# Scott Complexity of Reduced Abelian $p$-Groups

Rachael Alvir, Barbara F. Csima, Luke MacLean

###### Abstract

Given a reduced abelian $p$-group, we give an upper bound on the Scott
complexity of the group in terms of its Ulm invariants. For limit ordinals, we
show that this upper bound is tight. This gives an explicit sequence of such
groups with arbitrarily high Scott complexity below $\omega_{1}$. Along the
way, we give a largely algebraic characterization of the back-and-forth
relations on reduced abelian $p$-groups, making progress on an open problem of
Ash and Knight’s from [4].

##  1 Introduction

Scott’s isomorphism theorem [13] states that any countable structure can be
described $\omega$-categorically by a single sentence $\varphi$ of
$L_{\omega_{1}\omega}$, known as the _Scott sentence_ of that structure. In
the proof of this result, there arises an interesting invariant known as the
_Scott rank_ , which contains a wealth of metamathematical information about
the structure. Unfortunately for many years non-equivalent definitions of
Scott rank were considered in the literature. More recently, Antonio Montalbán
[12] argued to standardize the definition of a structure’s _categoricity Scott
rank:_ the least ordinal $\alpha$ such that the structure has a
$\Pi_{\alpha+1}$ Scott sentence. Though a useful notion, there is a little
arbitrariness in Montalbán’s definition as Scott sentences of other syntactic
forms can be considered. It is argued in Alvir, Greenberg, Harrison-Trainor,
and Turetsky [2] that we should consider the _Scott complexity_ of a
structure: the least complexity ($\Pi_{\alpha},\Sigma_{\alpha}$, or
$d$-$\Sigma_{\alpha}$) of a Scott sentence for that structure. Here, a formula
of $L_{\omega_{1}\omega}$ is said to be $d$-$\Sigma_{\alpha}$ if it is a
conjunction of a $\Sigma_{\alpha}$ and a $\Pi_{\alpha}$ formula.

In this paper, we investigate the Scott complexity of reduced abelian
$p$-groups. In many cases we calculate the Scott complexity of a given group
exactly, and in all other cases we give an upper bound. In particular, we show
in Theorem 3.1 that if ${\mathcal{A}}$ has length
$\lambda_{\mathcal{A}}=\omega\cdot\alpha$ where $\alpha$ is a limit ordinal,
then ${\mathcal{A}}$ has Scott complexity $\Pi_{2\alpha+1}$. That is, we give
an explicit sequence of structures with arbitrarily high Scott complexity.
Similar results about structures with arbitrarily high Scott complexity were
obtained for scattered linear orders in Alvir and Rossegger [3]. Optimal
complexity Scott sentences of groups in particular are considered in Knight
and Saraph [11], Ho [9], and Harrison-Trainor and Ho [8].

To obtain our results, we first characterize the back-and-forth relations on
reduced abelian $p$-groups. The standard back-and-forth relations on arbitrary
structures can be defined as follows [AK00].

######  Definition 1.1.

Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be structures, with finite tuples
$\overline{a}\in{\mathcal{A}}$ and $\overline{b}\in{\mathcal{B}}$. Then:

  * •

$({\mathcal{A}},\overline{a})\leq_{0}({\mathcal{B}},\overline{b})$ iff
$\overline{a}$ and $\overline{b}$ satisfy the same quantifier-free
formulas.111Note that for certain applications when the language is infinite,
the defintion of $\leq_{0}$ may vary from the one we use here.

  * •

For $\alpha>0$,
$({\mathcal{A}},\overline{a})\leq_{\alpha}({\mathcal{B}},\overline{b})$ if,
for each $\beta<\alpha$ and for each $\overline{d}\in{\mathcal{B}}$, there is
a $\overline{c}\in{\mathcal{A}}$ such that
$({\mathcal{B}},\overline{b}\overline{d})\leq_{\beta}({\mathcal{A}},\overline{a}\overline{c})$.

Giving a characterization of the back-and-forth relations within a class of
structures is an interesting result in its own right with a wide array of
possible uses. For example, in addition to results about Scott complexity,
they are often used to apply general results on intrinsically
$\Sigma_{\alpha}$ relations and on $\Delta_{\alpha}$ categoricity and
stability. Our characterization also makes progress on a long-standing open
problem from Ash and Knight’s book [4]. There, it was asked which algebraic
conditions characterize the back-and-forth relations on reduced abelian
$p$-groups. The characterization we give is largely algebraic.

There is a connection between Scott sentences and descriptive set theory. For
example, a result of Lopez-Escobar with several strengthenings due to Vaught
and D. Miller states that a structure $A$ has a $\Pi_{\alpha+1}$ Scott
sentence iff the set of presentations of $A$ is boldface $\Pi_{\alpha+1}$ in
the Borel hierarchy; see the discussion in Montalban’s paper [12]. It is also
known that for a class of countable structures $K$, the isomorphism relation
on $K$ is not a Borel equivalence relation iff structures in that class attain
arbitrarily high Scott complexity below $\omega_{1}$. There are several common
classes of groups for which is it known that the isomorphism relation is not a
Borel equivalence relation; however, it is not usually known exactly which
structures of that class achieve arbitrarily high Scott complexity and why,
which is one thing our paper accomplishes. In Friedman and Stanley [7], for
example, it is shown that isomorphism as an equivalence relation on the class
of abelian $p$-groups is complete analytic and thus not Borel. This is
accomplished by giving a reduction from pseudo-well-orders to $p$-groups; thus
to find an explicit description of abelian $p$-groups that achieve arbitrarily
high Scott complexity via this reduction, it would be necessary to do so first
for pseudo-well-orders. In addition, the reduction given in Friedman and
Stanley’s paper is accomplished by building an abelian $p$-group from a tree
via a construction of Feferman’s, a construction which is omitted. Thus, for
many reasons the explicit construction of abelian $p$-groups for each prime
$p$ with arbitrarily high Scott complexity, and the direct reason why the
Scott complexity must increase, is obscured. Our results show, roughly, that
Scott complexity increases with the length of the group at limit ordinals. In
addition our results improve those in [7] slightly, showing that for each
prime $p$, isomorphism restricted to the class of _reduced_ abelian $p$-groups
is still not Borel.

##  2 Preliminaries

For a standard resource covering the basics of reduced abelian $p$-groups and
Ulm’s Theorem, see [10]. Throughout this paper, all groups are assumed to be
countable abelian $p$-groups. For a fixed prime $p$, a group $G$ is said to be
primary (or $p$-primary) if for each group element $x\in G$ $p^{n}x=0$ for
some $n\geq 1$. A group $G$ is divisible if, for every $x\in G$, there exists
a $y\in G$ and $n\in\omega$ such that $x=ny$. A reduced group is one with no
non-trivial divisible subgroups.

Let $G_{0}=G$, and for any ordinal $\alpha$ we define
$G_{\alpha+1}=pG_{\alpha}=\\{x\in G:\exists y\in G_{\alpha}\,py=x\\}$. If
$\alpha$ is a limit ordinal, then we define
$G_{\alpha}=\bigcap_{\beta<\alpha}G_{\beta}$. Notice that if $\alpha<\beta$
then $G_{\alpha}\supseteq G_{\beta}$. We define the largest $\alpha$ such that
$x\in G_{\alpha}$ to be the _height_ of $x$ and denote it $h(x)$. This means
that $G_{\alpha}$ is the set of elements of height at least $\alpha$. The
height function does not always play nicely with respect to sums. For $x,y\in
G$ if $h(x)<h(y)$ then $h(x+y)=h(x)$, but if $h(x)=h(y)$ then $h(x+y)\geq
h(x)$. Given a subgroup $S$ of $G$, we say that $x$ is proper with respect to
$S$ if $h(x)\geq h(x+s)$ for all $s\in S$. This is the same as the height of
$x$ being maximal in its coset $mod\,S$. For such proper elements,
$h(x+s)=\min\\{h(x),h(s)\\}$. We also know that for all $x\neq 0$ we have
$h(px)>h(x)$. We define $h(0)=\infty$ to be greater than any ordinal to ensure
this property.

We similarly define, for all $\alpha$, $P_{\alpha}=P\cap G_{\alpha}$, where
$P=\\{g\in G:pg=0\\}$. Viewing $P_{\alpha}/P_{\alpha+1}$ as a vector space
over $\mathbb{Z}_{p}$, we define the $\alpha$-th Ulm invariant to be its
dimension, and denote it $f_{G}(\alpha)$. In addition, since $G$ is countable,
there must be an ordinal $\lambda$ called the length of $G$ such that
$G_{\lambda}=G_{\xi}$ for all $\xi\geq\lambda$. Since $G$ is reduced, we can
conclude that $G_{\lambda}=\\{0\\}$.

Our descriptions of reduced abelian $p$-groups up to isomorphism rely heavily
on the following important theorem:

######  Theorem 2.1 (Ulm [10]).

Two reduced countable primary abelian groups are isomorphic if and only if
they have the same Ulm invariants.

Thus, to describe a given a group $G$ of length $\lambda$ up to isomorphism,
we wish to say for each $\alpha<\lambda$ that there is a basis of
$P_{\alpha}/P_{\alpha+1}$ of size $k$.

Barker [5] has shown that

######  Proposition 2.2.

Let $\alpha$ be an ordinal. The group $G_{\omega\cdot\alpha}$ is definable by
a $\Pi^{0}_{2\alpha}$ formula. The group $G_{\omega\cdot\alpha+n}$ for $n>0$
is definable by a $\Sigma^{0}_{2\alpha+1}$ formula.

Given an $x\in G_{\alpha}$ we have that $h(px)>h(x)$. Note that it is not
always the case that $h(px)=h(x)+1$, as there are some $x$ for which the
height of $px$ increases more. Let $p^{-1}G_{\alpha+2}=\\{z\in G:pz\in
G_{\alpha+2}\\}$. Then, given a subgroup $S$, we define $S_{\alpha}=S\cap
G_{\alpha}$ and look at the subgroup $S_{\alpha}^{*}=S_{\alpha}\cap
p^{-1}G_{\alpha+2}$. For any subgroup $S$ of $G$, it turns out that
$S_{\alpha}^{*}/S_{\alpha+1}$ is isomorphic to a subgroup of
$P_{\alpha}/P_{\alpha+1}$.

Let us look at the map that does this. First, let us try to find a map from
$S_{\alpha}^{*}$ to $P_{\alpha}$. Now, for any $x\in S_{\alpha}^{*}$ $px\in
G_{\alpha+2}=pG_{\alpha+1}$ so $px=py$ for some $y\in G_{\alpha+1}.$ In
particular, $p(x-y)$ = 0, so the element $x-y$ has order $p$. Since $x$ is in
$S_{\alpha}$, and $y$ is in $G_{\alpha+1}$ and thus $G_{\alpha}$, we have that
$x-y$ is in $G_{\alpha}$ (since this is a subgroup.) Thus $x-y\in P_{\alpha}$.
Now the “map” that sends $x$ to $x-y$ is not actually well-defined, since $y$
is not unique. However, the map from $S_{\alpha}^{*}$ to
$P_{\alpha}/P_{\alpha+1}$ which sends $x$ to $\overline{x-y}$ (where
$\overline{x-y}$ is the image of $x-y$ under the natural homomorphism) is
actually well-defined. Let us call this map $u$. It turns out that $u$ is a
homomorphism whose kernel is exactly $S_{\alpha+1}$. This induces an
embedding, which we will also call $u$, that shows that
$S_{\alpha}^{*}/S_{\alpha+1}$ is isomorphic to a subgroup of
$P_{\alpha}/P_{\alpha+1}$.

The usefulness of this map lies in the following lemma.

######  Lem 2.3 (Kaplansky [10]).

Let $S$ be a subgroup of the group $G$ and $u$ as defined above. Then the
following are equivalent:

  1. 1.

The range of $u$ is not all of $P_{\alpha}/P_{\alpha+1}$.

  2. 2.

There exists an $x\in P_{\alpha}$ proper with respect to $S$.

##  3 Upper Bounds on Scott Complexity

Our first theorem gives a Scott sentence for an arbitrary reduced abelian
$p$-group, depending on the length of the group and whether certain Ulm
invariants are infinite or finite. This yields an upper bound on the Scott
complexity of the group.

In this paper, when writing formulas note that we do not have the natural
numbers as part of our language. Given a group $G$, when we write $n\cdot x$
for $n\in\omega,x\in G$, we mean $x+\overset{n}{\cdots}+x$.

######  Theorem 3.1.

Let $G$ be a group with length $\lambda>0$ and Ulm invariants given by $f$.
Then for $0<n<\omega$,

  * (i)

If $\lambda=\omega\cdot\gamma+n$ and $f(\omega\cdot\gamma+k)<\infty$ for
$0\leq k<n$ then $G$ has a Scott Sentence of complexity
$d$-$\Sigma_{2\gamma+2}$.

  * (ii)

If $\lambda=\omega\cdot\gamma+n$ and $f(\omega\cdot\gamma+i)=\infty$ for some
$i$ with $0\leq i<n$ then $G$ has a Scott Sentence of complexity
$\Pi_{2\gamma+3}$.

  * (iii)

If $\lambda=\omega\cdot\gamma$ then $G$ has a Scott Sentence of complexity
$\Pi_{2\gamma+1}$.

We will show at the end of Section 4 in Theorem 4.5 that for every group in
case (iii) above, the Scott sentence we give here is best possible. We also
give a class of groups in case (ii) for which the the Scott sentences here are
best possible. However, we show in Lemma 3.2 that such groups can have simpler
Scott sentences. It is not known if the sentences in case (i) are best
possible or not.

###### Proof.

Notice that, for an element $x\in G$, $x\in P$ iff $p\cdot x=0$. Thus,
$P_{\alpha}$ is definable with the same complexity as $G_{\alpha}$. Letting
$\varphi_{\beta}$, $\psi_{\beta}$ define $G_{\beta}$,$P_{\beta}$ respectively,
the following says $P_{\alpha}/P_{\alpha+1}$ has dimension at least $m$ for
$m\in\omega$, by saying that there exists $m$ linearly independent elements:

$\displaystyle\hat{\theta}_{\alpha}^{m}\equiv\exists
g_{1},\ldots,g_{m}\left(\bigwedge_{i\leq
m}\psi_{\alpha}(g_{i})\land\bigwedge_{(n_{1},\ldots,n_{m})\in\mathbb{Z}_{p}\setminus\\{\vec{0}\\}}\neg\psi_{\alpha+1}(n_{1}\cdot
g_{1}+\cdots+n_{m}\cdot g_{m})\right)$

Note that if $\alpha=\omega\cdot\beta+r$ then $\psi_{\alpha},\psi_{\alpha+1}$
both have complexity $\Sigma_{2\beta+1}$. Thus the above formula has
complexity $\Sigma_{2\beta+2}$.

Next, note that the formula

$\theta_{\alpha}^{m}\equiv\hat{\theta}_{\alpha}^{m}\land\neg\hat{\theta}_{\alpha}^{m+1}$

says that $P_{\alpha}/P_{\alpha+1}$ has dimension exactly $m$. This formula
has complexity $d$-$\Sigma_{2\beta+2}$.

If $P_{\alpha}/P_{\alpha+1}$ has dimension $\infty$ over $\mathbb{Z}_{p}$ then
we must say

$\theta_{\alpha}^{\infty}\equiv\bigwedge_{m\in\omega}\hat{\theta}_{\alpha}^{m}$

which has complexity $\Pi_{2\beta+3}$.

We also need to know the length $\lambda$ of the group $G$, so that we can
distinguish it from a group $G^{\prime}$ of longer length, but with all the
same Ulm invariants up to $\lambda$ as $G$. The length of the group is the
first ordinal $\lambda$ such that $G_{\lambda}=0$, so a formula saying this
fact is

$L_{\lambda}\equiv\forall x(\varphi_{\lambda}(x)\rightarrow
x=0)\wedge\bigwedge_{\alpha<\lambda}\exists x(\varphi_{\alpha}(x)\wedge x\neq
0).$

If $\lambda=\omega\gamma$ is a limit ordinal, then this formula is
$\Pi_{2\gamma+1}$. If $\lambda=\omega\gamma+n$ for $n>0$ then this formula has
complexity $d$-$\Sigma_{2\gamma+1}$.

Thus, the group $G$ with length $\lambda$ and Ulm invariants given by $f$ has
Scott sentence

$\Phi_{G}\equiv
L_{\lambda}\wedge\bigwedge_{\alpha<\lambda}\theta_{\alpha}^{f(\alpha)}\wedge\Omega$

where $\Omega$ is a $\Pi^{0}_{2}$ sentence saying that $G$ is an Abelian
$p$-group. Any structure that is isomorphic to $G$ will trivially satisfy this
formula, and if a group satisfies this formula, then Ulm’s Theorem tells us
that it is isomorphic to $G$.

We now consider the complexity of the formula $\Phi_{G}.$

_Case (i):_ Suppose that $\lambda=\omega\cdot\gamma+n$ and
$f(\omega\cdot\gamma+k)<\infty$ for $0\leq k<n$. Then
$\theta_{\alpha}^{f(\alpha)}$ for $\omega\cdot\gamma\leq\alpha<\lambda$ has
complexity $d$-$\Sigma_{2\gamma+2}$. For $\alpha<\omega\cdot\gamma$,
$\theta_{\alpha}^{f(\alpha)}$ is at most $\Pi_{2\gamma^{\prime}+3}$ for
$\gamma^{\prime}<\gamma$. Thus $\Phi_{G}$ has complexity
$d$-$\Sigma_{2\gamma+2}$.

_Case (ii):_ Suppose that $\lambda=\omega\cdot\gamma+n$ and
$f(\omega\cdot\gamma+i)=\infty$ for some $i$ with $0\leq i<n$. Then
$\theta_{\omega\cdot\gamma+i}^{f(\omega\cdot\gamma+i)}$ has complexity
$\Pi_{2\gamma+3}$. For all other $\alpha$, $\theta_{\alpha}^{f(\alpha)}$ has
the same complexity if infinite and lower complexity otherwise. Thus
$\Phi_{G}$ has complexity $\Pi_{2\gamma+3}$.

_Case (iii):_ Suppose that $\lambda=\omega\cdot\gamma$. For
$\alpha<\omega\cdot\gamma$, $\theta_{\alpha}^{f(\alpha)}$ is at most
$\Pi_{2\gamma^{\prime}+3}$ for $\gamma^{\prime}<\gamma$. Thus
$\bigwedge_{\alpha<\lambda}\theta_{\alpha}^{f(\alpha)}$ is $\Pi_{2\gamma}$.
However, $L_{\lambda}$ has complexity $\Pi_{2\gamma+1}$. Thus $\Phi_{G}$ has
complexity $\Pi_{2\gamma+1}$. ∎

The following example shows that when $G$ is an infinite group with one Ulm
invariant, the Scott sentence given above may not be best possible. This is
because there may be values other than the Ulm invariants that can determine
the isomorphism type of the group, and are easier syntactically to describe.
In this example, we also compute the Scott complexity of the group exactly.

######  Lem 3.2.

The group $G=(\mathbb{Z}_{p^{n}})^{\omega}$ has Scott complexity $\Pi_{2}$

###### Proof.

It is well-known that an infinite structure cannot have a Scott sentence
simpler than $\Pi_{2}$.

We claim that the conjunction of the following statements, along with axioms
stating that $G$ is an abelian group, is a Scott sentence for $G$:

  1. 1.

$\forall x\ p^{n}\cdot x=0$

  2. 2.

$\forall x\bigvee_{m\in\omega}p^{m}\cdot x=0\land\bigwedge_{k<p^{m}}k\cdot
x\neq 0.$

  3. 3.

$\bigwedge_{m\in\omega}\exists x_{1},\ldots,x_{m}\bigwedge_{i\neq j}x_{i}\neq
x_{j}\land\bigwedge_{i=1}^{n}\bigwedge_{k<p^{n}}k\cdot x_{i}\neq 0.$

  4. 4.

$\forall x\neq 0(p\cdot x=0\rightarrow\exists y\ x=p^{n-1}y.)$

Clause (1) says that every element has order at most $p^{n}$ ($\Pi_{1})$.
Clause (2) says that $G$ is a $p-$group ($\Pi_{2}$). Clause (3) says the group
has infinitely many elements of order $p^{n}$ ($\Pi_{2}$). Clause (4) says
that every element of order $p$ has height at least $n-1$.

Clearly all of these statements are true about $G$. Next, we show that they
determine $G$ up to isomorphism. Let $H$ be a group satisfying the conjunction
of the above clauses. Then since $H$ is of bounded order and a $p$-group, $H$
is a direct sum of cyclic groups of the form $\mathbb{Z}_{p^{m}}$ for $m\leq
n$. Since $H$ has infinitely many elements of order $p^{n}$, $H$ must contain
infinitely many copies of $\mathbb{Z}_{p^{n}}$. It remains to show that $H$
contains no group of the form $\mathbb{Z}_{p^{m}}$ for $m<n$ as a direct
summand. However, if it did it would have an element of order $p$ of height
less than $n-1$, which is impossible by clause (4). ∎

##  4 Scott Complexity

We saw earlier that when the length of the group $G$ is finite and there is
exactly one infinite nonzero Ulm invariant, that the Scott sentence given 3.1
may not be best possible. However, if $G$ has at least two infinite nonzero
Ulm invariants, this is no longer the case.

To prove this, we will need the following results. Recall that
$G\preceq_{\Sigma_{1}}H$ iff $G,H$ agree on all $\Sigma_{1}$ sentences with
parameters from $G$. Eklof [6] proved that, for abelian groups $G$ and $H$,
$G\preceq_{\Sigma_{1}}H$ iff $G$ is a pure subgroup of $H$ and every
$\Sigma_{1}$ sentence true of $H$ is true of $G$. Alvir proved the following:

######  Theorem 4.1 (Alvir, [1]).

Suppose that $(A_{i})_{i\in\omega}$ is a chain such that
$A_{i}\preceq_{\Sigma_{\alpha}}A_{i+1}$, and $A=\cup A_{i}$ where $A$ is not
isomorphic to any $A_{i}$. Then $A$ has no $d$-$\Sigma_{\alpha+1}$ Scott
sentence.

Using these results, we can begin to calculate Scott complexity exactly.

######  Theorem 4.2.

Suppose that $G$ is a reduced abelian $p$-group of finite length with at least
two infinite nonzero Ulm invariants. Then $G$ has Scott complexity $\Pi_{3}$.

###### Proof.

We have seen in 3.1 that $G$ has a $\Pi_{3}$ Scott sentence. It remains to
show that $G$ has no $d$-$\Sigma_{2}$ Scott sentence.

Assuming that $G$ has two infinite Ulm invariants $f(n)=\infty$ and
$f(m)=\infty$ for $n<m$ we can write $G$ as
$H\oplus(\mathbb{Z}_{p^{n}})^{\omega}$ or more importantly as a chain
$\bigcup_{i}H\oplus(\mathbb{Z}_{p^{n}})^{i}$. We seek to show that at each
stage
$H\oplus(\mathbb{Z}_{p^{n}})^{i}\preceq_{\Sigma_{1}}H\oplus(\mathbb{Z}_{p^{n}})^{i+1}$.
It is well-known that if $G$ is a primary group and $H$ a direct summand, then
$H$ is a pure subgroup of $G$. Thus we have that
$H\oplus(\mathbb{Z}_{p^{n}})^{i}$ is a pure subgroup of
$H\oplus(\mathbb{Z}_{p^{n}})^{i+1}$. By Eklof’s result, it remains to show
that
$H\oplus(\mathbb{Z}_{p^{n}})^{i}\leq_{1}H\oplus(\mathbb{Z}_{p^{n}})^{i+1}$.

This is true because we can embed
$\mathbb{Z}_{p^{n}}\oplus(\mathbb{Z}_{p^{m}})^{\omega}$ into
$(\mathbb{Z}_{p^{m}})^{\omega}$ since $\mathbb{Z}_{p^{m}}$ contains a subgroup
isomorphic to $\mathbb{Z}_{p^{n}}$. Since $H$ contains
$({\mathbb{Z}}_{p^{m}})^{\omega}$ as a direct summand, we can embed
$H\oplus(\mathbb{Z}_{p^{n}})^{i+1}$ into $H\oplus(\mathbb{Z}_{p^{n}})^{i}$
using the same trick. ∎

For groups whose length is a limit ordinal, we will later show the Scott
sentence given in 3.1 is best possible. This means that the sentences we give
are best possible for groups of arbitrarily high Scott complexity.

To prove this, it is helpful to first characterize the back-and-forth
relations for abelian $p$-groups. For example, to show that a group
${\mathcal{A}}$ has no lower Scott complexity than $\Pi_{\alpha+1}$, it is
enough to exhibit a group ${\mathcal{B}}\not\cong{\mathcal{A}}$ such that
${\mathcal{A}}\equiv_{\alpha}{\mathcal{B}}$. In this case, ${\mathcal{A}}$
will have no $d$-$\Sigma_{\alpha}$ Scott sentence since any
$d$-$\Sigma_{\alpha}$ sentence true of ${\mathcal{A}}$ must be true of
${\mathcal{B}}$. It will be easy to exhibit such a ${\mathcal{B}}$ if we have
a characterization of when the relation
${\mathcal{A}}\equiv_{\alpha}{\mathcal{B}}$ holds.

To characterize the back-and-forth relations for reduced Abelian $p$-groups in
particular is a longstanding open problem mentioned, e.g., in [4]. Here, we
characterize the back-and-forth relations that hold between two arbitrary
groups ${\mathcal{A}},{\mathcal{B}}$. This is a strengthening of a result of
Barker’s [5], who was able to characterize the back-and-forth relations only
for when ${\mathcal{A}}\cong{\mathcal{B}}$.

An exact statement of Barker’s results are as follows:

######  Proposition 4.3 (Barker [5]).

Let $G$ be a reduced abelian $p$-group, and suppose that
$\overline{a},\overline{b}\in G^{<\omega}$. Let
$g:\overline{b}\rightarrow\overline{a}$ map corresponding members of the
tuples to one another.

  1. 1.

$\overline{a}\leq_{2\alpha}\overline{b}$ if and only if

    1. (a)

$g$ extends to an isomorphism
$g:\langle\overline{b}\rangle\to\langle\overline{a}\rangle$ and

    2. (b)

for every $b\in\langle\overline{b}\rangle$ and $a=g(b)$ we have

$h(b)=h(a)<\omega\cdot\alpha\quad\text{ or }\quad
h(b),h(a)\geq\omega\cdot\alpha.$

  2. 2.

$\overline{a}\leq_{2\alpha+1}\overline{b}$ if and only if

    1. (a)

$g$ extends to an isomorphism
$g:\langle\overline{b}\rangle\to\langle\overline{a}\rangle$ and

    2. (b)(i)

If $P_{\omega\cdot\alpha+k}$ is infinite for every $k<\omega$ then for every
$b\in\langle\overline{b}\rangle$ and $a=g(b)$ we have

$h(b)=h(a)<\omega\cdot\alpha\quad\text{ or }\quad
h(b)\geq\omega\cdot\alpha\text{ and
}h(a)\geq\min\\{h(b),\omega\cdot\alpha+\omega\\}.$

    3. (b)(ii)

If $P_{\omega\cdot\alpha+k}$ is infinite and $P_{\omega\cdot\alpha+k+1}$ is
finite, then for every $b\in\langle\overline{b}\rangle$ and $a=g(b)$ we have

$h(b)=h(a)<\omega\cdot\alpha\text{ or }\omega\cdot\alpha\leq h(b)\leq
h(a)\leq\omega\cdot\alpha+k\text{ or }h(a)=h(b)>\omega\cdot\alpha+k$

    4. (b)(iii)

If $P_{\omega\cdot\alpha}$ is finite, then for every
$b\in\langle\overline{b}\rangle$ and $a=g(b)$ we have

$h(b)=h(a).$

We are now ready to characterize when
$({\mathcal{A}},\overline{a})\leq_{\alpha}({\mathcal{B}},\overline{b})$ for
arbitrary reduced abelian $p$-groups ${\mathcal{A}},{\mathcal{B}}$.
Unfortunately, our characterization will not be entirely in algebraic terms.
However, we can obtain an entirely algebraic characterization with the
additional assumption that $|P_{\beta}|=\infty$ for all $\beta$ strictly less
than $\min\\{\lambda,\omega\cdot\alpha+\omega\\}$ for one of the groups, which
is easily seen to be the case, for example, when the last Ulm invariant is
infinite or if the length is a limit ordinal.

######  Theorem 4.4.

Let ${\mathcal{A}},{\mathcal{B}}$ be reduced Abelian p-groups of lengths
$\lambda_{\mathcal{A}}$, $\lambda_{\mathcal{B}}$ respectively. Let
$\overline{a}\in{\mathcal{A}}^{<\omega},\overline{b}\in{\mathcal{B}}^{<\omega}$
and let $g:\overline{b}\rightarrow\overline{a}$ map corresponding members of
the tuples to one another.

  1. 1.

$({\mathcal{A}},\overline{a})\leq_{2\alpha+1}({\mathcal{B}},\overline{b})$ if
and only if

    1. (a)

for all $\beta<\omega\cdot\alpha$ we have
$f_{{\mathcal{A}}}(\beta)=f_{{\mathcal{B}}}(\beta)$,

    2. (b)

$\lambda_{{\mathcal{A}}}=\lambda_{{\mathcal{B}}}<\omega\cdot\alpha$, or
$\lambda_{\mathcal{B}}\geq\omega\cdot\alpha$ and
$\lambda_{\mathcal{A}}\geq\min\\{\lambda_{\mathcal{B}},\omega\cdot\alpha+\omega\\}$,

    3. (c)

$|P^{{\mathcal{A}}}_{\beta}|=\infty$ for all
$\beta<min\\{\lambda_{{\mathcal{A}}},\omega\cdot\alpha+\omega\\}$, or any
$\Sigma_{1}$ sentence with quantifiers relativized to $G_{\omega\cdot\alpha}$
true of $({\mathcal{B}},\overline{b})$ is true of
$({\mathcal{A}},\overline{a})$,

    4. (d)

$g$ extends to an isomorphism
$g:\langle\overline{b}\rangle\to\langle\overline{a}\rangle$, and

    5. (e)

for every $b\in\langle\overline{b}\rangle$ and $a=g(b)$ we have

$h(a)=h(b)<\omega\cdot\alpha\quad\text{or}\quad\left(h(b)\geq\omega\cdot\alpha\quad\text{and}\quad
h(a)\geq\min\\{h(b),\omega\cdot\alpha+\omega\\}\right).$

  2. 2.

$({\mathcal{A}},\overline{a})\leq_{2\alpha+2}({\mathcal{B}},\overline{b})$ if
and only if

    1. (a)

for all $\beta<\omega\cdot\alpha$ we have
$f_{{\mathcal{A}}}(\beta)=f_{{\mathcal{B}}}(\beta)$
and for $\omega\cdot\alpha\leq\beta<\omega\cdot\alpha+\omega$ we have
$f_{{\mathcal{B}}}(\beta)\leq f_{{\mathcal{A}}}(\beta)$,

    2. (b)

$\lambda_{{\mathcal{A}}}=\lambda_{{\mathcal{B}}}<\omega\cdot\alpha+\omega$ or
$\omega\cdot\alpha+\omega\leq\lambda_{{\mathcal{A}}},\lambda_{\mathcal{B}}$,

    3. (c)

$|P^{{\mathcal{B}}}_{\beta}|=\infty$ for all
$\beta<min\\{\lambda_{{\mathcal{B}}},\omega\cdot\alpha+\omega\\}$, or every
$\Sigma_{2}$ sentence with quantifiers relativized to $G_{\omega\cdot\alpha}$
true of $({\mathcal{B}},\overline{b})$ is true of
$({\mathcal{A}},\overline{a})$.

    4. (d)

$g$ extends to an isomorphism
$g:\langle\overline{b}\rangle\to\langle\overline{a}\rangle$, and

    5. (e)

for every $b\in\langle\overline{b}\rangle$ and $a=g(b)$ we have

$h(b)=h(a)<\omega\cdot\alpha+\omega\quad\text{or}\quad
h(b),h(a)\geq\omega\cdot\alpha+\omega.$

  3. 3.

$({\mathcal{A}},\overline{a})\leq_{\alpha}({\mathcal{B}},\overline{b})$ for
$\alpha$ a limit ordinal or zero if and only if

    1. (a)

for all $\beta<\omega\cdot\alpha$ we have
$f_{{\mathcal{A}}}(\beta)=f_{{\mathcal{B}}}(\beta)$,

    2. (b)

$\lambda_{{\mathcal{A}}}=\lambda_{{\mathcal{B}}}<\omega\cdot\alpha$ or
$\omega\cdot\alpha\leq\lambda_{{\mathcal{A}}},\lambda_{\mathcal{B}}$,

    3. (c)

$g$ extends to an isomorphism
$g:\langle\overline{b}\rangle\to\langle\overline{a}\rangle$, and

    4. (d)

for every $b\in\langle\overline{b}\rangle$ and $a=g(b)$ we have

$h(b)=h(a)<\omega\cdot\alpha\quad\text{or}\quad
h(b),h(a)\geq\omega\cdot\alpha.$

Before we prove this, as a sanity check, let us pause and note how our results
imply Barker’s in the case where ${\mathcal{A}}\cong{\mathcal{B}}$. It is
clear that our conditions in the even and limit cases imply Barker’s, and it
is also clear how our conditions on the $(2\alpha+1)$-relations imply Barker’s
in the case where $|P_{\beta}|=\infty$ for all
$\beta<min\\{\lambda,\omega\cdot\alpha+\omega\\}$.

Now, let us look at the case where $P_{\omega\cdot\alpha+k+1}$ is finite. It
is known that for any reduced abelian $p$-group $G$, if the subgroup $P$ of
elements of order $p$ is finite, then so is the whole group $G$. Thus, in this
case we have that $G_{\omega\cdot\alpha+k+1}$ is finite as well. We can then
produce a relativized $\Sigma_{1}$ sentence that says $b$ has height less than
or equal to $\omega\cdot\alpha+k$. It says $b$ cannot be any one of the $n$
elements of height higher than $\omega\cdot\alpha+k+1$. Let $n$ be the number
of elements in $G_{\omega\cdot\alpha+k+1}$:

$\exists x_{1}\neq x_{2}\neq\cdots\neq x_{n}\in
G_{\omega\cdot\alpha+k+1}\bigwedge_{i\leq n}b\neq x_{i}$

A similar $\Sigma_{1}$ sentence to that above says $b$ has height less than or
equal to $\omega\cdot\alpha+k+n$, for $n\geq 1$. In addition, since
$G_{\omega\cdot\alpha+k+n}$ is definable by a relativized $\Sigma_{1}$
formula, we can express that $b$ has height greater than or equal to
$\omega\cdot\alpha+k+n$ for every $n\geq 1$ with the same complexity. Thus it
is relativized $\Sigma_{1}$ to specify exactly the height of $b$ when it is
greater than $\omega\cdot\alpha+k$ and less than $\omega\cdot\alpha+\omega$.
When $P_{\omega\cdot\alpha}$ is finite, a similar arguments shows we can
express that $b$ has height $\omega\cdot\alpha+n$ for every $n\in\omega$.

Now we claim next that in addition, the length of the group is no more than
$\omega\cdot\alpha+\omega$. This would mean that $0$ is the only element of
height above $\omega\cdot\alpha+\omega$. For this, just note that since
$G_{\omega\cdot\alpha+k+1}$ is finite and the $G_{\beta}$’s form a decreasing
chain, the chain must stabilize before $G_{\omega\cdot\alpha+\omega}$.

These considerations together show that our results imply Barker’s in every
case.

###### Proof.

(Theorem 4.4)

$(\Rightarrow)$ Suppose that the relevant back-and-forth conditions hold.

Recall from Theorem 3.1, that the formula expressing that the $\beta$’th Ulm
invariant is at least $m$ is $\Sigma_{2\beta^{\prime}+2}$ where
$\beta=\omega\cdot\beta^{\prime}+r$. Now, if $\beta<\omega\cdot\alpha$ then
$\beta^{\prime}<\alpha$ which implies $\beta^{\prime}+1\leq\alpha$ so
$2\beta^{\prime}+2<2\alpha+1$. Thus any formula which is
$\Sigma_{2\beta^{\prime}+2}$ is both $\Pi_{2\alpha+1}$ and
$\Sigma_{2\alpha+1}$. Because of this, if
$({\mathcal{A}},\overline{a})\leq_{2\alpha+1}({\mathcal{B}},\overline{b})$
then condition (1a) holds. If $\alpha$ is a limit ordinal then
$2\beta^{\prime}<2\alpha=\alpha$ so $2\beta^{\prime}+2<\alpha$ and thus any
formula which is $\Sigma_{2\beta^{\prime}+2}$ is both $\Pi_{\alpha}$ and
$\Sigma_{\alpha}$. Therefore, if
$({\mathcal{A}},\overline{a})\leq_{\alpha}({\mathcal{B}},\overline{b})$ then
(3a) holds. If $\omega\cdot\alpha\leq\beta\leq\omega\cdot\alpha+\omega$ then
$\beta^{\prime}=\alpha$ and thus if
$({\mathcal{A}},\overline{a})\leq_{2\alpha+2}({\mathcal{B}},\overline{b})$
condition (2a) holds.

Again, recall from Theorem 3.1 that the formula expressing the fact that a
group has length $\lambda=\omega\gamma+r$ is $d$-$\Sigma_{2\gamma+1}$. Thus in
cases (1) and (3), if $\lambda_{{\mathcal{A}}}$ or
$\lambda_{{\mathcal{B}}}<\omega\cdot\alpha$ then $\gamma<\alpha$. Thus
$2\gamma+1<2\alpha+1$ (or $2\gamma+1<\alpha$ if $\alpha$ is limit), so
$\lambda_{{\mathcal{A}}}=\lambda_{{\mathcal{B}}}$. In case (2), if
$\lambda_{{\mathcal{A}}}$ or $\lambda_{{\mathcal{B}}}<\omega(\alpha+1)$ then
$\gamma<\alpha+1$ and $2\gamma+1<2\gamma+2$ and again
$\lambda_{{\mathcal{A}}}=\lambda_{{\mathcal{B}}}$. Thus in cases (2) and (3),
conditions (2b) and (3b) hold. It remains to show in case (1) that
$\lambda_{{\mathcal{A}}}\geq\lambda_{{\mathcal{B}}}$ if
$\lambda_{{\mathcal{A}}},\lambda_{{\mathcal{B}}}<\omega\cdot\alpha+\omega$.
However, if $G_{\omega\cdot\alpha+k}\neq\\{0\\}$ then
$\lambda>\omega\cdot\alpha+k$. Recall that $G_{\omega\cdot\alpha+k}$ is
$\Sigma_{2\alpha+1}$, so the formula expressing the fact that there is some
nonzero element in $G_{\omega\cdot\alpha+k}$ is as well.

Since $G_{\omega\cdot\alpha}$ is $\Pi_{2\alpha}$-definable, any $\Sigma_{1}$
formula with quantifiers relativized to this subgroup is $\Sigma_{2\alpha+1}$
and any $\Sigma_{2}$ formula with relatives quantifiers is
$\Sigma_{2\alpha+2}$. Thus in cases (1) and (2), conditions (1c) and (2c)
hold.

In each case, conditions (1d), (2d), and (3c) clearly hold.

If $h(a)$ or $h(b)<\omega\cdot\alpha$ then WLOG $h(a)=\omega\gamma+r$ for some
$\gamma<\alpha$. However, the formula $x\in G_{\omega\gamma+r}\land x\not\in
G_{\omega\gamma+r}$ is $d$-$\Sigma_{2\gamma+1}$. But $2\gamma+1<2\alpha+1$
and, if $\alpha$ is limit, $2\gamma+1<\alpha$. Thus in cases (1) and (3) we
have that $h(a)=h(b)$. If $h(a)$ or $h(b)<\omega\cdot\alpha+1$ then
$\gamma<\alpha+1$ and similarly in case (2) we have that $h(a)=h(b)$. It
remains to show in case (1) that if $h(a),h(b)\leq\omega\cdot\alpha+\omega$
then $h(a)\geq h(b)$. The formula $x\in G_{\omega\cdot\alpha+k}$ expresses
that $h(x)\geq\omega\cdot\alpha+k$ for an element $x$, however, and is
$\Sigma_{2\alpha+1}$.

$(\Leftarrow)$ We show that the relevant conditions imply that
$({\mathcal{A}},\overline{a})\leq_{\gamma}({\mathcal{B}},\overline{b})$ by
induction on $\gamma$. When $\gamma=0$, conditions 1(c), 2(d), and 3(c)
guarantee that the relevant back and forth relations hold. For $\gamma>0$, we
proceed in cases depending on whether $\gamma$ is odd, even, or a limit
ordinal.

Case 1: Suppose that $\gamma=2\alpha+1$, and assume that
$({\mathcal{A}},\overline{a}),({\mathcal{B}},\overline{b})$ satisfy the above
conditions for part 1. We wish to show that for any
$\overline{d}\in{\mathcal{B}}^{<\omega}$ we can find a
$\overline{c}\in{\mathcal{A}}^{<\omega}$ such that
$({\mathcal{B}},\overline{b}\overline{d})\leq_{2\alpha}({\mathcal{A}},\overline{a}\overline{c})$.
By induction it is enough to show that the relevant conditions we claim are
equivalent to the back and forth relations hold.

That is, in the case that $\alpha=\delta+1$, by induction and the fact that
$\omega\cdot\delta+\omega=\omega\cdot\alpha$ it is enough to show the
following:

  1. (i)

for all $\beta<\omega\cdot\delta$ we have
$f_{{\mathcal{A}}}(\beta)=f_{{\mathcal{B}}}(\beta)$
and for $\omega\cdot\delta\leq\beta<\omega\cdot\alpha$ we have
$f_{{\mathcal{B}}}(\beta)\leq f_{{\mathcal{A}}}(\beta)$,

  2. (ii)

$\lambda_{{\mathcal{A}}}=\lambda_{{\mathcal{B}}}<\omega\cdot\alpha$ or
$\omega\cdot\alpha\leq\lambda_{{\mathcal{A}}},\lambda_{\mathcal{B}}$,

  3. (iii)

$|P^{{\mathcal{A}}}_{\beta}|=\infty$ for all
$\beta<min\\{\lambda_{A},\omega\cdot\alpha\\}$

  4. (iv)

$g$ extends to an isomorphism
$g:\langle\overline{b}\overline{d}\rangle\to\langle\overline{a}\overline{c}\rangle$

  5. (v)

for every $b\in\langle\overline{b}\overline{d}\rangle$ and $a=g(b)$ we have

$h(b)=h(a)<\omega\cdot\alpha\quad\text{or}\quad
h(b),h(a)\geq\omega\cdot\alpha.$

Note that (i) follows immediately by assumption (1a), and (ii) follows
immediately by assumption (1b).

We now claim condition (iii) is met. If
$\lambda_{\mathcal{A}}<\omega\cdot\alpha$ then by assumption (1c) the claim
follows. Using a result of Kaplansky’s ([10] Problem 36) if
$\lambda_{\mathcal{A}}\geq\omega\cdot\alpha$ then
$|P^{{\mathcal{A}}}_{\beta}|=\infty$ for all $\beta<\omega\cdot\alpha$.

In the case that $\alpha$ is a limit ordinal, we must show that:

  1. (a)

for all $\beta<\omega\cdot\alpha$ we have
$f_{{\mathcal{A}}}(\beta)=f_{{\mathcal{B}}}(\beta)$, which follows from
assumption (1a)

  2. (b)

conditions on $g$, $\overline{b}\overline{d}$, $\overline{a}\overline{c}$
identical to (iv) and (v) above, as well as a condition identical to (ii)
above.

Thus, whether or not $\alpha$ is a limit ordinal, it remains only to show
(iv), (v) above hold.

Let $A=\langle\overline{a}\rangle,B=\langle\overline{b}\rangle$ be subgroups
of ${\mathcal{A}},{\mathcal{B}}$ respectively. Given a
$\overline{d}\in{\mathcal{B}}$ we seek to extend the isomorphism $g:B\to A$ to
an isomorphism $g:B^{\prime}=\langle\overline{b}\overline{d}\rangle\to
C^{\prime}=\langle\overline{a}\overline{c}\rangle$ for a suitable
$\overline{c}\in{\mathcal{A}}$ such that it meets (v) above. We will do this
by making a sequence of extensions $B=B_{0}\subseteq
B_{1}\subseteq\cdots\subseteq B^{\prime}$ such that for each $i$, $x\in
B^{\prime}\setminus B_{i}$ with $px\in B_{i}$, and at each stage extending $g$
to $B_{i+1}=\langle x,B_{i}\rangle$.

Case 1(a): If
$\lambda_{{\mathcal{A}}}=\lambda_{\mathcal{B}}<\omega\cdot\alpha$ then our
assumption of condition 1(a) forces the groups to be isomorphic by Ulm’s
theorem. By Barker’s results, since $g$ is height-preserving, the tuples
$\overline{a},\overline{b}$ are actually automorphic and so we can certainly
extend $g$ as needed.

Case 1(b): For this case, assume that
$\lambda_{{\mathcal{A}}},\lambda_{{\mathcal{B}}}\geq\omega\cdot\alpha$ and
that $|P^{{\mathcal{A}}}_{\beta}|=\infty$ for all
$\beta<min\\{\lambda_{{\mathcal{A}}},\omega\cdot\alpha+\omega\\}$.

Let $N_{0}=|B^{\prime}\setminus B|$. We shall ensure that we extend $g$ to
$B_{i}$ such that for all $b\in B_{i}$ we have $h(a)=h(b)<\omega\cdot\alpha$
or $\omega\cdot\alpha\leq h(b)\leq h(a)$. If this is accomplished for all
$i\leq N_{0}$ then we are done. We shall call the above condition $\star_{i}$,
and, by our assumption of condition 1(d), $g$ meets $\star_{0}$.

Now by elongating and reordering the tuple $\overline{d}$ if necessary we may
assume for all $i$ that if $x\in B^{\prime}\setminus B_{i}$ then $px\in
B_{i}$, $x$ is proper with respect to $B_{i}$, and $h(px)$ is maximal amongst
all such elements of $x+B_{i}$. Suppose now that we are at the stage $i$ where
$g,A_{i}$ has been constructed as desired and let $g(px)=y$.

Case 1(b)(i): If $h(y)=h(px)=h(x)+1<\omega\cdot\alpha$ then we choose a $w$
such that $pw=y$ and $h(w)=h(x)$. Since $h(y)=h(x)+1$ we can choose such a $w$
with $h(w)\geq h(x)$; such a $w$ will have height exactly $h(x)$ since
otherwise $h(y)=h(pw)>h(w)$ would be greater than $h(x)+1$. We know that
$w\not\in A_{i}$ because if $w=g(z)$ for some $z\in A_{i}$ then $pz=px$ since
$g$ is a group isomorphism. This would mean that $h(px-pz)=h(0)=\infty>h(px)$
which contradicts our choice of $x$.

We also get that $w$ is proper with respect to $A_{i}$, for if there were some
$a\in A_{i}$ such that $h(w+a)\geq h(w)+1=h(x)+1$ and $a=g(b)$ for $b\in
B_{i}$, then since $w+a\neq 0$ we must have $h(p(w+a))\geq h(x)+2$ which
forces $h(p(x+b))\geq h(x)+2$. This also contradicts the maximal height of
$px\in B_{i}$.

We can now extend $g$ to $B_{i+1}=\langle x,B_{i}\rangle$ by mapping
$g(rx+b)=rw+a$ for $0<r<p,b\in B_{i}$ and $a=g(b)$.

To see that $g$ meets condition $\star_{i+1}$ take an arbitrary element $rx+b$
of $B_{i+1}$ where $b\in B_{i}$ and $0\leq r<p$ since $px\in B_{i}$. Then
since $h(w)=h(x)$ and $h(a)\geq h(b)$

$h(rw+a)=min\\{h(w),h(a)\\}\geq min\\{h(x),h(b)\\}=h(rx+b).$

Case 1(b)(ii): A similar and slightly stronger version of this case is proven
in case 2(b)(ii).

Case 1(b)(iii): If $h(x)\geq\omega\cdot\alpha$ then $h(x)<h(px)\leq h(y)$.
Thus $y\in G_{h(x)+1}^{{\mathcal{A}}}$ so there exists $w_{1}\in{\mathcal{A}}$
such that $pw_{1}=y$ and $h(w_{1})\geq h(x)$.

Now $G_{h(x)}^{{\mathcal{A}}}$ is nonempty so $P_{h(x)}^{{\mathcal{A}}}$ is
infinite. Thus there exists an element $w_{2}\in
P_{h(x)}^{{\mathcal{A}}}-A_{i}$. Let $w=w_{1}+w_{2}$ and extend $g$ to
$B_{i+1}$ by defining $g(x)=w$. It is easy to check that the extension remains
an isomorphism since $pw=y$ and $w\not\in A_{i}$.

To see that $g$ meets condition $\star_{i+1}$ take an arbitrary element $rx+b$
of $B_{i+1}$ where $b\in B_{i}$ and $0\leq r<p$ since $px\in B_{i}$. We have
that $g(rx+b)=rw+a$. If $h(b)<\omega\cdot\alpha$ then $h(rx+b)=h(b)$ and
$h(b)=h(a)<\omega\cdot\alpha$ by the fact that $g$ satisfies $\star_{i}$. Now
$h(rw)\geq min\\{h(w_{1}),h(w_{2})\\}\geq h(x)\geq\omega\cdot\alpha$ so
$h(rw+a)=h(a)=h(b)=h(rx+b)$. Thus $g$ preserves heights below
$\omega\cdot\alpha$.

If $h(b)\geq\omega\cdot\alpha$ then $h(a)\geq h(b)$. Moreover,

$h(rw+a)\geq min\\{h(w),h(a)\\}\geq min\\{h(w_{1}),h(w_{2}),h(a)\\}\geq
min\\{h(x),h(b)\\}$

However, since $x$ is proper with respect to $B_{i}$ we have that
$h(rx+b)=min\\{h(x),h(b)\\}$. Thus $h(rw+a)\geq h(rx+b)$, as desired. Thus $g$
is non-decreasing for heights above $\omega\cdot\alpha$ as well, and condition
$\star_{i+1}$ is met.

Case 1(c): For this case, assume that
$\lambda_{{\mathcal{A}}},\lambda_{{\mathcal{B}}}\geq\omega\cdot\alpha$ and
that for some
$\beta<min\\{\lambda_{{\mathcal{A}}},\omega\cdot\alpha+\omega\\}$,
$|P^{{\mathcal{A}}}_{\beta}|\neq\infty$. Then we must have that
$\omega\cdot\alpha\leq\lambda_{{\mathcal{B}}}\leq\lambda_{{\mathcal{A}}}\leq\omega\cdot\alpha+\omega$.
As in case 1(b), we will extend $g$ to $B_{i}$ such that condition $\star_{i}$
holds, and may assume that for all $i$, if $x\in B^{\prime}\setminus B_{i}$
then $px\in B_{i}$, $x$ is proper with respect to $B_{i}$, and $h(px)$ is
maximal amongst all such elements of $x+B_{i}$.

We now claim that we can add the additional simplifying assumption that
$h(x)\leq\omega\cdot\alpha$ for any $x\in B^{\prime}-B_{0}$. Note that the
$B_{i}$’s can be built before the $A_{i}$’s and the associated extensions of
$g$. The goal is to deal first with all of the elements in $B^{\prime}\cap
G^{{\mathcal{B}}}_{\omega\cdot\alpha}$ before we begin the construction. Let
$\overline{d}^{\prime}$ be the sub-tuple of $\overline{d}$ of elements in
$G^{{\mathcal{B}}}_{\omega\cdot\alpha}$. We claim that we can extend $g$ to a
height non-decreasing isomorphism.

Now, by (1c) any $\Sigma_{1}$ formula where the quantifiers are relativized to
$G_{\omega\cdot\alpha}$ true of $\overline{b}$ is true of $\overline{a}$. Let,
for any group $G$ and tuple $\overline{s}\in G$, the formula
$\varphi^{G}_{\overline{s}}(\overline{x})$ define the full atomic type of
$\overline{s}$ in $G$. Note this is $\Delta_{0}$ since the language of groups
is finite. Now consider the formula

$\exists\overline{u}\in
G_{\omega\cdot\alpha}(\varphi_{\overline{b}\overline{d}^{\prime}}^{B}(\overline{b}\overline{u})\land\bigwedge_{i=0}^{|\overline{d}^{\prime}|-1}u_{i}\in
G_{h(d_{i}^{\prime})})$

Note that $h(d_{i}^{\prime})$ is strictly less than $\alpha+\omega$ since
$\omega\cdot\alpha\leq\lambda_{{\mathcal{B}}}\leq\lambda_{{\mathcal{A}}}\leq\omega\cdot\alpha+\omega$.
Thus $G_{h(d_{i}^{\prime})}$ is $\Sigma_{1}$-definable in
$G_{\omega\cdot\alpha}$ so is $\Sigma_{1}$-definable in $G$ with quantifiers
relativized to $G_{\omega\cdot\alpha}$.

This formula is true of $\overline{b}$ so is true of $\overline{a}$. If
$\overline{c^{\prime}}$ is the witness for this formula in ${\mathcal{A}}$,
then $\overline{a}\overline{c^{\prime}}$ will have the same atomic type as
$\overline{b}\overline{d}^{\prime}$, and the map sending the corresponding
tuples to each other is height non-decreasing. In other words,

$({\mathcal{A}},\overline{a}\overline{c}^{\prime})\equiv_{0}({\mathcal{B}},\overline{b}\overline{d}^{\prime})$

We now consider the associated extension of $g$ which includes
$\overline{d}^{\prime}$ in its domain. Replace $B_{0}$, $A_{0}$ with the
groups generated by the domain and range, respectively, of this extension.
Since the extension is height non-decreasing above $\omega\cdot\alpha$,
condition $\star_{0}$ is met.

However, we need to show that our simplifying conditions still hold if we
reorder the tuple $\overline{d}$ so that all of the elements of height
$\geq\omega\cdot\alpha$ appear first, generating $B_{0}$. It is easily seen
that we may continue to assume that $px\in B_{i}$.

Now if $x$ is proper with respect to $B_{i}$, with $h(x)<\omega\cdot\alpha$
and we consider $B_{i}^{\prime}=\langle B_{i},z\rangle$ where $z$ has height
$\geq\omega\cdot\alpha$. Then for arbitrary $b\in B_{i}$ and $n\in\omega$, we
have that $h(x)\geq h(x+b)$ so $h(x+b)<\omega\cdot\alpha\leq h(nz)$. This
means that

$h(x+b+nz)=h((x+b)+nz)=h(x+b)\leq h(x)$

so $x$ remains proper with respect to $B_{i}^{\prime}$. This argument
generalizes to show that after reordering the tuple we may continue to assume
that $x$ is proper with respect to $B_{i}$.

Finally, note that the simplifying assumption that $h(px)$ is maximal is
achieved by possibly replacing $x$ with an element of the same height as $x$
so this may still be assumed after reordering the tuple.

Let $g(px)=y$.

Since $g$ meets condition $\star_{i}$, we know that either
$h(px)=h(y)<\omega\cdot\alpha$ or $h(px)\geq\omega\cdot\alpha$ and $h(y)\geq
h(px)$.

Now we continue with the construction. Assume $A_{i}$ and $g:B_{i}\rightarrow
A_{i}$ has been constructed and condition $\star_{i}$ is met.

Case 1(c)(i): In the case where $h(y)=h(px)=h(x)+1<\omega\cdot\alpha$, we
proceed as in Case 1(b)(i).

Case 1(c)(ii): In the case where $h(x)=\gamma<\omega\cdot\alpha,h(x)>\gamma+1$
we proceed as in Case 1(b)(ii).

Case 2: Suppose that $\gamma=2\alpha+2$, and assume that
$({\mathcal{A}},\overline{a}),({\mathcal{B}},\overline{b})$ satisfy the above
conditions for part 2. Now we must do something very similar to before to show
that given any $\overline{d}$ we can find a $\overline{c}$ such that
$({\mathcal{B}},\overline{b}\overline{d}),({\mathcal{A}},\overline{a}\overline{c})$
meet the conditions for part 1. As before, most of the conditions follow by
assumption.

Our remaining concern is extending the isomorphism $g$ to a map
$\langle\overline{b},\overline{d}\rangle\to\langle\overline{a},\overline{c}\rangle$.
Let $\overline{d}\in{\mathcal{B}}^{<\omega}$ and we once again let
$B_{0}=B=\langle\overline{b}\rangle$,
$B^{\prime}=\langle\overline{b}\overline{d}\rangle$, and
$A_{0}=A=\langle\overline{a}\rangle$.

Case 2(a): If $\lambda_{\mathcal{A}}=\lambda_{\mathcal{B}}<\omega\cdot\alpha$
then, as in case 1(a), we conclude that ${\mathcal{A}}$ and ${\mathcal{B}}$
are isomorphic.

Case 2(b): For this case, assume that
$\lambda_{{\mathcal{A}}},\lambda_{{\mathcal{B}}}\geq\omega\cdot\alpha$ and
that $|P^{B}_{\beta}|=\infty$ for all
$\beta<min\\{\lambda_{{\mathcal{A}}},\omega\cdot\alpha+\omega\\}$. Let

$M=\max\\{m:b^{\prime}\in B^{\prime},h(b^{\prime})=\omega\cdot\alpha+m\text{
for }m\in\omega\\}+1,$ $N_{0}=|B^{\prime}/B|+M,\text{ and }N_{i+1}=N_{i}-1.$

For each $0\leq i\leq N_{0}$ we would like to extend $g$ to $B_{i}$ in such a
way that it satisfies

$h(b)=h(a)<\omega\cdot\alpha+M\quad\text{or}\quad\left(h(b)\geq\omega\cdot\alpha+\omega\quad\text{and}\quad
h(a)\geq\omega\cdot\alpha+N_{i}\right)$

for all $b\in B_{i}$, $a=g(b)$. We will call this condition $\star_{i}$. If
this is met for all $i$ then $g$ will satisfy

$h(b^{\prime})=h(a^{\prime})\leq\omega\cdot\alpha\quad\text{or}\quad
h(a^{\prime})\geq\omega\cdot\alpha\text{ and
}h(b^{\prime})\geq\min\\{h(a^{\prime}),\omega\cdot\alpha+\omega\\}$

for all $b^{\prime}\in B^{\prime}$, which would complete our proof that
$({\mathcal{B}},\overline{b}\overline{d})\leq_{2\alpha+1}({\mathcal{A}},\overline{a}\overline{c})$.
By assumption, $g$ meets condition $\star_{0}$.

Now by induction on $i$, we take an $x\in B^{\prime}\setminus B_{i}$ such that
$px\in B_{i}$, $x$ is proper with respect to $B_{i}$, and $h(px)$ is maximal
amongst all such elements of $x+B_{i}$. Let $g(px)=y$. Since $g$ meets
condition $\star_{i}$ we know that either $h(px)=h(y)<\omega\cdot\alpha+M$ or
$h(px)\geq\omega\cdot\alpha+\omega$ and $h(y)\geq\omega\cdot\alpha+N_{i}$.

Case 2(b)(i): If $h(y)=h(px)=h(x)+1<\omega\cdot\alpha+M$ then we choose a $w$
such that $pw=y$ and $h(w)=h(x)$. Since $h(y)=h(x)+1$ we can choose such a $w$
with $h(w)\geq h(x)$; such a $w$ will have height exactly $h(x)$ since
otherwise $h(y)=h(pw)>h(w)$ would be greater than $h(x)+1$.

We know that $w\not\in A_{i}$ because if $w=g(z)$ for some $z\in A_{i}$ then
$pz=px$ since $g$ is a group isomorphism. This would mean that $h(px-
pz)=h(0)=\infty>h(px)$ which contradicts our choice of $x$.

We also get that $w$ is proper with respect to $A_{i}$, for if there were some
$a\in A_{i}$ such that $h(w+a)\geq h(w)+1=h(x)+1$ and $a=g(b)$ for $b\in
B_{i}$, then since $w+a\neq 0$ we must have $h(p(w+a))\geq h(x)+2$ which
forces $h(p(x+b))\geq h(x)+2$. This also contradicts the maximal height of
$px\in B_{i}$.

We can now extend $g$ to $B_{i+1}=\langle x,B_{i}\rangle$ by mapping
$g(rx+b)=rw+a$ for $0<r<p,b\in B_{i}$ and $a=g(b)$.

To see that $g$ meets condition $\star_{i+1}$ take an arbitrary element $rx+b$
of $B_{i+1}$ where $b\in B_{i}$ and $0<r<p$ since $px\in B_{i}$. Then since
$x,w$ are proper, and $h(w)=h(x)$ we have for $r\neq 0,r<p$

$h(rw+a)=min\\{h(w),h(a)\\}=min\\{h(x),h(b)\\}=h(rx+b).$

for if $h(a)>h(w)$ then $min\\{h(w),h(a)\\}=h(w)<\omega\cdot\alpha+M$, and if
$h(w)\geq h(a)$ then $h(a)<\omega\cdot\alpha+M$ and so in both cases
$h(a)=h(b)$. If $r=0$ in $rw+a$ then since $g$ already meets $\star_{i}$ so
also does our extension of $g$.

Case 2(b)(ii): Suppose $h(px)<\omega\cdot\alpha+M$ and let $h(x)=\gamma$ with
$h(px)>\gamma+1$. Since $h(px)>\gamma+1$, there is a
$v\in{\mathcal{B}}_{\gamma+1}$ such that $pv=px$. The element $x-v$ is then in
$P^{{\mathcal{B}}}_{\gamma}$ and it also has height $\gamma$ and is proper
with respect to $B_{i}$. We can apply Lemma 2.3 to see that the range of $u$
is not all of $P^{{\mathcal{B}}}_{\gamma}/P^{{\mathcal{B}}}_{\gamma+1}$.

Now, by assumption and induction on $i$, $g$ preserves heights for elements of
height $<\omega\cdot\alpha+M$ and is an isomorphism between $A_{i}$ and
$B_{i}$. Therefore $g$ maps $B_{i}\cap{\mathcal{B}}_{\gamma+1}$ in bijection
with $A_{i}\cap{\mathcal{A}}_{\gamma+1}$, $B_{i}\cap{\mathcal{B}}_{\gamma}$ in
bijection with $A_{i}\cap{\mathcal{A}}_{\gamma}$ and
$B_{i}\cap{\mathcal{B}}_{\gamma}\cap p^{-1}{\mathcal{B}}_{\gamma+2}$ in
bijection with $A_{i}\cap{\mathcal{A}}_{\gamma}\cap
p^{-1}{\mathcal{A}}_{\gamma+2}$.

Since $u$ is not surjective and $(B_{i}\cap{\mathcal{B}}_{\gamma}\cap
p^{-1}{\mathcal{B}}_{\gamma+2})/(B_{i}\cap{\mathcal{B}}_{\gamma+1})$ is
finite, its dimension as a vector space over $\mathbb{Z}_{p}$ is strictly less
than $f_{{\mathcal{B}}}(\gamma).$ Since $(A_{i}\cap{\mathcal{A}}_{\gamma}\cap
p^{-1}{\mathcal{A}}_{\gamma+2})/(A_{i}\cap{\mathcal{A}}_{\gamma+1})$ has the
same cardinality and thus the same dimension since it is finite, its dimension
is also less than $f_{{\mathcal{B}}}(\gamma)\leq f_{{\mathcal{A}}}(\gamma)$.
Thus we can apply Lemma 2.3 in reverse, giving us an element
$w_{1}\in{\mathcal{A}}$ such that $pw_{1}=0$, $h(w_{1})=\gamma$, and which is
proper with respect to $A_{i}$.

Since $h(y)>\gamma+1$ we know that $y=pw_{2}$ for
$w_{2}\in{\mathcal{A}}_{\gamma+1}$. Let $w=w_{1}+w_{2}$. Then $pw=0+y$,
$h(w)=h(w_{1})=\gamma$, and $w$ is proper with respect to $A_{i}$. If we
extend $g$ by $g(x)=w$ as before, we see that $g$ satisfies $\star_{i+1}$
since $w$ was chosen to be proper as in the previous case.

Case 2(b)(iii): $h(px)\geq\omega\cdot\alpha+\omega$ and
$h(y)\geq\omega\cdot\alpha+N_{i}$. As in case 1(b)(iii), we can find a
$w_{1}\in{\mathcal{A}}$ such that $pw_{1}=y$ and
$h(w_{1})=h(y)-1\geq\omega\cdot\alpha+N_{i}-1$. Furthermore, if we are in this
case, then $\lambda_{{\mathcal{B}}}\geq\omega\cdot\alpha+\omega$ which forces
$\lambda_{{\mathcal{A}}}\geq\omega\cdot\alpha+\omega$. Thus, since
$P^{{\mathcal{A}}}_{\omega\cdot\alpha+N_{i}}$ is infinite, we can find a
$w_{2}\in P^{{\mathcal{A}}}_{\omega\cdot\alpha+N_{i}}\setminus A_{i}$. Set
$w=w_{1}+w_{2}$ and extend $g$ by sending $x$ to $w$. Note that $w\not\in
A_{i}$ and $h(w)\geq\omega\cdot\alpha+N_{i+1}>\omega\cdot\alpha+M$.

Let $b\in B_{i}$. If $h(b)\leq\omega\cdot\alpha+M$ then by assumption
$h(a)=h(b)$ and so for $0\leq r<p$, $h(rx+b)=h(b)$ and $h(rw+a)=h(a)$. If
$h(b)\geq\omega\cdot\alpha+\omega$ then by induction on $i$,
$h(a)\geq\omega\cdot\alpha+N_{i}>\omega\cdot\alpha+N_{i+1}$. Thus,
$h(rx+b)\geq\omega\cdot\alpha+\omega$ and
$h(rw+a)\geq\min\\{h(w),h(a)\\}\geq\omega\cdot\alpha+N_{i+1}$. This shows that
$g$ meets condition $\star_{i+1}$.

Case 2(c): For this case, assume that
$\lambda_{{\mathcal{A}}},\lambda_{{\mathcal{B}}}\geq\omega\cdot\alpha$ and
that every $\Sigma_{2}$ sentence with quantifiers relativized to
$G_{\omega\cdot\alpha}$ true of $(B,\overline{b})$ is true of
$({\mathcal{A}},\overline{a})$. Then we show first that as in Case 1 we may
assume that $h(x)<\omega\cdot\alpha.$ We do this just as before by reordering
the tuple $\overline{d}$ so that the sub-tuple $\overline{d}^{\prime}$ of
elements of height greater than $\omega\cdot\alpha$ appears first.

We claim that we can extend $g$ to a height-preserving isomorphism including
$\overline{d}^{\prime}$. Consider the formula

$\exists\overline{u}\in
G_{\omega\cdot\alpha}(\varphi_{\overline{b}\overline{d}^{\prime}}^{B}(\overline{b}\overline{u})\land\bigwedge_{i=0}^{|\overline{d}^{\prime}|-1}u_{i}\in
G_{h(d_{i}^{\prime})}\land u_{i}\not\in G_{h(d_{i}^{\prime})+1})$

This is a $\Sigma_{2}$ formula true of $\overline{b}$ so is true of
$\overline{a}$. If $\overline{c^{\prime}}$ is the witness for this formula in
${\mathcal{A}}$, then $\overline{a}\overline{c^{\prime}}$ will have the same
atomic type as $\overline{b}\overline{d}^{\prime}$, and the map sending the
corresponding tuples to each other is height preserving. In other words,

$({\mathcal{A}},\overline{a}\overline{c}^{\prime})\equiv_{0}({\mathcal{B}},\overline{b}\overline{d}^{\prime})$

We now consider the associated extension of $g$ which includes
$\overline{d}^{\prime}$ in its domain. Replace $B_{0}$, $A_{0}$ with the
groups generated by the domain and range, respectively, of this extension. As
before, we may assume that $px\in B_{i}$, $x$ is proper with respect to
$B_{i}$, and $h(px)$ is maximal amongst all such elements of $x+B_{i}$.

Assume that we have built $g:B_{i}\rightarrow A_{i}$ such that $h(b)=h(a)$ for
all $b\in B_{i}$, $a=g(b)$. We would like to extend $g$ to $B_{i+1}$ such that
$g$ preserves heights. We will call this condition $\star_{i+1}$.

Case 2(c)(i): We extend $g$ as in case 1(b)(i). Since we begin with a height-
preserving $g$ and $w$ is proper with respect to $A_{i}$, our extension of $g$
will be height-preserving.

Case 2(c)(ii): We extend $g$ as case 2(b)(ii). Since we begin with a height-
preserving $g$ and $w$ is proper with respect to $A_{i}$, our extension of $g$
will be height-preserving.

Case 3: Assume that
$({\mathcal{A}},\overline{a}),({\mathcal{B}},\overline{b})$ meet the
conditions of 3 and let $\overline{d}\in{\mathcal{B}}^{<\omega}$ and
$\beta<\alpha$.

Case 3 (a): If
$\lambda_{{\mathcal{A}}}=\lambda_{{\mathcal{B}}}<\omega\cdot\alpha$ then as
before the groups are isomorphic.

Case 3 (b): If
$\lambda_{{\mathcal{A}}},\lambda_{{\mathcal{B}}}\geq\omega\cdot\alpha$ then we
know that $|P^{\mathcal{B}}_{\omega\cdot\beta+k}|=\infty$ (Kaplansky [10]
Problem 36) and so we meet the conditions for
$({\mathcal{A}},\overline{a})\leq_{\beta+1}({\mathcal{B}},\overline{b})$
regardless of whether $\beta$ is odd, even, or a limit ordinal. Hence, by
induction there exists a $\overline{c}\in{\mathcal{A}}^{<\omega}$ such that
$({\mathcal{B}},\overline{b}\overline{d})\leq_{\beta}({\mathcal{A}},\overline{a}\overline{c})$
for all $\beta<\omega\cdot\alpha$. ∎

Using the characterization of the back-and-forth relations, we can show that
the Scott sentences we give in Theorem 3.1 are optimal for arbitrarily high
$\alpha$.

######  Theorem 4.5.

Suppose that ${\mathcal{A}}$ is a structure with
$\lambda_{\mathcal{A}}=\omega\cdot\alpha$ where $\alpha$ is a limit ordinal.
Then ${\mathcal{A}}$ has Scott complexity $\Pi_{2\alpha+1}$.

###### Proof.

By Theorem 3.1, ${\mathcal{A}}$ has a Scott sentence of complexity
$\Pi_{2\alpha+1}$. It remains to show that ${\mathcal{A}}$ has no lower-
complexity Scott sentence. To this end, it is enough to exhibit
${\mathcal{A}}\not\cong{\mathcal{B}}$ such that
${\mathcal{A}}\equiv_{2\alpha}{\mathcal{B}}$. If so, then it is impossible for
${\mathcal{A}}$ to have a $d$-$\Sigma_{2\alpha}$ Scott sentence.

Consider ${\mathcal{B}}$ of strictly longer length than ${\mathcal{A}}$, such
that the Ulm invariants of both structures agree below $\omega\cdot\alpha$. It
is known that any Ulm sequence with infinitely many nonzero Ulm invariants
“between” each relevant pair of limit ordinals is realizable as the Ulm
sequence of a reduced abelian $p$-group (see e.g. [10]) so such a group
exists. Then ${\mathcal{A}}\equiv_{2\alpha}{\mathcal{B}}$ by 4.4, part (3). ∎

##  5 Open Questions and Problems

We were able to prove that when $G$ is finite-length and has at least two
infinite Ulm invariants that our upper bounds on Scott sentences were best
possible. One can ask the same question about $G$ of arbitrary length:

######  Question 5.1.

Are the Scott sentences given in 3.1 optimal in every case where there are at
least two infinite Ulm invariants?

It is often of interest whether certain kinds of Scott complexities are
realizable in a class of structures. In particular, it difficult to construct
an example of structure of Scott complexity $\Sigma_{\lambda+1}$ where
$\lambda$ is a limit ordinal. Our examples only show that the Scott complexity
$\Pi_{\alpha}$ is realizable for various $\alpha$.

######  Question 5.2.

Are the Scott complexities $\Sigma_{\alpha}$ or $d$-$\Sigma_{\alpha}$
realizable within the class of abelian $p$-groups? If so, for which $\alpha$?

Finally, it should be noted that the best characterization of the back-and-
forth relations, an open problem from [4], should be an entirely algebraic
characterization.

######  Question 5.3.

Can one give a completely algebraic characterization of the back-and-forth
relations on abelian $p$-groups?

## References

  * [1] Rachael Alvir  “Scott Analysis of Countable Structures”, 2022  DOI: 10.7274/3t945q50s14
  * [2] Rachael Alvir, Noam Greenberg, Matthew Harrison-Trainor and Dan Turetsky  “Scott Complexity of Countable Structures”  In _Journal of Symbolic Logic_ 86.4, 2021, pp. 1706–1720  DOI: 10.1017/jsl.2021.4
  * [3] Rachael Alvir and Dino Rossegger  “The Complexity of Scott Sentences of Scattered Linear Orders”  In _The Journal of Symbolic Logic_ 85.3 Cambridge University Press, 2020, pp. 1079–1101  DOI: 10.1017/jsl.2020.46
  * [4] C.J. Ash and J. Knight  “Computable Structures and the Hyperarithmetical Hierarchy”, ISSN  Elsevier Science, 2000  URL: https://books.google.ca/books?id=A4RQI1NdS-UC
  * [5] Ewan J. Barker  “Back and Forth Relations for Reduced Abelian P-Groups”  In _Annals of Pure and Applied Logic_ 75.3 Elsevier, 1995, pp. 223–249  DOI: 10.1016/0168-0072(94)00045-5
  * [6] Paul C. Eklof  “Some model theory of abelian groups”  In _Journal of Symbolic Logic_ 37, 1972, pp. 335–342
  * [7] Harvey Friedman and Lee Stanley  “A Borel reductibility theory for classes of countable structures”  In _Journal of Symbolic Logic_ 54.3, 1989, pp. 894–914  DOI: 10.2307/2274750
  * [8] Matthew Harrison-Trainor and Meng-Che Ho  “On optimal Scott sentences of finitely generated algebraic structures”  In _Proceedings of the American mathematical society_ 146.10, 2018, pp. 4473–4485
  * [9] Meng-Che Ho  “Describing groups”  In _Proceedings of the American Mathematical Society_ 145.5, 2017, pp. 2223–2239
  * [10] Irving Kaplansky  “Infinite Abelian groups”  In _Infinite Abelian groups_ Courier Dover Publications, 1954, pp. 26–29
  * [11] Julia F Knight and Vikram Saraph  “Scott sentences for certain groups”  In _Archive for Mathematical Logic_ 57.3 Springer, 2018, pp. 453–472
  * [12] Antonio Montalbán  “A robuster Scott rank”  In _Proceedings of the American mathematical society_ 143, 2015
  * [13] Dana Scott  “Logic with denumerably long formulas and finite strings of quantifiers. (Proceedings of the 1963 International Symposium at Berkeley, edited by J. W. Addison, Leon Henkin, and Alfred Tarski)”  In _The theory of models_ North-Holland Publishing Company, 1965, pp. 329–341

AK00


# Observations of Existence and Instability Dynamics of Near-Zero-Dispersion
Temporal Kerr Cavity Solitons

Zongda Li1,2<EMAIL_ADDRESS>Yiqing Xu1,2 Caleb Todd1,2 Gang Xu1,2
Stéphane Coen1,2 Stuart G. Murdoch1,2 Miro Erkintalo1,2 1Department of
Physics, University of Auckland, Auckland 1010, New Zealand  2The Dodd-Walls
Centre for Photonic and Quantum Technologies, New Zealand

###### Abstract

Dissipative Kerr cavity solitons (CSs) are persisting pulses of light that
manifest themselves in driven optical resonators and that have attracted
significant attention over the last decade. Whilst the vast majority of
studies have revolved around conditions where the resonator exhibits strong
anomalous dispersion, recent studies have shown that solitons with unique
characteristics and dynamics can arise under conditions of near-zero-
dispersion driving. Here we report on experimental studies of the existence
and stability dynamics of Kerr CSs under such conditions. In particular, we
experimentally probe the solitons’ range of existence and examine how their
breathing instabilities are modified when group-velocity dispersion is close
to zero, such that higher-order dispersion terms play a significant role. On
the one hand, our experiments directly confirm earlier theoretical works that
predict (i) breathing near-zero-dispersion solitons to emit polychromatic
dispersive radiation, and (ii) that higher-order dispersion can extend the
range over which the solitons are stable. On the other hand, our experiments
also reveal a novel cross-over scenario, whereby the influence of higher-order
dispersion changes from stabilising to destabilising. Our comprehensive
experiments sample soliton dynamics both in the normal and anomalous
dispersion regimes, and our results are in good agreement with numerical
simulations and theoretical predictions.

##  I Introduction

Temporal Kerr cavity solitons (CSs) are localized pulses of light that can
circulate in a driven passive optical resonator without distortion
wabnitz_suppression_1993 ; leo_temporal_2010 . They underpin the highly-
coherent Kerr frequency combs that can be generated in low-loss
microresonators coen_modeling_2013 ; erkintalo_coherence_2014 ;
herr_temporal_2014 ; kippenberg_dissipative_2018 ; pasquazi_micro-combs_2018
and have enabled ground-breaking advances in a wide array of applications,
including telecommunication marin-palomo_microresonator-based_2017 , frequency
synthesis spencer_optical-frequency_2018 , optical ranging
trocha_ultrafast_2018  and many more suh_microresonator_2016 ;
hu_reconfigurable_2020 ; liang_high_2015 ; papp_microresonator_2014 . CSs are
also dynamically rich. In particular, for certain system parameters, CSs are
known to destabilise through a Hopf bifurcation, giving rise to oscillatory
behaviour leo_dynamics_2013 ; lucas_breathing_2017 ; yu_breather_2017 ;
yi_imaging_2018 . Also more complex instability behaviours
johansson_stability_2019 ; xu_spontaneous_2020 ; yang_stokes_2017 ;
jang_controlled_2016 , including spatiotemporal chaos
anderson_observations_2016 , soliton binding wang_universal_2017 ;
weng_heteronuclear_2020  and soliton crystals cole_soliton_2017 ;
guo_universal_2017 ; karpov_dynamics_2019 , are possible.

Most of the studies concerning CSs have, to date, revolved around the scenario
where the cavity driving field experiences strongly anomalous group-velocity
dispersion, with higher-order dispersion terms acting as small perturbations
only. In this regime, the most conspicuous perturbative effect of higher-order
dispersion is the generation of dispersive radiation that manifests itself as
a single narrow-band Lorentzian peak in the spectral domain
jang_observation_2014 ; milian_soliton_2014 ; brasch_photonic_2016 ;
cherenkov_dissipative_2017 . More recently, there has however been growing
interest in exploring the dynamics of CSs under conditions of near-zero-
dispersion driving, i.e., with the wavelength of the input laser coincident
(or close to coincident) with the zero-dispersion point of the resonator. In
this regime, higher-order dispersion dominates the dynamics, giving rise to
novel bright localized structures both in the normal and anomalous dispersion
regimes parra-rivas_coexistence_2017 ; li_experimental_2020 ; anderson_zero-
dispersion_2020 . Theoretical studies have also suggested that near-zero-
dispersion conditions can significantly affect the solitons’ instability
dynamics and range of existence melchert_multi-frequency_2020 ; parra-
rivas_third-order_2014 . In particular, it has been theoretically predicted
that (i) strong third-order dispersion combined with soliton breathing can
lead to the emission of comb-like, _polychromatic_ dispersive radiation
melchert_multi-frequency_2020  – in stark contrast to the sharp,
_monochromatic_ dispersive radiation that is commonly observed with stable
(i.e. non-oscillatory) solitons jang_observation_2014 ; li_experimental_2020
– and that (ii) third-order dispersion can stabilize CSs, i.e., restrict the
range of parameters over which the solitons are oscillatorily unstable parra-
rivas_third-order_2014 . However, as of yet, no comprehensive experimental
tests of these predictions have been reported.

In this Article, we experimentally explore the existence and stability
dynamics of Kerr CSs under conditions of near-zero-dispersion driving. Our
experiments are performed in a coherently driven fibre ring resonator, where
the dispersion conditions are systematically controlled by tuning the driving
wavelength li_experimental_2020 . Our results provide experimental
confirmation of the theoretical predictions mentioned above melchert_multi-
frequency_2020 ; parra-rivas_third-order_2014 . First, we observe the emission
of polychromatic dispersive radiation emitted by breathing solitons. Second,
we confirm that third-order dispersion indeed can stabilise CSs parra-
rivas_third-order_2014 . Interestingly, however, our experiments also reveal
for the first time that the latter behaviour is not universal: the impact of
higher-order dispersion switches from stabilising to destabilising for high
driving power levels. Our results provide significant new insights into the
dynamics of CSs under conditions of near-zero-dispersion driving and could
facilitate the design of broadband frequency combs or novel sources of ultra-
short pulse trains.

##  II Experimental and numerical methods

Figure 1 shows a schematic illustration of our experimental setup, which is
overall similar to that used in Ref. li_experimental_2020 . The experiment is
built around a 5-m-long optical fibre ring resonator with a free-spectral
range of 41.5 MHz and Kerr nonlinearity coefficient
$\gamma=1.8~{}\mathrm{W}^{-1}\mathrm{km}^{-1}$. The zero-dispersion wavelength
(ZDW) of the resonator was measured to be 1564.5 nm with third- and fourth-
order dispersion coefficients
$\beta_{3,\mathrm{zdw}}=0.13~{}\mathrm{ps}^{3}\mathrm{km}^{-1}$ and
$\beta_{4,\mathrm{zdw}}=-6.65\times 10^{-4}~{}\mathrm{ps}^{4}\mathrm{km}^{-1}$
at the ZDW, respectively. The cavity is comprised of a single 99/1 fibre
coupler spliced to form a ring, yielding a large measured finesse of
$\mathcal{F}=400$. We synchronously drive the resonator with a train of
nanosecond pulses carved from a tunable external cavity diode laser (ECDL). As
described in Ref. li_experimental_2020 , the synchronization between the
nanosecond driving pulses and the sub-picosecond solitons inside the resonator
is actively maintained by using a computer-controlled feedback loop. The
driving pulses are amplified and injected into the resonator via the input
port of the 99/1 coupler and the intracavity dynamics are monitored both in
the temporal and spectral domains at the output port of that coupler. Because
the output consists of a superposition between the intracavity field and the
reflected part (99%) of the driving field, a tunable bandstop filter is used
to remove the latter prior to temporal measurements so as to improve the
signal-to-noise ratio. To overcome fluctuations in the frequencies of the
driving laser and the cavity modes, we actively lock the driving laser to a
cavity resonance by using the schemes introduced in nielsen_invited_2018  and
li_experimental_2020 . This allows us to systematically control the detuning
between the laser and the cavity, and hence investigate the detuning-dependent
properties of the CSs.

Figure 1: Simplified depiction of the experimental setup. AM, amplitude
modulator; DSF, dispersion-shifted fiber; ECDL, external-cavity diode laser;
EDFA, Erbium-doped fiber amplifier; OSA, optical spectrum analyzer; OSC,
oscilloscope.

To compare our experimental observations with theoretical predictions, we
model our system using the generalized Lugiato-Lefever equation (LLE) that
includes higher-order dispersion and delayed Raman nonlinearity
wang_stimulated_2018 :

$\small\begin{split}t_{R}\frac{\partial E(t,\tau)}{\partial
t}&=\left[-\alpha-i\delta_{\mathrm{0}}+\sum_{k=2}^{4}\frac{i^{k+1}}{k!}L\beta_{k}\frac{\partial^{k}}{\partial\tau^{k}}\right]E+\sqrt{\theta}E_{\text{in}}\\\
&+i\gamma
L\left[(1-f_{R})|E|^{2}+f_{R}h_{R}(\tau)*|E|^{2}\right]E.\end{split}$ (1)

Here, $t_{\mathrm{R}}$ is the round trip time, $E(t,\tau)$ is the slowly
varying intracavity electric field envelope with units of $W^{1/2}$, $t$ is a
slow time that describes the evolution of $E$ at the scale of the photon
lifetime, $\tau$ is a corresponding fast-time which describes the intracavity
field temporal profile over a single roundtrip, $\alpha=\pi/\mathcal{F}$ is
half of the fraction of power lost per round trip, $\delta_{0}$ is the phase
detuning of the driving field from the closest cavity resonance, $\beta_{k}$
are the dispersion coefficients at the pump frequency, $L$ is the cavity
length, $\theta$ is the power transmission coefficient of the input coupler
($\theta=0.01$ in our experiments), $E_{\mathrm{in}}=\sqrt{P_{\mathrm{in}}}$
is the amplitude of the driving field with power $P_{\mathrm{in}}$,
$f_{\mathrm{R}}=0.18$ is the Raman fraction of the nonlinearity,
$h_{\mathrm{R}}(\tau)$ is the corresponding time-domain Raman response
function hollenbeck_multiple-vibrational-mode_2002  and $*$ denotes
convolution. Note that our model includes fourth-order dispersion and Raman
nonlinearity to improve quantitative agreement with our experiments, but we
have carefully verified that they do not affect the overall trends that are
instead set by the third-order dispersion $\beta_{3}$. Furthermore, in what
follows, we will discuss and contrast the results obtained from Eq. 1 with
corresponding results of the ‘pure’ LLE that is obtained from Eq. 1 by setting
all higher-order terms to zero, i.e., $\beta_{k}=0$ for $k>2$ and
$f_{\mathrm{R}}=0$.

In our analysis below, we will refer to the following normalized parameters
leo_temporal_2010 ; jang_observation_2014 : driving power $X=\gamma
P_{\mathrm{in}}L\theta/\alpha^{3}$, detuning $\Delta=\delta_{0}/\alpha$, and
third-order dispersion $d_{3}=\sqrt{2\alpha/L}\beta_{3}/(3|\beta_{2}|^{3/2})$.
We stress that the normalized parameter $d_{3}$ describes the relative
strength of third-order dispersion; for $|d_{3}|\geq 1$, the impact of third-
order dispersion is comparable or dominates over that of second-order
dispersion jang_observation_2014 . Thanks to the use of a widely tunable ECDL,
we are able in our experiments to systematically control the magnitude of
$d_{3}$ and to investigate the soliton dynamics both when driving in the
anomalous and normal dispersion regimes li_experimental_2020 .

Figure 2: (a) Red curve shows CS bifurcation curve for X = 150 and
$d_{3}=1.12$; green curve shows corresponding results in the absence of
higher-order dispersion [$\beta_{3}=\beta_{4}=0$] and stimulated Raman
scattering [$f_{\mathrm{R}}=0$]. Vertical dashed-dotted lines labelled as
$\Delta_{\mathrm{H}2}$ and $\Delta_{\mathrm{H}3}$ indicate the detunings at
which breathing solitons emerge in the absence and presence of higher-order
terms, respectively. Blue curves show CW solutions for X = 150. Solid (dashed)
curves correspond to stable (unstable) solutions. (b) and (c) blue (orange)
curves show experimentally measured (numerically simulated) CS energy per
round-trip and time-averaged spectrum at $\Delta=44.5$, respectively. (d) and
(e) are as in (b) and (c), respectively, but with $\Delta=18$. The positions
of both measurements along the CS bifurcation curves are indicated in (a) by
green circles. Inserts in (c) and (e) show the dispersive wave spectral
components in more detail.

##  III Results

###  III.1 Anomalous-dispersion driving

We first consider experiments that probe the CSs’ instability dynamics under
near-zero-dispersion conditions when driving in the anomalous dispersion
regime. To this end, we set the pump power to $0.97$ W such that $X=150$, and
we set the driving wavelength to $1566.2$ nm (i.e., 1.7 nm away from the ZDW),
where dispersion is anomalous and $d_{3}=1.12$. In Fig. 2(a), we show a
theoretically predicted CS bifurcation curve [red curve] obtained by finding
the steady-state solutions of Eq. (1) using a Newton-Raphson method with our
experimental parameters [see also Figure caption]. Also shown in green is the
corresponding curve in the absence of higher-order terms [i.e., $\beta_{k}=0$
for $k>2$ and $f_{\mathrm{R}}=0$], as well as the intensity levels of the
homogeneous CW state of the system. (Note that the stability of all the
solutions was deduced via a linear stability analysis kuznecov_elements_2010
.) We see that, both in the presence and absence of higher-order dispersion
terms, the CSs are stable (solid curves) for large detunings, but then become
unstable (dashed curves) through a Hopf bifurcation at detuning
$\Delta_{\mathrm{H}3}$ and $\Delta_{\mathrm{H}2}$, respectively.

Figure 3: The right panel shows a pseudo-colour plot of experimentally
measured spectra at various detuning values, obtained through vertically
concatenating 72 individual spectral measurements with parameters as in Fig.
2. The left panel shows a corresponding plot of high resolution spectral
measurements around the dispersive wave spectral profile; note that the latter
measurements are centred at the wavelength of the strongest dispersive
radiation peak and encompass a spectral window of 20 nm. In both panels, the
predicted positions of the CS Hopf bifurcation points in the presence and
absence of higher-order terms, $\Delta_{\mathrm{H}3}$ and
$\Delta_{\mathrm{H}2}$ [same values shown in Fig. 2(a)], are indicated by the
yellow arrows, respectively.

In our experiments, we stabilize the detuning at different values and examine
the soliton characteristics. Blue curves in Figs. 2(b)–(e) show the
experimentally measured CS energy per roundtrip as well as time-averaged
optical spectra when the detuning is stabilised at $\Delta=44.5$ [Figs. 2(b,
c)] and $\Delta=18$ [Figs. 2(d, e)]. In agreement with the bifurcation curve
shown in Fig. 2(a), the CSs at $\Delta=44.5$ are dynamically stable,
exhibiting a typical spectrum with a strong monochromatic dispersive wave peak
at about 1537 nm. In stark contrast, at $\Delta=18$, the solitons exhibit
regular oscillations with a period of about
$T_{\mathrm{osc}}=1.59~{}\mu\mathrm{s}$ (or equivalently 70 roundtrips), with
the dispersive wave peak exhibiting a manifestly polychromatic character. The
latter feature was recently predicted by Melchert et al. melchert_multi-
frequency_2020  and can be explained as follows. Efficient dispersive wave
emission requires that the CS spectrally overlaps with the phase-matched
dispersive wave wavelengthwebb_efficiency_2014 ; when the (spectral width of
the) CS oscillates, the efficiency of dispersive wave emission is accordingly
modulated. Compounded by the fact that the emitted dispersive waves walk off
from the soliton due to dispersion, this results in the generation of a quasi-
periodic train of phase coherent dispersive waves with decaying amplitude. The
temporal spacing between two consecutive dispersive waves, $\Delta\tau$, is
equal to the dispersive walk-off during one soliton oscillation period
$T_{\mathrm{osc.}}$, i.e.,

$\Delta\tau=L\frac{T_{\mathrm{osc}}}{t_{\mathrm{R}}}\left[\beta_{\mathrm{2,s}}\Omega+\frac{\beta_{\mathrm{3,s}}}{2}\Omega^{2}+\frac{\beta_{\mathrm{4,s}}}{6}\Omega^{3}\right],$
(2)

where $\Omega$ is the angular frequency shift of the dispersive wave from the
soliton and the dispersion coefficients $\beta_{k,\mathrm{s}}$ are evaluated
at the center of the soliton spectrum. For our parameters (estimating the
soliton and the dispersive wave to be centred at 1573 nm 1543 nm,
respectively) $\Delta\tau=\Delta\tau^{-1}\approx 4.9~{}\mathrm{ps}$, yielding
a corresponding spectral modulation frequency $\Delta f\approx
204~{}\mathrm{GHz}$, which is in reasonable agreement with 198 GHz extracted
from the measured spectrum (average frequency spacing between the peaks that
are not equidistant due to dispersion). To the best of our knowledge, the
results shown in Figs. 2(d) and (e) represent the first experimental
confirmation of polychromatic dispersive wave emission by oscillating near-
zero-dispersion CSs.

The experimental observations in Figs. 2(b)–(e) are in good agreement with
numerical simulations [see orange curves in Fig. 2] and predictions of the
theoretical bifurcation curve [Fig. 2(a)]. Somewhat surprisingly, the
bifurcation curve predicts that, in the presence of higher-order terms, the
Hopf bifurcation that marks the transition between stable and unstable CSs
occurs at a larger detuning than in the pure LLE model, i.e.,
$\Delta_{\mathrm{H}3}>\Delta_{\mathrm{H}2}$. Arising predominantly due to
third-order dispersion (we again emphasize that fourth-order dispersion and
Raman scattering play a minor role only), this observation seemingly
contradicts the notion that third-order dispersion stabilises CSs parra-
rivas_third-order_2014 . To experimentally test this prediction, we measured
the soliton characteristics as a function of detuning over the entire range of
the soliton existence. More specifically, we excite and stabilise a soliton at
a given detuning, and then adiabatically adjust the detuning stabilisation
point in small steps while recording the soliton spectral and temporal
profiles. (Of note: occasionally the soliton dies due to environmental
perturbations during the process, in which case we re-excite the soliton and
resume the measurement.) Figure 3 shows the measured soliton spectral
evolution with $X=150$ [i.e. parameters as in Fig. 2(a)], and reveals that the
dispersive wave component transforms from poly- to monochromatic at a detuning
$\Delta\approx 26$. This transformation is indicative of the transition
between unstable and stable CSs, and the detuning at which it occurs is in
good agreement with the predicted Hopf bifurcation point
$\Delta_{\mathrm{H}3}=25.9$. (Note that stability is inferred from the
presence of polychromatic dispersive waves in the spectrum, which provides a
better signal-to-noise ratio than temporal measurements.) The observed
transition point is noticeably larger (the uncertainty in our measurement of
$\Delta$ is about unity) than the value of $\Delta_{\mathrm{H}2}=21.9$
predicted in the pure LLE model, thus confirming that, for the parameters
considered in Fig. 2 and 3, third-order dispersion does _not_ stabilize the
CS.

To gain more insight and explain the apparent contradiction between our
findings and results reported in Ref. parra-rivas_third-order_2014 , we
computed the soliton’s bifurcation characteristics for a range of driving
powers and detunings. The results are shown in Fig. 4 for $d_{3}=1.12$, where
various coloured regions indicate parameters space over which solitons exhibit
distinct dynamical behaviours, superimposed with the bifurcation points
predicted in the absence of higher-order effects. We have carefully verified
that neglecting fourth-order dispersion and Raman scattering yields
qualitatively similar existence and stability boundaries, thus allowing us to
draw two major conclusions with regards to the effect of third-order
dispersion. First, third-order dispersion reduces the range of detunings over
which the CS exists – especially on the large detuning side. Second, third-
order dispersion induces a non-trivial change in solitons’ stability
boundaries. For low driving power level ($X<80$), the solitons are stable at
all detunings – even when instabilities would otherwise manifest themselves in
the pure LLE model. Thus, for low driving powers, third-order dispersion
stabilises the solitons, as predicted in Ref. parra-rivas_third-order_2014 .
As the driving power increases, we find however that the solitons’ immunity
against instabilities vanishes, and we observe a cross-over where solitons are
unstable over a larger range of detunings in the presence of third-order
dispersion than in its absence. The results in Fig. 4 suggest that the
stabilising effect of third-order dispersion (as reported in Ref. parra-
rivas_third-order_2014 ) only holds for low driving power levels, whilst for
larger power levels, the instability range is, in fact, extended [congruent
with results reported in Fig. 2 and Fig. 3].

Figure 4: Bifurcation characteristics of CSs with parameters as in Fig. 2 over
a range of normalized detunings and driving powers; solutions within the
green, orange and purple patches are stable, oscillatory and spatio-temporally
chaotic, respectively. Black solid lines delineate the existence boundaries of
CSs in the absence of higher-order terms (pure LLE model) whilst the black
dashed line shows the corresponding Hopf bifucation point
$\Delta_{\mathrm{H}2}$. Blue and red dots correspond to experimental
measurements of stable and unstable CSs, respectively.

To more comprehensively test the bifurcation predictions, we performed
experiments as in Fig. 3 for a range of driving power levels. The results are
shown in Fig. 4 with blue and red solids circles representing stable and
oscillatory solitons, respectively. Relating to the instability transition,
the experimental results are in good agreement with the theoretical
predictions. There is only one discrepancy: a theoretically predicted small
island of stability around $X=100$ is not observed in our experiment. We
suspect this is due to experimental imperfections that prevent access to the
small island predicted. (Of note: we have observed the corresponding island in
experiments performed for larger values of $d_{3}$ where the island extends
over a larger range of detuning.) Relating to the range of existence, our
experiments show good qualitative agreement with numerical predictions, but
quantitative discrepancies arise at the low detuning side. We believe this may
be due to inhomogeneities in the driving pulse that locally trigger modulation
instability that then pervades the entire intracavity field. Nonetheless, our
experiments confirm the salient theoretical predictions: third-order
dispersion limits the range of soliton existence and switches from stabilizing
to destabilizing for large driving power levels.

###  III.2 Normal-dispersion driving

The results reported above pertain to the (common) situation where the
resonator exhibits anomalous dispersion at the driving wavelength. However,
recent studies have shown that, due to higher-order dispersion, the existence
of bright solitons can extend into the regime of normal-dispersion-driving
milian_soliton_2014 ; li_experimental_2020 . As a matter of fact, a number of
different bright structures can exist when driving close to the ZDW in the
normal dispersion regime, distinguished by a different number of prominent
peaks in their temporal profile parra-rivas_coexistence_2017 . Here we are
interested in the “single-peak” structures that are the normal-dispersion
counterparts of the standard CSs that exist in the anomalous dispersion regime
li_experimental_2020 ; we will henceforth refer to such structures simply as
CSs.

To explore the existence and stability of CSs with normal-dispersion driving,
we adjust the driving wavelength to 1562.8 nm (i.e., 1.6 nm away from the ZDW)
where the group-velocity dispersion is normal and $d_{3}=1.07$. We then
repeated the theoretical and experimental analyses reported in Fig. 4 and
summarise the results in Fig. 5. Also shown for comparison are the theoretical
boundaries in the presence of higher-order terms as extracted from Fig. 4.

Figure 5: Bifurcation characteristic of single-peak CSs with normal-dispersion
driving and $d_{3}=1.07$ over a range of normalized detunings and driving
powers. The green patch indicates the parameter range within which normal-
dispersion CS exist and are predicted to be stable; note that there are no
instabilities for the considered parameters. For comparison, the solid grey
curves delineate the existence boundaries of anomalous-dispersion CSs with
$d_{3}=1.12$, whilst dashed and dotted grey curves indicate the corresponding
instability boundaries (data extracted from Fig. 4). Blue dots correspond to
experimental measurements of stable CSs (no unstable CSs were observed for the
considered parameters). The insert shows a pseudo-colour plot of spectral
measurements at $X=120$ and for various detunings (corresponding to data
points enclosed by the dashed orange rectangle). Note that, due to
experimental error, a data point near $X=150$ and $\Delta=20$ is missing.
However, by observing the stability state of its neighbours, it is reasonable
to assume the CS at this position will be stable.

Several conclusion can be drawn. First, the solitons’ range of existence is
further reduced from the case of anomalous-dispersion driving. Second, in
contrast to CSs in the anomalous dispersion regime, the solitons are stable
for all the detunings and driving powers considered. (Of note: simulations
predict that instabilities can arise if one consider even larger driving
levels.) Third, our experiments are again in good qualitative agreement with
the theoretical results, showing no signs of instability and reproducing the
theoretically predicted upper-detuning limit of soliton existence. As for the
anomalous dispersion results (Fig. 4), discrepancies arise especially at the
low-detuning limit. It is interesting to note that normal-dispersion driving
appears to permit full immunity against instabilities over a larger range of
driving powers than anomalous-dispersion driving. However, it is important to
stress that, at each driving power, the latter regime is still associated with
a larger overall range of detunings where the solitons are stable thanks to
the larger range of existence of CSs in the anomalous dispersion regime.

###  III.3 Zero-dispersion driving

The results reported in Figs. 4 and 5 show that, despite similar relative
third-order dispersion parameters $d_{3}\approx 1$, the existence and
stability of CSs differ markedly between anomalous and normal dispersion
driving. On the one hand, this is within expectation since CSs are commonly
associated with anomalous dispersion driving only (and do no exist at all with
normal dispersion driving in the limit of $d_{3}\rightarrow 0$). But on the
other hand, one may expect that in the limit $d_{3}\rightarrow\infty$, which
occurs as the pump approaches the ZDW of the resonator, the two driving
scenarios must converge to one; indeed, in this case, the second-order
dispersion can be considered negligible and the driving scenarios physically
merge.

To elucidate whether the soliton dynamics evolve smoothly as the driving
wavelength is tuned across the ZDW (or whether a separatrix exists that
divides the two regions), we theoretically calculated the soliton solutions of
Eq. (1) over a wide range of driving wavelengths on either side of the ZDW for
a constant $X=100$. To compare the soliton properties, we plot in Fig. 6(a)
the normalized cavity detuning where the solitons cease to exist (the upper
limit of existence) as a function of pump wavelength (and hence relative
third-order dispersion). As can be seen, the upper-detuning-limit varies
smoothly across the ZDW, monotonically decreasing as the pump wavelength is
moved from anomalous to normal dispersion regime, until the soliton eventually
ceases to exist in an abrupt fashion when the pump is tuned sufficiently deep
into the normal dispersion regime. This result corroborates the observation
made in Ref. li_experimental_2020  that there is no qualitative change in the
characteristic of near-zero-dispersion CSs across the ZDW.

Before closing, we finally note that when the pump wavelength is close to the
ZDW, the dynamics and characteristics of the CSs can of course be well-
predicted by Eq. (1) with second-order dispersion identically set to zero,
i.e. $\beta_{2}=0$. The dashed red lines in Fig. 6(a) highlight the upper-
detuning-limit of existence of solitons that exist when $\beta_{2}=0$, whilst
Fig. 6(b) shows the corresponding soliton temporal profile compared with
selected profiles for normal and anomalous dispersion driving with $\Delta=20$
and $d_{3}=1$. A thorough analysis of the characteristics and dynamics of such
single-peak, zero-dispersion CS is beyond the scope of our present work, yet
we envisage that such future analysis can provide valuable insights and
facilitate the design of ultra-broadband microresonator frequency combs and
sources of ultra-short cavity solitons.

Figure 6: (a) Upper-detuning-limit of soliton range of existence as a function
of deviation from the ZDW, $\lambda_{\mathrm{p}}-\lambda_{\mathrm{zdw}}$,
where $\lambda_{\mathrm{p}}$ and $\lambda_{\mathrm{zdw}}$ are the driving
wavelength and ZDW, respectively. The dashed red line indicates the detuning-
limit of the zero-dispersion soliton that exists when $\beta_{2}=0$ (or
equivalently $d_{3}=\infty$). (b) Temporal profiles of simulated CSs at
$\Delta=20$ for normal-dispersion driving (orange solid curve), anomalous-
dispersion driving (blue dashed curve), and driving exactly at the ZDW (grey
circles)

##  IV Conclusions

In conclusion, we have experimentally studied the impact of third-order
dispersion on the existence and stability boundaries of temporal Kerr cavity
solitons. This has been achieved by using an experimental configuration that
allows operation (arbitrarily) close to the ZDW of the resonator, thus
enabling systematic control over the relative impact of third-order
dispersion. Our results include the first observations of polychromatic
dispersive waves emitted by breathing near-zero-dispersion CSs, as predicted
by earlier theories melchert_multi-frequency_2020 . Moreover, we have shown
that, for low driving powers, third-order dispersion can stabilize CSs (as
predicted by earlier theories parra-rivas_third-order_2014 ), but that for
high-power levels, the effect of third-order dispersion switches from
stabilizing to destablizing, expanding the range of parameters over which
solitons are unstable. In addition to experiments performed under conditions
of anomalous-dispersion driving, we have also considered solitons that
manifest themselves with the driving field in the normal dispersion regime,
and examined how the two pumping conditions merge as the driving wavelength
approaches the ZDW. Our experiments are in good agreement with numerical
simulations of the generalized Lugiato-Lefever equations, with some
discrepancies likely stemming from experimental imperfections. Our work
provides convincing experimental confirmation of earlier theoretical
predictions melchert_multi-frequency_2020 ; parra-rivas_third-order_2014  and
provides new insights on CSs with (near-)zero-dispersion driving.

##  V Acknowledgements

The authors acknowledge financial support from the Marsden Fund and the
Rutherford Discovery Fellowships of the Royal Society of New Zealand.

## References

  * (1) S. Wabnitz, “Suppression of interactions in a phase-locked soliton optical memory,” Optics Letters 18, 601–603 (1993).
  * (2) F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nature Photonics 4, 471–476 (2010).
  * (3) S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato–Lefever model,” Optics Letters 38, 37 (2013).
  * (4) M. Erkintalo and S. Coen, “Coherence properties of Kerr frequency combs,” Optics Letters 39, 283 (2014).
  * (5) T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photonics 8, 145–152 (2014).
  * (6) T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, “Dissipative Kerr solitons in optical microresonators,” Science 361, eaan8083 (2018).
  * (7) A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, “Micro-combs: A novel generation of optical sources,” Physics Reports 729, 1–81 (2018).
  * (8) P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546, 274–279 (2017).
  * (9) D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
  * (10) P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359, 887–891 (2018).
  * (11) M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
  * (12) J. Hu, J. He, J. Liu, A. S. Raja, M. Karpov, A. Lukashchuk, T. J. Kippenberg, and C.-S. Brès, “Reconfigurable radiofrequency filters based on versatile soliton microcombs,” Nature Communications 11, 4377 (2020).
  * (13) W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nature Communications 6, 7957 (2015).
  * (14) S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1, 10 (2014).
  * (15) F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Optics Express 21, 9180 (2013).
  * (16) E. Lucas, M. Karpov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Breathing dissipative solitons in optical microresonators,” Nature Communications 8, 736 (2017).
  * (17) M. Yu, J. K. Jang, Y. Okawachi, A. G. Griffith, K. Luke, S. A. Miller, X. Ji, M. Lipson, and A. L. Gaeta, “Breather soliton dynamics in microresonators,” Nature Communications 8, 14569 (2017).
  * (18) X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Imaging soliton dynamics in optical microcavities,” Nature Communications 9, 3565 (2018).
  * (19) M. Johansson, V. E. Lobanov, and D. V. Skryabin, “Stability analysis of numerically exact time-periodic breathers in the Lugiato-Lefever equation: Discrete vs continuum,” Physical Review Research 1, 033196 (2019).
  * (20) G. Xu, A. Nielsen, B. Garbin, L. Hill, G.-L. Oppo, J. Fatome, S. G. Murdoch, S. Coen, and M. Erkintalo, “Spontaneous symmetry breaking of dissipative optical solitons in a two-component Kerr resonator,” arXiv:2008.13776 [physics] (2020).
  * (21) Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, “Stokes solitons in optical microcavities,” Nature Physics 13, 53–57 (2017).
  * (22) J. K. Jang, M. Erkintalo, K. Luo, G.-L. Oppo, S. Coen, and S. G. Murdoch, “Controlled merging and annihilation of localised dissipative structures in an AC-driven damped nonlinear Schrödinger system,” New Journal of Physics 18, 033034 (2016).
  * (23) M. Anderson, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Observations of spatiotemporal instabilities of temporal cavity solitons,” Optica 3, 1071 (2016).
  * (24) Y. Wang, F. Leo, J. Fatome, M. Erkintalo, S. G. Murdoch, and S. Coen, “Universal mechanism for the binding of temporal cavity solitons,” Optica 4, 855–863 (2017).
  * (25) W. Weng, R. Bouchand, E. Lucas, E. Obrzud, T. Herr, and T. J. Kippenberg, “Heteronuclear soliton molecules in optical microresonators,” Nature Communications 11, 2402 (2020).
  * (26) D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nature Photonics 11, 671–676 (2017).
  * (27) H. Guo, M. Karpov, E. Lucas, A. Kordts, M. Pfeiffer, V. Brasch, G. Lihachev, V. Lobanov, M. Gorodetsky, and T. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nature Physics 13, 94–102 (2017).
  * (28) M. Karpov, M. H. P. Pfeiffer, H. Guo, W. Weng, J. Liu, and T. J. Kippenberg, “Dynamics of soliton crystals in optical microresonators,” Nature Physics 15, 1071–1077 (2019).
  * (29) J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Observation of dispersive wave emission by temporal cavity solitons,” Optics Letters 39, 5503 (2014).
  * (30) C. Milián and D. Skryabin, “Soliton families and resonant radiation in a micro-ring resonator near zero group-velocity dispersion,” Optics Express 22, 3732 (2014).
  * (31) V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip–based optical frequency comb using soliton Cherenkov radiation,” Science 351, 357–360 (2016).
  * (32) A. V. Cherenkov, V. E. Lobanov, and M. L. Gorodetsky, “Dissipative Kerr solitons and Cherenkov radiation in optical microresonators with third-order dispersion,” Physical Review A 95, 033810 (2017).
  * (33) P. Parra-Rivas, D. Gomila, and L. Gelens, “Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators,” Physical Review A 95, 053863 (2017).
  * (34) Z. Li, Y. Xu, S. Coen, S. G. Murdoch, and M. Erkintalo, “Experimental observations of bright dissipative cavity solitons and their collapsed snaking in a Kerr resonator with normal dispersion driving,” Optica 7, 1195–1203 (2020).
  * (35) M. H. Anderson, G. Lihachev, W. Weng, J. Liu, and T. J. Kippenberg, “Zero-dispersion Kerr solitons in optical microresonators,” arXiv:2007.14507 [physics] (2020).
  * (36) O. Melchert, A. Demircan, and A. Yulin, “Multi-frequency radiation of dissipative solitons in optical fiber cavities,” Scientific Reports 10, 8849 (2020).
  * (37) P. Parra-Rivas, D. Gomila, F. Leo, S. Coen, and L. Gelens, “Third-order chromatic dispersion stabilizes Kerr frequency combs,” Optics Letters 39, 2971 (2014).
  * (38) A. U. Nielsen, B. Garbin, S. Coen, S. G. Murdoch, and M. Erkintalo, “Invited Article: Emission of intense resonant radiation by dispersion-managed Kerr cavity solitons,” APL Photonics 3, 120804 (2018).
  * (39) Y. Wang, M. Anderson, S. Coen, S. G. Murdoch, and M. Erkintalo, “Stimulated Raman Scattering Imposes Fundamental Limits to the Duration and Bandwidth of Temporal Cavity Solitons,” Physical Review Letters 120, 053902 (2018).
  * (40) D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function,” Journal of the Optical Society of America B 19, 2886 (2002).
  * (41) J. A. Kuznecov, _Elements of applied bifurcation theory_ , number 112 in Applied mathematical sciences, Springer, New York, NY, 3. ed edition (2010).
  * (42) K. E. Webb, M. Erkintalo, Y. Q. Xu, G. Genty, and S. G. Murdoch, “Efficiency of dispersive wave generation from a dual-frequency beat signal,” Optics Letters 39, 5850 (2014).