The loop integrals can depend in principle on the scales $m_H$, $m_L$, and $q$, where all external momenta $q$ are assumed to be of order $q^2\sim m_L^2$. Thus we separate all loop integrals coming from eq. (<ref>) into a hard and soft region, where the loop momentum $k$ is assumed to be hard $k\sim m_H$ and soft $k\sim m_L$, respectively. We then expand each region in the quantities that are small, i.e. $m_L/k$ for the hard region and $k/m_H$ in the soft region, and consequently integrate both regions over the full $d$-dimensional space of loop momenta.
Summing both expanded integrals then yields the full original loop integral.

Following this procedure we can write

\begin{eqnarray}
\Gamma_\mathrm{UV}^{(1)} &= \left. \Gamma_\mathrm{UV}^{(1)} \right|_\mathrm{hard} + \left. \Gamma_\mathrm{UV}^{(1)} \right|_\mathrm{soft}.
\end{eqnarray}

An essential simplification arrises by realizing that $\smash{\Gamma_\mathrm{UV}^{(1)} \big|_\mathrm{soft}}$ contains the same contributions as one-loop contributions by the tree-level effective Lagrangian $\smash{\mathcal{L}_\mathrm{EFT}^{(0)}}$. Therefore, the one-loop contributions to the EFT Wilson coefficients in $\smash{\mathcal{L}_\mathrm{EFT}^{(1)}}$ are entirely encoded in the hard region of the effective action

\begin{eqnarray}
\left. \Gamma_\mathrm{UV}^{(1)} \right|_\mathrm{hard}
\int \mathrm{d}^d x \, \mathcal{L}_\mathrm{EFT}^{(1)} \,.
\label{eq:one-loop_lagrangian}
\end{eqnarray}

Thus, only supertraces with at least one heavy propagator in the loop can contribute to $\mathcal{L}_\mathrm{EFT}^{(1)}$, since loops without heavy propagators only contribute to the soft region.

Combining our results in eqs. (<ref>) and (<ref>) we obtain the one-loop EFT matching condition

\begin{eqnarray}
\int\mathrm{d}^d x \, \mathcal{L}_\mathrm{EFT}^{(1)}
\frac{i}{2} \left. \mathrm{STr} \log \Delta^{-1} \right|_\mathrm{hard}
- \frac{i}{2} \sum_{n=0}^{\infty} \frac{1}{n} \left. \mathrm{STr}\left[\left(\Delta X\right)^n\right] \right|_\mathrm{hard} \,,
\label{eq:master}
\end{eqnarray}

where we expanded the logarithm in the last term since in the hard region $\Delta X$ is at most $\mathcal{O}(m_H^{-1})$. We find that $\smash{\mathcal{L}_\mathrm{EFT}^{(1)}}$ is obtained by computing two types of supertraces:

* Log-type supertraces only depend on heavy particle propagators $\Delta_i$ as the light particles only contribute to the soft region. Furthermore, they are model independent as they do not depend on the interactions $X_{ij}$.

* Power-type supertraces contain contributions by both heavy and light fields, but at least one of the propagators in $(\Delta X)^n$ must correspond to a heavy particle for it to contribute to the hard region. Since $\Delta X$ is at most $\mathcal{O}(m_H^{-1})$ only a finite number of STr has to be computed in eq. (<ref>) when working up to a fixed EFT order.

The supertraces can be evaluated in a manifestly gauge covariant form using a covariant derivative expansion [10, 12]. A detailed description of the calculation of supertraces is beyond the scope of these proceedings and we refer to refs. [1, 9, 17] for more details.

§  – MATCHING EFFECTIVE THEORIES EFFICIENTLY

Due to the great variety of UV models, and due to the complexity of the supertrace evaluation for a particular model, an automation is required. In recent years there was an increased effort in the community to built tools automating at least partially the EFT matching [1, 18, 19, 17, 20].
Here we show a preview of the  package  [2], that we are currently developing.  implements the functional matching procedure outlined in sec. <ref> and is built upon  [1] which handles the identification and evaluation of the suptertraces. However, when made public,  will offer further functionalities such as automatic tree-level matching, derivation of the fluctuation operator, and EFT operator simplification, providing a complete software suite for fully automatic one-loop matching.

The functional matching procedure is well suited for this task, as it can be applied to generic weakly coupled UV theories, which have a mass power counting that allows for the construction of the corresponding low-energy effective theory. Thus,  can be used with a wide range of theories that can, but do not necessarily have to, match to the SMEFT. Furthermore, the matching can be performed to in principle arbitrary EFT dimension. However, in practice the procedure is limited by computation time.

The workflow of  is illustrated in fig. <ref> and a working example is presented in sec. <ref>. The starting point is a UV Lagrangian $\mathcal{L}_\mathrm{UV}$ that has to be provided by the user. This must include the specification of all symmetries, the field content, the interactions, and the power counting of the theory. Starting from this input and by applying functional derivatives on $\mathcal{L}_\mathrm{UV}$, the code derives and solves the EOMs for the heavy fields, thus obtaining the tree-level EFT Lagrangian. The program then derives the fluctuation operator $\mathcal{O}_{ij}$, and with that all propagators $\Delta_i$ and interaction terms $X_{ij}$ of the theory, as defined in eq. (<ref>). These are then passed on to the routines of the  [1] package, [ will be fully integrated into , and not be available as a standalone after the release of .] which identify and evaluate all relevant supertraces.

The one-loop EFT Lagrangian thus obtained includes, in general, redundant operators, i.e. operators that can be related to other operators present in the Lagrangian by operations like: integration by parts, field redefinitions, Fierz identities, Dirac algebra identities, etc. These identities and relations are implemented in , allowing for a further simplification of the Lagrangian obtained from the supertraces. The eventual goal is to obtain the EFT Lagrangian in a minimal basis without any redundancies.

To our knowledge, so far there is only one other fully automatic matching tool available called  [18]. This code performs the matching of effective theories based on the conventional diagrammatic matching procedure.
Since  and  are based on different techniques, having two programs available will allow for a more precise validation of both, although  and  have already been checked against results available in the literature such as, e.g., refs. [21, 22].

A well-known advantage of the functional matching prescription is that the EFT operators are directly obtained by evaluating the supertraces in eq. (<ref>). Thus, contrary to the diagrammatic matching procedure, no a priori knowledge of an EFT basis is required for the matching calculation. This can significantly simplify the matching in the case where the UV theory does not match onto an EFT with a well known basis for matching, such as the SMEFT, since the construction of a basis can be a non-trivial task.

Illustration of the functional EFT matching procedure. The blue (red) box show the parts of the procedure that are handled by the  [2] ( [1]) program.

§ ONE-LOOP MATCHING OF A VECTORLIKE FERMION TOY MODEL

In this section we reconsider a toy model with vectorlike fermions, already discussed in ref. [1], to illustrate the functional matching procedure. However, here we will focus on the application of  to this example. Notice that the code examples shown below are preliminary and can be subject to change in the version published eventually.

Consider a theory with a $U(1)$ gauge symmetry, where the gauge bosons is labeled $A_\mu$. In addition we include two vectorlike fermions that both have charge $1$ under the $U(1)$ group. One of the fermions is heavy $\Psi$ with mass $M$, whereas the other fermion $\psi$ is massless. Furthermore, we add a massless real scalar singlet $\phi$ to the theory that interacts with the fermions through a Yukawa interaction with coupling $y$. The Lagrangian of this UV theory reads

\begin{align}
\label{eq:example_Lagrangian}
\mathcal{L}_\mathrm{UV}
-\frac{1}{4} A_{\mu\nu} A^{\mu\nu}
+ \frac{1}{2} (\partial_\mu \phi)(\partial^\mu \phi)
+ i \bar{\psi} \slashed{D} \psi
+ i \bar{\Psi} \slashed{D} \Psi - M \bar{\Psi}\Psi
- (y \bar{\psi}_L \phi \Psi_R + \mathrm{h.c.})
\,,
\end{align}

with the covariant derivative $D_\mu = \partial_\mu - i g A_\mu$, and the field-strength tensor $A_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$. This theory is specified in  with the commands outlined below.

We can define the $U(1)$ gauge group labeled []InputU1e by
DefineGaugeGroup[U1e, U[1], e, A]

which also defines the associated coupling []Inpute, gauge field and field-strength tensor labeled []InputA. The remaining fields can then be defined with the commands
DefineField[\(\Psi\), Fermion, Charges -> {U1e[1]}, Mass -> {Heavy, M}]
DefineField[\(\psi\), Fermion, Charges -> {U1e[1]}, Mass -> 0]
DefineField[\(\phi\), Scalar, Mass -> 0, SelfConjugate -> True]

Finally, we define the Yukawa coupling []Inputy, which is of order $\mathcal{O}(m_L^0)$, by
DefineCoupling[y, EFTorder -> 0]

After these specifications  can automatically construct the free Lagrangian of the theory with the routine []InputFreeLag which needs to be combined with the Yukawa interactions to obtain the full Lagrangian shown in eq. (<ref>)
LUV = FreeLag[] - PlusHc[y[] \(\phi\)[] Bar[\(\psi\)[]]**PR**\(\Psi\)[]];
LUV // NiceForm

where []InputPR is a right-chirality projector, the Hermitian conjugate is included through []InputPlusHc, and all objects carrying spinor indices must be contracted using []Input**, due to their non-commutativity. In the second line the Lagrangian is printed in a readable format using the []InputNiceForm routine.

The dimension-six tree-level EFT Lagrangian can be derived with the []InputMatch routine
LEFT0 = Match[LUV, LoopOrder -> 0, EFTorder -> 6];
LEFT0 // NiceForm

The contribution by the last two terms vanishes on-shell, as can be shown using integration by parts and field redefinitions. These redundancies are removed in  by
LEFT0min = EoMSimplify[IBPSimplify[LEFT0]];
LEFT0min // NiceForm

-\(\frac{1}{4}\)A\(\mu\nu\)2 +\(\frac{1}{2}\)(D\(\mu\)\(\phi\))2 +i(\(\bar{\psi}\)\(\cdot\)\(\gamma\)\(\mu\)\(\cdot\)D\(\mu\)\(\cdot\)\(\psi\))

where the EFT contribution vanishes since $\psi$ is massless leading to the equation of motion $\slashed{D}\psi=0$.
Integrating out the heavy fermion $\Psi$ at one loop and dimension six in the EFT, can be achieved similarly
LEFT1 = Match[LUV, LoopOrder -> 1, EFTorder -> 6];
LEFT1min = EoMSimplify[IBPSimplify[LEFT1]];
LEFT1min // NiceForm

-14A\(\mu\nu\)2 +12(D\(\mu\)\(\phi\))2 +i(\(\psi\)_\(\cdot\)\(\gamma\)\(\mu\)\(\cdot\)D\(\mu\)\(\psi\))
-2\(\hbar\)y_yM2(1+Log[\(\mu\)2M2])\(\phi\)2 -\(\hbar\)y_2y2Log[\(\mu\)2M2]\(\phi\)4 +13\(\hbar\)y_3y3M2\(\phi\)6 +13\(\hbar\)y_ye2M2\(\phi\)2A\(\mu\nu\)2
-215\(\hbar\)e4M2(\(\psi\)_\(\cdot\)\(\gamma\)\(\mu\)\(\cdot\psi\))2 +736\(\hbar\)y_ye2M2(\(\psi\)_\(\cdot\)\(\gamma\)\(\mu\)\(\cdot\psi\))(\(\psi\)_\(\cdot\)\(\gamma\)\(\mu\)PL\(\cdot\psi\))

where we directly applied the simplification routines. Here $\hbar$ represents a loop factor, i.e. $\hbar\to\frac{\hbar}{16\pi^2}$. This gives the final matching result for the EFT Lagrangian at one loop. The results of this computation have been partially cross checked in ref. [1] with a diagrammatic by hand calculation. The automatic implementation, however, is considerably less time consuming and also less prone to errors than the derivation by hand, showing clearly the advantages of using codes such as  for one-loop matching computations, even for fairly simple models such as the one considered here.

§ CONCLUSIONS

The construction of EFTs describing the same low-energy dynamics as a given NP model is a common problem when studying BSM theories. Since EFT matching often has to be performed at the one-loop level and even several matching steps might be involved, this derivation is a challenging although very technical task. Thus, an automated solution for these calculations would significantly simplify the analyses of possible BSM theories. In these proceeding we presented and summarized the functional matching procedure, in which the heavy particles of a theory are removed from the theory by integrating over them in the path integral. This method is particularly suited for an automation in a computer algebra code, as the path integral yields both the effective operators and their coefficients, removing the need to first derive the EFT operator basis before starting the matching calculation, as is necessary for the diagrammatic matching procedure.
The functional method is currently being implemented in the  code  and we illustrated the functionalities of this package, by performing the fully automatic one-loop matching for a toy model with vectorlike fermions in this talk. When released,  will significantly simplify the analysis of a wide range of different BSM theories.

§ ACKNOWLEDGMENTS

I would like to thank the organizers of the “Les Rencontres de Physique de la Vallée d'Aoste” for the invitation. I am very grateful to my colleagues Javier Fuentes-Martín, Matthias König, Julie Pagès, and Anders E. Thomsen for their collaboration on this project.
The author has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under grant agreement 833280 (FLAY). The participation in the conference was supported by a scholarship awarded by the INFN.

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