# Mixed identities for oligomorphic automorphism groups

Manuel Bodirsky  M. Bodirsky, TU Dresden, 01062 Dresden, Germany
<EMAIL_ADDRESS>,  Jakob Schneider  J. Schneider, TU Dresden,
01062 Dresden, Germany<EMAIL_ADDRESS>and  Andreas Thom  A.
Thom, TU Dresden, 01062 Dresden, Germany<EMAIL_ADDRESS>
###### Abstract.

We study mixed identities for oligomorphic automorphism groups of countable
relational structures. Our main result gives sufficient conditions for such a
group to not admit a mixed identity without particular constants. We study
numerous examples and prove in many cases that there cannot be a non-singular
mixed identity.

###### Contents

  1. 1 Introduction
  2. 2 General homogeneous relational structures
  3. 3 Examples with mixed identities
  4. 4 Examples without mixed identities
  5. 5 Ruling out non-singular mixed identities for ${\rm Aut}(\mathbb{Q},<)$

##  1\. Introduction

In this paper, we study so-called _mixed identities_ for oligomorphic
automorphism groups of _countable relational structures_. This continues the
work of [bradfordschneiderthom2023non, bradfordschneiderthom2023length] and we
shall use the same notation as there: By a _word with constants_ in the group
$G$ and in $r$ variables $x_{1},\ldots,x_{r}$, we mean an element

$c_{0}x_{\iota(1)}^{\varepsilon(1)}c_{1}\cdots
c_{l-1}x_{\iota(l)}^{\varepsilon(l)}c_{l}\in G\ast\mathbf{F}_{r},$

of the free product, where $\mathbf{F}_{r}=\langle x_{1},\ldots,x_{r}\rangle$
denotes the free group of rank $r$; $\varepsilon(j)\in\\{\pm 1\\}$
($j\in\\{1,\ldots,l\\}$), and $c_{0},\ldots,c_{l}\in G$. We call $w$ _reduced_
if $\iota(j)=\iota(j+1)$ and $\varepsilon(j)=-\varepsilon(j+1)$ implies that
$c_{j}\notin\mathbf{Z}(G)$ for $j\in\\{1,\ldots,l-1\\}$. In this article, all
words with constants are reduced. The _length_ of the (reduced) word $w$ is
defined to be $\lvert w\rvert\coloneqq l$. We call the constants
$c_{1},\ldots,c_{l-1}$ _intermediate constants_. Moreover, we define the sets
of indices

$\displaystyle
J_{0}(w)\coloneqq\\{j\in\\{1,\ldots,l-1\\}\mid\iota(j)\neq\iota(j+1)\\},$
$\displaystyle
J_{+}(w)\coloneqq\\{j\in\\{1,\ldots,l-1\\}\mid\iota(j)=\iota(j+1)\text{ and
}\varepsilon(j)=\varepsilon(j+1)\\},$ $\displaystyle
J_{-}(w)\coloneqq\\{j\in\\{1,\ldots,l-1\\}\mid\iota(j)=\iota(j+1)\text{ and
}\varepsilon(j)=-\varepsilon(j+1)\\}.$

The elements of $J_{-}(w)$ are called _critical indices_ and the elements
$c_{j}$ with $j\in J_{-}(w)$ are called _critical constants_. The latter will
play a crucial role in the course of the article. We call $w$ _strong_ if
there are no critical indices. In other words, $w$ is strong if it does not
contain a subword of the form $x^{-1}cx$ or of the form $xcx^{-1}$ for a
variable $x$ and a constant $c$. (In [tomanov1985generalized] the term
_strict_ is used; however, we used the term _strong_ in our recent articles
[schneiderthom2022word, bradfordschneiderthom2023length].)

From $w$ we obtain the corresponding word map $w\colon G^{r}\to G$ by
substitution, which is also denoted by $w$: Write $w(g_{1},\ldots,g_{r})\in G$
for the image of $w$ under the unique homomorphism $G\ast\mathbf{F}_{r}\to G$
which fixes $G$ elementwise and maps $x_{i}\mapsto g_{i}$
($i\in\\{1,\ldots,r\\})$. The word $w$ is called a _mixed identity_ for $G$
iff $w\neq 1_{\mathbf{F}_{r}}$ and $w(G^{r})=\\{1_{G}\\}$. Strong mixed
identities arise naturally from coset identities on finite index subgroups and
deserve special attention.

We consider the _natural augmentation_ $\epsilon\colon
G\ast\mathbf{F}_{r}\to\mathbf{F}_{r}$ that sends the elements of $G$ to
$1_{\mathbf{F}_{r}}$ and fixes $\mathbf{F}_{r}.$ For a word with constants, we
call $\epsilon(w)$ the _content_ of $w$. A word is called _singular_ if
$\epsilon(w)=1_{\mathbf{F}_{r}}$ and _non-singular_ otherwise. Note that every
strong word of positive length is non-singular.

While the study of finite groups [bradfordschneiderthom2023non,
schneiderthom2024simple], finite-dimensional unitary groups
[klyachkothom2017new], and algebraic groups
[gordeevkunyavskiiplotkin2018wordmaps] shows that non-singular word maps tend
to have a larger word image, our understanding of non-singular word maps is
limited for a number of naturally occurring groups. In view of a resolution of
Kervaire’s Conjecture and its generalization [klyachkothom2017new]*Conjecture
1.1 it is of considerable importance to identify interesting groups so that
non-singular word maps are surjective or, as a first step, at least non-
trivial. A key example is the automorphism group of $(\mathbb{Q},<).$ It
follows from [alonsobrum2018spaces]*Lemma 3.3 that word maps without constants
are surjective for ${\rm Aut}(\mathbb{Q},<)$.

Apart from the case $\text{Sym}(\omega)$ [hullosin2016transitivity] and ${\rm
GL}(\mathbb{F}_{q}^{\oplus\omega})$ [bradfordschneiderthom2023non], word maps
for words with constants in oligomorphic automorphism groups of relational
structures have not been studied much in the literature. However, in
[etedadialiabadigaolemaitremelleray2021dense], Etedadialiabadi, Gao, Le
Maitre, and Melleray studied dense locally finite subgroups of automorphism
groups and obtained some results on the non-existence of mixed identities for
the automorphism groups of various versions of the Urysohn metric space and
related structures.

Our main motivation for this article is the following conjecture:

######  Conjecture 1.

Let $(X,R)$ be an countably infinite relational structure with oligomorphic
automorphism group. Every mixed identity for the group ${\rm Aut}(X,R)$ is
singular.

In Section 3, we prove Conjecture 1 for the groups $\text{Sym}(\omega)$, ${\rm
GL}(\mathbb{F}_{q}^{\oplus\omega})$, ${\rm Sp}(\mathbb{F}_{q}^{\oplus\omega})$
and ${\rm O}(\mathbb{F}_{q}^{\oplus\omega})$ and related examples. For the
automorphism group of the Rado graph and, more generally, structures whose age
has the free amalgamation property, as well as the random poset we prove that
there do not exist any mixed identities at all. More boldly, one could ask
whether every word map for a non-singular word with constants in a simple
oligomorphic automorphism group as above is surjective. This holds in a
different setting for the compact Lie group ${\rm U}(n)$ and in one variable
by the famous result of Gerstenhaber–Rothaus [gerstenhaberrothaus1962solution,
klyachkothom2017new] and for $\text{Aut}(\mathbb{Q},<)$ for words without
constants as mentioned above; but any positive result in this direction is
beyond the scope of this article.

Conjecture 1 is interesting for the group $(\mathbb{Q},<)$ and non-singularity
must be crucial, since it follows from work of Zarzycki [zarzycki2010limits]
that ${\rm Aut}(\mathbb{Q},<)$ admits a singular mixed identity. We will prove
the conjecture above for $(\mathbb{Q},<)$ in Section 5 under various
additional assumptions on the mixed identity $w$, for example that
$\epsilon(w)\not\in[[\mathbf{F}_{2},\mathbf{F}_{2}],[\mathbf{F}_{2},\mathbf{F}_{2}]]$,
see Theorem 5.1.2, or that $w$ is strong. Another sufficient condition
involves additional constraints on the critical constants. Indeed, the
conjecture also holds for words with constants if none of the critical
constants fixes a non-trivial interval in $\mathbb{Q}$, see Theorem 2.2.1.

The methods to deal with these cases and variations thereof have extensions to
automorphism groups of homogeneous relational structures with finite
signature, or, more generally, to oligomorphic automorphism groups; see
Section 2. Indeed, in Theorem 2.2.1 we identify various natural conditions
that ensure that a word with constants is not a mixed identity and explain
numerous examples of relational structures, see Section 3.

In parallel, we collect results from the literature and explain how they imply
the existence of mixed identities for particular automorphism groups, such as
the ones of the random permutation, the generic cyclic order, the generic
semi-linear order and the Ważewski dendrite. Many more examples can be treated
with the same methods, but we leave it there.

##  2\. General homogeneous relational structures

In this section we first recall basic concepts from model theory, then present
the main theorem about non-existence of certain mixed identities.

###  2.1. Preliminaries

All definitions in this section are standard in model theory (see, e.g.,
[hodges1997shorter]), except for Definition 2.1.12, which appears to be new.
Let $(X,R)$ be a pair consisting of a countable set $X$ and a set
$R=(R_{i})_{i\in I}$ of relations on $X$, with $R_{i}\subseteq X^{m_{i}}$
where $m_{i}\in\mathbb{N}$, for $i\in I$, is called the _arity_ of $R_{i}$. We
call such a pair $(X,R)$ a _relational structure_ , $X$ its _domain_ , and
$(m_{i})_{i\in I}$ its _signature_. We say that a relational structure $(X,R)$
is _finite_ if its domain $X$ is finite, and that it has _finite signature_ if
$I$ is finite. Subsequently, we will consider the automorphism group ${\rm
Aut}(X,R)$, which is the set of bijections $g$ of $X$ such that both $g$ and
$g^{-1}$ preserve all the relations of $R$, i.e., for all $i\in I$ we have
that $(x_{1},\ldots,x_{m_{i}})\in R_{i}$ holds if and only if
$(x_{1}.g,\ldots,x_{m_{i}}.g)\in R_{i}$.

For $Y\subseteq X$, we also consider the pair
$(Y,(\left.R\right\rvert_{Y})_{i\in I})$ and call it a _substructure_ of
$(X,R)$. We define the stabilizer of $Y$ by

${\rm stab}(Y)\coloneqq\\{g\in{\rm Aut}(X,R)\mid y.g=y\text{ for all }y\in
Y\\}.$

We start out with some definitions and lemmas, which will be necessary to
state and prove our main results.

######  Definition 2.1.1.

A relational structure $(X,R)$ is called _homogeneous_ if every isomorphism
between finite substructures can be extended to an automorphism of $(X,R)$.

######  Remark 2.1.2.

A subgroup of ${\rm Sym}(X)$ (which is equipped with the product topology
where $X$ is taken to be discrete) is closed if and only if it is the
automorphism group of a homogeneous relational structure.

Every countable homogeneous relational structure is (up to isomorphism)
uniquely given by its _age_ , i.e., by the class $\mathcal{C}$ of all finite
structures that are isomorphic to a substructure of $(X,R)$. Then, $(X,R)$ is
called the _Fraïssé limit_ of ${\mathcal{C}}$. By Fraïssé’s theorem, a class
$\mathcal{C}$ of finite structures of the same signature $(m_{i})_{i\in I}$ is
the age of a countable homogeneous structure if and only if ${\mathcal{C}}$ is
closed under isomorphism and substructures, has countably many isomorphism
classes, and has the so-called _amalgamation property_ : for all
$(X_{1},R_{1}),(X_{2},R_{2})\in{\mathcal{C}}$ such that the identity on
$X_{0}\coloneqq X_{1}\cap X_{2}$ is an isomorphism between the finite
substructures of $(X_{1},R_{1})$ and $(X_{2},R_{2})$ with domain $X_{0}$,
respectively, there exists $(Y,S)\in{\mathcal{C}}$ and for every
$i\in\\{1,2\\}$ and isomorphism $f_{i}$ between $(X,R_{i})$ and a substructure
of $(Y,S)$ such that $f_{1}(x)=f_{2}(x)$ for all $x\in X_{0}$. The structure
$(X,S)$ is then called the _amalgam_. Some stronger forms of the amalgamation
property will be relevant when discussing concrete examples in later sections.
For the _strong amalgamation property_ we additionally require that
$f_{1}(X_{1})\cap f_{2}(X_{2})=f_{1}(X_{0})=f_{2}(X_{0})$. For the _free
amalgamation property_ we even require that the structure $(X_{1}\cup
X_{2},((R_{1})_{i}\cup(R_{2}))_{i\in I})$ is in ${\mathcal{C}}$. Clearly, the
free amalgamation property implies the strong amalgamation property.

######  Definition 2.1.3.

${\rm Aut}(X,R)$ is called _oligomorphic_ if the componentwise action of ${\rm
Aut}(X,R)$ on $n$-tuples has for every $n\in\mathbb{N}$ only finitely many
orbits.

Throughout the text _definable_ will always mean _first-order definable with
parameters_ and if the parameters are restricted to some subset of the domain
it will be mentioned explicitly.

######  Remark 2.1.4.

Note that the automorphism group of every homogenenous relational structure
with finite signature is oligomorphic. Conversely, if ${\rm Aut}(X,R)$ is
oligomorphic, and we add all relations to $(X,R)$ that are first-order
definable without parameters in $(X,R)$, then the resulting expanded structure
is homogeneous and has the same automorphism group as the original structure
$(X,R)$.

######  Remark 2.1.5.

A relational structure has an oligomorphic automorphism group if and only if
it is _$\omega$ -categorical_, i.e., if its first-order theory has at most one
countable model up to isomorphism (a theorem of Engeler, Svenonius, and Ryll-
Nardzewski; see, e.g., [hodges1997shorter]). Moreover, a relation $S$ is
definable in an $\omega$-categorical structure $(X,R)$ without parameters if
and only if $S$ is preserved by ${\rm Aut}(X,R)$. We do not make real use of
the logic perspective here, because with the facts mentioned above all the
arguments can be translated into the language of permutation groups.

The following notions are standard in model theory.

######  Definition 2.1.6.

Let $(X,R)$ be a relational structure with an oligomorphic automorphism group
and let $Y$ be a subset of $X$. We set

$\displaystyle{\rm acl}(Y)\coloneqq\\{x\in X\mid\text{there exists a finite
set }W\subseteq Y$ $\displaystyle\text{ such that }x.{\rm stab}(W)\text{ is
finite}\\}$

and call it the _algebraic closure_.

For $Y$ finite, we have ${\rm acl}(Y)=\\{x\in X\mid x.{\rm stab}(Y)\mbox{ is
finite}\\}$. One verifies that ${\rm acl}$ is a closure operator. Note the
following simple observation.

######  Lemma 2.1.7.

Let $(X,R)$ be a relational structure with oligomorphic automorphism group. If
$Y\subseteq X$ is finite, then ${\rm acl}(Y)$ is also finite.

###### Proof.

Indeed, for $n=\lvert Y\rvert$, ${\rm Aut}(X,R)$ acts with finitely many
orbits on $X^{n+1}$. Hence the stabilizer ${\rm stab}(Y)\leq{\rm Aut}(X,R)$
also has at most finitely many finite orbits on the elements of $X\setminus
Y$. Hence ${\rm acl}(Y)$ is finite. ∎

######  Definition 2.1.8.

We say that $c\in{\rm Aut}(X,R)$ is _small_ if $c\in{\rm stab}(Y)$ for some
infinite subset $Y$ of $X$ which is definable in $(X,R)$.

The notion of small elements will play a crucial role in Theorem 2.2.1 below.

######  Lemma 2.1.9.

Let $(X,R)$ be a homogeneous relational structure with oligomorphic
automorphism group. The following are equivalent:

  1. $(i)$

If $Y\subseteq X$ is a finite set, $x\in X\setminus{\rm acl}(Y)$, and
$Z\subseteq x.{\rm stab}(Y)$ is a finite subset, then ${\rm stab}(x.{\rm
stab}(Y)\setminus Z)$ is trivial.

  2. $(ii)$

If $Y\subseteq X$ is a finite set, $x\in X\setminus{\rm acl}(Y)$, then ${\rm
stab}(x.{\rm stab}(Y))$ is trivial.

  3. $(iii)$

No non-trivial element of ${\rm Aut}(X,R)$ is small.

###### Proof.

We start by proving the implication (iii)$\Rightarrow$(i). Since $(X,R)$ is
homogeneous and ${\rm Aut}(X,R)$ is oligomorphic, the set $x.{\rm
stab}(Y)\setminus Z$ is infinite and definable with parameters $Y\cup Z$.
Thus, if (iii) holds, we conclude that ${\rm stab}(x.{\rm stab}(Y)\setminus
Z)$ is trivial. The implication (i)$\Rightarrow$(ii) is trivial.

Let’s now prove (ii)$\Rightarrow$(iii). Suppose that $W$ is an infinite set
which is definable in $(X,R)$ by a formula with parameters from a finite set
$Y$. Since $W$ is infinite and ${\rm acl}(Y)$ is finite by Lemma 2.1.7, there
exists $x\in W\setminus{\rm acl}(Y)$. Then $x.{\rm stab}(Y)\subseteq W$ is
infinite. Now, ${\rm stab}(x.{\rm stab}(Y))$ being trivial implies that ${\rm
stab}(W)$ is trivial. Hence, we conclude that (iii) holds. ∎

######  Definition 2.1.10.

We say that $(X,R)$ has _no algebraicity_ if for every finite subset $Y$ of
$X$, we have ${\rm acl}(Y)=Y$, i.e., all finite subsets of $X$ are
_algebraically closed_.

######  Remark 2.1.11.

It is well known that a homogeneous structure has no algebraicity if and only
if its age has the strong amalgamation property (see, e.g.,
[cameron1990oligomorphic]).

The following definition describes an algebraic property a relational
structure can have which allows us to prove very strong results on the non-
existence of mixed identities for the corresponding automorphism group.

######  Definition 2.1.12.

We say that a relational structure $(X,R)$ is _algebraically convex_ if for
every finite set $Y\subseteq X$, the following implication holds: if $x\in{\rm
acl}({\rm acl}(Y\cup\\{x\\})\setminus\\{x\\}),$ then $x\in{\rm acl}(Y)$.

The name of this property is inspired by the fact that the _convex hull_ as a
closure operator has the same property. Indeed, removing an extreme point from
a convex set preserves convexity. It is clear that no algebraicity implies
algebraical convexity. Further examples will be discussed in Sections 3.7 and
3.8. Examples which are not algebraically convex can be found in Section 3.2
and Section 3.3.

######  Definition 2.1.13.

We say that $c\in{\rm Aut}(X,R)$ is _slender_ if there exists a finite subset
$Y$ of $X$ and an infinite definable set $Z\subseteq X$, such that

$Z\subseteq\\{x\in X\mid x.c\in{\rm acl}(Y\cup\\{x\\})\\}.$

######  Remark 2.1.14.

Every small automorphism is slender. Indeed, if $c\in{\rm stab}(Z)$ for some
infinite definable set $Z$, then the definition of slender is satisfied for
$Y=\varnothing$ and $Z$. The converse holds if $(X,R)$ has no algebraicity.

Indeed, assume that there are a finite set $Y\subseteq X$ and an infinite
definable set $Z\subseteq X$ such that

$\displaystyle Z$ $\displaystyle\subseteq\\{x\in X\mid x.c\in{\rm
acl}(Y\cup\\{x\\})\\}$ $\displaystyle=\\{x\in X\mid x.c\in
Y\cup\\{x\\}\\}=Y.c^{-1}\cup\\{x\in X\mid x.c=x\\}.$

Then $c\in{\rm stab}(Z\setminus Y.c^{-1})$ and $Z\setminus Y.c^{-1}$ is an
infinite definable set, proving that $c$ is small.

Hence, we can think of being slender as a generalization of smallness. Indeed,
we think that the definition of slender assures in some way that a set of
‘generalized fixed points’ is large in the sense that it contains an infinite
definable subset. The concept of a slender automorphism will prove to be
useful beyond the algebraically convex setting, see Section 3.3.

######  Definition 2.1.15.

Let $(X,R)$ be a relational structure. We say that a tuple
$(x_{1},\dots,x_{n})\in X^{n}$ is _staggered algebraically independent_ if
$x_{i}\notin{\rm acl}(\\{x_{1},\dots,x_{i-1}\\})$ for every
$i\in\\{1,\dots,n\\}$. In particular, $x_{1}\notin{\rm acl}(\varnothing).$

Staggered algebraically independent tuples have a concrete meaning in many
circumstances. For example, a sequence of vectors $(v_{1},\dots,v_{n})$ in
$\mathbb{F}_{q}^{\omega}$ (see Section 3.3) is staggered algebraically
independent if and only if it is linearly independent.

###  2.2. The main theorem

At this point we are prepared to state and prove our main result.

######  Theorem 2.2.1.

Suppose that $(X,R)$ is a homogeneous relational structure with oligomorphic
automorphism group. Let $w\in{\rm Aut}(X,R)\ast\mathbf{F}_{r}$ be a word with
constants of positive length. Assume that one of the following conditions
holds true:

  1. $(i)$

$w$ is strong,

  2. $(ii)$

$w$ has no small critical constants and $(X,R)$ is algebraically convex,

  3. $(iii)$

$w$ has no slender critical constants.

Then, for all $n\in\mathbb{N}$, there exists a staggered algebraically
independent tuple $(\alpha_{1},\beta_{1},\dots,\alpha_{n},\beta_{n})\in
X^{2n}$ and $h_{1},\dots,h_{r}\in{\rm Aut}(X,R)$ such that

$\alpha_{i}.w(h_{1},\dots,h_{r})=\beta_{i},\quad\forall i\in\\{1,\dots,n\\}.$

In particular, $w$ is not a mixed identity.

###### Proof.

We prove (i), (ii), and (iii) simultaneously. Since the property of mapping
some staggered algebraically independent tuple in the required way is
invariant under conjugation, we may assume that the first constant $c_{0}$ is
trivial. Thus, without loss of generality, $w$ can be written as

$w=x_{\iota(1)}^{\varepsilon(1)}c_{1}x_{\iota(2)}^{\varepsilon(2)}\cdots
x_{\iota(l-1)}^{\varepsilon(l-1)}c_{l-1}x_{\iota(l)}^{\varepsilon(l)}c_{l}\in{\rm
Aut}(X,R)\ast\mathbf{F}_{r}.$

We want to use induction on $n$ to establish the claim. Subsequently, assume
we are in the $n$th step of the induction for some $n\geq 1$. Let
$\Lambda_{k}^{\varepsilon}$ be some finite sets specified later which contain
elements from the previous steps (if any) of the induction
($\varepsilon\in\\{\pm 1\\}$, $k\in\\{1,\ldots,r\\}$). In the case $n=1$, we
set $\Lambda_{k}^{\varepsilon}\coloneqq\varnothing$ for all $\varepsilon$ and
$k$. Let $\alpha_{n}\in X\setminus({\rm
acl}(\Lambda_{\iota(1)}^{\varepsilon(1)})\cup{\rm
acl}(\\{\alpha_{1},\beta_{1},\ldots,\alpha_{n-1},\beta_{n-1}\\}))$ be
arbitrary and $\beta_{n}\in X\setminus{\rm
acl}(\\{\alpha_{1},\beta_{1},\ldots,\alpha_{n-1},\beta_{n-1},\alpha_{n}\\})$
to be defined later. For $j\in\\{1,\ldots,l\\}$ we will choose elements
$\omega_{j}^{-\varepsilon(j)}\in X$ with the aim to define maps
$h_{1},\ldots,h_{r}\in{\rm Aut}(X,R)$ such that

$\displaystyle\alpha_{n}\eqqcolon\omega_{1}^{\varepsilon(1)}\stackrel{{\scriptstyle
h^{\varepsilon(1)}_{\iota(1)}}}{{\mapsto}}\omega_{1}^{-\varepsilon(1)}\stackrel{{\scriptstyle
c_{1}}}{{\mapsto}}\omega_{2}^{\varepsilon(2)}\stackrel{{\scriptstyle
h^{\varepsilon(2)}_{\iota(2)}}}{{\mapsto}}\cdots\stackrel{{\scriptstyle
c_{l-1}}}{{\mapsto}}\omega_{l}^{\varepsilon(l)}\stackrel{{\scriptstyle
h^{\varepsilon(l)}_{\iota(l)}}}{{\mapsto}}\omega_{l}^{-\varepsilon(l)}\stackrel{{\scriptstyle
c_{l}}}{{\mapsto}}\beta_{n}.$

We see that the elements $\omega_{j}^{\varepsilon(j)}$ are determined by
$\omega_{1}^{\varepsilon(1)}\coloneqq\alpha_{n}$ and the equations
$\omega_{j+1}^{\varepsilon(j+1)}\coloneqq\omega_{j}^{-\varepsilon(j)}.c_{j}$
for $j\in\\{1,\ldots,l-1\\}$. For $k\in\\{1,\ldots,r\\}$ and
$\varepsilon\in\\{\pm 1\\}$, define

$\displaystyle\Omega^{\varepsilon}_{k}$
$\displaystyle\coloneqq\\{\omega^{\varepsilon}_{j}\mid
j\in\\{1,\ldots,l\\},\iota(j)=k\\}\cup\Lambda_{k}^{\varepsilon}.$

For $j\in\\{0,1,\ldots,l\\}$, define

$\Omega_{j,k}^{\varepsilon}\coloneqq\\{\omega_{j^{\prime}}^{\varepsilon}\mid
j^{\prime}\leq j,\iota(j^{\prime})=k\\}\cup\Lambda_{k}^{\varepsilon}.$

Note that here $\Omega_{0,k}^{\varepsilon}\coloneqq\Lambda_{k}^{\varepsilon}$
for all $k$. For better readability, in exponents we write in the following
$+$ instead of $+1$ and $-$ instead of $-1$.

In order for the $h_{k}$ to exist, we want to achieve in the course of the
inductive process that

  * •

the substructures induced by $\Omega^{+}_{k}$ and $\Omega^{-}_{k}$ are
isomorphic (for all $k\in\\{1,\ldots,r\\}$) via the natural map sending
$\omega^{+}_{j}$ to $\omega^{-}_{j}$ for $j\in\\{1,\ldots,l\\}$ with
$\iota(j)=k$, and mapping the set $\Lambda_{k}^{+}$ bijectively onto
$\Lambda_{k}^{-}$;

  * •

$\omega_{j}^{\varepsilon}\neq\omega_{j^{\prime}}^{\varepsilon}$ and
$\omega_{j}^{\varepsilon}\notin\Lambda_{\iota(j)}^{\varepsilon}$, for
$\varepsilon\in\\{\pm 1\\}$ and $j,j^{\prime}\in\\{1,\ldots,l\\}$ such that
$j\neq j^{\prime}$ and $\iota(j)=\iota(j^{\prime})$.

The proof proceeds by induction on $j\in\\{1,\ldots,l\\}$. In the $j$th step
we extend the isomorphism between the substructures on $\Omega_{j-1,k}^{+}$
and $\Omega_{j-1,k}^{-}$ from the previous step to an isomorphism between the
substructures on $\Omega_{j,k}^{+}$ and $\Omega_{j,k}^{-}$ (for all
$k\in\\{1,\ldots,r\\}$). For $k\neq\iota(j)$, there is nothing to do and we
just define $\Omega_{j,k}^{\varepsilon}\coloneqq\Omega_{j-1,k}^{\varepsilon}$
for $\varepsilon\in\\{\pm 1\\}$. If $k=\iota(j)$, we will add the arrow
$\omega_{j}^{+}\mapsto\omega_{j}^{-}$ to our partial isomorphism from the
previous step and set
$\Omega_{j,k}^{\varepsilon}=\Omega_{j,\iota(j)}^{\varepsilon}\coloneqq\Omega_{j-1,\iota(j)}^{\varepsilon}\cup\\{\omega_{j}^{\varepsilon}\\}$
for $\varepsilon\in\\{\pm 1\\}$. During the inductive process, we want to
ensure that $\omega_{j}^{\varepsilon}\notin{\rm
acl}(\Omega_{j-1,\iota(j)}^{\varepsilon})$ for any $\varepsilon$. For $j=1$,
we have by definition

$\omega^{\varepsilon(1)}_{1}\coloneqq\alpha_{n}\notin{\rm
acl}(\Omega_{0,\iota(1)}^{\varepsilon(1)})\cup{\rm
acl}(\\{\alpha_{1},\beta_{1},\ldots,\alpha_{n-1},\beta_{n-1}\\}).$

For the inductive step, we assume that

$\omega_{1}^{+},\omega_{1}^{-},\ldots,\omega_{j-1}^{+},\omega_{j-1}^{-},\omega_{j}^{\varepsilon(j)}$

are already defined and we want to determine a suitable choice for
$\omega_{j}^{-\varepsilon(j)}$. By the inductive assumption,
$\omega_{j}^{\varepsilon(j)}\notin{\rm
acl}(\Omega_{j-1,\iota(j)}^{\varepsilon(j)})$ and there is an isomorphism
between the substructures on $\Omega_{j-1,\iota(j)}^{\varepsilon(j)}$ and
$\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}$. Let $g$ be an extension of this
partial isomorphism to an automorphism of $(X,R).$ A priori, the set of valid
choices for $\omega_{j}^{-\varepsilon(j)}$ lies in the definable set

$\Sigma_{j}\coloneqq\omega_{j}^{\varepsilon(j)}.g{\rm
stab}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}).$

Since $\omega_{j}^{\varepsilon(j)}.g\notin{\rm
acl}(\Omega_{j-1,\iota(j)}^{\varepsilon(j)}).g={\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)})$, the set $\Sigma_{j}$ is
infinite.

Apart from this, we need that if $j\in\\{1,\ldots,l-1\\}$, then
$\omega_{j+1}^{\varepsilon(j+1)}=\omega_{j}^{-\varepsilon(j)}.c_{j}\notin{\rm
acl}(\Omega_{j,\iota(j+1)}^{\varepsilon(j+1)})$. In order to achieve this, we
distinguish the following three cases.

_Case 1: $j\in J_{0}(w)$._ In this case, we have $\iota(j)\neq\iota(j+1)$ and
hence
$\Omega_{j,\iota(j+1)}^{\varepsilon(j+1)}=\Omega_{j-1,\iota(j+1)}^{\varepsilon(j+1)}$
which is already fixed. Then we must require that

$\omega_{j}^{-\varepsilon(j)}\in\Sigma_{j}\setminus({\rm
acl}(\Omega_{j-1,\iota(j+1)}^{\varepsilon(j+1)}).c_{j}^{-1}),$

which is clearly possible since $\Sigma_{j}$ is infinite and ${\rm
acl}(\Omega_{j-1,\iota(j+1)}^{\varepsilon(j+1)})$ is finite.

_Case 2: $j\in J_{+}(w)$._ Then $\iota(j)=\iota(j+1)$ and
$\varepsilon(j)=\varepsilon(j+1)$, so that
$\Omega_{j,\iota(j+1)}^{\varepsilon(j+1)}=\Omega_{j,\iota(j)}^{\varepsilon(j)}$
which is also already fixed. Then we must have that

$\omega_{j}^{-\varepsilon(j)}\in\Sigma_{j}\setminus({\rm
acl}(\Omega_{j,\iota(j)}^{\varepsilon(j)}).c_{j}^{-1}),$

which is again possible since $\Sigma_{j}$ is infinite and ${\rm
acl}(\Omega_{j,\iota(j)}^{\varepsilon(j)})$ is finite.

Thus, the proof of (i) is complete, since Case 1 and Case 2 are the only ones
that occur when $w$ is strong as $J_{-}(w)=\varnothing$. Indeed, note that in
the last step we have infinitely many choices for
$\omega_{l}^{-\varepsilon(l)}$. In particular, we can choose
$\omega_{l}^{-\varepsilon(l)}$ such that
$\omega_{l}^{-\varepsilon(l)}.c_{l}=\beta_{n}\notin{\rm
acl}(\\{\alpha_{1},\beta_{1},\ldots,\alpha_{n}\\})$ which is a finite set by
Lemma 2.1.7. As discussed before, by homogeneity, we obtain suitable
$h_{1},\ldots,h_{k}\in{\rm Aut}(X,R)$ with
$\alpha_{n}.w(h_{1},\ldots,h_{k})=\beta_{n}.$

To finish the proof of (ii), we also need to consider the case
$J_{-}(w)\neq\varnothing$:

_Case 3: We have $j\in J_{-}(w)$._ Then $\iota(j)=\iota(j+1)$ and
$\varepsilon(j)=-\varepsilon(j+1)$. In this case we obtain

$\Omega_{j,\iota(j+1)}^{\varepsilon(j+1)}=\Omega_{j,\iota(j)}^{-\varepsilon(j)}=\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega_{j}^{-\varepsilon(j)}\\}.$

We need to find $\omega_{j}^{-\varepsilon(j)}$ satisfying

$\omega_{j}^{-\varepsilon(j)}\in\Sigma_{j}\quad\mbox{and}\quad\omega_{j}^{-\varepsilon(j)}.c_{j}\not\in{\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega_{j}^{-\varepsilon(j)}\\}).$

Let’s fix $\omega\in\Sigma_{j}$ and recall that by definition
$\omega\notin{\rm acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)})$. By the
assumption that $(X,R)$ is algebraically convex we have

$\omega\not\in{\rm acl}\left({\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega\\})\setminus\\{\omega\\}\right).$

Consider now

$\Sigma_{j}^{\prime}\coloneqq\omega.{\rm stab}\left({\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega\\})\setminus\\{\omega\\}\right)\subseteq\Sigma_{j},$

which is definable and infinite by assumption on $\omega$. For all
$\omega^{\prime}\in\Sigma^{\prime}_{j},$ we have

${\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega^{\prime}\\})\setminus\\{\omega^{\prime}\\}={\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega\\})\setminus\\{\omega\\},$

since $\omega^{\prime}=\omega.k$ for some $k\in{\rm stab}({\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega\\})\setminus\\{\omega\\})$
(so that $k$ fixes ${\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega\\})\setminus\\{\omega\\})$
pointwise, while it also maps $\omega$ to $\omega^{\prime}$). Since
$\Sigma_{j}^{\prime}$ is infinite, the definable set

$\Sigma^{\prime}_{j}\setminus\left({\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega\\})\setminus\\{\omega\\}\right).c_{j}^{-1}$

is also infinite. Since $c_{j}$ is not small, we find

$\omega_{j}^{-\varepsilon(j)}\in\Sigma_{j}^{\prime}\setminus\left({\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega\\})\setminus\\{\omega\\}\right).c_{j}^{-1}$

which is not fixed by $c_{j}.$ Then,
$\omega_{j}^{-\varepsilon(j)}.c_{j}\not\in{\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega_{j}^{-\varepsilon(j)}\\})$
as required. Indeed, this follows from the decomposition

${\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega_{j}^{-\varepsilon(j)}\\})=\left({\rm
acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{\omega\\})\setminus\\{\omega\\}\right)\cup\\{\omega_{j}^{-\varepsilon(j)}\\}.$

This completes the proof of (ii), since again in the last step there are
infinitely many choices for
$\beta_{n}=\omega_{l}^{-\varepsilon(l)}.c_{l}\notin{\rm
acl}(\\{\alpha_{1},\beta_{1},\ldots,\alpha_{n}\\})$ as required.

Similarly, we need a special argument to finish the proof of (iii). Indeed,
for $j\in\\{1,\ldots,l-1\\}$, in Case 3 of the proof, we need that the set of
all $x\in X$ such that

$x.c_{j}\notin{\rm acl}(\Omega_{j-1,\iota(j)}^{-\varepsilon(j)}\cup\\{x\\})$

intersects with the infinite definable set $\Sigma_{j}$. By definition, this
is the case if $c_{j}$ is not slender. The case $j=l$ is dealt with as above.

This completes the argument for (i), (ii) and (iii) in step $n\geq 1$ of the
induction. For the next step $n+1$, the proof is the same, but we replace
$\Lambda_{k}^{\varepsilon}$ by the set $\Omega_{k}^{\varepsilon}$ from step
$n$. ∎

We record the following three corollaries:

######  Corollary 2.2.2.

A closed oligomorphic subgroup of ${\rm Sym}(\mathbb{N})$ does not admit
strong mixed identities.

######  Corollary 2.2.3.

Every mixed identity of the automorphism group of an $\omega$-categorical
structure which is algebraically convex must have a small critical constant.

######  Corollary 2.2.4.

Every mixed identity of the automorphism group of an $\omega$-categorical
structure must have a slender critical constant.

######  Remark 2.2.5.

Despite numerous examples in Section 3 we do not know whether the existence of
non-trivial small or slender automorphisms for some $\omega$-categorical
relational structure $(X,R)$ implies the existence of a mixed identity for
${\rm Aut}(X,R)$.

##  3\. Examples with mixed identities

Now we will provide some examples of homogeneous relational structures with
oligomorphic automorphism group for which we can apply Theorem 2.2.1. We will
also provide mixed identities in the examples in this section; needless to
say, those mixed identities are singular in accordance with Conjecture 1.

###  3.1. The countably infinite set

The simplest example of a homogeneous structure with an oligomorphic
automorphism group is a _countably infinite set_ $X$ equipped with no
relations at all. In this case, $(X,\varnothing)$ has no algebraicity, and its
automorphisms are small if and only if they are slender if and only if they
have finite support. Thus, we can apply Theorem 2.2.1 to deduce that every
mixed identity for ${\rm Sym}(X)$ has a critical constant with finite support,
a fact which was originally proved by Hull–Osin in [hullosin2016transitivity].
Note that for $\tau$ a transposition, $w(x)=[\tau,x]^{6}$ is such an identity
and there are many more of the same kind. Despite the abundance of mixed
identities, we have the following consequence of Theorem 2.2.1 or
[schneiderthom2022word]*Theorem 1:

######  Theorem 3.1.1.

There are no non-singular mixed identities for ${\rm Sym}(X)$.

###### Proof.

The proof proceeds as the proof of [bradfordschneiderthom2023non]*Corollary 1.
Let $w\in{\rm Sym}(X)\ast\mathbf{F}_{r}$ be a non-singular word with
constants. We replace the critical constants of finite support in $w$ one by
one with the identity element. This leads to cancellation of the variables,
changes of the neighboring constants, and reduces the word length in each step
by at least two. Hence, this process ends after finitely many steps and leaves
us with $w^{\prime}\in{\rm Sym}(X)\ast\mathbf{F}_{r}$ without critical
constants of finite support. It is clear that the content is unchanged in this
process, i.e., $\epsilon(w)=\epsilon(w^{\prime}).$ In particular, the word
length of $w^{\prime}$ is positive. Moreover, it is clear from the
construction that there exists a constant $f\in\mathbb{N}$ such that

${\rm dist}(w(h_{1},\ldots,h_{r}),w^{\prime}(h_{1},\ldots,h_{r}))\leq f$

for all $h_{1},\ldots,h_{r}\in{\rm Sym}(X),$ where ${\rm dist}$ denotes the
Hamming distance of permutations. Indeed, $f$ can be taken to be the sum of
the support sizes of all critical constants that have been replaced. Now, by
Theorem 2.2.1(ii), there exists a particular choice $h_{1},\ldots,h_{r}\in H$
such that the support of $w^{\prime}(h_{1},\ldots,h_{r})$ is larger than
$f+1.$ This implies that $w(h_{1},\ldots,h_{r})\neq{\rm id}_{X}$ and the proof
is complete. ∎

###  3.2. The equivalence relation with $k$-element classes

Let $X$ be a countably infinite set and let $k\in{\mathbb{N}}$. On $X$ there
is an _equivalence relation $E_{k}\subseteq X^{2}$ with $k$-element
equivalence classes_; note that this generalizes the example from the previous
section, because ${\rm Aut}(X,E_{1})={\rm Sym}(X)$. The structure $(X,E_{k})$
is homogeneous and ${\rm acl}(Y)=\bigcup_{y\in Y}{\rm acl}(\\{y\\})$, but the
structure is not algebraically convex. The automorphism group of $(X,E_{k})$
is isomorphic to $S_{k}\wr{\rm Sym}(X/E_{k})$, and the slender automorphisms
are those in $S_{k}\wr{\rm Sym}_{\rm fin}(X/E_{k})$, where ${\rm Sym}_{\rm
fin}(X/E_{k})$ denotes the subgroup of permutations of finite support. Theorem
2.2.1(iii) implies that every mixed identity for $S_{k}\wr{\rm Sym}(X/E_{k})$
has a critical constant in $S_{k}\wr{\rm Sym}_{\rm fin}(X/E_{k})$. While there
are non-trivial singular mixed identities as $S_{k}^{X/E_{k}}$ is a normal
subgroup of ${\rm Aut}(X,E_{k})$ and this group has many such identities, we
obtain the following as an application of Theorem 2.2.1(iii) or a direct
consequence of Theorem 3.1.1.

######  Corollary 3.2.1.

The group ${\rm Aut}(X,E_{k})$ does not have any non-singular mixed identites.

###  3.3. The countably infinite dimensional vector space

The _countably infinite dimensional vector space_
$\mathbb{F}_{q}^{\oplus\omega}$ over the finite field $\mathbb{F}_{q}$ is
another interesting case. Further complications arise if we add a non-
degenerate symmetric or alternating bilinear form
$f\colon\mathbb{F}_{q}^{\oplus\omega}\times\mathbb{F}_{q}^{\oplus\omega}\to\mathbb{F}_{q}$
to the structure. We denote the respective automorphism groups by ${\rm
GL}(\mathbb{F}_{q}^{\oplus\omega}),{\rm O}(\mathbb{F}_{q}^{\oplus\omega}),$
and ${\rm Sp}(\mathbb{F}_{q}^{\oplus\omega})$. They are oligomorphic; see,
e.g., [evans1991small].

A concrete relational structure for $\mathbb{F}_{q}^{\oplus\omega}$ such that
the automorphism group is homogeneous is given by the infinite set of
relations

$R_{\lambda_{1},\ldots,\lambda_{n}}\coloneqq\left\\{(x_{1},\ldots,x_{n})\in(\mathbb{F}_{q}^{\oplus\omega})^{n}\,\middle|\,\sum_{i=1}^{n}\lambda_{i}x_{i}=0\right\\}$

for $\lambda_{1},\ldots,\lambda_{n}\in\mathbb{F}_{q}$ and $n\in\mathbb{N}.$ In
the presence of a bilinear form $f$, we also add the relations
$B_{\lambda}\coloneqq\\{(x,y)\in(\mathbb{F}_{q}^{\oplus\omega})^{2}\mid
f(x,y)=\lambda\\}$ for $\lambda\in\mathbb{F}_{q}.$

The algebraic closure coincides with the linear hull for these examples. Note
that these structures are not algebraically convex so that Theorem 2.2.1(ii)
cannot be applied. Instead, we will apply Theorem 2.2.1(iii). Before we do so,
let’s characterize slender automorphisms in more familiar terms.

######  Lemma 3.3.1.

An element $c\in{\rm GL}(\mathbb{F}_{q}^{\oplus\omega})$, ${\rm
O}(\mathbb{F}_{q}^{\oplus\omega})$, and ${\rm
Sp}(\mathbb{F}_{q}^{\oplus\omega})$ is slender if and only if ${\rm
rk}(c-\lambda{\rm id})<\infty$ for some $\lambda\in\mathbb{F}_{q}$.

###### Proof.

Let’s first describe the infinite definable subsets in sufficient detail. In
the case of no bilinear form, the infinite definable sets contain the
complement of a subspace of finite dimension; in particular, they are co-
finite. In the other two cases, further constraints arise from the bilinear
forms encoded in the relations $B_{\lambda}$ for $\lambda\in\mathbb{F}_{q}$.
In any case, an infinite definable set contains an affine subspace of finite
co-dimension.

Assume that ${\rm rk}(c-\lambda{\rm id})$ is finite and set
$g_{\lambda}\coloneqq c-\lambda{\rm id}$. Then $W_{\lambda}\coloneqq{\rm
im}(g_{\lambda})$ is finite and for all $x\in\mathbb{F}_{q}^{\oplus\omega}$,
we get $x.(c-\lambda{\rm id})\in W_{\lambda}$ and hence $x.c\in\langle
W_{\lambda},x\rangle={\rm acl}(W_{\lambda}\cup\\{x\\})$. Hence, $c$ is
slender.

Conversely, if ${\rm rk}(c-\lambda{\rm id})=\infty$ for all
$\lambda\in\mathbb{F}_{q}$, then $Y.g_{\lambda}^{-1}\subseteq X$ has infinite
codimension for all finite-dimensional $Y$. Now, an infinite definable sets
contains an affine subspace of finite codimension; hence, it cannot be
contained in a subspace of infinite codimension. Whence, we conclude that $c$
is not slender. ∎

We can now apply Theorem 2.2.1(iii).

######  Theorem 3.3.2.

Every mixed identity for the groups ${\rm
GL}(\mathbb{F}_{q}^{\oplus\omega}),{\rm O}(\mathbb{F}_{q}^{\oplus\omega})$,
and ${\rm Sp}(\mathbb{F}_{q}^{\oplus\omega})$ contains a critical constant $c$
such that ${\rm rk}(c-\lambda{\rm id})<\infty$ for some
$\lambda\in\mathbb{F}_{q}$.

A stronger result was already proven along with quantitative forms in finite
dimensions in [bradfordschneiderthom2023non] for ${\rm
GL}(\mathbb{F}_{q}^{\oplus\omega})$ and will appear in a forthcoming paper
[schneiderthom2024simple] in the other cases. While it is well-known that the
groups ${\rm GL}(\mathbb{F}_{q}^{\oplus\omega})$, ${\rm
O}(\mathbb{F}_{q}^{\oplus\omega})$, ${\rm Sp}(\mathbb{F}_{q}^{\oplus\omega})$
admit singular mixed identities, we obtain the following consequence:

######  Corollary 3.3.3.

The groups ${\rm GL}(\mathbb{F}_{q}^{\oplus\omega}),{\rm
O}(\mathbb{F}_{q}^{\oplus\omega})$, and ${\rm
Sp}(\mathbb{F}_{q}^{\oplus\omega})$ do not admit non-singular mixed
identities.

###### Proof.

Note that Theorem 2.2.1(iii) implies that for every $n\in\mathbb{N}$ there
exist $h_{1},\ldots,h_{k}$ with

$\min\\{{\rm rk}(w(h_{1},\ldots,h_{k})-\lambda{\rm
id})\mid\lambda\in\mathbb{F}_{q}^{\times}\\}\geq n.$

The proof proceeds as the proof of [bradfordschneiderthom2023non]*Corollary 1.
∎

######  Remark 3.3.4.

Note that ${\rm GL}(\mathbb{F}_{q}^{\oplus\omega})$ does not have non-trivial
small elements, since any cofinite set of vectors spans the entire vector
space. Hence, from this example we also see that Theorem 2.2.1(ii) does not
hold without the assumption of algebraic convexity.

###  3.4. The countable unbounded dense linear order

Another interesting case to consider is the (up to isomorphism unique)
_countable unbounded dense linear order_ $(\mathbb{Q},<)$, which is the
Fraïssé limit of all finite linear orders.

######  Lemma 3.4.1.

The structure $(\mathbb{Q},<)$ has no algebraicity and its small automorphisms
are precisely those that fix a non-trivial interval pointwise.

This follows from the fact that every infinite definable set contains a non-
trivial interval. Then Lemma 3.4.1 and Theorem 2.2.1(ii) implies the
following.

######  Corollary 3.4.2.

Every mixed identity for ${\rm Aut}(\mathbb{Q},<)$ has a critical constant
which fixes a non-trivial interval pointwise.

A variation of the proof of [zarzycki2010limits]*Proposition 2.2 shows that
the group ${\rm Aut}(\mathbb{Q},<)$ has singular mixed identities. We denote
by ${\rm supp}(g)=\\{x\in\mathbb{Q}\mid x.g\neq x\\}$ the support of an
element $g\in{\rm Aut}(\mathbb{Q},<)$.

######  Lemma 3.4.3.

There exist singular mixed identities for ${\rm Aut}(\mathbb{Q},<)$.

###### Proof.

Let $g_{1},g_{2},g_{3}\in{\rm Aut}(\mathbb{Q},<)$ be three non-trivial
elements having their support in non-trivial intervals
$I_{1}=(a_{1},b_{1}),I_{2}=(a_{2},b_{2}),I_{3}=(a_{3},b_{3})$ with the
property that $a_{1}<b_{1}<a_{2}<b_{2}<a_{3}<b_{3}$. Then,

$w\coloneqq[[g_{1}^{x},g_{3}],[g_{2}^{x},g_{2}]]\in{\rm
Aut}(\mathbb{Q},<)\ast\langle x\rangle$

is a mixed identity for ${\rm Aut}(\mathbb{Q},<).$ To see this, we need two
basic facts:

  1. $(i)$

${\rm supp}(h^{g})={\rm supp}(h).g$ for any $g,h\in{\rm Aut}(\mathbb{Q},<)$,
and

  2. $(ii)$

$[g,h]={\rm id}_{\mathbb{Q}}$ if ${\rm supp}(g)\cap\sup(h)=\varnothing.$

If $I_{1}.g\cap I_{3}=\varnothing$, then by (i) ${\rm
supp}(g_{1}^{g})\subseteq I_{1}.g$ and by (ii) we conclude
$[g_{1}^{g},g_{3}]={\rm id}_{\mathbb{Q}}$ and hence $w(g)={\rm
id}_{\mathbb{Q}}.$ On the other side, if $I_{1}.g\cap I_{3}\neq\varnothing$
then by the monotonicity of $g$, we must have $I_{2}.g\cap I_{2}=\varnothing$.
Hence, we conclude in a similar way $[g_{2}^{g},g_{2}]={\rm id}_{\mathbb{Q}}$
and hence $w(g)={\rm id}_{\mathbb{Q}}.$ This finishes the proof. ∎

We will come back to the study of non-singular mixed identities for the group
${\rm Aut}(\mathbb{Q},<)$ in Section 5.

###  3.5. The random permutation

The _random permutation_ is the (up to isomorphism unique) countable
homogeneous structure such that every finite structure with two linear orders
is isomorphic to a substructure of $(P,<_{1},<_{2})$. Note that $(P,<_{1})$
and $(P,<_{2})$ are each isomorphic to $({\mathbb{Q}},<)$, so we may identify
$P$ with a subset of $\mathbb{Q}\times\mathbb{Q}$, together with $<_{1}$ and
$<_{2}$ induced by the two orderings coming from the coordinate projections.

######  Lemma 3.5.1.

The random permutation $(P,<_{1},<_{2})$ has non-trivial small automorphisms
and its automorphism group admits a mixed identity.

###### Proof.

The set $Y\coloneqq\\{x\in P\mid x<_{1}0\wedge x<_{2}0\\}\cup\\{0\\}$ is an
infinite set which is definable with the parameter $0$. We claim that ${\rm
stab}(Y)$ acts transitively on $Z\coloneqq\\{x\mid 0<_{1}x\wedge 0<_{2}x\\}$,
i.e., if $u,v\in Z$ then there exists a (small) automorphism $\alpha$ of
$(P,<_{1},<_{2})$ which fixes $Y$ pointwise and maps $u$ to $v$. Indeed, by
the homogeneity of $(P,<_{1},<_{2})$ there exists an automorphism $\beta$ of
$(P,<_{1},<_{2})$ which maps $u$ to $v$ and fixes $0$. Define $\alpha$ to be
the map that fixes all elements of $Y$ pointwise and agrees with $\beta$ on
$P\setminus Y$; then $\alpha$ is an automorphism of $(P,<_{1},<_{2})$.

We set $p<q$ iff $p<_{1}q$ and $p<_{2}q$. Choose six elements

$a_{1},b_{1},a_{2},b_{2},a_{3},b_{3}\in P$

such that $a_{1}<b_{1}<a_{2}<b_{2}<a_{3}<b_{3}$. Similarly as above, for each
$i\in\\{1,2,3\\}$ we may find a non-trivial automorphism $g_{i}$ whose support
lies in $I_{i}\coloneqq\\{x\mid a_{i}<x<b_{i}\\}$. If $I_{1}.g\cap
I_{3}\neq\varnothing$, then $I_{2}.g$ cannot intersect $I_{2}$. We now argue
as in the proof of Theorem 3.6.1. ∎

###  3.6. The generic cyclic order

Consider now ${\rm Aut}(\mathbb{Q}/\mathbb{Z},t)$, where $t$ is the ternary
betweenness relation, i.e., $t(\alpha,\beta,\gamma)$ holds if and only if
$\beta$ lies in the counter-clockwise oriented segment from $\alpha$ to
$\gamma$, where we view $\mathbb{Q}/\mathbb{Z}$ as a subset of
${\mathbb{S}}^{1}$ in a natural way. We can also consider the group ${\rm
Aut}^{\pm}(\mathbb{Q}/\mathbb{Z},t)$ of automorphisms of
$\mathbb{Q}/\mathbb{Z}$ that either preserve or reverse the betweenness
relation. This group contains ${\rm Aut}(\mathbb{Q}/\mathbb{Z},t)$ as a
subgroup of index $2$ and acts 3-transitively on $\mathbb{Q}/\mathbb{Z}$. The
result follows also from work of Le Boudec–Matte Bon; see
[leboudecmattebon2022triple]*Lemma 4.6.

######  Theorem 3.6.1.

Let $g_{1},g_{2},g_{3},g_{4},g_{5}\in{\rm Aut}(\mathbb{Q}/\mathbb{Z},t)$ be
five non-trivial elements having their support in cyclically counter-clockwise
ordered, pairwise disjoint intervals
$I_{1},I_{2},I_{3},I_{4},I_{5}\subseteq\mathbb{Q}/\mathbb{Z}.$ Then,

$w(x)\coloneqq[[[g_{1}^{x},g_{4}],[g_{3}^{x},g_{5}]],[g_{2}^{x},g_{2}]]$

is a mixed identity for ${\rm Aut}(\mathbb{Q}/\mathbb{Z},t)$ and
$w^{\prime}(x)\coloneqq w(x^{2})$ is a mixed identity for ${\rm
Aut}^{\pm}(\mathbb{Q}/\mathbb{Z},t).$

###### Proof.

The proof uses the same idea as above and the geometric fact that for
$g\in{\rm Aut}(\mathbb{Q}/\mathbb{Z},t)$ the condition $I_{1}.g\cap
I_{4}\neq\varnothing$ together with $I_{3}.g\cap I_{5}\neq\varnothing$ implies
that $I_{2}.g\cap I_{2}=\varnothing.$ Thus, $[g_{2}^{g},g_{2}]$ is trivial
unless $[g_{1}^{g},g_{4}]$ or $[g_{3}^{g},g_{5}]$ are trivial. We conclude
that $w(g)={\rm id}_{\mathbb{Q}/\mathbb{Z}}$ for all $g\in{\rm
Aut}(\mathbb{Q}/\mathbb{Z},t).$ Thus, $w(x)$ is a mixed identity for ${\rm
Aut}(\mathbb{Q}/\mathbb{Z},t)$ and, clearly, $w^{\prime}(x)=w(x^{2})$ is a
mixed identity for ${\rm Aut}^{\pm}(\mathbb{Q}/\mathbb{Z},t)$. ∎

###  3.7. The generic semi-linear order with joins

A _semilinear order_ [macpherson2011survey] is a partially ordered set $(S,<)$
such that

  * •

for each $a\in S$ the set $\\{x\in S\mid a<x\\}$ is linearly ordered by $<$,
and

  * •

for all $x,y\in S$ there exists $z\in S$ such that $x<z$ and $y<z$.

Countable semilinear orders where all maximal chains are isomorphic to
$({\mathbb{Q}},<)$ have been classified by Droste [droste1985structure]. There
exists such a semilinear order $(S,<)$ where for any two $x,y$

  * •

there exists a supremum of $x$ and $y$ in $S$, i.e., a (unique) smallest upper
bound $\sup(x,y)$ for $x$ and $y$, and

  * •

the equivalence relation $E_{x,y}$ on $T_{x,y}\coloneqq\\{z\in S\mid
z\leq\sup(x,y)\\}$ which contains all pairs $(u,v)\in T_{x,y}^{2}$ such that
there exists $w\in T_{x,y}$ with $u,v<w<\sup(x,y)$, has infinitely many
equivalence classes.

The semilinear order $(S,<)$ is unique up to isomorphism, and has an
oligomorphic automorphism group, but is not homogeneous. However, as explained
in Remark 2.1.4 it has a homogeneous expansion with the same automorphism
group. In model theory, it is a frequently studied example in the context of
structures that are NIP but not stable; for recent results, see
[kaplanrzepeckisiniora2021automorphism]. In the following, we call $(S,<)$ the
_generic semilinear order with joins_.

######  Lemma 3.7.1.

The generic semi-linear order with joins $(S,<)$ does have algebraicity and is
algebraically convex. The small elements $c\in{\rm Aut}(S,<)$ are those fixing
a non-trivial interval.

###### Proof.

For the first two claims, let $Y$ be a finite subset of $S$. Note that the
algebraic closure of $Y$ is the smallest subset of $S$ containing $Y$ which is
closed under joins. Moreover, it is easy to see that
$\text{acl}(Y\cup\\{x\\})=\text{acl}(Y)\cup\\{\bar{x},x\\}$ for some
$\bar{x}\in S$ (we may have $x=\bar{x}$). Note that
$\text{acl}(Y)\cup\\{\bar{x}\\}$ is algebraically closed. In particular,
$(S,<)$ is algebraically convex.

For the third claim, one can prove that each infinite definable set contains a
non-trivial interval $(a,b)=\\{x\in S\mid a<x<b\\}$ ($a<b$, $a,b\in S$). Hence
$c\in{\rm Aut}(S,<)$ is small if and only if it fixes such an interval
pointwise (in analogy to Section 3.4 above). ∎

We obtain from Lemma 3.7.1 and Theorem 2.2.1:

######  Corollary 3.7.2.

Every mixed identity of ${\rm Aut}(S,<)$ contains a critical constant fixing a
non-trivial interval.

We can also prove the existence of mixed identities. The following
construction is inspired by a result in [leboudecmattebon2022triple].

######  Lemma 3.7.3.

There exist mixed identities for $(S,<)$.

###### Proof.

Suppose that $g,h\in{\rm Aut}(S,<)$ are supported in disjoint half-trees. In
order to see that such $g$ and $h$ exist, we consider two incomparable
elements $y,z\in S$. Let $y_{1}$ and $y_{2}$ be incomparable below $y$. Then,
there exists an automorphism $g^{\prime}$ that fixes $y$ and interchanges
$y_{1}$ and $y_{2}$. Define $g\in{\rm Aut}(S,<)$ to fix $y$, everything above
$y$ and everything incomparable to $y$; and to be equal to $g^{\prime}$ below
$y$. This defines an automorphism of the generic semi-linear order. Similarly,
you find $h\in{\rm Aut}(S,<)$ which only acts below $z$. Now, $g$ and $h$ are
supported on the disjoint half-trees $\\{x\in S\mid x<y\\}$ and $\\{x\in S\mid
x<z\\}$. Consider an arbitrary element $k\in{\rm Aut}(S,<)$. There are only
three possible cases. In each of them we use that two automorphisms with
disjoint support commute:

  1. $(i)$

If $y.k$ is incomparable to $y$, then $g^{k}$ and $g$ commute.

  2. $(ii)$

If $y.k$ is greater than or equal to $y$, then $z.k$ is incomparable to $y$,
and so $h^{k}$ and $g$ commute.

  3. $(iii)$

If $y.k$ is below $y$, then $z$ is incomparable to $y.k$, and $g^{k}$ and $h$
commute.

We conclude that

$w(x)=[[g^{x},g],[[h^{x},g],[g^{x},h]]]\in{\rm Aut}(S,<)\ast\langle x\rangle$

is a mixed identity for the automorphism group. ∎

###  3.8. The Ważewski dendrite

A _dendrite_ $X$ is a compact, metrizable, connected and locally connected
topological space without simple closed curves. Then, for each pair of
distinct points $x,y\in X$, there exists a unique simple path from $x$ to $y$
in $X.$ We say that a set $W\subseteq X$ is _arcwise dense_ , if it intersects
with each such simple path. See [duchesnemonod2019structural] for more
background and references.

For a dendrite $X$, we call $x\in X$ a _branch point_ , if $X\setminus\\{x\\}$
consists of at least three connected components. The cardinality of the set of
connected components of $X\setminus\\{x\\}$ is called the _order_ of $x.$ We
denote by ${\rm Br}(X)$ the set of branch points of $X$.

Throughout, let $S\subseteq\\{3,4,\ldots\\}\cup\\{\infty\\}$ be a finite
subset. The Ważewski dendrite $D_{S}$ is uniquely characterized by the
property that each branch point has order in $S$ and that the set of branch
points of order $s\in S$ is arcwise dense in $D_{S}$. It follows that the set
${\rm Br}(D_{S})$ is countable.

It was proved in [duchesnemonod2019structural]*Proposition 2.4 that the
natural group homomorphism $\varphi\colon{\rm Homeo}(D_{S})\to{\rm Sym}({\rm
Br}(D_{S}))$ is a homeomorphism onto its image and it follows from
[duchesnemonod2019structural]*Corollary 6.8 that its image is an oligomorphic
permutation group. A natural relation that is preserved by this permutation
action is the relation $R$ on triples with $(x,y,z)\in R$ if and only if $y$
lies on the simple path from $x$ to $z$. Moreover, one can identify the image
of $\varphi$ with ${\rm Aut}({\rm Br}(D_{S}),R).$ From this relation, one can
define uniquely the midpoint of the tripod spanned by three points
$x,y,z\in{\rm Br}(D_{S})$. We see from this that the algebraic closure of
three points can be non-trivial. More generally, the algebraic closure of a
finite set $F\subseteq{\rm Br}(D_{S})$ consists of the vertices of the finite
tree spanned by $F.$

######  Lemma 3.8.1.

The homogeneous relational structure $({\rm Br}(D_{S}),R)$ is algebraically
convex. The small automorphisms are those fixing a non-trivial simple path
point-wise.

######  Corollary 3.8.2.

Every non-trivial mixed identity of ${\rm Aut}({\rm Br}(D_{S}),R)$ contains a
critical constant that fixes a non-trivial simple path.

We say that $A\subseteq{\rm Br}(D_{S})$ is a _half-tree_ if there exists a
point $x\in D_{S}\setminus{\rm Br}(D_{S})$ of order $2$ and $A={\rm
Br}(D_{S})\cap B$, where $B$ is one of the two connected components of
$D_{S}\setminus\\{x\\}.$ Using [leboudecmattebon2022triple]*Lemma 3.5, we can
now construct a mixed identity for the group ${\rm Aut}({\rm
Br}(D_{S}),R)={\rm Homeo}(D_{S}).$ The following is restating this lemma in
our setting.

######  Lemma 3.8.3.

Let $A_{1},A_{2},A_{3}\subseteq{\rm Br}(D_{S})$ be three disjoint half-trees
and let $g\in{\rm Aut}({\rm Br}(D_{S}),R).$ Then either

  1. (1)

there exists $i\in\\{1,2,3\\}$ with $A_{i}$ and $A_{i}.g$ disjoint, or

  2. (2)

$A_{i}.g=A_{i}$ for all $i\in\\{1,2,3\\}.$

We can now follow the arguments in [leboudecmattebon2022triple]*Lemma 3.6 and
the discussion after its proof to obtain the following corollary:

######  Corollary 3.8.4.

Let $h_{1},h_{2},h_{3}\in{\rm Aut}({\rm Br}(D_{S}),R)$ be automorphisms that
have support in disjoint half-trees and let $g\in{\rm Aut}({\rm Br}(D_{S}),R)$
be arbitrary. Then, either $[h_{i}^{g},h_{i}]=1$ for one $i\in\\{1,2,3\\}$ or
$[h_{i}^{g},h_{j}]=1$ for all $i,j\in\\{1,2,3\\}$ with $i\neq j.$ In
particular, the word

$w(x)=[[[h_{1}^{x},h_{1}],[h_{2}^{x},h_{2}]],[[h_{3}^{x},h_{3}],[h_{1}^{x},h_{2}]]]$

is a mixed identity for ${\rm Aut}({\rm Br}(D_{S}),R)$.

##  4\. Examples without mixed identities

This section contains examples of oligomorphic automorphism groups for which
there do not exist any mixed identities at all.

###  4.1. The Rado graph

The celebrated Rado graph $R=(V,E)$ is the Fraïssé limit of the class of all
finite graphs. Here $V$ is the set of vertices and $E$ the set of edges.

######  Lemma 4.1.1.

The Rado graph $R=(V,E)$ has no algebraicity and no non-trivial small
automorphisms.

###### Proof.

The class of all finite graphs even has the free amalgamation property, and
hence has no algebraicity by Remark 2.1.11. To establish that $R$ has no non-
trivial small automorphisms, we prove Property (ii) of Lemma 2.1.9. Let $Y$ be
finite and $x\in V\setminus Y$. Set $Z\coloneqq x.{\rm stab}(Y)$. Consider
$c\in{\rm stab}(Z).$ Let $u,v\in V\setminus Z$ be distinct vertices. By the
defining property of the Rado graph applied to $Y\cup\\{u,v\\}$, there exists
a vertex $z\in V$ that behaves to $Y$ as $x$ does, and such that $u$ is
connected to $z$, but $v$ is not. Thus by homogeneity, we have that $z\in
x.{\rm stab}(Y)$ and hence $z.c=z$. But then $u$ cannot be mapped to $v$ by
$c$ by the choice of $z$. Hence, $c$ must be trivial. ∎

Lemma 4.1.1 and Theorem 2.2.1 imply:

######  Corollary 4.1.2.

The group ${\rm Aut}(R)$ has no mixed identities.

###  4.2. Free amalgamation

Examples of homogeneous structures whose age has the free amalgamation
property is the random graph from the previous section, the random $k$-uniform
hypergraph, for some $k\geq 3$, the homogeneous graph whose age consists of
all finite triangle-free graphs (or, more generally, $K_{n}$-free graphs, for
some fixed $n\geq 3$), the uncountable family of homogeneous directed graphs
constructed by Henson [henson1972countable], and many more. Already for the
random triangle-free graph, the easy proof that we presented for the Rado
graph in the previous section fails, since for some choices of $Y$ and $x$,
there might not exist a vertex $y\in V$ that has the required properties
because $R$ does not contain triangles. All of the listed examples above also
satisfy the additional assumptions made in the following theorem.

######  Theorem 4.2.1.

Let $(X,R)$ be a homogeneous relational structure whose age has the free
amalgamation property such that ${\rm Aut}(X,R)$ is transitive and ${\rm
Aut}(X,R)\neq{\rm Sym}(X)$. Then ${\rm Aut}(X,R)$ has no mixed identities.

###### Proof.

By Remark 2.1.11 we have no algebraicity in case that the age has the free
amalgamation property. By Theorem 2.2.1, it suffices to show that $(X,R)$ has
no non-trivial small automorphisms. Let $c\in{\rm Aut}(X,R)$ be such that
$c\in{\rm stab}(Z)$ for some infinite $Z\subseteq X$ which is definable in
$(X,R)$ with parameters from some finite $Y\subseteq X$. Let $x\in X$ and
suppose for contradiction that $x.c\neq x$.

By the assumption that $\text{Aut}(X,R)\neq\text{Sym}(X)$ and that
$\text{Aut}(X,R)$ is transitive, there exists $k\geq 2$ such that ${\rm
Aut}(X,R)$ is $(k-1)$-transitive, but not $k$-transitive. By the homogeneity
of $(X,R)$, there exist tuples $(s_{1},\dots,s_{k})$ and $(t_{1},\dots,t_{k})$
such that $s_{i}\mapsto t_{i}$ is not an isomorphism between the substructures
${\mathfrak{S}}$ and ${\mathfrak{T}}$ of $(X,R)$ with domain
$S=\\{s_{1},\dots,s_{k}\\}$ and $T=\\{t_{1},\dots,t_{k}\\}$, respectively.

Note that by the transitivity of $\text{Aut}(X,R)$, there is precisely one
orbit of $k$-tuples $(r_{1},\dots,r_{k})$ such that the substructure of
$(X,R)$ induced by $\\{r_{1},\dots,r_{k}\\}$ equals the free amalgam of the
1-element substructures of $(X,R)$ with domains
$\\{r_{1}\\},\dots,\\{r_{k}\\}$, respectively; this orbit will be called the
_free orbit_. Without loss of generality, we may assume that
$(s_{1},\dots,s_{k})$ is _not_ the free orbit.

Note that if ${\mathfrak{A}}$ and ${\mathfrak{B}}$ are structures with the
finite domains $A$ and $B$, respectively, and
${\mathfrak{A}}\cup{\mathfrak{B}}$ is a substructure of $(X,R)$, then every
tuple $(r_{1},\dots,r_{k})\in(A\cup B)^{k}$ with distinct entries and an entry
from $A\setminus B$ and from $B\setminus A$ must be the free orbit. Indeed,
suppose that $r_{1}\in A\setminus B$ and $r_{k}\in B\setminus A$. Then by
$(k-1)$-transitivity the substructure with domain $\\{r_{1},\dots,r_{k-1}\\}$
is the union of its $1$-element substructures. The substructure with domain
$\\{r_{1},\dots,r_{k}\\}$ is the union of the substructure with domain
$\\{r_{1},\dots,r_{k-1}\\}$ and the substructure with domain
$\\{r_{2},\dots,r_{k}\\}$, and hence it is the union of its $1$-element
substructures as well.

Let ${\mathfrak{B}}$ be the finite substructure of $(X,R)$ with domain
$B\coloneqq Y\cup\\{x,x.c\\}$. Since $\text{Aut}(X,R)$ is transitive, we may
assume without loss of generality that $s_{1}=x$. Since $(X,R)$ has no
algebraicity, we may even assume that $B\cap S=\\{x\\}$. Since the age of
$(X,R)$ has the free amalgamation property, the structure
${\mathfrak{B}}\cup{\mathfrak{S}}$ has an isomorphism $e$ to a substructure of
$(X,R)$, and by the homogeneity of $(X,R)$ we may assume that $b.e=b$ for all
$b\in B$. Note that there exists $j\in\\{2,\dots,k\\}$ such that $s_{j}.e\neq
s_{j}.ec$, because otherwise the following tuple

$\displaystyle(s_{1}.ec,s_{2}.ec,\dots,s_{k}.ec)$
$\displaystyle=(x.c,s_{2}.e,\dots,s_{k}.e)$

would lie in the free orbit, since $x.c\in B.e\setminus S.e$ and
$s_{2}.e,\dots,s_{k}.e\in S.e\setminus B.e$. This leads to a contradiction,
because $(s_{1},\dots,s_{k})$ does not lie in the free orbit by assumption. An
illustration of the proof for the special case where $(X,R)$ is the random
triangle-free graph can be found in Figure 1.

${s_{2}.e^{\prime}}$${s_{2}.e}$${s_{2}.ec}$${z}$${z.gh}$${x}$${x.c}$${y_{1}}$${y_{2}}$$\scriptstyle{g}$$\scriptstyle{c}$$\scriptstyle{c}$
Figure 1. An illustration of the proof of Theorem 4.2.1 for the special case
of the random triangle-free graph; that is, we have $k=2$. We consider the
case where $Y=\\{y_{1},y_{2}\\}$ and where $Z$ consists of all vertices that
are adjacent to both $y_{1}$ and $y_{2}$, i.e., $z,z.gh\in Z$, and where $x$
is adjacent to $y_{2}$ but not $y_{1}$. Solid black lines indicate edges,
dashed lines indicate non-edges (the free orbit).

Let $z\in Z$ and let ${\mathfrak{C}}$ be the substructure of $(X,R)$ with
domain $C\coloneqq B\cup S.e\cup S.ec\cup\\{z\\}$. Similarly as above one can
see that there exists an isomorphism $e^{\prime}$ between ${\mathfrak{S}}$ and
a substructure ${\mathfrak{S}}^{\prime}$ of $(X,R)$ such that

  * •

${\mathfrak{S}}^{\prime}\cup{\mathfrak{C}}$ is a substructure of $(X,R)$,

  * •

$s_{1}.e^{\prime}=z$ and $S.e^{\prime}\cap C=\\{z\\}$.

The map that fixes $Y$ and sends $(s_{2}.e^{\prime},\dots,s_{k}.e^{\prime})$
to $(s_{2}.e,\dots,s_{k}.e)$ is an isomorphism between substructures of
$(X,R)$ and hence can be extended to an automorphism $g$ of $(X,R)$. Let
${\mathfrak{D}}$ be the free amalgam of the substructures of $(X,R)$ with
domain $A\coloneqq Y\cup\\{s_{2}.e,\dots,s_{k}.e,s_{2}.ec,\dots,s_{k}.ec\\}$
and with domain $B^{\prime}\coloneqq
Y\cup\\{s_{1}.e^{\prime}g,\dots,s_{k}.e^{\prime}g\\}$ with $A\cap
B^{\prime}=Y$. There exists an isomorphism $h$ between ${\mathfrak{D}}$ and a
substructure of $(X,R)$, and by the homogeneity of $(X,R)$ we may assume that
the isomorphism $h$ is the identity on $A$. Since $h$ and $g$ fix the elements
of $Y$ pointwise, we have $s_{1}.e^{\prime}gh=z.gh\in Z$ since $z\in Z$ and
$Z$ is definable with parameters from $Y$. Thus, $z.ghc=z.gh$. Also note that
$s_{j}.e^{\prime}ghc=s_{j}.ehc=s_{j}.ec\notin B^{\prime}.h$. The tuple

$\displaystyle(s_{1}.e^{\prime}ghc,s_{2}.e^{\prime}ghc,\dots,s_{k}.e^{\prime}ghc)=(z.gh,s_{2}.ec,\dots,s_{k}.ec)$

lies in the free orbit, because $z.gh\in B^{\prime}.h\setminus A.h$, and
$s_{2}.ec,\dots,s_{k}.ec\in A.h$, and $s_{j}.ec\notin B.h$. But since
$e^{\prime}ghc$ is an isomorphism, the tuple is in the same orbit as
$(s_{1},\dots,s_{k})$, which is not the free orbit, a contradiction. ∎

###  4.3. The random poset

The _random poset_ $(P,<)$ is defined as the Fraïssé limit of the class of all
finite posets; note that this class does not have the free amalgamation
property, but the strong amalgamation property, and hence $(P,<)$ has no
algebraicity. For this relational structure the following holds.

######  Lemma 4.3.1.

The random poset $(P,<)$ has no non-trivial small automorphisms.

###### Proof.

Let $a,b\in P$ with $a<b$. We will prove that every automorphism that fixes
$Y\coloneqq\\{x\in P\mid a<x<b\\}$ is trivial. We will write $x\perp y$ if
neither $x<y$ nor $y<x,$ i.e., $x$ and $y$ are incomparable.

Suppose that $c$ and $c^{\prime}$ are incomparable to $b$ and both larger than
$a$. Then, by homogeneity, there exists $x\in P$ with $a<x<b$, $x<c$ but
$x\not<c^{\prime}.$ Now, any automorphism that fixes $Y$ cannot map $c$ to
$c^{\prime}$. It follows that ${\rm stab}(Y)$ fixes $\\{c\mid c\perp b,a<c\\}$
pointwise.

Similarly, ${\rm stab}(Y)$ fixes $\\{c\mid c<b,a\perp c\\}$ and $\\{c\mid
c\perp b,c\perp a\\}$ pointwise. Consider now $c$ and $c^{\prime}$ with $b<c$
and $b<c^{\prime}$. By homogeneity, there exists $x\in P$ incomparable to $b$
such that $x<c$ but $x\not<c^{\prime}.$ Since $x$ is fixed by ${\rm stab}(Y)$
by the previous arguments, no element of ${\rm stab}(Y)$ can map $c$ to
$c^{\prime}$. It follows that ${\rm stab}(Y)$ fixes the set $\\{c\mid b<c\\}$
pointwise. Similarly, it fixes $\\{c\mid c<a\\}$ pointwise. This covers all
cases and implies that ${\rm stab}(Y)$ is trivial. Hence the claim is proven.
∎

As a consequence of Lemma 4.3.1 and Theorem 2.2.1 we obtain:

######  Corollary 4.3.2.

The group ${\rm Aut}(P,<)$ has no mixed identities.

###  4.4. Parametrised families of linear orders

Consider $k\in\mathbb{N}$ and consider the class $\mathcal{R}_{k}$ of finite
structures $(X,R)$, where $X$ is a finite set and $R$ is a $(k+2)$-ary
relation with the properties

  1. (1)

$(x_{1},\ldots,x_{k},a,b)\in R$ only if $x_{1},\ldots,x_{k},a,b$ are pairwise
distinct,

  2. (2)

for a fixed $k$-tuple $x=(x_{1},\ldots,x_{k})\in X^{k}$ of pairwise distinct
elements, the binary relation

$L_{k}(x)\coloneqq\;\\{(a,b)\in(X\setminus\\{x_{1},\ldots,x_{k}\\})^{2}\mid(x_{1},\ldots,x_{k},a,b)\in
R\\}$

defines a linear order on $X(x)\coloneqq X\setminus\\{x_{1},\ldots,x_{k}\\}.$

######  Lemma 4.4.1.

The class $\mathcal{R}_{k}$ is closed under substructures, isomorphisms, and
has the strong amalgamation property.

###### Proof.

Only the strong amalgamation property requires a proof. Let
${\mathfrak{A}},{\mathfrak{B}}_{1},{\mathfrak{B}}_{2}\in{\mathcal{R}}_{k}$ be
such that $A=B_{1}\cap B_{2}$ and ${\mathfrak{A}}$ is a substructure of both
${\mathfrak{B}}_{1}$ and ${\mathfrak{B}}_{2}$. We need to find a structure
${\mathfrak{C}}\in{\mathcal{R}}_{k}$ and isomorphisms $e_{i}$, for
$i\in\\{1,2\\}$, between ${\mathfrak{B}}_{i}$ and a substructure of
${\mathfrak{C}}$ such that $e_{1}(a)=e_{2}(a)$ for every $a\in A$.

The domain of ${\mathfrak{C}}$ will be $C\coloneqq B_{1}\cup B_{2}$. For every
$x\in C^{k}$, we define a linear order $L$ on
$C\setminus\\{x_{1},\dots,x_{k}\\}$ as follows. If $x\in(B_{i})^{k}$, for
$i\in\\{1,2\\}$, then $L_{i}$ denotes the restriction of the linear order
$L_{k}(x)$ to $B_{i}\setminus\\{x_{1},\dots,x_{k}\\}$. If $x\in A^{k}$, then
$L$ is any linear extension of $L_{1}\cup L_{2}$. Otherwise, if
$x\in(B_{i})^{k}$ for $i\in\\{1,2\\}$, then $L$ is any linear extension of
$L_{i}$ to all of $C\setminus\\{x_{1},\dots,x_{k}\\}$. Finally, if $x$ is
neither in $(B_{1})^{k}$ nor in $(B_{2})^{k}$, then $L$ is any linear order on
$C\setminus\\{x_{1},\dots,x_{k}\\}$. The relation $R$ of ${\mathfrak{C}}$ then
consists of all tuples of the form $(x,a,b)$ for $(a,b)\in L$; clearly,
${\mathfrak{C}}\in{\mathcal{R}}_{k}$. It is also clear that the identity map
on $B_{i}$ is an isomorphism between ${\mathfrak{B}}_{i}$ and a substructure
of ${\mathfrak{C}}$, which proves the strong amalgamation property. ∎

Consider now the Fraïssé limit ${\mathfrak{X}}_{k}=(X_{k},R_{k})$ of the class
$\mathcal{R}_{k}$ and its automorphism group $G_{k}={\rm
Aut}({\mathfrak{X}}_{k})$ (see, e.g., [hodges1997shorter]). Note that the
action of $G_{k}$ on $X_{k}$ is $(k+1)$-transitive and oligomorphic.

Let $x=(x_{1},\ldots,x_{k})$ be a $k$-tuple of distinct elements of $X_{k}$.
The homogeneity of $(X_{k},R_{k})$ implies that $(X_{k}(x),L_{k}(x))$ is dense
and unbounded, and hence isomorphic to $(\mathbb{Q},<)$. We write $y<_{x}z$
for $(y,z)\in L_{k}(x).$

######  Lemma 4.4.2.

Every infinite definable set contains a set of the form $\\{y\in X_{k}\mid
a<_{x}y<_{x}b\\}$ for some $k$-tuple $x=(x_{1},\ldots,x_{k})$ of distinct
elements of $X_{k}$ and some $a,b\in X_{k}(x)$ with $a<_{x}b$.

######  Lemma 4.4.3.

The structure ${\mathfrak{X}}_{k}$ has no algebraicity, and for $k\geq 1$, it
has no non-trivial small automorphisms.

###### Proof.

No algebraicity follows from the strong amalgamation property and Remark
2.1.11. It remains to exclude the existence of small automorphisms. Let $Z$ be
an infinite set which is definable in ${\mathfrak{X}}_{k}$ and let
$c\in\text{stab}(Z)$. Note that $Z$ contains a set of the form

$\bigcap_{j=1}^{d}\left\\{y\in X_{k}\mid
a_{j}<_{x^{(j)}}y<_{x^{(j)}}b_{j}\right\\}$

for some $d\in{\mathbb{N}}$, some $k$-tuples
$x^{(j)}=(x^{(j)}_{1},\ldots,x^{(j)}_{k})$ of distinct elements of $X_{k}$,
and some $a_{j},b_{j}\in X_{k}(x^{(j)})$ with $a_{j}<_{x^{(j)}}b_{j}$.

Let now $z_{1},z_{2}\in\bigcap_{j=1}^{d}X_{k}(x^{(j)}).$ By homogeneity and
since $k\geq 1$, there exists a tuple
$x^{\prime}=(x^{\prime}_{1},\ldots,x^{\prime}_{k})\in(X_{k})^{k}$ satisfying

  * •

$a_{j}<_{x^{(j)}}x^{\prime}_{i}<_{x^{(j)}}b_{j}$ for all
$i\in\\{1,\ldots,k\\}$ and $j\in\\{1,\ldots,d\\}$,

  * •

$z_{1}<_{x^{\prime}}a$, and

  * •

$a<_{x^{\prime}}z_{2}$.

Note that $x^{\prime}\in Z^{k}$. Thus $x^{\prime}.c=x^{\prime}$, and since
$z_{1}<_{x^{\prime}}z_{2}$ we have $z_{1}.c\neq z_{2}$. This implies that $c$
fixes every element in $\bigcap_{j=1}^{d}X_{k}(x^{(j)})$. Let now
$y=(y_{1},\ldots,y_{k})$ be a $k$-tuple of distinct elements of
$\bigcap_{j=1}^{d}X_{k}(x^{(j)})$ satisfying
$a_{j}<_{x^{(j)}}<y_{i}<_{x^{(j)}}b_{j}$. Then $y.c=y$ and the linear order
$L_{k}(y)$ on $\bigcup_{j=1}^{d}\\{x^{(j)}_{1},\ldots,x^{(j)}_{k}\\}$ needs to
be preserved by $c$. Hence, $x^{(j)}_{i}.c=x^{(j)}_{i}$ for all
$i\in\\{1,\dots,k\\}$ and $j\in\\{1,\dots,d\\}$. This proves that $c$ is
trivial and completes the proof. ∎

######  Corollary 4.4.4.

For $k\geq 1$, the group ${\rm Aut}({\mathfrak{X}}_{k})$ has no mixed
identities.

##  5\. Ruling out non-singular mixed identities for ${\rm Aut}(\mathbb{Q},<)$

In this section, we continue to consider the group ${\rm Aut}(\mathbb{Q},<)$
of automorphisms of $(\mathbb{Q},<)$, see also Section 3.4. Our aim is to
prove Conjecture 1 under some assumption on the content of the word with
constants.

###  5.1. ${\rm Homeo}^{+}([0,1])$

Our technique applies to the slightly larger group ${\rm Homeo}^{+}([0,1])$ of
orientation-preserving homeomorphisms of the closed interval
$[0,1]\subseteq{\mathbb{R}}$. We will need the following folklore result.

######  Lemma 5.1.1.

There exists an injective continuous homomorphism

$\varphi\colon{\rm Aut}(\mathbb{Q},<)\to{\rm Homeo}^{+}([0,1])$

with dense image.

Now, every mixed identity of ${\rm Aut}(\mathbb{Q},<)$ extends by continuity
to a mixed identity of ${\rm Homeo}^{+}([0,1])$ and every mixed identity of
${\rm Homeo}^{+}([0,1])$ with constants in ${\rm Aut}(\mathbb{Q},<)$ restricts
to a mixed identity of ${\rm Aut}(\mathbb{Q},<).$ We will therefore restrict
attention to ${\rm Homeo}^{+}([0,1])$ from here on.

The main result of this section is the following theorem.

######  Theorem 5.1.2.

Let $w\in{\rm Homeo}^{+}([0,1])\ast\mathbf{F}_{2}$ be a word with constants
such that
$\epsilon(w)\not\in[[\mathbf{F}_{2},\mathbf{F}_{2}],[\mathbf{F}_{2},\mathbf{F}_{2}]]$.
Then $w$ is not a mixed identity for ${\rm Homeo}^{+}([0,1])$.

We also consider the subgroup ${\rm Homeo}^{+}_{\rm Lip}([0,1])$ of bi-
Lipschitz homeomorphisms, which serves as an important tool in the proof. The
following result can be found as Théorème D in
[deroinkleptsynnavas2007dynamique].

######  Theorem 5.1.3 (Deroin–Kleptsyn–Navas).

Every countable subgroup of the group ${\rm Homeo}^{+}([0,1])$ is conjugate to
a subgroup of ${\rm Homeo}^{+}_{\rm Lip}([0,1])$.

Note that there is a natural action of the group ${\rm Homeo}^{+}([0,1])$ on
$L^{\infty}([0,1],\mathbb{R})$ by composition. This turns
$L^{\infty}([0,1],\mathbb{R})$ into a left-module over the integral group ring
of the group ${\rm Homeo}^{+}([0,1])$ by the formula $(g\xi)(t)=\xi(t.g),$ for
$g\in{\rm Homeo}^{+}([0,1])$ and $\xi\in L^{\infty}([0,1],\mathbb{R}).$

Recall that a cocycle $\mu\colon{\rm Homeo}^{+}([0,1])\to
L^{\infty}([0,1],\mathbb{R})$ is a map that satisfies the equation

$\mu(gh)=g\mu(h)+\mu(g),\quad\forall g,h\in{\rm Homeo}^{+}([0,1]).$

An important example of a cocycle is the _derivative cocycle_ or _Rademacher
cocycle._

######  Lemma 5.1.4 (Rademacher).

There exists a cocycle

$\mu\colon{\rm Homeo}^{+}_{\rm Lip}([0,1])\to L^{\infty}([0,1],\mathbb{R}),$

which is uniquely determined by the equation

$g(t)=\int_{0}^{t}\exp(\mu(g)(s))ds$

for all $g\in{\rm Homeo}^{+}_{\rm Lip}([0,1]).$

Note that $\mu(f)=\log\left(\frac{df}{dx}\right)$ if $f$ is continuously
differentiable. Moreover, the same formula applies, when we note that a
Lipschitz function is almost everywhere differentiable and interpret the terms
in a suitable form. The cocycle equation $\mu(gh)=g\mu(h)+\mu(g)$ is then a
simple consequence of the chain rule for differentiation.

As a warm-up, we prove that there are no non-singular mixed identities in one
variable. This already covers the case where
$\epsilon(w)\notin[\mathbf{F}_{2},\mathbf{F}_{2}]$ by a simple reduction.

######  Theorem 5.1.5.

The group ${\rm Homeo}^{+}([0,1])$ does not satisfy a non-singular mixed
identity in one variable.

###### Proof.

Suppose that

$w(x)=x^{\varepsilon(1)}h_{1}x^{\varepsilon(2)}\cdots x^{\varepsilon(l)}h_{l}$

is a non-singular identity with constants in ${\rm Homeo}^{+}([0,1])$. We are
assuming that $\varepsilon(i)\in\\{\pm 1\\}$ for all $i\in\\{1,\ldots,l\\}$.
Because $w$ is non-singular, we know that
$e\coloneqq\sum_{i=1}^{l}\varepsilon(i)\neq 0$ and we may assume without loss
of generality that $e$ is positive.

Consider a countable subgroup $\Lambda$ of ${\rm Homeo}^{+}([0,1])$ containing
$h_{1},\ldots,h_{l}$. By Theorem 5.1.3, we may assume without loss generality
that $\Lambda$ is contained in ${\rm Homeo}^{+}_{\rm Lip}([0,1])$. We set
$C\coloneqq\max_{1\leq i\leq l}\|\mu(h_{i})\|_{\infty}$ so that

$k\mu(h_{i})\geq-\|\mu(h_{i})\|_{\infty}\geq-C$

for every $k\in{\rm Homeo}^{+}([0,1])$. From the cocycle identity, we obtain
for all $g\in{\rm Homeo}^{+}_{\rm Lip}([0,1])$:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\mu(w(g))$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{l}g^{\varepsilon(1)}h_{1}\cdots
g^{\varepsilon(i-1)}h_{i-1}(\mu(g^{\varepsilon(i)})+g^{\varepsilon(i)}\mu(h_{i}))$
$\displaystyle\geq$ $\displaystyle-
lC+\sum_{i=1}^{l}g^{\varepsilon(1)}h_{1}\cdots
g^{\varepsilon(i-1)}h_{i-1}\mu(g^{\varepsilon(i)})$ $\displaystyle=$
$\displaystyle-lC+\alpha\mu(g)$

with (using that $\mu(g^{-1})=-g^{-1}\mu(g)$)

$\alpha\coloneqq\sum_{i:e(i)=+1}g^{\varepsilon(1)}h_{1}\cdots
g^{\varepsilon(i-1)}h_{i-1}-\sum_{i:\varepsilon(i)=-1}g^{\varepsilon(1)}h_{1}\cdots
g^{\varepsilon(i-1)}h_{i-1}g^{-1}$

considered as an element in the integral group ring of ${\rm Homeo}^{+}_{\rm
Lip}([0,1])$. Note that the augmentation of $\alpha$ is equal to $e$.

For $\lambda\geq 1$, we consider now the specific bi-Lipschitz homeomorphism

$t.g_{\lambda}\coloneqq\begin{cases}\lambda t&0\leq t\leq 1/(2\lambda)\\\
\frac{1+t-1/\lambda}{2-1/\lambda}&1/(2\lambda)\leq t\leq 1.\end{cases}$

Then, the germ of the function $\alpha\mu(g_{\lambda})$ near $0$ is equal to
$e\log(\lambda)$. But this is incompatible with the inequality
$\alpha\mu(g_{\lambda})\leq lC$ for $\lambda>\exp(lC/e).$ Thus, we arrive at a
contradiction to our assumption that $w$ was a non-singular mixed identity.
This finishes the proof. ∎

###  5.2. The commutator

In order to deal with the many variable case under the assumptions of Theorem
5.1.2, we start with the simplest case, i.e., when $\epsilon(w)=[x,y]\coloneqq
xyx^{-1}y^{-1}\in\mathbf{F}_{2}=\langle x,y\rangle.$

Consider the homeomorphism $f\colon[0,1]\to[0,1]$ given by $t.f=t^{1/2}$. Let
$g_{\lambda}\colon[0,1]\to[0,1]$ be a homeomorphism with germ
$t.g_{\lambda}=\lambda t$ near zero as considered above. Then
$t.[f,g_{\lambda}]=t.fg_{\lambda}f^{-1}g_{\lambda}^{-1}=\lambda x$ near zero.

Consider now $w(x,y)=h_{1}xh_{2}yh_{3}x^{-1}h_{4}y^{-1}$. Let’s assume that
all constants $h_{1},\ldots,h_{4}$ are bi-Lipschitz with derivative bounded
below by $\eta>0$ near zero. We will show that $w(f,g_{\lambda})$ is not
trivial for $\lambda$ large enough. We write

$\displaystyle w(x,y)$ $\displaystyle=$
$\displaystyle[x,y]yxy^{-1}x^{-1}h_{1}xh_{2}yh_{3}x^{-1}h_{4}y^{-1}$
$\displaystyle=$
$\displaystyle[x,y]yx(y^{-1}((x^{-1}h_{1}x)h_{2})yh_{3})x^{-1}h_{4}y^{-1}.$

We will now construct lower bounds for the derivative of $w(f,g_{\lambda})$
near zero, starting from the innermost bracket. The derivative of
$f^{-1}h_{1}f$ is bounded below by $\eta^{1/2}$ near zero. Similarly, the
derivative of $f^{-1}h_{1}fh_{2}$ is bounded below by $\eta^{1/2+1}$. Hence,
the derivative of $g_{\lambda}^{-1}f^{-1}h_{1}fh_{2}g_{\lambda}$ is bounded
below by $\eta^{3/2}$ near zero as well. We obtain a similar bound of
$\eta^{5/2}$ for $g_{\lambda}^{-1}f^{-1}h_{1}fh_{2}g_{\lambda}h_{3}$ and of
$\eta^{5}$ for $fg_{\lambda}^{-1}f^{-1}h_{1}fh_{2}g_{\lambda}h_{3}f^{-1}$.
Finally we get the bound of $\eta^{6}$ for

$g_{\lambda}fg_{\lambda}^{-1}f^{-1}h_{1}fh_{2}g_{\lambda}h_{3}f^{-1}h_{4}g_{\lambda}^{-1}.$

This implies that the derivative of $w(f,g_{\lambda})$ near zero is bounded
below by $\lambda\eta^{6}.$ This shows that $w(f,g_{\lambda})\neq{\rm
id}_{[0,1]}$ for $\lambda$ large enough.

###  5.3. Proof of Theorem 5.1.2

Our aim is now to generalize the technique from the previous section and to
apply it to an arbitrary element $w\in{\rm
Homeo}^{+}([0,1])\ast\mathbf{F}_{2}$ with
$\epsilon(w)\not\in[[\mathbf{F}_{2},\mathbf{F}_{2}],[\mathbf{F}_{2},\mathbf{F}_{2}]]$.
Without loss of generality, we may assume that
$\epsilon(w)\in[\mathbf{F}_{2},\mathbf{F}_{2}]$. Indeed, if
$\epsilon(w)\notin[\mathbf{F}_{2},\mathbf{F}_{2}],$ then $w(x,1)$ or $w(1,x)$
are non-singular words in one variable and the result follows from Theorem
5.1.5.

Let $\lambda,\mu\in(0,\infty)$ and consider $f_{\mu},g_{\lambda}\in{\rm
Homeo}([0,1])$ with germ $t.f_{\mu}=t^{\mu}$ and $t.g_{\lambda}=\lambda t$
near zero. Then, $[f^{n}_{\mu},g^{m}_{\lambda}]$ has germ
$t\mapsto\lambda^{m(\mu^{-n}-1)}t.$ In general, for $\mathbf{F}_{2}=\langle
x,y\rangle$, the derived subgroup $[\mathbf{F}_{2},\mathbf{F}_{2}]$ is equal
to the free group on the set $\\{[x^{n},y^{m}]\mid nm\neq 0\\}.$

We define a group homomorphism
$\alpha\colon[\mathbf{F}_{2},\mathbf{F}_{2}]\to\mathbb{R}[\mu^{\pm}]$ defined
on generators by

$\alpha([x^{n},y^{m}])\coloneqq\log(\lambda)m(\mu^{-n}-1)\in\mathbb{R}[\mu^{\pm}].$

By the result above, the map $\alpha$ computes the logarithm of the derivative
near zero of $w(f_{\mu},g_{\lambda}).$ For arbitrary $w$, we need to find a
choice of $\mu$ and $\lambda$ so that this logarithm of the derivative near
zero does not vanish. This is clearly possible if the polynomial $\alpha(w)$
is not identically zero. Now, if
$w\not\in[[\mathbf{F}_{2},\mathbf{F}_{2}],[\mathbf{F}_{2},\mathbf{F}_{2}]]$,
then the proof of [elkasapythom2014goto]*Corollary 3.8 yields the existence of
a basis of $\mathbf{F}_{2}$ such that $\alpha(w(f_{\mu},g_{\lambda}))$ is not
the zero-polynomial when considered in this basis. Since a change of variable
does not affect the question whether $w$ is a mixed identity, we therefore may
assume that there exists $\mu_{0}>0$ so that
$\alpha(w(f_{\mu_{0}},g_{\lambda}))$ is a non-constant affine function in
$\log(\lambda)$. Then, the argument in the previous paragraph can be repeated.
This concludes the proof of Theorem 5.1.2.

In analogy to ${\rm Aut}(\mathbb{Q},<)$, we conjecture that there do not exist
non-singular mixed identities for ${\rm Homeo}^{+}([0,1]).$

## Acknowledgments

The second and third author want to thank Henry Bradford for interesting
discussions on the subject. The third author acknowledges funding by the
Deutsche Forschungsgemeinschaft (SPP 2026). Manuel Bodirsky received funding
from the European Union (project POCOCOP, ERC Synergy grant No. 101071674).
Views and opinions expressed are however those of the author(s) only and do
not necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them.

## References