# On groups interpretable in various valued fields

Yatir Halevi  Department of Mathematics
University of Haifa
199 Abba Khoushy Avenue
Haifa
Israel  Department of Mathematics, Ben Gurion University of the Negev, Be’er-
Sheva 84105, Israel<EMAIL_ADDRESS>,  Assaf Hasson  Department
of Mathematics, Ben Gurion University of the Negev, Be’er-Sheva 84105, Israel
<EMAIL_ADDRESS>and  Ya’acov Peterzil  Department of Mathematics,
University of Haifa, Haifa, Israel<EMAIL_ADDRESS>
(Date: January 2022; Date: March 2024)

###### Abstract.

We study infinite groups interpretable in three families of valued fields:
$V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show
that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal
then it is abelian-by-finite.

Along the way, we associate with any infinite interpretable group an infinite
type-definable subgroup which is definably isomorphic to a group in one of
four distinguished sorts: the underlying valued field $K$, its residue field k
(when infinite), its value group $\Gamma$, or $K/{\mathcal{O}}$, where
${\mathcal{O}}$ is the valuation ring.

Our work uses and extends techniques developed in [11] to circumvent
elimination of imaginaries.

The first author was partially supported by ISF grant No. 555/21 and 290/19.
The second author was supported by ISF grant No. 555/21. The third author was
supported by ISF grant No. 290/19.

###### Contents

  1. 1 Introduction
  2. 2 Background and preliminaries
  3. 3 Some results on $p$-adically closed fields
  4. 4 Vicinic sorts and their properties
  5. 5 Infinitesimal groups
  6. 6 The case of a strongly minimal D
  7. 7 The main theorems
  8. 8 Examples
  9. 9 The distinguished sorts are foreign
  10. A Endowing an infinitesimal group in RCVF with a linear order

##  1\. Introduction

We continue our work from [11], where we studied fields interpretable in a
variety of valued fields, and extend our investigation to interpretable
groups. To recall, in [11] we considered interpretable objects, namely
quotients of definable sets by definable equivalence relations, in valued
fields, with a focus on interpretable fields in three families of valued
fields: (i) V-minimal (i.e. certain expansions of algebraically closed valued
fields of residue characteristic $0$), (ii) $T$-convex expansions of power
bounded o-minimal structures (i.e. certain expansions of real closed valued
fields) and (iii) certain expansions of $p$-adically closed fields, namely of
fields that are elementarily equivalent to finite extensions of
$\mathbb{Q}_{p}$).

The key idea from [11] was to bypass results on elimination of imaginaries and
replace them with a reduction to four distinguished sorts: the valued field
$K$ itself, the residue field k (when infinite), the value group $\Gamma$, and
the quotient $K/{\mathcal{O}}$, where ${\mathcal{O}}$ is the valuation ring.
Applying techniques similar to those developed in [11] we introduce here the
following notion: We say that $G$ is locally strongly internal to a set $D$ if
there exists a definable infinite $X\subseteq G$ and a definable injective
$f:X\to D^{n}$, for some $n$.

Using this notion we state our main theorem111For brevity, we shall use “power
bounded $T$-convex expansions” instead of “$T$-convex expansions of power
bounded o-minimal structures”:

######  Theorem 1.1 (Section 7).

Let $K$ be a valued field of characteristic $0$ and assume that either (i)
${\mathcal{K}}$ is a $V$-minimal expansion of $K$ (ii) ${\mathcal{K}}$ is a
power bounded $T$-convex expansion of $K$, or (iii) $K$ is a $p$-adically
closed field and ${\mathcal{K}}=K$.

Let $G$ be an infinite group interpretable in ${\mathcal{K}}$. Then, after
possibly replacing $G$ with a quotient by a finite normal subgroup, there
exists a distinguished sort $D\in\\{K,\textbf{k},\Gamma,K/{\mathcal{O}}\\}$,
such that $G$ is locally strongly internal to $D$ and for every such $D$ there
is an infinite type-definable subgroup $\nu_{D}$ of $G$, such that

  1. (1)

If $D=K$ or $D=\textbf{k}$ then $\nu_{D}$ is definably isomorphic to a type-
definable group $\nu_{D}^{\prime}\vdash K^{n}$, or
$\nu_{D}^{\prime}\vdash\textbf{k}^{n}$, respectively, for some $n$.

  2. (2)

If $D=\Gamma$ then $\nu_{D}$ is definably isomorphic to a type-definable
subgroup of $\langle\Gamma^{n},+\rangle$, for some $n$.

  3. (3)

If $D=K/{\mathcal{O}}$ then there is a definable subgroup $H$ of G,
$\nu_{D}\vdash H$, such that $H$ is definably isomorphic to a subgroup of
$\langle(K/{\mathcal{O}})^{n},+\rangle$, for some $n$.

######  Remark 1.2.

  1. (1)

By a “type-definable” subgroup of $G$ we mean a partial type over $K$
(consisting possibly of infinitely many formulas) whose realization in every
elementary extension $\widehat{\mathcal{K}}$ of ${\mathcal{K}}$ is a subgroup
of $G(\widehat{\mathcal{K}})$. A type-definable group $\nu$ is infinite  if
$\nu(\widehat{\mathcal{K}})$ is infinite for some
$\widehat{K}\succ{\mathcal{K}}$. We write $\nu\vdash S$ to mean that the
formula defining $S$ belongs to the type $\nu$.

  2. (2)

As we show in Proposition 5.11, the group $\nu_{D}$ above has various
invariance properties. In addition, its dp-rank equals
$\max\mathrm{dp\text{-}rk}(X)$, as $X\subseteq G$ varies over all definable
subsets of $G$ for which there is some finite-to-one $f:X\to D^{n}$, for some
$n$,

Though the above theorem may seem technical, it gives a new set of invariants
for studying interpretable groups. In a subsequent paper, [12], we use this
result to classify definably simple non-abelian and definably semisimple
groups interpretable in the same classes of fields. In the present paper, we
give more direct global applications of the local data provided by our main
result:

######  Theorem 1.3 (Section 7).

Let $K$ be a valued field of characteristic $0$ assume that either (i)
${\mathcal{K}}$ is a $V$-minimal expansion of $K$, (ii) ${\mathcal{K}}$ is a
power bounded $T$-convex expansion of $K$, or (iii) ${\mathcal{K}}=K$ is
$p$-adically closed. Let $G$ be an infinite group interpretable in
${\mathcal{K}}$. Then (1) $G$ has unbounded exponent and (2) If $G$ is dp-
minimal then it is abelian-by-finite.

######  Remark 1.4.

Theorem 1.3 implies, in particular, that every 1-dimensional group $G\subseteq
K^{n}$ definable in any of the above structures ${\mathcal{K}}$ is abelian-by-
finite. In $p$-adically closed fields, this was proved by Pillay and Yao,
[28].

Since the value group in a $p$-adically closed field is a ${\mathbb{Z}}$-group
(i.e., a model of Presburger Arithmetic) the above theorem also covers the
work of Onshuus and Vicaria [24, Theorem 1.1] on dp-minimal groups
interpretable in ${\mathbb{Z}}$-groups.

Finally, the above theorem complements the work of Simonetta, [35], who gave
an example of a dp-minimal group interpretable in an algebraically closed
valued field of characteristic $p$ which is nilpotent of class $2$ (so not
abelian-by-finite).

Finding a framework that would allow us to avoid repetition of proofs, as we
move across the different settings and the different distinguished sorts,
turned out to be one of the challenges of the present work. In [11] much of
the work was carried out in SW-uniformities, a framework of dp-minimal uniform
structures introduced by Simon and Walsberg in [34] (see below the precise
definition). SW-uniformities generalise o-minimal expansions of groups and
share several well known tameness properties of the latter setting. In
particular, the topology associated with these structures is Hausdorff and
non-discrete. This framework fits the sorts $K$ in all settings, the sorts
$\Gamma$ and $K/{\mathcal{O}}$ in the non-$p$-adically closed setting and the
sort k in the power bounded $T$-convex setting. The remaining cases are
divided into the unstable sorts of $K/{\mathcal{O}}$ and $\Gamma$ in the
$p$-adically close setting, and the algebraically closed (hence stable) sort k
in the $V$-minimal setting.

In order to uniformly treat all the unstable distinguished sorts (namely, all
sorts except k in the $V$-minimal setting), we introduce in Section 4 the
notion of _a vicinity_ , generalising the notion of a topological
neighbourhood, and the associated _vicinic sort_ , generalising here the
notion of an SW-uniform sort.

Without getting into technical details, let us give a brief overview of the
strategy of the proof and through it also some of the intuition underlying
these new definitions. As in [11], our method is local, in the sense that the
subgroup $\nu$ in the conclusion of Theorem 1.1 appears as a generalised
infinitesimal group (as in, e.g., [25] or [18]).

The first step in the proof is to find, after possibly passing to a quotient
by a finite normal subgroup, a definable infinite $X\subseteq G$ with a
definable injection from $X$ into $D^{n}$, for $D$ one of the four
distinguished sorts. Roughly, for a fixed such $D$, if $X$ has maximal dp-rank
with respect to this property, then $X$ is called a $D$-set (see Definition
4.16 for a precise formulation).

Most of the work in the paper is devoted to showing that each $D$-set gives
rise to a type definable group $\nu_{D}$ as in Theorem 1.1 (as it turns out,
this $\nu_{D}$ does not depend on the set $X$).

When $D$ is one of the SW-uniform sorts and $X$ is a $D$-set, the latter
inherits a topology from $D$. Using ideas going back to Pillay, [26] (in a
version due to Maříková, [21]), we show that the topology can be extended to a
group topology on $G$. The subgroup $\nu$ in the conclusion of Theorem 1.1 is
precisely the infinitesimal subgroup of $G$ with respect to this topology,
namely the intersection of all definable open neighbourhoods of $e$.

More generally, in all the unstable distinguished sorts, the notion of a
vicinity is a good generalization of that of a neighbourhood, and it allows us
to define the subgroup $\nu$ of Theorem 1.1 similarly to the above definition
of the infinitesimal group.

The remaining case, of the strongly minimal k in the V-minimal setting, is
different and is handled using methods of stable groups.

As can be seen in the statement of Theorem 1.1, when $D$ is either the value
group $\Gamma$ or the sort of closed balls $K/{\mathcal{O}}$, we obtain a
stronger result, asserting that $\nu$ can be identified with a type-definable
subgroup of $(D^{n},+)$ (here $n$ can be taken to be, in the above notation,
$\mathrm{dp\text{-}rk}(X)$). This follows, essentially, from the fact that in
ordered vector spaces and in ${\mathbb{Z}}$-groups definable functions are
piece-wise affine with respect to the additive structure. For the sort
$K/{\mathcal{O}}$ this follows from an analogous result proved in [11] when
$K/{\mathcal{O}}$ is an SW-uniformity, and in Section 3.2 below in the
$p$-adic case.

On the technical level, there are several themes around which the paper is
focused:

  * •

Properties of definable subsets and functions in $K/{\mathcal{O}}$, in the
setting of $p$-adically closed fields (see Section 3).

  * •

The definition of vicinic sorts and their basic properties (see Section 4),
and the verification that all the unstable distinguished sorts, in all our
settings, are vicinic. In SW-uniformities and in ${\mathbb{Z}}$-groups these
properties are known (or easy), and most of the work is dedicated to settling
them for the sort of closed balls, $K/{\mathcal{O}}$, in the $p$-adically
closed setting.

  * •

To carry out the above strategy, we have to work with definable injections
from (definable subsets of) $G$ into the relevant distinguished sorts. In
general, we can only assure the existence of finite-to-one such functions. We
show that this can be resolved by replacing $G$ with a quotient by a finite
normal subgroup (see Section 4.6).

To deduce the main part of Theorem 1.3 we first show that the type definable
group $\nu_{D}$ obtained in Theorem 1.1 has unbounded exponent, namely for
every $\widehat{\mathcal{K}}\succ{\mathcal{K}}$, the group
$\nu_{D}(\widehat{\mathcal{K}})$ has unbounded exponent. This easily implies
that so if $G$.

By a result of Simon, [31, Proposition 3.1], if $G$ is dp-minimal then it has
a definable abelian normal subgroup $H$, such that $G/H$ has bounded exponent.
Since $G/H$ is itself an interpretable group, the above implies that it must
be finite.

We conclude the introduction with a word on the setting: Case (iii) of Theorem
1.1 is stated for $p$-adically closed fields (by which we mean models
elementarily equivalent, in the MacIntyre language, to a finite field
extension of $\mathbb{Q}_{p}$). However, the proofs work without any changes
in an expansion of $K$, a finite field extension of $\mathbb{Q}_{p}$, by
analytic functions on ${\mathcal{O}}^{n}$ (in the sense of [8]). In fact, our
proofs work in $P$-minimal expansions of $p$-adically closed fields that are
$1$-h-minimal and have definable Skolem functions. However, for the sake of
clarity of exposition, we state the results in the $p$-adically closed
setting. As some of the results may be of independent interest and for ease of
future reference, we will specify throughout the level of generality at which
the same proof works.

###  1.1. Previous work on groups in valued fields

We list briefly additional work on groups in valued fields, beyond what has
been mentioned above. Definable groups in $\mathbb{Q}_{p}$ were first studied
by Pillay, [27]. Hrushovski and Pillay, [16], discuss connections between
definable groups in local fields and algebraic groups. In [2] Acosta gives an
exhaustive list of all 1-dimensional groups definable in $\mathbb{Q}_{p}$. In
[1] he extends this result to 1-dimensional _commutative_ groups definable in
algebraically closed valued fields222By Theorem 1.3 the commutativity
assumption is redundant in algebraically closed valued fields of equi-
characteristic $0$. Johnson and Yao, [20], study definable, definably compact,
groups in $\mathbb{Q}_{p}$. Montenegro, Onshuus and Simon, [23, Theorem 2.19],
work under general assumptions, applicable in several examples of our setting
(e.g., ACVF0,0, RCVF, $p$-adically closed fields). A classification of
definably simple groups definable in henselian fields of characteristic $0$
(or, more generally, in 1-h-minimal fields of characteristic 0) is obtained in
an unpublished work of Gismatullin, Halupczok and Macpherson.

While the above results mostly study definable groups, the work of Hrushovski
and Rideau-Kikuchi, [17], covers interpretable, stably dominated groups in
algebraically closed valued fields, using the results on elimination of
imaginaries in such fields. In a recent pre-print, Johnson [19] studies
topology on groups interpretable in $p$-adically closed fields.

Finally, although we do not make use of these here, we note the seminal papers
of [13], [22], and [15], where elimination of imaginaries with appropriate
sorts is proved for algebraically closed valued fields, real closed valued
fields and $p$-adically closed fields, respectively.

###  1.2. Structure of the paper

In Section 2 we review the notion of SW-uniformities and the distinguished
sorts introduced in [11] and collect some of their useful properties. In
Section 3.1, we study the definable sets and functions in $K/{\mathcal{O}}$ in
$p$-adically closed fields, isolating (Section 4.1) key geometric properties
which they share with SW-uniformities.

Based on these properties, we introduce in Section 4.2 the framework of
_vicinic structures_ where most of our work takes place. It is a
generalisation of SW-uniformities, encompassing also the distinguished sorts
of $p$-adically closed fields.

The rest of Section 4 is dedicated to collecting the tools needed for the
construction in Section 5 of infinitesimal subgroups of interpretable groups.
Much of the work in Section 4 is devoted to dealing with a technical issue not
arising in the study of fields: in [11] we have shown that every interpretable
field can be locally injected into one of the distinguished sorts. In the
present setting, we are required to work with finite-to-one functions, and
this requires some additional work. Also, in the V-minimal case, the sort
$D=\textbf{k}$ is a pure algebraically field, and thus has a different nature.
It is treated in Section 6.

In Section 7 we collect all the results of previous sections to prove Theorem
1.1 and Theorem 1.3. Section 8 is dedicated to the study of several natural
examples of interpretable groups in light of the results of this paper. In
Section 9 we apply the techniques developed thus far to show that, in our
setting, there are no definable finite-to-finite correspondences between two
distinct distinguished sorts. In the Appendix, we study the special case of a
dp-minimal group in a power bounded T-convex setting.

Acknowledgements We thank Pablo Cubides Kovacsics, Amnon Besser and Dugald
Macpherson for several discussions during the preparation of the article. We
thank Yair Glasner for pointing out a mistake in an earlier draft, and the
anonymous referee for the thorough reading and useful suggestions.

##  2\. Background and preliminaries

###  2.1. Notation, terminology and some preliminaries

Throughout, structures will be denoted by calligraphic letters,
${\mathcal{M}}$, ${\mathcal{N}}$, ${\mathcal{K}}$ etc., and their respective
universes by the corresponding Latin letters, $M$, $N$ and $K$. We reserve
${\mathcal{K}}$ to denote expansions (possibly trivial) of valued fields, and
$K$ will always be a valued field. All structures may be multi-sorted. A
valued field ${\mathcal{K}}=(K,v,\ldots)$ is always considered with a single
home sort (for the ground field $K$) with all other sorts coming from
${\mathcal{K}}^{eq}$. All sets are definable using parameters unless
specifically mentioned otherwise.

Tuples from a structure ${\mathcal{M}}$ are denoted by small Roman characters
$a,b,c,\dots$ and are always assumed to be finite. We apply the standard model
theoretic abuse of notation writing $a\in M$ for $a\in M^{|a|}$. Variables
will be denoted $x,y,z,\dots$ with the same conventions as above. We do not
distinguish notationally tuples and variables belonging to different sort,
unless some ambiguity can arise. Capital Roman letters $A,B,C,\ldots$ usually
denote small subsets of parameters from $M$. As is standard in model theory,
we write $Ab$ as a shorthand for $A\cup\\{b\\}$. In the context of definable
groups we will, whenever confusion can arise, distinguish between, e.g.,
$Agh:=A\cup\\{g,h\\}$ and $Ag\cdot h:=A\cup\\{g\cdot h\\}$.

By a partial type we mean a consistent collection of formulas. Two partial
types $\rho_{1},\rho_{2}$ are equal, denoted $\rho_{1}=\rho_{2}$, if they are
logically equivalent, i.e., if they have the same realizations in some
sufficiently saturated elementary extension.

We use standard model theoretic terminology and notation, providing references
as we proceed. The introductory sections of [14] provide a quick, more or less
self-contained, guide to much of the model theory of valued fields used in the
present paper, as well as many relevant examples. Below we give some basic
terminology and notation.

Valued fields. When $(K,v,\ldots)$ is an expansion of a valued field, we let
${\mathcal{O}}_{K}$ (or just ${\mathcal{O}}$, if the context is clear) denote
its valuation ring. Its maximal ideal is $\textbf{m}_{K}$ (or m) and
$\textbf{k}:={\mathcal{O}}/\textbf{m}$ its residue field. The value group is
$\Gamma_{K}$ (or just $\Gamma$). As in [11], $K$, k, $\Gamma$ and
$K/{\mathcal{O}}$ are the _distinguished sorts_. We shall occasionally
(especially in Section 8) also use the sorts
$\mathrm{RV_{\gamma}}:=K^{\times}/(1+\textbf{m}_{\gamma})\cup\\{0\\}$ where,
for a non-negative $\gamma\in\Gamma$ we denote $\textbf{m}_{\gamma}:=\\{x\in
K:v(x)>\gamma\\}$. We also let $\mathrm{rv}_{\gamma}:K\to\mathrm{RV}_{\gamma}$
denote the quotient map extended by $0$ at $0$.

A closed ball in $K$ is a set of the form $B_{\geq\gamma}(a):=\\{x\in
K:v(x-a)\geq\gamma\\}$ and $\gamma$ is its (valuative) radius, denoted $r(B)$.
Open balls are defined similarly and denoted by $B_{>\gamma}(a)$. Throughout
by "ball" we mean either an open or a closed ball. An open (closed) ball in
$K^{n}$ is a product of $n$ open (closed) balls of equal radius, i.e.,
$B_{1}\times\dots\times B_{n}$ where $B_{i}:=B_{>r}(a_{i})$ (resp.
$B_{i}:=B_{\geq r}(a_{i})$) for some $r\in\Gamma$ and $a_{i}\in K$. A ball in
$K^{n}$ is either a closed or an open ball, it has radius $r$ if it is a
product of $n$ closed (or $n$ open) balls in $K$ each of radius $r$.

Throughout Section 2 ${\mathcal{K}}$ is either a $V$-minimal field, a power
bounded $T$-convex expansion of a real closed field or a $p$-adically closed
field (see [11, Section 2.3] and references therein for the definitions and
basic properties of such fields). All these fields are 1-h-minimal, and though
this notion is used, throughout, as a black box, for the sake of completeness
we recall the definition (in a formulation suitable for mixed characteristic
as well). We first recall the following definition from [6].

######  Definition 2.1.

[6, Definition 2.1.6] For an integer $m\in\mathbb{N}$, a ball $B\subseteq K$
is $m$-next to $c\in K$ if $B=\\{x\in K:\operatorname{rv}_{m}(x-c)=t\\}$ for
some non-zero $t\in\mathrm{RV}_{m}$.

A ball $B$ is $m$-next to a finite non-empty set $C\subseteq K$ if
$B=\bigcap_{c\in C}B_{c}$ where $B_{c}$ is a ball $m$-next to $c$ for all
$c\in C$.

######  Remark 2.2.

Note that if a ball B is $m$-next to some $c\in K$ then it is an open ball.
Furthermore, an open ball $B$ is $m$-next to $c$ if and only if $c\notin B$
and the radius of $B$ is equal to $v(x-c)+m$ for every (equivalently, some)
$x\in B$. Thus, an open ball $B$ is $m$-next to a finite set $C$ if and only
if $B\cap C=\emptyset$ and $r(B)=\max\\{v(x-c)+m:c\in C\\}$, for every
(equivalently, some) $x\in B$.

######  Definition 2.3.

A theory $T$ of valued fields of characteristic $0$ is _1-h-minimal_ if for
any set of parameters $A$, and any $A$-definable $f:K\to K$ the following
hold:

  1. (1)

There exists a finite $A$-definable $C$ and a positive integer $m$ such that
for any ball $B$, $m$-next to $C$, there exists $\gamma_{B}\in\Gamma$ such
that or all $x,y\in B$ we have

$\gamma_{B}-v(m)\leq v(f(x)-f(y))-v(x-y)\leq\gamma_{B}+v(m).$

  2. (2)

The set $\\{y\in K:|f^{-1}(y)|=\infty\\}$ is finite.

######  Remark 2.4.

  1. (1)

The original formulation from [7] has some restrictions on the parameter set
$A$, that is $A\subseteq K\cup\mathrm{RV}_{n}$ for some $n$. In our setting,
since all the sorts are from $K^{eq}$ the definitions coincide.

  2. (2)

The above geometric definition is equivalent to the model theoretic
definition, by [7, Theorem 2.2.7].

It is known that in all the settings we are working in, the sorts k and
$\Gamma$ are stably embedded (for $p$-adically closed fields, or more
generally $P$-minimal fields this is [4, Theorem 6, Lemma 2], for $V$-minimal
field, or more generally $C$-minimal fields this is [14, Lemma 3.30] and for
the power bounded $T$-convex setting it is [39, Theorem A, Theorem B]). It is,
therefore, clear what the induced structure is on any of these sorts: $\Gamma$
is a pure ordered vector space when dense, and a ${\mathbb{Z}}$-group in the
$p$-adic setting, k is a model of $T$ in the $T$-convex setting (for our needs
it suffices that it is an o-minimal expansion of a real closed field) and a
pure algebraically closed field of characteristic $0$ in the $V$-minimal case.
We note, however, that the sort $K/{\mathcal{O}}$ is not stably embedded in
any of the structures we are studying,

We assume the reader is familiar with the notion of dp-rank, see [11, §2.1]
for the definition and a quick survey of dp-rank and its main properties. Let
us point out that in the field $K$, the value group $\Gamma$ and the residue
field k, dp-rank coincides with the classical notions of dimension associated
with each of these structures. It may also help the reader to note (and this
will be proved in some cases below) that in every distinguished sort $D$ in
our setting the dp-rank of $a\in D^{n}$ equals its
$\operatorname{acl}$-dimension, even if $\operatorname{acl}$ does not satisfy
exchange. For most applications below it suffices to know that dp-rank is
finite and sub-additive;

$\mathrm{dp\text{-}rk}(ab/A)\leq\mathrm{dp\text{-}rk}(a/A)+\mathrm{dp\text{-}rk}(b/Aa).$

Finally, when saying that a valued field has _definable Skolem functions_ , we
mean that the valued field sort $K$ has definable Skolem functions. Power
bounded $T$-convex valued fields have definable Skolem function by [39, Remark
2.4] (after adding a constant). So do $p$-adically closed fields [38, Theorem
3.2] and their expansion by all sub-analytic sets [8, Theorem 3.6].

###  2.2. Almost strong internality

The starting point of the present work is the following result that can be
deduced from [11, Proposition 5.5] (see also §7, loc. cit. for the details).
By a finite-to-finite correspondence between $X_{1}$ and $X_{2}$ we mean a
definable relation $C\subseteq X_{1}\times X_{2}$ such that both projections
$\pi_{i}:C\to X_{i}$ ($i=1,2$) are surjective with finite fibres.

######  Fact 2.5.

If $X$ is infinite and interpretable in ${\mathcal{K}}$ then there exist a
distinguished sort $D$, infinite definable $T\subseteq X,S\subseteq D$
(possibly defined over additional parameters) and a definable finite-to-finite
correspondence $C\subseteq T\times S$.

When – in the above notation – $X$ was a field, we were able to eliminate the
discrete sorts $K/{\mathcal{O}}$ and $\Gamma$ in the $p$-adically closed case
(in fact, there this was done for any P-minimal field), and the remaining
distinguished sorts turned out to be rather tame topological structures (that
we called SW-uniformities – to be described in more detail below). In that
setting (and for $X$ a field) we were able to strengthen Fact 2.5 to obtain a
definable bijection between $T$ and a subset of one of the distinguished
sorts. In the present setting some extra work is needed to obtain a weaker
result, i.e. a finite-to-one function from an infinite definable subset of our
group $G$ into one of the distinguished sorts. Extending our terminology from
[11] we define:

######  Definition 2.6.

A definable set $X$ is _$A$ -almost strongly internal to a definable set $D$_
if there exists an $A$-definable finite-to-one function $f:X\to D^{k}$, for
some $k\in{\mathbb{N}}$. The set $X$ is _locally almost strongly internal to
$D$_ if there exists a parameter set $A$ and an infinite $A$-definable
$X^{\prime}\subseteq X$ that is $A$-almost strongly internal to $D$.

Recall that $X$ is _strongly internal_ to $D$ if we can find an injective
$f:X\to D^{k}$ and _locally strongly internal_ to $D$ if some definable
infinite $X^{\prime}\subseteq X$ is strongly internal to $D$. As we shall see
in Example 3.33, a set which is locally almost strongly internal to $D$ need
not be locally strongly internal to $D$. The upshot of this situation is that
we have to develop the theory, in parallel, for subsets of $G$ almost strongly
internal to a distinguished sort $D$, and for subsets of $G$ strongly internal
(possibly to the same sort $D$). Since the statements and the proofs are, as a
rule, similar in both cases we usually state both results simultaneously; e.g.

> “If $X$ is locally (almost) strongly internal to $D$ … then $Y$ is (almost)
> strongly internal…”

with the convention that either all brackets are included or all brackets are
omitted. I.e., the two statements included in the above formulation are “If
$X$ is locally strongly internal to $D$ … then $Y$ is strongly internal…” and
“If $X$ is locally almost strongly internal to $D$ … then $Y$ is almost
strongly internal…”.

###  2.3. Simon-Walsberg uniformities and group topologies

Though the setting of the present paper requires that we take into account the
discrete sorts $\Gamma$ and $K/{\mathcal{O}}$, in $p$-adically closed fields,
SW-uniformities still have a significant role to play, and even in these two
cases some arguments are modelled after the analogous arguments in the
topological setting. So we remind:

######  Definition 2.7.

A definable set $D$ in a multi-sorted structure ${\mathcal{M}}$ is an SW-
uniformity if:

  1. (1)

$D$ is dp-minimal.

  2. (2)

$D$ has a definable uniform structure (or, uniformity) giving rise to a
Hausdorff topology (see [9, IX.11]). I.e., there exists a formula
$\varphi(x,y,z)$ and a definable set $S$ in ${\mathcal{M}}$ such that
$\\{\varphi(D^{2},s):s\in S\\}$ is a uniform structure on $D$, and the
intersection of all $\varphi(D^{2},s)$, for $s\in S$, is the diagonal.

  3. (3)

Every definable subset of $D$ has non-empty interior with respect to the
uniform topology.

  4. (4)

$D$ has no isolated points in the uniform topology.

As we already noted, the sorts $K$ in all settings, $\Gamma$ and
$K/{\mathcal{O}}$ in the power bounded T-convex and V-minimal settings and k
in the power bounded $T$-convex setting, are all SW uniformities (the fact
that $K/{\mathcal{O}}$ is not stably embedded is not a problem since the
definition allows for that).

One of the most important tools in [11] was the technical fact below. We shall
prove here an a analogue of this result in other settings as well, and
eventually include a variant of it as one of the axioms of our _ad hoc_
setting of vicinic sorts.

######  Fact 2.8 ([11, Corollary 3.13]).

Let $D$ be an SW-uniformity and $X\subseteq D^{n}$, $Y\subseteq X$ be
definable sets. Let $b\in D^{k}$ be any element, $a\in Y$ in the relative
interior of $Y$ in $X$ and $A$ a small set of parameters. Then there exist
$B\supseteq A$ and a $B$-definable open subset $U=U_{1}\times\dots\times
U_{n}\subseteq D^{n}$ such that $a\in U\cap X\subseteq Y$ and
$\mathrm{dp\text{-}rk}(a,b/B)=\mathrm{dp\text{-}rk}(a,b/A)$.

We also require the following result (implicit in [11]) that plays a key role
in our definition of $D$-groups, the main object of study of Section 4 and
Section 5 :

######  Lemma 2.9.

Let ${\mathcal{M}}$ be some structure and let $D$ be an SW-uniformity
definable in ${\mathcal{M}}$, $W,Z$ definable sets, and $f:Z\to W$ a definable
finite-to-one surjection.

  1. (1)

If $Z$ is strongly internal to $D$ then there exists a definable
$W_{1}\subseteq W$, with
$\mathrm{dp\text{-}rk}(W_{1})=\mathrm{dp\text{-}rk}(W)$ such that $W_{1}$ is
strongly internal to $D$.

  2. (2)

If $Z$ is almost strongly internal to $D$ then there exists a definable
$W_{1}\subseteq W$, $\mathrm{dp\text{-}rk}(W_{1})=\mathrm{dp\text{-}rk}(W)$,
such that $W_{1}$ is almost strongly internal to $D$.

###### Proof.

Assume everything is definable over some parameter set $A$.

(1) Since $Z$ is strongly internal to $D$ we may assume that $Z\subseteq
D^{k}$ for some $k$. We fix $z_{0}\in Z$ such that
$\mathrm{dp\text{-}rk}(z_{0}/A)=\mathrm{dp\text{-}rk}(Z)$. We first find a
definable open set $V\subseteq D^{k}$ such that $|[z_{0}]_{f}\cap V|=1$, where
$[z_{0}]_{f}=f^{-1}(f(z_{0}))$. By Fact 2.8 there exists an open set $U$,
$z_{0}\in U\subseteq V$, definable over some $B\supseteq A$, such that
$\mathrm{dp\text{-}rk}(z_{0}/A)=\mathrm{dp\text{-}rk}(Z)$, and
$[z_{0}]_{f}\cap U=\\{z_{0}\\}$. But now, the set $Z_{1}=\\{z\in Z:[z]_{f}\cap
U=\\{z\\}\\}$ is defined over $B$ and contains $z_{0}$ hence
$\mathrm{dp\text{-}rk}(Z_{1})=\mathrm{dp\text{-}rk}(Z)=\mathrm{dp\text{-}rk}(W)$.
Now $f\restriction Z_{1}$ is injective, and its image $W_{1}$ is our desired
set.

(2) Since $Z$ is almost strongly internal to $D$ there exists $\sigma:Z\to
D^{k}$ with finite fibres. Let

$C=\\{(w,y)\in W\times D^{k}:y\in\sigma(f^{-1}(w))\\};$

it is a finite-to-finite correspondence. Choose $y_{0}\in\pi_{2}(C)$ (the
projection of $C$ into $D^{k}$) with
$\mathrm{dp\text{-}rk}(y_{0}/A)=\mathrm{dp\text{-}rk}(\pi_{2}(C))$. Since $C$
is finite-to-finite the set of all $y\in D^{k}$ such that $C^{y}\cap
C^{y_{0}}\neq\emptyset$ is finite, so there exists an open $V\ni y_{0}$ such
that for all $y\in V$, $C^{y}\cap C^{y_{0}}=\emptyset$.

As above, we may replace $V$ with an open $U$ defined over $B$ such that
$\mathrm{dp\text{-}rk}(y_{0}/B)=\mathrm{dp\text{-}rk}(y_{0}/A)$. Then
$C\cap(W\times U)$ is a graph of a finite-to-one function, $F$. Since $y_{0}$
is in the range of that function and
$\mathrm{dp\text{-}rk}(y_{0}/A)=\mathrm{dp\text{-}rk}(W)$, necessarily
$\mathrm{dp\text{-}rk}(\operatorname{dom}(F))\geq\mathrm{dp\text{-}rk}(W)$, so
equality must hold, as required. ∎

Much of Section 3.1 is dedicated to making sense of the above results for
$K/{\mathcal{O}}$ in $p$-adically closed fields and proving an appropriate
analogue. Similar results for ${\mathbb{Z}}$-groups are easier and follow from
known properties of such groups.

When $G$ is locally strongly internal to an SW-uniformity, the group $\nu$ we
obtain in Theorem 1.1 admits a group topology. This will allow us to
topologise $G$, using the following two easy observations.

######  Lemma 2.10.

Let $G$ be a group and $H$ a subgroup endowed with a Hausdorff group topology,
with $\mathcal{B}_{H}$ a basis for the $e$ neighbourhoods in $H$. For
$V\in\mathcal{B}_{H}$, let $U_{V}=\\{(x,y)\in G\times G:x^{-1}y\in V\\}$.

Then the collection $\mathcal{U}^{G}=\\{U_{V}:V\in\mathcal{B}_{H}\\}$ is a
left-invariant uniformity on $G$ extending the associated uniformity on $H$.

The uniformity $\mathcal{U}^{G}$ induces a topology on $G$, call it
$\tau^{G}$, whose basis is the collection of all sets $U_{V}(h,G)$, as $h$
varies in $G$ and $V$ varies in $\mathcal{B}_{H}$. Though this need not in
general be a group topology, for every $g\in G$ the map $x\mapsto gx$ is
continuous. We can show, however:

######  Lemma 2.11.

Let $G$ be a group and $H\leq G$ a topological group such that,

  1. (i)

For every $g\in G$ there is an open $V\subseteq H$ such that $V\subseteq
H^{g}\cap H$.

  2. (ii)

For every $g\in G$, the function $x\mapsto x^{g}=gxg^{-1}$, from $H$ into
$H^{g}$, is continuous at $e$ with respect to the topology on $H$ (this makes
sense due to (i)).

Then the uniformity $\mathcal{U}^{G}$ defined above makes $G$ a topological
group.

###### Proof.

Let ${\mathcal{B}}_{H}$ denote a neighbourhood base for $H$ at $e$. We may
assume that each $W\in{\mathcal{B}}_{H}$ is symmetric. Note also that by Lemma
2.10 for each $g\in G$, the family $\\{gV:V\in{\mathcal{B}}_{H}\\}$ is a basis
for the $\tau^{G}$-neighbourhoods at $g$.

We prove first that group inverse is continuous. Assume that $g^{-1}V$ is an
open neighbourhood of $g^{-1}$. We need to find an open neighbourhood $gW$ of
$g$ such that $W^{-1}g^{-1}\subseteq g^{-1}V$. By our assumptions, the map
$x\mapsto gxg^{-1}$ is continuous at $e$ and hence there is
$W\in{\mathcal{B}}_{H}$ (in particular $W^{-1}=W$), such that
$gWg^{-1}\subseteq V$. It follows that $Wg^{-1}\subseteq g^{-1}V$.

To prove continuity of multiplication let $g_{1},g_{2}\in G$, and assume that
$g=g_{1}g_{2}$, and $gV$ is a basic neighbourhood of $g$ and fix some
$V^{\prime}\in{\mathcal{B}}_{H}$ such that $V^{\prime}\subseteq H^{g}\cap H$,
as provided by (i). Using the continuity at $e$ of $x\mapsto x^{g}$, we can
shrink $V^{\prime}$ so that $V^{\prime}\subseteq V^{g}$. Using the continuity
of $x\mapsto x^{g_{1}}$ and the fact that $H$ is a topological group, we can
find $W_{1},W_{2}\in{\mathcal{B}}_{H}$, such that

$(g_{1}W_{1}g_{1}^{-1})(gW_{2}g^{-1})\subseteq V^{\prime}.$

It follows that $g_{1}W_{1}g_{2}W_{2}g^{-1}\subseteq gVg^{-1}$, and hence
$g_{1}W_{1}g_{2}W_{2}\subseteq gV$, as needed. Thus $\mathcal{U}^{G}$ gives
rise to a group topology. ∎

##  3\. Some results on $p$-adically closed fields

An important tool in [11] is the analysis of interpretable fields via subsets
that are strongly internal to SW-uniform sorts. Such an analysis is not
available to us when studying interpretable groups in a $p$-adically closed
field $K$, as neither $\Gamma$ nor $K/{\mathcal{O}}$ are SW-uniform sorts. In
the present section we endow $K/{\mathcal{O}}$ with a topology – neither
definable nor Hausdorff – where, nonetheless, many of the tools available in
SW-uniformities can still be applied.

We prove two geometric properties that play a crucial role in the sequel. The
first, comprising the heart of Section 3.1, is an analogue of Fact 2.8, that
can be interpreted as asserting the existence of generic neighbourhoods (of
generic points) with respect to the above topology. This is formulated
precisely in Proposition 3.8(3). Using this result we then proceed to proving
an analogue of Lemma 2.9. Section 3.2 is dedicated to the second main property
that we need: local linearity of definable functions in $K/{\mathcal{O}}$,
culminating in Corollary 3.32. The strategy of proof is similar to that
carried out in [11, §6.2], using results on 1-h-minimal fields, but working in
mixed characteristic requires a little more care.

While the results of Section 3.2 may be of independent interest, the results
of Section 3.1 are more technical. Readers unfamiliar with the analogous
results from [11] may wish to first read Section 4.1 for a more detailed
discussion of the underlying geometric motivation.

From this point up to Section 4, ${\mathcal{K}}:=(K,+,\cdot,v)$ is a
$p$-adically closed field. We note, however, that the results of Section 3.1
hold in any $P$-minimal field. For Section 3.2 we need, in addition,
1-h-minimality and definable Skolem functions (as discussed in the
introduction).

We remind that $p$-adically closed fields are elementarily equivalent (in the
Macintyre language) to a finite extension, denoted $\mathbb{F}$, of
$\mathbb{Q}_{p}$. We work in a large saturated $p$-adically closed field,
${\mathcal{K}}=(K,v)$, and so we may assume that $(K,v)$ is an elementary
extension of $(\mathbb{F},v)$. We add constants for all elements of
$\mathbb{F}$; since its value group $\Gamma_{\mathbb{F}}$ is isomorphic to
$\mathbb{Z}$ as an ordered abelian group for simplicity of notation we
identify $\Gamma_{\mathbb{F}}$ with $\mathbb{Z}$, and note that
$\textbf{k}_{K}=\textbf{k}_{\mathbb{F}}$. It is well known that, as an abelian
group, $\mathbb{F}/\mathcal{O}_{\mathbb{F}}$ is isomorphic to
$\bigoplus_{i=1}^{n}\mathbb{Z}(p^{\infty})$, where $\mathbb{Z}(p^{\infty})$ is
the Prüfer $p$-group and $n=[\mathbb{F}:\mathbb{Q}_{p}]$. Throughout, we will
use without further reference the fact that $v$ is well-defined on
$K/{\mathcal{O}}\setminus\\{0\\}$ and denote $v(0)=\infty$ (for $0\in
K/{\mathcal{O}}$). As usual $v$ extends to $(K/O)^{n}$ by setting
$v(a_{1},\dots,a_{n})=\min\\{v(a_{i}):i=1,\dots,n\\}.$

######  Fact 3.1.

  1. (1)

For every $a\in K$ with $v(a)\in\mathbb{Z}$, there is an element
$r\in\mathbb{F}$ with $v(a-r)>\mathbb{Z}$.

  2. (2)

For every $b\in K/{\mathcal{O}}$, if $v(b)\in\mathbb{Z}\cup\\{\infty\\}$ then
$b\in\mathbb{F}/{\mathcal{O}}_{\mathbb{F}}\leq K/{\mathcal{O}}$.

  3. (3)

For all $n\in{\mathbb{N}}$, there are only finitely many $b\in
K/{\mathcal{O}}$ such that $nb=0$; in fact every such $b$ is in
$\mathbb{F}/\mathcal{O}_{\mathbb{F}}$.

###### Proof.

(1) This follows from the completeness of $\mathbb{F}$ as a valued field.
Indeed, for $t\in\mathbb{F}$ with $v(at)=0$ we can find $c\in\mathbb{F}$ with
$v(at-c)>0$ thus $v(a-ct^{-1})>-v(t)$; continuing inductively we construct a
pseudo-Cauchy sequence in $\mathbb{F}$, converging, by completeness, to a
limit in $\mathbb{F}$. This limit has the desired properties.

(2) Every $b\in K/{\mathcal{O}}$ with $v(b)\in\mathbb{Z}\cup\\{\infty\\}$ is
of the form $b=B_{\geq 0}(a)$ for some $a\in K$ with
$v(a)\in\mathbb{Z}\cup\\{\infty\\}$. By (1) there is $r\in\mathbb{F}$ with
$v(a-r)>\mathbb{Z}\geq 0$, so $r+{\mathcal{O}}=a+{\mathcal{O}}$ as required.

(3) It suffices to show that if $nb=0$ then
$b\in\mathbb{F}/{\mathcal{O}}_{\mathbb{F}}$. If $nb=0$ and
$b=b^{\prime}+{\mathcal{O}}$, for some $b^{\prime}inK$, then
$v(nb^{\prime})=v(n)+v(b^{\prime})\geq 0$. So $v(b^{\prime})\in\mathbb{Z}$,
i.e., $b\in\mathbb{F}/{\mathcal{O}}_{\mathbb{F}}$. ∎

###  3.1. Local analysis in $K/{\mathcal{O}}$

As noted above, the valuation on $K$ is well-defined on $K/{\mathcal{O}}$, but
the topology it induces on $K/{\mathcal{O}}$ is discrete. For that reason we
introduce a coarser (non-Hausdorff) topology retaining some properties of SW-
uniformities. The main technical property we need regarding this topology is
the existence of generic neighbourhoods (Proposition 3.8), but we also take
the opportunity to single out several other properties that will be of use in
the sequel. As an application we obtain Corollary 3.10 on the torsion of
definable subgroups of $K/{\mathcal{O}}$, that is used in the sequel, but may
be of independent interest. We first need some preliminaries:

######  Definition 3.2.

A ball $B\subseteq K^{n}$ is _large_ if $r(B)$, the valuative radius of $B$,
satisfies $r(B)<\mathbb{Z}$.

Let $\pi:K\to K/{\mathcal{O}}$ be the quotient map, we also write
$\pi:K^{n}\to(K/{\mathcal{O}})^{n}$ for any integer $n$.

######  Definition 3.3.

A _ball_ $U\subseteq(K/{\mathcal{O}})^{n}$ is the image under $\pi$ of a large
ball in $K$.

We consider the ball topology  on $(K/{\mathcal{O}})^{n}$ whose basis is the
family of balls. Thus, for $n>1$, this is the product topology on
$(K/{\mathcal{O}})^{n}$.

######  Remark 3.4.

The ball topology on $K/{\mathcal{O}}$ is not Hausdorff, in fact, not even
$T_{0}$: for $b_{1}\neq b_{2}\in K/{\mathcal{O}}$, if
$v(b_{1}-b_{2})\in\mathbb{Z}$ then every open set containing $b_{1}$ also
contains $b_{2}$. In addition, the ball topology has no definable basis (since
the set of balls of infinite radius in $K$ is not definable),

Below, all topological notions in $K/{\mathcal{O}}$ and
$(K/{\mathcal{O}})^{n}$ refer to the ball topology. Despite the above remark,
some amount of local analysis in $K/{\mathcal{O}}$ is possible using the lemma
below. For $a=(a_{1},\ldots,a_{n})\in(K/{\mathcal{O}})^{n}$, we let
$\dim_{\operatorname{acl}}(a)$ be the maximal $k$ such that some sub-tuple
$a^{\prime}$ of $a$ of length $k$ is algebraically independent and
$a\in\operatorname{acl}(a^{\prime})$ (this is well-defined even when
$\operatorname{acl}$ does not satisfy exchange).

######  Lemma 3.5.

Let $a=(a_{1},\dots,a_{m})$ be a tuple of elements in $K$ or $K/{\mathcal{O}}$
and $A$ an arbitrary parameter set such that
$\dim_{\operatorname{acl}}(a/A)=n$. Then for any
$\gamma_{0}<\gamma_{1}<\mathbb{Z}$ with $\gamma_{0}-\gamma_{1}<\mathbb{Z}$
there exists $\gamma\in\Gamma$ such that
$\gamma_{0}<\gamma<\gamma_{1}<\mathbb{Z}$ and
$\dim_{\operatorname{acl}}(a/A\gamma)=n$.

###### Proof.

By passing to a sub-tuple of $a$ which is $\operatorname{acl}$-independent
over $A$, we may assume that $|a|=n$.

By replacing $\gamma_{0}$ with $\gamma_{0}-\gamma_{1}<\mathbb{Z}$ it is enough
to find $\gamma\in\Gamma$, $\gamma_{0}<\gamma<\mathbb{Z}$ such that
$\dim_{\operatorname{acl}}(a/A\gamma)=n$.

For $1\leq i\leq n$, let
$\hat{a}_{i}:=(a_{1},\dots,a_{i-1},a_{i+1},\dots,a_{n})$. Assume towards a
contradiction that for every infinite $\gamma>\gamma_{0}$, there is $1\leq
i\leq n$ such that $a_{i}\in\operatorname{acl}(\hat{a}_{i},A,\gamma)$. We may
assume, without loss of generality, that
$a_{1}\in\operatorname{acl}(\hat{a}_{1},A,\gamma)$ for arbitrarily large
$\gamma$ below $\mathbb{Z}$. We now consider all formulas $\varphi(x,y)$ over
$\hat{a}_{1},A$ with $y$ a $\Gamma$-variable and $x$ a
$K/{\mathcal{O}}$-variable.

By compactness, there are formulas $\varphi_{1}(x,y),\dots,\varphi_{k}(x,y)$
over $\hat{a}_{1},A$, and some $k\in\mathbb{N}$, such that
$|\varphi_{i}(K/{\mathcal{O}},\gamma)|\leq k_{i}$ for every $i\leq k$, and for
every $\gamma_{1}\in\Gamma$, if $\gamma_{0}<\gamma_{1}<\mathbb{Z}$ then there
exists $\gamma_{1}<\gamma<{\mathbb{Z}}$, such that
$\bigvee_{i=1}^{k}\varphi_{i}(a_{1},\gamma)$.

Without loss of generality we may assume that for $i=1$ the set of
$\gamma>\gamma_{0}$ such that $\models\varphi_{1}(a_{1},\gamma)$, is cofinal
below $\mathbb{Z}$. Since $\Gamma$ is a pure ${\mathbb{Z}}$-group, we may
further assume that $\varphi_{1}(a_{1},\Gamma)$ is a definable set of the form
$\\{t\in\Gamma:\alpha<t<\beta\wedge x\equiv_{N}c\\}$ for some $N\in\mathbb{N}$
and $0\leq c<N$. By the cofinality assumption, we cannot have
$\beta<\mathbb{Z}$, so there exists $m\in\mathbb{Z}$, such that
$\varphi_{1}(a_{1},m)$, contradicting the assumption that
$a_{1}\notin\operatorname{acl}(A,a_{2},\dots,a_{n})$. ∎

In terms of the ball topology on $K/{\mathcal{O}}$, the above lemma says that
if $a\in(K/{\mathcal{O}})^{n}$ is acl-generic then any neighbourhood of $a$
contains a ball of “generic radius”. In Proposition 3.8 we strengthen this to
provide a “generic neighbourhood” of $a$ (inside any neighbourhood of $a$).
But we first need some preliminary results.

######  Lemma 3.6.

Assume that $b\in(K/{\mathcal{O}})^{n}$ is such that
$\dim_{\operatorname{acl}}(b/A)=n$, for some parameter set $A$.

  1. (1)

For every $A$-definable $X\subseteq(K/{\mathcal{O}})^{n}$, if $b\in X$ then
$b$ is in the interior of $X$.

  2. (2)

If $p=\operatorname{tp}(b/A)$ then $p({\mathcal{K}})$ is open in
$(K/{\mathcal{O}})^{n}$

  3. (3)

$\mathrm{dp\text{-}rk}(b/A)=n$.

###### Proof.

Let $a\in K^{n}$ be such that $\pi(a)=b$.

(1) We proceed by induction on $n$. For $n=1$, we first prove that every
$A$-definable set $Y\subseteq K$ containing $a$ must contain a large ball.
Indeed, since $b\in\pi(Y)$ is non-algebraic $Y$ intersects infinitely many
$0$-balls. Since $K$ is P-minimal, by [11, Proposition 5.8, Lemma 6.26], $Y$
contains a large ball.

Let $q=\operatorname{tp}(a/A)$; by compactness we conclude that
$q({\mathcal{K}})$ contains a large ball, but then, as $\mathbb{Z}$ is
automorphism invariant, every $\alpha\models q$ belongs to a large ball
$B\subseteq q({\mathcal{K}})$. In particular, $a\in B\subseteq Y$ for some
large ball $B$.

Now assume that $b\in X$ for an $A$-definable set $X\subseteq K/{\mathcal{O}}$
and apply the above to $a\in Y=\pi^{-1}(X)$. We conclude that there is a ball
$U\subseteq K/{\mathcal{O}}$ with $b\in U\subseteq X$.

For $n>1$, write $b=(b^{\prime},b_{n})\in X$ for
$b^{\prime}\in(K/{\mathcal{O}})^{n-1}$. Let $a=(a^{\prime},a_{n})\in K^{n}$ be
such that $\pi(a)=b$. Then $\dim_{\operatorname{acl}}(a)=n$ and
$\dim_{\operatorname{acl}}(a^{\prime}/a_{n}A)=n-1$. Let $Y=\pi^{-1}(X)$.

Because $b_{n}\notin\operatorname{acl}(b^{\prime}A)$, it follows that
$X_{b^{\prime}}=\\{(b^{\prime},t)\in X:t\in K/{\mathcal{O}}\\}$ is infinite,
so by the case $n=1$, $b_{n}$ is in the interior of $X_{b^{\prime}}$. It
follows that $a_{n}$ is in the interior of
$Y_{a^{\prime}}=\\{(a^{\prime},t)\in Y:t\in K\\}$ in the large ball topology.
Since $\dim_{\operatorname{acl}}(a^{\prime}/a_{n}A)=n-1$, by Lemma 3.5 we can
find $\gamma<\mathbb{Z}$ such that
$\dim_{\operatorname{acl}}(a^{\prime}/a_{n}A\gamma)=n-1$ and
$B_{\geq\gamma}(a_{n})\subseteq Y_{a^{\prime}}$.

We consider the $a_{n}A\gamma$-definable set $W=\\{x\in
K^{n-1}:B_{\gamma}(a_{n})\subseteq Y_{x}\\}$. It contains $a^{\prime}$, so by
induction it contains a large ball $B_{\geq\gamma^{\prime}}(a^{\prime})$.
Denoting $\widehat{\gamma}=\max\\{\gamma,\gamma^{\prime}\\}$ the ball
$B_{\geq\widehat{\gamma}}(a^{\prime})\times B_{\geq\widehat{\gamma}}(a_{n})$
is centred at $a$ and contained in $Y$. It follows that $b=\pi(a)$ is in the
interior of $X$.

(2) Follows by compactness from (1).

(3) By (2), for $p=\operatorname{tp}(b/A)$, $p({\mathcal{K}})$ contains a ball
$U$. Such a ball is a Cartesian product of $n$ balls, each of which has dp-
rank $1$. So $\mathrm{dp\text{-}rk}(p)\geq n$, and equality must hold. ∎

We conclude:

######  Corollary 3.7.

For $b\in(K/{\mathcal{O}})^{n}$ and any parameter set $A$ we have
$\mathrm{dp\text{-}rk}(b/A)=\dim_{\operatorname{acl}}(b/A)$.

###### Proof.

By sub-additivity of the dp-rank we just need to show that
$\mathrm{dp\text{-}rk}(b/A)\geq\dim_{\operatorname{acl}}(b/A)$. If
$\dim_{\operatorname{acl}}(b/A)=k$ then by passing to a sub-tuple we may
assume that $\dim_{\operatorname{acl}}(b_{1},\dots,b_{k}/A)=k$. By Lemma
3.6(3) we have that $\mathrm{dp\text{-}rk}(b_{1},\dots,b_{k}/A)=k$ but then
$\mathrm{dp\text{-}rk}(b/A)\geq k$, as needed. ∎

We can now prove the desired generalisation of Lemma 3.5:

######  Proposition 3.8.

Let $b\in(K/{\mathcal{O}})^{n}$, $A$ any set of parameters and
$c\in(K/{\mathcal{O}})^{m}$.

  1. (1)

Assume that $\mathrm{dp\text{-}rk}(b/A)=n$ and that $b\in W$ for some ball
$W\subseteq(K/{\mathcal{O}})^{n}$. Then there is a ball $V\subseteq W$,
defined over some $B\supseteq A$ such that $b\in V$ and
$\mathrm{dp\text{-}rk}(b/B)=\mathrm{dp\text{-}rk}(b/A)=n$.

  2. (2)

For any $\gamma<\mathbb{Z}$ there exists $B\supseteq A$ and a $B$-definable
ball $V\subseteq B_{\geq\gamma}(b)$ such that $b\in B$ and
$\mathrm{dp\text{-}rk}(b,c/A)=\mathrm{dp\text{-}rk}(b,c/B)$.

  3. (3)

Assume that $b\in X\subseteq(K/{\mathcal{O}})^{n}$ for some $B$-definable $X$
with $\mathrm{dp\text{-}rk}(b/B)=n$ then there exists $C\supseteq A$ and a
$C$-definable ball $V\subseteq X$ such that $b\in V$ and
$\mathrm{dp\text{-}rk}(b,c/C)=\mathrm{dp\text{-}rk}(b,c/A)$.

###### Proof.

In the proof, we repeatedly use – without further mention – Corollary 3.7,
i.e., that $\dim_{\operatorname{acl}}=\mathrm{dp\text{-}rk}$.

(1) By Lemma 3.6(2) $\operatorname{tp}(b/A)$ is open and thus contains a ball,
$Z$, centred at $b$. Without loss of generality, $Z=W$. By Lemma 3.5 we can
find $\gamma$, $r(W)<\gamma<\mathbb{Z}$, such that $B_{\geq\gamma}(b)\subseteq
W\vdash\operatorname{tp}(b/A)$ and $\mathrm{dp\text{-}rk}(b/A\gamma)=n$. We
let $s=B_{\geq\gamma}(b)$, $r(s)=\gamma$ its radius and $[s]$ its code.

######  Claim 3.8.1.

For any parameter set $A$, any $b\in(K/{\mathcal{O}})^{n}$ and any
$K/{\mathcal{O}}$-ball $s$ containing $b$, if $r(s)\in\operatorname{dcl}(A)$
and $\mathrm{dp\text{-}rk}(b/A)=n$ then $\mathrm{dp\text{-}rk}(b/A[s])=n$.

###### Proof.

Fix $A,b,s$ as in the statement and let $\gamma=r(s)$. By Lemma 3.6(2),
$q:=\operatorname{tp}(b/A)$ is open in $(K/{\mathcal{O}})^{n}$. It follows
that $q$ contains a large ball $U$, that we may assume to be contained in $s$.

The proof is based on the case $n=1$ which we prove first: Since $U$ is large,
there are $b_{m}\in U\subseteq s$, $m\in\mathbb{N}$, such that
$b_{m^{\prime}}\neq b_{m}$ for $m^{\prime}\neq m$. Obviously, $b_{m}\models q$
for all $m$. Thus, for every $b_{m}$, there is
$\sigma_{m}\in\operatorname{Aut}({\mathcal{K}}/A)$ mapping $b$ to $b_{m}$.
Because $b_{m}\in s=B_{\geq\gamma}(b)$ and $\gamma\in\operatorname{dcl}(A)$
also $\sigma_{m}(B_{\geq\gamma}(b))=B_{\geq\gamma}(b_{m})=B_{\geq\gamma}(b)$;
so $\sigma_{m}(s)=s$. As a result, we get $b\notin\operatorname{acl}(A[s])$;
i.e. $\mathrm{dp\text{-}rk}(b/A[s])=1$.

Assume now $b=(b_{1},\ldots,b_{n})$, and $s\subseteq(K/{\mathcal{O}})^{n}$ is
a ball containing $b$ as above. We shall see that for every $i=1,\ldots,n$,
$b_{i}$ is not in the algebraic closure of $A[s]b^{\prime}$, where
$b^{\prime}=(b_{1},\dots,\widehat{b_{i}},\dots,b_{n})$. By Corollary 3.7, this
implies $\mathrm{dp\text{-}rk}(b/A[s])=n$. For simplicity of notation, we
treat the case $i=n$, so $b^{\prime}=(b_{1},\ldots,b_{n-1})$.

We write $s^{\prime}=B_{\geq\gamma}(b^{\prime})$ and
$s_{n}=B_{\geq\gamma}(b_{n})$. By applying the case $n=1$ to
$\operatorname{tp}(b_{n}/Ab^{\prime})$, we get that
$\mathrm{dp\text{-}rk}(b_{n}/Ab^{\prime}[s_{n}])=1$. Because
$r(s^{\prime})=r(s)\in\operatorname{dcl}(A)$ (by assumption) we get
$[s^{\prime}]\in\operatorname{dcl}(Ab^{\prime})$ and as $s=s^{\prime}\times s$
also $[s]\in\operatorname{dcl}(Ab^{\prime}[s_{n}])$. Therefore
$\mathrm{dp\text{-}rk}(b_{n}/Ab^{\prime}[s])=1$, as required. ∎

To conclude, recall that we have fixed some $\gamma<\mathbb{Z}$ such that
$\mathrm{dp\text{-}rk}(b/A\gamma)=n$ and such that $B_{\geq\gamma}(b)\subseteq
W$. Applying the claim to $b$ over $A\gamma$ we obtain the desired conclusion.

(2) To simplify notation, assume that $A=\emptyset$. We start with the case
where $m=0$, i.e. there is no $c$. Let $b=(b_{1},\dots,b_{n})$ and
$k=\mathrm{dp\text{-}rk}(b)$; without loss of generality, assume that
$\mathrm{dp\text{-}rk}(b_{1},\dots,b_{k})=k$. Fix any
$\gamma<\gamma^{\prime}<{\mathbb{Z}}$ such that
$\gamma-\gamma^{\prime}<{\mathbb{Z}}$. Choose $\gamma_{1}$ be as provided by
Lemma 3.5 applied to $\gamma^{\prime}$, so $\gamma<\gamma_{1}$,
$\mathrm{dp\text{-}rk}(b/\gamma_{1})=k$ and $\gamma-\gamma_{1}<\mathbb{Z}$.
Let $U_{1}=B_{\geq\gamma_{1}}(b)$ and let
$b^{\prime}=(b_{1},\dots,b_{k},b_{k+1}^{\prime},\dots,b_{n}^{\prime})\in
U_{1}$ be such that $\mathrm{dp\text{-}rk}(b^{\prime}/\gamma_{1})=n$.

By (1), there exists $B\supseteq A$ and a $B$-definable ball $V\subseteq
U_{1}$ with $V\ni b^{\prime}$ and $\mathrm{dp\text{-}rk}(b^{\prime}/B)=n$.
Since $\mathrm{dp\text{-}rk}(b^{\prime}/B)=n$ and
$\gamma-\gamma_{1}<\mathbb{Z}$ we get by Lemma 3.5 that there exists
$\gamma_{2}$, $\gamma<\gamma_{2}<\gamma_{1}$ with
$\mathrm{dp\text{-}rk}(b^{\prime}/B\gamma_{2})=n$. Consider the ball

$V^{\prime}=\\{x\in(K/{\mathcal{O}})^{n}:v(x-y)\geq\gamma_{2}\text{ for some
$y\in V$}\\}.$

Observe that $b^{\prime}\in V^{\prime}$ as
$v(b-b^{\prime})\geq\gamma_{1}>\gamma_{2}$. Our choice of
$b^{\prime},\gamma_{2}$ assures that
$\mathrm{dp\text{-}rk}(b^{\prime}/B\gamma_{2})=n$ and that
$\mathrm{dp\text{-}rk}(b/B\gamma_{2})=k$. Since $V^{\prime}\subseteq
B_{\geq\gamma}(b)$ it satisfies the requirements.

Now, given $c\in(K/{\mathcal{O}})^{m}$ and a ball $U\ni b$, we apply the above
result to the tuple $(b,c)$ and the open set $U\times(K/{\mathcal{O}})^{m}$ to
obtain the desired conclusion.

(3) By Lemma 3.6(1), $b$ is in the interior of $X$. Hence, there exists some
$B_{\geq\gamma}(b)\subseteq X$ for $\gamma<\mathbb{Z}$. By (2), there exists
$C\supseteq A$ and a $C$-definable ball $V\subseteq(K/{\mathcal{O}})^{n}$,
$b\in V\subseteq B_{\geq\gamma}(b)$, with
$\mathrm{dp\text{-}rk}(b,c/C)=\mathrm{dp\text{-}rk}(b,c/A)$. ∎

We conclude this section with a couple of applications of the tools developed
thus far. The first result is known for SW-uniformities (see [11, §6.2]). The
present proof is similar, but some extra care is needed, since the ball
topology on $K/{\mathcal{O}}$ is not Hausdorff.

######  Lemma 3.9.

  1. (1)

Let $\\{X_{t}:t\in T\\}$ be a definable family of finite subsets of
$(K/{\mathcal{O}})^{n}$, such that for all $b_{1}\neq b_{2}\in X_{t}$ we have
$v(b_{1}-b_{2})\in\mathbb{Z}$ and every $b\in(K/{\mathcal{O}})^{n}$ belongs to
finitely many $X_{t}$. Then there is a finite-to-one function from $T$
into$(K/{\mathcal{O}})^{n}$.

  2. (2)

Let $f:X\to T$ be a definable finite-to-one surjective map, and assume that
$X$ is almost strongly internal to $K/{\mathcal{O}}$. Then there exists a
definable subset $T_{1}\subseteq T$ with
$\mathrm{dp\text{-}rk}(T_{1})=\mathrm{dp\text{-}rk}(T)$, such that $T_{1}$ is
almost strongly internal to $K/{\mathcal{O}}$.

###### Proof.

(1) By saturation,

$Z:=\\{v(b_{1}-b_{2}):b_{1},b_{2}\in X_{t},t\in T\\}$

is finite. Let $m_{0}=\min\\{Z\\}\in\mathbb{Z}$; then for every $t\in T$ the
set $\bigcup X_{t}$ is contained in a single ball of radius $m_{0}$.

Let $U\subseteq K^{n}$ be a closed ball of valuational radius $m_{0}$ centred
at $0$. Since $(K,v)$ is $p$-adically closed, for each $a\in K^{n}$, $a+U$
contains only finitely many balls of radius $0$. Hence, the map sending $t$ to
the unique coset of $U$ containing $X_{t}$ is finite-to-one, so we have
constructed a finite-to-one definable map from $T$ into $K^{n}/U$. Since
$(K/{\mathcal{O}})^{n}$ and $K^{n}/U$ are in definable bijection, we are done.

(2) Let $g:X\to(K/{\mathcal{O}})^{n}$ be a finite-to-one definable map. For
any $t\in T$ let $X_{t}=g(f^{-1}(t))$. Then $\\{X_{t}:t\in T\\}$ is a
definable family of finite subsets of $(K/{\mathcal{O}})^{n}$ such that each
$b\in(K/{\mathcal{O}})^{n}$ belongs to only finitely many $X_{t}$.

By saturation, there is a uniform bound $m$ on $|X_{t}|$ for $t\in T$. We
proceed by induction on $m$. If $m=1$, we are done; so assume otherwise. For
simplicity, assume everything is defined over $\emptyset$.

Let $t_{0}\in T$ be with
$\mathrm{dp\text{-}rk}(t_{0})=\mathrm{dp\text{-}rk}(T)$ and assume first that
there are distinct $b,b^{\prime}$ in $X_{t}$ with
$v(b-b^{\prime})<\mathbb{Z}$; so there exists a ball $U\ni b$ containing $b$
and not $b^{\prime}$. By Proposition 3.8(2), there exists a ball $V\subseteq
U$ containing $b$ defined over parameters $B$ satisfying that
$\mathrm{dp\text{-}rk}(b/B)=\mathrm{dp\text{-}rk}(b)$. As $b$ and $t_{0}$ are
inter-algebraic over $\emptyset$, we also have
$\mathrm{dp\text{-}rk}(t_{0}/B)=\mathrm{dp\text{-}rk}(t_{0})=\mathrm{dp\text{-}rk}(T)$.
As $b^{\prime}\notin V$, clearly $|X_{t_{0}}\cap V|<|X_{t_{0}}|$.

The set $T_{1}=\\{t\in T:|X_{t}\cap V|<|X_{t}|\\}$ is defined over $B$ and
contains $t_{0}$, thus
$\mathrm{dp\text{-}rk}(T_{1})=\mathrm{dp\text{-}rk}(T)$. We many now replace
each $X_{t}$ with $X_{t}\cap V$ and proceed by induction. This completes the
proof in the case where $v(b-b^{\prime})<\mathbb{Z}$ for some $b,b^{\prime}\in
X_{t_{0}}$.

So we assume now that $v(b-b^{\prime})\in\mathbb{Z}$ for all distinct
$b,b^{\prime}$ in $X_{t_{0}}$. By saturation there exists $m\in\mathbb{Z}$
such that $v(b-b^{\prime})\geq m$ for all distinct $b,b^{\prime}$ in
$X_{t_{0}}$. Thus, the set

$T_{1}=\\{t\in T:\mbox{for all distinct }b,b^{\prime}\in
X_{t},\,\,v(b-b^{\prime})\geq m\\}$

contains $t_{0}$ so $\mathrm{dp\text{-}rk}(T_{1})=\mathrm{dp\text{-}rk}(T)$,
and we conclude by (1). ∎

We end this section with a small observation that will be used later on.

######  Lemma 3.10.

  1. (1)

For any definable subgroup $H\subseteq(K/{\mathcal{O}})^{n}$ of full dp-rank,
$\mathrm{Tor}(H)=(\mathbb{F}/\mathcal{O}_{\mathbb{F}})^{n}$.

  2. (2)

Every definable subgroup $H\subseteq(K/{\mathcal{O}})^{r}$ has non-trivial
torsion.

###### Proof.

(1) The fact that every torsion element of $H$ is included in
$(\mathbb{F}/\mathcal{O}_{\mathbb{F}})^{n}$ is Fact 3.1(3). For the other
direction, assume that $H$ is $A$-definable and let $b\in H$ with
$\mathrm{dp\text{-}rk}(b/A)=1$. By Lemma 3.6, there exists some
$\gamma<\mathbb{Z}$ with $B_{>\gamma}(b)\subseteq H$; but then
$B_{>\gamma}(0)=B_{>\gamma}(b)-b\subseteq H$ as well. Now obviously,
$\mathbb{F}/\mathcal{O}_{\mathbb{F}}\subseteq B_{>\gamma}(0)\subseteq H$.

(2) Let $H\subseteq(K/{\mathcal{O}})^{r}$ be a definable subgroup, which we
may assume to be infinite, and let $n=\mathrm{dp\text{-}rk}(H)$. Since by
Lemma 3.7 the dp-rank is given by the $\operatorname{acl}$-dimension, there
exists a finite-to-one coordinate projection $\tau:H\to(K/{\mathcal{O}})^{n}$,
with $\mathrm{dp\text{-}rk}(\tau(H))=n$. By (1) above, $\tau(H)$ has non-
trivial torsion, and since $\ker(\tau)$ is a finite group $H$ has non-trivial
torsion as well. ∎

###  3.2. Definable functions in $K/{\mathcal{O}}$

We show that definable functions in $K/{\mathcal{O}}$ are locally affine at
generic points, with respect to the additive structure. With the tools
developed in the previous subsection, the proof is quite similar to the one in
[11, §6.2]. All results remain true in
$\mathbb{Q}_{p}^{an}:=({\mathbb{Q}}_{p},{\mathcal{L}}_{an})$, the expansion of
${\mathbb{Q}}_{p}$ by all convergent power series $f:{\mathcal{O}}^{n}\to K$,
since those are $P$-minimal, 1-h-minimal with definable Skolem functions [41]
and [8, Theorem 3.6], respectively.

######  Definition 3.11.

A partial type $P\subseteq K^{n}$ over a set of parameters $A$ is _large_ if
$\mathrm{dp\text{-}rk}(\pi_{*}P)=n$, where $\pi:K\to K/{\mathcal{O}}$ is the
quotient map, and $\pi_{*}P$ the push forward of $P$.

By Lemma 3.6 (and Corollary 3.7), a partial type $P$ is large if and only if
$(\pi_{*}P)({\mathcal{K}})$ has non-empty interior, if and only if $P$
contains a large ball. Note that to a ball in $K$ we can associate two notions
of largeness: as a partial type and as a ball. The above shows that the two
notions coincide, i.e., that a ball $B$ is large as a partial type if and only
if $r(B)<\mathbb{Z}$.

For the following, recall that by [11, Example 6.2] $K/{\mathcal{O}}$ is
opaque,333 Opacity is needed only in order to invoke [11, Lemma 6.4], so we
omit the definition. See [11, §6.1] for details. thus by [11, Lemma 6.4] for
every complete type $q$ concentrated om $(K/{\mathcal{O}})^{n}$ with
$\mathrm{dp\text{-}rk}(q)=n$ there exists a unique complete type $p$
concentrated on $K^{n}$ with $\pi_{*}p=q$; moreover
$\mathrm{dp\text{-}rk}(p)=n$ as well.

######  Lemma 3.12.

Let ${\mathcal{K}}=(K,v)$ be a $p$-adically closed field and $A$ an arbitrary
set of parameters.

  1. (1)

A partial type $P\vdash K^{n}$ over $A$ is large if and only if there is a
completion $p$ of $P$ (over $A$) that is large.

  2. (2)

If a partial type $P\vdash K^{n}$ is large, then $\mathrm{dp\text{-}rk}(P)=n$.

  3. (3)

If $\operatorname{tp}(a_{1},\dots,a_{n}/A)$ is large, then so is
$\operatorname{tp}(a_{1}/Aa_{2},\dots,a_{n})$.

  4. (4)

Let $B\subseteq K^{n}$ be a large open ball. Then $B+{\mathcal{O}}^{n}=B$.

###### Proof.

(1) We only need to show that if $P$ is large, it has a large completion.
Because $\mathrm{dp\text{-}rk}(\pi_{*}P)=n$ it has a completion $q$ of full
dp-rank, and since $K/{\mathcal{O}}$ is opaque, by [11, Lemma 6.4] there is a
unique complete type $p$ such that $\pi_{*}(p)=q$. Since dp-rank can only
decrease under definable maps, necessarily $\mathrm{dp\text{-}rk}(p)=n$.

The rest is as in [11, Lemma 6.9]. ∎

We need some results from the theory of 1-h-minimal fields in mixed
characteristic, as developed in [7]. Since, as in [11], we apply
1-h-minimality as a black box, we will not dwell on it beyond the definition
already given (Definitions 2.3 and Definition 2.1). It suffices for us that
$p$-adically closed fields, as well as the above-mentioned sub analytic
expansions, are P-minimal and 1-h-minimal ([6, Lemma 6.2.7]).

The main result concerning 1-h-minimality (Definition 2.3), that we need is:

######  Fact 3.13.

[7, Corollary 3.1.3] Let $T$ be a $1$-h-minimal theory, $K\models T$ and
$f:K\to K$ an $A$-definable function. Then there exists an $A$-definable
finite set $C$, and $m\in{\mathbb{N}}$ such that for any ball $B$ $m$-next to
$C$, $f$ is differentiable on $B$ and $v(f^{\prime})$ is constant on $B$.
Moreover:

  1. (1)

For all $x,x^{\prime}\in B$,

$v(f(x)-f(x^{\prime}))=v(f^{\prime}(x))+v(x-x^{\prime})$

  2. (2)

If $f^{\prime}\neq 0$ on $B$ then for any open ball $B^{\prime}\subseteq B$
the image $f(B^{\prime})$ is an open ball of radius
$v(f^{\prime})+r(B^{\prime})$ where $r(B^{\prime})$ is the valuative radius of
$B^{\prime}$.

To apply this result, we need the following observation:

######  Lemma 3.14.

Let $p\in S_{n}(A)$ be a large type in $K^{n}$, $C\subseteq K^{n}$ a finite
$A$-definable set. Then for any fixed $m\in{\mathbb{N}}$ and any $b\models p$
there exists a large ball $B$, $b\in B\subseteq p({\mathcal{K}})$, and a large
ball $B^{\prime}\supseteq B$ that is $m$-next to $C$.

###### Proof.

Fix $b\models p$, $m\in\mathbb{N}$ and a finite $A$-definable set $C$; by
Lemma 3.6, there exists an open large ball $B_{>r_{0}}(b)\subseteq
p({\mathcal{K}})$. Since $C$ is $A$-definable, $p({\mathcal{K}})\cap
C=\emptyset$ so also $B_{>r_{0}+m}(b)\cap C=\emptyset$.

Now, if $r=\max\\{v(b-c)+m:c\in C\\}$ then $r<r_{0}+m$, and by Remark 2.2, the
ball $B_{>r}(b)$ is $m$-next-to $C$ and contains $B_{r_{0}+m}(b)\subseteq
p({\mathcal{K}})$. ∎

Recall the following from [11].

######  Definition 3.15.

A (partial) function $f:K^{n}\to K$ _descends_ to $K/{\mathcal{O}}$ if
$\operatorname{dom}(f)+{\mathcal{O}}^{n}=\operatorname{dom}(f)$ and for every
$a,b\in\operatorname{dom}(f)$, if $a-b\in{\mathcal{O}}^{n}$, then
$f(a)-f(b)\in{\mathcal{O}}$. The function $f$ descends to $K/{\mathcal{O}}$ on
some (partial) type $P\vdash\operatorname{dom}(f)$ if $f\restriction P$
descends to $K/{\mathcal{O}}$.

Conversely, a function $F:(K/{\mathcal{O}})^{n}\to K/{\mathcal{O}}$ _lifts_ to
$K$, if it is the image under the natural quotient map of a definable function
$f:K^{n}\to K$ descending to $K/{\mathcal{O}}$.

Recall that a structure ${\mathcal{M}}$ has _definable Skolem functions_ (for
the sort $S$) if for any definable formula $\varphi(x,z)$ with $x$ ranging
over $S$, there exists a definable function $f_{\varphi}$ such that $\exists
y\varphi(x,z)\to\varphi(f_{\varphi}(z),z)$. Thus, if ${\mathcal{K}}$ has
definable Skolem functions (for the valued field sort $K$) then every
definable function $f:(K/{\mathcal{O}})^{n}\to K/{\mathcal{O}}$ lifts to $K$.
This is the only application of the existence of definable Skolem functions in
the present section.

The following basic example will play an important role:

######  Example 3.16.

When $a\in{\mathcal{O}}$ the linear function $x\mapsto a\cdot x$ descends to
an endomorphism $\lambda_{a}:(K/{\mathcal{O}},+)\to(K/{\mathcal{O}},+)$. If
$a\in\textbf{m}$, then $\lambda_{a}$ has an infinite kernel.

The next technical lemma is an analogue of [11, Lemma 6.14]. The proof is
similar with minor adaptations. For $f:K^{n}\to K$ we let $f_{x_{i}}$ denote
the partial derivative with respect to $x_{i}$.

######  Lemma 3.17.

Let $p\in S_{n}(A)$ be a large type, $p\vdash\operatorname{dom}(f)$ for some
$A$-definable $f:K^{n}\to K$. Then:

  1. (1)

$f$ is differentiable on $p$.

  2. (2)

If $f$ descends to $K/{\mathcal{O}}$ on $p$ then
$(f_{x_{1}},\dots,f_{x_{n}})(a)\in{\mathcal{O}}^{n}$ for all $a\models p$.

  3. (3)

Assume that $\mathrm{Im}(f)\subseteq{\mathcal{O}}$. Then for every $a\models
p$ there exists a large ball $B\ni a$ contained in $p({\mathcal{K}})$ such
that for all $b\in B$ and $1\leq i\leq n$

$v(f_{x_{i}}(b))+2r(B))>0.$

In particular $f_{x_{i}}(a)\in\mathfrak{m}$ for all $a\models p$.

###### Proof.

(1) By [7, Proposition 3.1.1], every definable function is generically
differentiable and $p$ is a generic type in $K^{n}$, so the result follows.

(2) For simplicity, assume that $A=\emptyset$. Fix
$a=(a_{1},\dots,a_{n})\models p$. We show that $f_{x_{1}}(a)\in{\mathcal{O}}$.
Set $g(t)=f(t,a_{2},\dots,a_{n})$. By Fact 3.13 there is a finite
$(a_{2},\dots,a_{n})$-definable set, $C$, and a natural number $m$ such that
$g^{\prime}$ is constant on any ball $B$ $m$-next to $C$. Since
$p_{1}:=\operatorname{tp}(a_{1}/a_{2},\dots,a_{n})$ is large, it follows from
Lemma 3.14 that there is a large ball $B\subseteq p_{1}({\mathcal{K}})$
containing $a_{1}$ where $g^{\prime}$ is constant. By Lemma 3.12, $B+1=B$ and
therefore by Fact 3.13 we have:

$v(g(a_{1}+1)-g(a_{1}))=v(g^{\prime}(a_{1}))=v(f_{x_{1}}(a)).$

Since $f$ descends to $K/{\mathcal{O}}$ on $p$ we get that
$v(f(a_{1}+1,a_{2},\dots,a_{n})-f(a))\geq 0$, so that $v(f_{x_{1}}(a))\geq 0$,
as required.

(3) Let $B_{0}\subseteq p({\mathcal{K}})$ be a large ball containing $a$,
$r_{0}:=r(B_{0})$. By $\aleph_{1}$-saturation of ${\mathcal{K}}$ there exists
$r_{0}<r<\mathbb{Z}$ such that $r_{0}-r<\mathbb{Z}$. For $B:=B_{>r}(a)$, let
$b\in B$; we claim that $v(f_{x_{i}}(b))+r\geq 0$.

To simplify the notation, we show it for $i=1$. Let
$g(t):=f(t,b_{2},\dots,b_{n})$. Note that since $b\in B_{>r_{0}}(a)$, for all
$r^{\prime}>r_{0}$ we have that $B_{>r^{\prime}}(b_{1})\subseteq
p_{1}({\mathcal{K}})$ where
$p_{1}:=\operatorname{tp}(b_{1}/b_{2},\dots,b_{n})$. In particular, if $C$ is
the $(b_{2},\dots,b_{n})$-definable finite set provided by Fact 3.13 (applied
to $g$) and $m\in{\mathbb{N}}$ is the corresponding natural number, then
$B_{>r+m}(b_{1})\subseteq p_{1}({\mathcal{K}})$ and therefore
$v(b_{1}-c)<r_{0}+m$ for all $c\in C$. In particular, $B_{>r}(b_{1})$ is
contained in a ball $m$-next to $C$. So $v(g^{\prime})$ is constant on
$B_{>r}(b_{1})$. If $g^{\prime}(t)\equiv 0$, the claim holds trivially.
Otherwise, by Fact 3.13 $g(B_{>r}(b_{1}))$ is an open ball of radius
$v(g^{\prime}(b_{1}))+r$. Since $g(B_{>r}(b_{1}))\subseteq{\mathcal{O}}$, the
claim follows.

Replacing $r$ with $r-1$, if needed, we may assume that $r$ is even. Applying
the claim to $r/2$ the result follows. ∎

######  Lemma 3.18.

Let $f:K\to K$ be an $A$-definable partial function and
$p\vdash\operatorname{dom}(f)$ a complete large type over $A$. If $f$ descends
to $K/{\mathcal{O}}$ on $p$ then for every $a\models p$ there is a large ball
$B$, $a\in B\subseteq p({\mathcal{K}})$, such that for all $x\in B$.

$f(x)-f(a)-f^{\prime}(a)(x-a)\in\textbf{m}.$

###### Proof.

By Lemma 3.17(2), $f^{\prime}(c)\in{\mathcal{O}}$ for all $c\models p$. We may
thus assume that $f^{\prime}(c)\in{\mathcal{O}}$ for all
$c\in\operatorname{dom}(f)$.

For every finite $A$-definable set $C\subseteq K$, $p({\mathcal{K}})\cap
C=\emptyset$. Let $a\models p$ and $B_{0}\subseteq p({\mathcal{K}})$ be a
large ball containing $a$. Fix $r<\mathbb{Z}$ such that
$r(B_{0})-r<\mathbb{Z}$. We let $B=B_{>r}(a)$, then for any $m\in{\mathbb{N}}$
we see (Remark 2.2) that $B$ is contained in a ball $m$-next to $C$. By
Taylor’s theorem [7, Theorem 3.1.2], for every $x\in B$.

(1)
$v(f(x)-f(a)-f^{\prime}(a)(x-a))=v(\frac{1}{2}f^{\prime\prime}(a)(x-a)^{2})$

As $f^{\prime}(B)\subseteq{\mathcal{O}}$, Lemma 3.17(3), applied to
$f^{\prime}$, gives a large open ball $B^{\prime}$, $a\in B^{\prime}\subseteq
p({\mathcal{K}})$, such that for all $b\in B^{\prime}$,

$v(f^{\prime\prime}(b))+2r(B^{\prime})>0.$

Thus for $B_{1}=B\cap B^{\prime}$, $v(f^{\prime\prime}(a))+2r(B_{1})>0$ and
for every $x\in B_{1}$, $v((x-a)^{2})>2r(B_{1})$. Since all of the above
remains true if we replace $r$ with $r+1$ we can ignore the contribution of
$v(\frac{1}{2})$. Thus, it follows from the above equation that
$v(f(x)-f(a)-f^{\prime}(a)(x-a))>0$, as required. ∎

To conveniently apply the result of the last lemma, we make the following
definition:

######  Definition 3.19.

A definable function $\lambda:K/{\mathcal{O}}\to K/{\mathcal{O}}$ is a scalar-
endomorphism of $K/{\mathcal{O}}$ if there exists $a\in{\mathcal{O}}$ such
that $\lambda(x+{\mathcal{O}})=\lambda_{a}(x+{\mathcal{O}}):=\pi(ax)$ where
$\pi:K\to K/{\mathcal{O}}$ is the quotient map. More generally,
$\lambda:(K/{\mathcal{O}})^{n}\to(K/{\mathcal{O}})$ is a scalar endomorphism
if $\lambda(x_{1},\dots,x_{n})=\sum\lambda_{i}(x_{i})$ where $\lambda_{i}$ are
scalar-endomorphisms in one variable.

We now turn to proving that definable functions on $(K/{\mathcal{O}})^{n}$ are
generically given by translates of scalar endomorphisms. We start by observing
that the germs (at 0) of definable scalar-endomorphism are
$\emptyset$-definable (recall that
$\mathbb{F}\subseteq\operatorname{dcl}(\emptyset)$):

######  Lemma 3.20.

For any scalar endomorphism $\lambda:K/{\mathcal{O}}\to K/{\mathcal{O}}$,
there exists $r\in{\mathcal{O}}_{\mathbb{F}}$ such that
$\lambda(x)=\lambda_{r}(x)$ for all $x$ in some open ball around $0$.

###### Proof.

By assumption $\lambda=\lambda_{a}$ for some $a\in{\mathcal{O}}$. By Fact 3.1,
there exists $a^{\prime}\in{\mathcal{O}}_{\mathbb{F}}$ such that
$v(a-a^{\prime})>\mathbb{Z}$ thus, $\lambda_{a-a^{\prime}}(b)=0$ for any $b\in
K/{\mathcal{O}}$ with $v(b)\in\mathbb{Z}$. By compactness, $\lambda$ and
$\lambda_{a^{\prime}}$ agree on some open ball around $0$. ∎

The proof of [11, Proposition 6.16] (for unary functions) goes through
unaltered using Lemmas 3.17 and 3.18 giving the following.

######  Proposition 3.21.

Let $f:K/{\mathcal{O}}\to K/{\mathcal{O}}$ be an $A$-definable partial
function with $\operatorname{dom}(f)$ open in the ball topology. If $f$ lifts
to $K$ then for every $a\in\operatorname{dom}(f)$ with
$\mathrm{dp\text{-}rk}(a/A)=1$ there exist a ball
$U\subseteq\operatorname{dom}(f)$ with $a\in U$ and a scalar endomorphism
$L:(K/{\mathcal{O}})\to(K/{\mathcal{O}})$ such that $f(x)=L(x-a)+f(a)$ for all
$x\in U$.

######  Corollary 3.22.

Let $f:K/{\mathcal{O}}\to K/{\mathcal{O}}$ be a partial $A$-definable
function, $p\in S_{1}(A)$ a non-algebraic type concentrated on
$\operatorname{dom}(f)$. Then there is $t\in{\mathcal{O}}_{\mathbb{F}}$ such
that $f(x)-\lambda_{t}(x)$ is locally constant on $p$, with respect to the
ball topology.

###### Proof.

Since ${\mathcal{K}}$ has definable Skolem functions, $f$ lifts to $K$ at $p$.
By Proposition 3.21, if $a\models p$ then there is some scalar endomorphism
$\lambda$ (that may depend on $a$) such that $f(x)=f(a)+\lambda(x-a)$ for all
$x$ in some ball $B\subseteq p({\mathcal{K}})$. By Lemma 3.20, we may assume
that $\lambda=\lambda_{t}$ for some $t\in{\mathcal{O}}_{\mathbb{F}}$. Hence,
$f(x)=\lambda_{t}(x)+d$ on some large sub-ball of $B$, for some $d\in
K/{\mathcal{O}}$ depending on $a$. As $t\in{\mathcal{O}}_{\mathbb{F}}$ and
$\mathbb{F}\subseteq\operatorname{dcl}(\emptyset)$, for every $\alpha\models
p$, $f(x)-\lambda_{t}(a)$ is locally constant at $\alpha$. ∎

Summing up all of the above, we get:

######  Proposition 3.23.

Let $f:(K/O)^{n}\to K/{\mathcal{O}}$ be a partial $A$-definable function,
$p\in S_{n}(A)$ concentrated on $\operatorname{dom}(f)$ with
$\mathrm{dp\text{-}rk}(p)=n$. Then there exists a scalar-endomorphism
$\lambda:(K/{\mathcal{O}})^{n}\to(K/O)$ definable over
${\mathcal{O}}_{\mathbb{F}}$ such that $f-\lambda$ is locally constant on
$p({\mathcal{K}})$.

###### Proof.

Without loss of generality $A=\emptyset$. Let $a=(a_{1},\dots,a_{n})\models p$
and consider the function $g_{1}(t):=f(t,a_{2},\dots,a_{n})$. By what we have
shown, there is $r_{1}\in{\mathcal{O}}_{\mathbb{F}}$ such that
$g_{1}(t)-\lambda_{r_{1}}$ is locally constant on
$\operatorname{tp}(a_{1}/a_{2},\dots,a_{n})$. Similarly, for all $1\leq i\leq
n$ we can find $\lambda_{r_{i}}$ such that $g_{i}(t)=f(a_{1},\dots
a_{i-1},t,a_{i+1}\dots,a_{n})$ is locally constant on
$\operatorname{tp}(a_{i}/a_{1},\dots,\hat{a_{i}},\dots.,a_{n})$. The result
follows. ∎

Our next goal is to extend the above result to types not necessarily of full
dp-rank in $(K/{\mathcal{O}})^{r}$. To do so, we introduce a notion that will
be of importance in the sequel as well:

######  Definition 3.24.

A set $S\subseteq(K/{\mathcal{O}})^{n}$ has _minimal fibres_ if
$\mathrm{dp\text{-}rk}(S)=k$ and there exists a coordinate projection
$\pi:S\to(K/{\mathcal{O}})^{k}$ and some $m\in\mathbb{N}$ such that for every
$y\in(K/{\mathcal{O}})^{k}$, $|\pi^{-1}(y)|\leq m$ and there is no definable
(possibly over additional parameters) $S_{1}\subseteq S$ such that
$\mathrm{dp\text{-}rk}(S_{1})=\mathrm{dp\text{-}rk}(S)$ and for every
$y\in(K/{\mathcal{O}})^{k}$, $|f^{-1}(y)\cap X_{1}|<m$.

######  Remark 3.25.

It is easy to see that a set $S\subseteq(K/{\mathcal{O}})^{n}$ has minimal
fibres if and only if $\mathrm{dp\text{-}rk}(S)=k$ and there exists a
coordinate projection $\pi:S\to(K/{\mathcal{O}})^{k}$ with finite fibres such
that for any definable $S^{\prime}\subseteq S$,
$\mathrm{dp\text{-}rk}(\pi^{-1}(\pi(S^{\prime})))\setminus
S^{\prime})<\mathrm{dp\text{-}rk}(S)$.

Notice that if some $S\subseteq(K/{\mathcal{O}})^{n}$ of rank $k$ projects
finite-to-one into $(K/{\mathcal{O}})^{k}$ then there exists
$S^{\prime}\subseteq S$,
$\mathrm{dp\text{-}rk}(S^{\prime})=\mathrm{dp\text{-}rk}(S)$ (possibly defined
over additional parameters) such that $S^{\prime}$ has minimal fibres with
respect to the same projection. Indeed, we just choose $S^{\prime}\subseteq S$
with the property that the fibres of $\pi\restriction S^{\prime}$ are of
minimal size among all definable subsets of $S$ of maximal dp-rank. The
following is a local version of this observation:

######  Lemma 3.26.

Let $X\subseteq(K/{\mathcal{O}})^{r}$ be an $A$-definable subset. For any
$a\in X$ with $\mathrm{dp\text{-}rk}(a/A)=\mathrm{dp\text{-}rk}(X)$ there
exists $X^{\prime}\subseteq X$ definable over some set $C\supseteq A$ such
that $a\in X^{\prime}$,
$\mathrm{dp\text{-}rk}(a/C)=\mathrm{dp\text{-}rk}(a/A)$ and $X^{\prime}$ has
minimal fibres.

###### Proof.

Let $p=\operatorname{tp}(a/A)$ and set
$\mathrm{dp\text{-}rk}(a/A)=\dim_{\operatorname{acl}}(a/A)=\mathrm{dp\text{-}rk}(X)=n$.
Let $\tau:(K/{\mathcal{O}})^{r}\to(K/{\mathcal{O}})^{n}$ be a coordinate
projection such that
$\dim_{\operatorname{acl}}(\tau(a)/A)=\dim_{\operatorname{acl}}(a/A)$. Letting
$q=\tau_{*}p$ it follows that $\mathrm{dp\text{-}rk}(q)=n$.

By Proposition 3.8, there exists a large ball $B\subseteq q({\mathcal{K}})$
containing $\tau(a)$; so without loss of generality $X=\tau^{-1}(B)\cap
p({\mathcal{K}})$, as it is a definable set of dp-rank $n$. Note that
$\tau\restriction X$ has finite fibres and as $\mathrm{dp\text{-}rk}(X)=n$,
there exists $X^{\prime}\subseteq X$ satisfying that
$\mathrm{dp\text{-}rk}(X^{\prime})=\mathrm{dp\text{-}rk}(X)$ with minimal
fibres for $\tau$; assume that $X^{\prime}$ is definable over some parameter
set $C^{\prime}$.

Let $a^{\prime}\in X^{\prime}$ be an element with
$\mathrm{dp\text{-}rk}(a^{\prime}/C^{\prime})=\mathrm{dp\text{-}rk}(X^{\prime})$.
Since $a^{\prime}\models p$ there exists an automorphism $\sigma$ over $A$
mapping $a^{\prime}$ to $a$. Then $\sigma(X^{\prime})\subseteq X$ contains $a$
and has minimal fibres for $\tau$ and is definable over $C:=\sigma(C)$. ∎

For any function $f:X\to Y$, we write $[x]_{f}=f^{-1}(f(x))$. For the next
result, recall that we view $\mathbb{F}/{\mathcal{O}}_{\mathbb{F}}$ as
embedded inside $K/{\mathcal{O}}$.

######  Lemma 3.27.

Let $X\subseteq(K/{\mathcal{O}})^{r}$ be an $A$-definable set with minimal
fibres. Assume that $\mathrm{dp\text{-}rk}(X)=n$ and let $a\in X$ be such that
$\mathrm{dp\text{-}rk}(a/A)=n$. Then there exists an $A$-definable subset
$X_{1}\subseteq X$ with $a\in X_{1}$, a finite subgroup
$G_{a}\subseteq(\mathbb{F}/{\mathcal{O}}_{\mathbb{F}})^{r}$ and a coordinate
projection $\tau:(K/{\mathcal{O}})^{r}\to(K/{\mathcal{O}})^{n}$ such that for
every $b\in X_{1}$, $[b]_{\tau}=G_{a}+b$.

###### Proof.

Let $\tau:X\to(K/{\mathcal{O}})^{n}$ be a coordinate projection witnessing
that $X$ has minimal fibres. For $b\in X$, set
$G_{b}=\\{x-b:x\in[b]_{\tau}\\}$.

Let $a\in X$, $\mathrm{dp\text{-}rk}(a/A)=n$. We claim that for all
$b\in[a]_{\tau}$, $v(b-a)\in\mathbb{Z}$. Indeed, assume towards a
contradiction that there is $b\in[a]_{\tau}$ with $v(b-a)<\mathbb{Z}$. By
Proposition 3.8 we can find a ball $U$ containing $a$, defined over some
parameters $C$ such that $\mathrm{dp\text{-}rk}(b/C)=n$ and $b\notin U$. Thus
$|[a]_{\tau}\cap U|<|[a]_{\tau}|$. Now the set $S^{\prime}=\\{x\in
X:|[x]_{\tau}\cap U|<|[a]_{\tau}]|\\}$ contradicts the assumption that $X$ has
minimal fibres.

Therefore, $v(G_{a})\subseteq\mathbb{Z}$ and by Fact 3.1(2),
$G_{a}\subseteq\mathbb{F}/{\mathcal{O}}_{\mathbb{F}}$ and so it is
$\emptyset$-definable. Thus, replacing $X$ by a subset $X_{1}$ of the same dp-
rank, we may assume that $G_{a}=G_{b}$ for all $b\in X$. In particular,
$G_{a}=G_{b}$ for any $b\in[a]_{\tau}$. It easily follows that $G_{a}$ is a
subgroup and $[a]_{\tau}$ its coset. ∎

######  Definition 3.28.

For $a\in(K/{\mathcal{O}})^{r}$, let

$Z(a):=\\{x\in(K/{\mathcal{O}})^{r}:v(x-a)\in\mathbb{Z}\cup\\{\infty\\}\\}.$

For $X\subseteq(K/{\mathcal{O}})^{r}$ and $a\in X$ we let $Z_{X}(a):=Z(a)\cap
X$.

Notice that $Z_{X}(a)$ is a $\bigvee$-definable set. The next lemma shows that
$Z_{X}(a)$ does not depend on the choice of $X$:

######  Lemma 3.29.

Let $X_{1}\subseteq X\subseteq(K/{\mathcal{O}})^{r}$ be $A$-definable subsets
with minimal fibres and $a\in X_{1}$ with
$\mathrm{dp\text{-}rk}(a/A)=n=\mathrm{dp\text{-}rk}(X)$. Then
$Z_{X}(a)=Z_{X_{1}}(a)$.

###### Proof.

Let $\tau:X\to(K/{\mathcal{O}})^{n}$ be a coordinate projection witnessing
minimal fibres. It follows that $[a]_{\tau}\cap X_{1}=[a]_{\tau}$. By
replacing $X_{1}$ with a subset of minimal fibres, still of full dp-rank we
may assume that for all $x\in X_{1}$ we have $[x]_{\tau}\cap
X_{1}=[x]_{\tau}$.

By Lemma 3.6(1), we have that $\tau(a)$ is in the interior of $\tau(X_{1})$.
Thus, there exists a ball $V\subseteq(K/{\mathcal{O}})^{n}$, with $\tau(a)\in
V\subseteq\tau(X_{1})$. By our choice of $X_{1}$, we have
$\tau^{-1}(V)\subseteq X_{1}$; the result follows. ∎

The next lemma is the main result of this section. It states that definable
sets are locally affine at all generic points:

######  Lemma 3.30.

Let $X\subseteq(K/{\mathcal{O}})^{r}$ be $A$-definable with minimal fibres,
$\mathrm{dp\text{-}rk}(X)=n$. Let $a\in X$ be such that
$\mathrm{dp\text{-}rk}(a/A)=n$. Then there exists $C\supseteq A$, with
$\mathrm{dp\text{-}rk}(a/C)=n$, a $C$-definable ball
$U\subseteq(K/{\mathcal{O}})^{r}$, $0\in U$, (hence $U$ is a subgroup), a
$\emptyset$-definable subgroup $H\subseteq(K/{\mathcal{O}})^{r}$, and a
$C$-definable $X_{1}\subseteq X$ containing $a$, such that $X_{1}=a+(H\cap
U)$. In particular, $Z_{X_{1}}(a)=a+H(\mathbb{F}/{\mathcal{O}}_{\mathbb{F}})$.

###### Proof.

By Lemma 3.27, after reducing $X$ and permuting the coordinates, we may assume
that the fibres of the projection $\tau$ on the first $n$-coordinates are all
cosets of the same finite subgroup
$G\subseteq(\mathbb{F}/{\mathcal{O}}_{\mathbb{F}})^{r}$. Since, by definition,
$\tau(G)=0$ we identify $G$ with a subgroup of
$(\mathbb{F}/{\mathcal{O}}_{\mathbb{F}})^{r-n}$. We also let
$\sigma:(K/{\mathcal{O}})^{r}\to(K/{\mathcal{O}})^{r-n}$ be the projection
onto the last $r-n$ coordinates.

Let $k=|G|$, $(x,x^{\prime})\in X$ such that $x\in\tau(X)$, denote

$f(x):=\sigma\left(\sum_{b\in\tau^{-1}(x)}b\right)\in(K/{\mathcal{O}})^{r-n},$

and note that $f(x)=kx^{\prime}$, as the sum over all elements of $G$ is $0$.
Let $p=\operatorname{tp}(\tau(a)/A)\vdash(K/{\mathcal{O}})^{n}$. Since in
${\mathcal{K}}$ every definable function $f:(K/{\mathcal{O}})^{n}\to
K/{\mathcal{O}}$ lifts to $K$, we may apply Proposition 3.23 to find some ball
$V\subseteq p({\mathcal{K}})$ with $\tau(a)\in V$, such that $f$ is a
translate of a scalar-endomorphism, on $V$. By Proposition 3.8(1), we may
assume that $V$ is defined over some $C\supseteq A$ with
$\mathrm{dp\text{-}rk}(\tau(a)/C)=n$. Let $Y:=\mathrm{Graph}(f\restriction
V)$. The graph of $f$ is a coset of a $\emptyset$-definable subgroup
$H_{1}\subseteq(K/{\mathcal{O}})^{r}$, and thus

$Y=(b+H_{1})\cap(V\times(K/{\mathcal{O}})^{r-n}),$

for some $b\in(K/{\mathcal{O}})^{r}$. Let $X_{1}:=\\{(x,y)\in X:(x,ky)\in
Y\\}$. Note that $X_{1}$ is definable over $C$ and contains $a$, and hence
$\mathrm{dp\text{-}rk}(X_{1})=n$.

Because the map $(x,y)\mapsto(x,ky)$ is a $\emptyset$-definable endomorphism
of $(K/{\mathcal{O}})^{n}\times(K/{\mathcal{O}})^{n-r}$, the pre-image of
$b+H_{1}$ under the map $(x,y)\to(x,ky)$ is of the form $a+H$ for some
$\emptyset$-definable subgroup $H$, and the pre-image of $V$ is a ball
$V^{\prime}\ni a$ in $(K/{\mathcal{O}})^{r}$, which we may write
$V^{\prime}=a+U$, for some ball $U\ni 0.$ Thus, $X_{1}=a+H\cap a+U=a+(H\cap
U)$. By Fact 3.1(2), we now get that
$Z_{X_{1}}(a)=a+H(\mathbb{F}/{\mathcal{O}}_{\mathbb{F}})$. ∎

######  Remark 3.31.

In the statement of the previous lemma, we cannot require that the ball $U$ is
$\emptyset$-definable. For example, if $X$ itself was a ball around $0$, which
is smaller than all $\emptyset$-definable balls, then we cannot find a
$\emptyset$-definable such $U$.

######  Corollary 3.32.

If $f:(K/{\mathcal{O}})^{r}\to K/{\mathcal{O}}$ is an $A$-definable partial
function and $a\in\operatorname{dom}(f)$ is such that
$\mathrm{dp\text{-}rk}(a/A)=\mathrm{dp\text{-}rk}(\operatorname{dom}(f))$ then
there exists $C\supseteq A$, a $C$-definable coset $X:=H+d$ and a
$\emptyset$-definable scalar endomorphism $\ell:(K/{\mathcal{O}})^{r}\to
K/{\mathcal{O}}$ such that

  1. (1)

$\mathrm{dp\text{-}rk}(a/C)=\mathrm{dp\text{-}rk}(a/A)$.

  2. (2)

$a\in X$.

  3. (3)

$f\restriction X=\ell(x-a)+\ell(a)$.

###### Proof.

By Lemma 3.26 we may assume that the graph of $f$ has minimal fibres. We can
now apply Lemma 3.30 to the graph of $f$. ∎

The next example shows that a group almost locally strongly internal to a
distinguished sort $D$ need not be strongly internal to it. This distinction,
that apparently can not be avoided, will have to be taken into account in our
work here:

######  Example 3.33.

Let $\mathbb{F}$ be a residual quadratic extension of $\mathbb{Q}_{p}$ and let
${\mathcal{K}}\succ\mathbb{F}$ be a sufficiently saturated extension. Let
$C_{p}\leq K/{\mathcal{O}}$ be a cyclic subgroup of order $p$, and
$G:=(K/{\mathcal{O}})/C_{p}$. Then $G$ is locally almost strongly internal to
$K/{\mathcal{O}}$ but not locally strongly internal to $K/{\mathcal{O}}$.

On the other hand, letting $H=\\{x\in G:px=0\\}$ the group $G/H$ is strongly
internal to $K/{\mathcal{O}}$.

###### Proof.

Since the quotient map $\pi:K/{\mathcal{O}}\to G$ is finite-to-one it follows
directly from Lemma 3.9(2) that $G$ is almost strongly internal to
$K/{\mathcal{O}}$.

We now verify that $G$ is not locally strongly internal to $K/{\mathcal{O}}$.
Assume towards a contradiction that there exists a definable injection
$f:X\to(K/{\mathcal{O}})^{r}$ with $X$ infinite. Let $V=\pi^{-1}(X)$ and
$\widehat{f}:V\to(K/{\mathcal{O}})^{r}$ the (clearly, definable) lifting of
$f$ to $K/{\mathcal{O}}$. As $\mathrm{dp\text{-}rk}(V)=1$, we may assume –
shrinking $V$ if needed – that $V$ is an open ball. Assume everything is
definable over $\emptyset$. Let $a\in V$ with $\mathrm{dp\text{-}rk}(a)=1$.
Shrinking $V$ further, we may apply Proposition 3.21, to deduce that
$\widehat{f}:=(\widehat{f}_{1},\dots,\widehat{f}_{r})$ where each
$\widehat{f}_{i}$ is of the form $L_{i}(x-a)+\widehat{f}_{i}(a)$ on $V$, for
some scalar endomorphism, $L_{i}$. By Fact 3.1(3) and Lemma 3.10, $V+C_{p}=V$.
Because $\widehat{f}$ factors through $C_{p}$ on $V$, it follows that the
scalar endomorphisms $L_{i}$ are all invariant under $C_{p}$, and therefore
$C_{p}\subseteq\ker(L_{i})$ for $i=1,\ldots,r$.

Since $\mathbb{F}$ is a residual extension of $\mathbb{Q}_{p}$, the subgroup
$H=\\{x\in K/{\mathcal{O}}:v(x)\geq-1\\}$, where $1\in\Gamma$ is the minimal
positive element, is readily seen to contain $C_{p}$, and
$|H|=|\textbf{k}_{K}|=p^{2}$. It is also easy to verify that for every scalar
endomorphism $\lambda$ of $K/{\mathcal{O}}$ with non-trivial kernel, we have
$H\subseteq\ker(\lambda)$, so $C_{p}\subsetneq H\subseteq\ker(L_{i})$ for
every $L_{i}$.

Thus, the function $\widehat{f}$ is invariant under the group $H\supseteq
C_{p}$, contradicting our assumption that $f$ was injective.

For the final statement, since $\mathbb{F}/{\mathcal{O}}_{\mathbb{F}}$ is
isomorphic to $\mathbb{Z}(p^{\infty})\oplus\mathbb{Z}(p^{\infty})$, $G/H$ is
definably isomorphic to $K/{\mathcal{O}}$ as witnessed by the map $x\mapsto
px$. ∎

######  Remark 3.34.

It can be shown that if $\mathbb{F}$ is a residual quadratic extension of
$\mathbb{Q}_{p}$, as above, and
$H\subseteq\mathbb{F}/{\mathcal{O}}_{\mathbb{F}}$ is a finite subgroup, then
$(\mathbb{F}/{\mathcal{O}}_{\mathbb{F}})/H$ is abstractly isomorphic to
$\mathbb{F}/{\mathcal{O}}_{\mathbb{F}}$. However, the above example shows that
a definable isomorphism need not exist.

In contrast, it can be shown that for every finite subgroup
$H\subseteq\mathbb{Q}_{p}/\mathbb{Z}_{p}$, the group
$(\mathbb{Q}_{p}/\mathbb{Z}_{p})/H$ is definably isomorphic to
$\mathbb{Q}_{p}/\mathbb{Z}_{p}$. Furthermore, this remains true in elementary
extensions of $\mathbb{Q}_{p}$.

##  4\. Vicinic sorts and their properties

In the present section we develop an _ad hoc_ axiomatic setting allowing us to
unify the treatment of all unstable distinguished sorts considered in the
sequel. We then apply the technical tools we develop to show that, given a
group $G$ almost strongly internal to such a sort $D$, there exists a finite
normal subgroup $H\trianglelefteq G$ such that $G/H$ is strongly internal to
$D$ and furthermore, it is “a $D$-group” (to be defined below). This suffices
for the construction of an infinitesimal subgroup in $G/H$.

Throughout Section 4 we let ${\mathcal{M}}$ be a multi-sorted,
$|T|^{+}$-saturated structure and $D$ a dp-minimal definable set.

###  4.1. The key geometric properties

To motivate our axiomatic setup, we isolate key geometric properties shared by
SW-uniformities, ${\mathbb{Z}}$-groups and $K/{\mathcal{O}}$, for
${\mathcal{K}}$ a $p$-adically closed field. Those are, in the latter setting,
Corollary 3.7 (asserting that dp-rank is the same as
$\operatorname{acl}(\cdot)$-dimension) and Proposition 3.8 (asserting the
existence of generic neighbourhoods). These two properties constitute our
axiomatisation of vicinic sorts (Section 4.2).

An important property shared by stable groups, groups definable in o-minimal
structures and other tame settings is that the product of two independent
generic elements of the group is itself generic. It turns out that a variant
of this property can be deduced, in the $p$-adically closed case, from Lemma
3.9(2). Statements analogous to Lemma 3.9(2) (Lemma 2.9 for SW-uniformities,
Lemma 4.3 for ${\mathbb{Z}}$-groups) imply the same property of groups in the
remaining cases covered by our setting. This geometric property of groups is
encapsulated in our notion of $D$-groups (Section 4.5)

In all of our settings, the induced structure on $K/{\mathcal{O}}$ does not
satisfy Exchange (there are definable locally constant functions with infinite
image). Nevertheless, the fact below, together with its analogue, Corollary
3.7, for $K/{\mathcal{O}}$ in the $p$-adic case, give dp-rank enough geometric
flavour to get us going in all the unstable distinguished sorts:

######  Fact 4.1.

If $D$ is an SW-uniformity (in some structure) or a stably embedded
${\mathbb{Z}}$-group then
$\dim_{\operatorname{acl}}(a/A)=\mathrm{dp\text{-}rk}(a/A)$ for all $a\in
D^{n}$ and any parameter set $A$.

###### Proof.

For SW-uniformities this follows from [34, Propsition 2.4] noting that by
Simon’s work dp-rank is local in dp-minimal theories, [32, Theorem 0.3]. If
$D$ is a ${\mathbb{Z}}$-group, then this follows from [32, Theorem 0.3] since
$D$ satisfies exchange. ∎

The existence of generic neighbourhoods, originally proved for SW-uniformities
(Fact 2.8), and extended above to $K/{\mathcal{O}}$ in the $p$-adically closed
setting (Proposition 3.8), is crucial for carrying out local analysis at a
generic point. The following may be viewed as the ${\mathbb{Z}}$-group
analogue:

######  Lemma 4.2.

Let $(D,+,<)$ be a ${\mathbb{Z}}$-group. Let $A$ be any set of parameters,
$g\in D^{n}$, $h\in D^{m}$. For any set of parameters $B$ and $B$-definable
$X\subseteq D^{n}$, if $g\in X$ and $\mathrm{dp\text{-}rk}(g/B)=n$ then there
exists $C\supseteq A$ and a $C$-definable set $X_{1}\subseteq X$ such that
$\mathrm{dp\text{-}rk}(X_{1})=n$, $g\in X_{1}$ and
$\mathrm{dp\text{-}rk}(g,h/A)=\mathrm{dp\text{-}rk}(g,h/C)$.

###### Proof.

By [11, Lemma 3.14(2)], we can find a small model $L\supseteq B$ for which
$\mathrm{dp\text{-}rk}(g/L)=n$. By [24, Lemma 3.4], there exists an
$L$-definable subset $X_{0}\subseteq X$ containing $g=(g_{1},\dots,g_{n})$ of
the form $I_{1}\times\dots\times I_{n}$, where for each $i\leq n$, $g_{i}\in
I_{i}=\\{\alpha_{i}\leq x\leq\beta_{i}:x\equiv_{N_{i}}c_{i}\\}$, for some
$N_{i}\in\mathbb{N}$, where both intervals $[\alpha_{i},g_{i}]$ and
$[g_{i},\beta_{i}]$ are infinite and $0\leq c_{i}<N_{i}$.

For simplicity, we prove the result for $n=1$. The general case follows by
induction. We use compactness to find $\alpha_{1}<\alpha_{1}^{\prime}<g_{1}$
such that $\mathrm{dp\text{-}rk}(\alpha_{1}^{\prime}/g_{1},hL)=1$ and
$[\alpha_{1}^{\prime},g_{1}]$ infinite and then
$g_{1}<\beta_{1}^{\prime}<\beta_{1}$ such that
$\mathrm{dp\text{-}rk}(\beta_{1}^{\prime}/g_{1},h,\alpha_{1}^{\prime}L)=1$ and
$[g_{1},\beta_{1}^{\prime}]$ infinite. We have $g_{1}\in
I_{1}^{\prime}=\\{\alpha_{1}\leq
x\leq\beta_{1}^{\prime}:x\equiv_{N_{i}}c_{i}\\}\subseteq I_{1}$ and by
exchange
$\mathrm{dp\text{-}rk}(g_{1},h/\alpha_{1}^{\prime},\beta_{1}^{\prime},L)=\mathrm{dp\text{-}rk}(g_{1},h/L)$.
∎

The next lemma is a rather simple first approximation to more technical
results which follow:

######  Lemma 4.3.

Let ${\mathcal{M}}$ be any structure, and $S$ a stably embedded definable set.
If $S$ has elimination of imaginaries, $X$ is (almost) strongly internal to
$S$ and $g:X\to Y$ is a definable surjection then $Y$ is (almost) strongly
internal to $S$.

###### Proof.

We consider the almost strongly internal case only, as the strongly internal
case is similar and easier. Let $f:X\to S^{n}$ be a finite-to-one definable
function. For any $y\in Y$ consider the definable set
$W_{y}=f(g^{-1}(y))\subseteq S^{n}$. By elimination of imaginaries in $S$,
there exists a definable function $h:Y\to S^{m}$ such that $h(y_{1})=h(y_{2})$
if and only if $W_{y_{1}}=W_{y_{2}}$. It will suffice to show that $h$ is
finite-to-one. So assume towards a contradiction that this is not the case.
I.e., there exists $b\in S$ with $H:=h^{-1}(b)$ infinite. Fix $y_{1}\in H$.
Then for any other $y_{2}\in H$, obviously, $g^{-1}(y_{1})\cap
g^{-1}(y_{2})=\emptyset$ (but $g^{-1}(y_{2})\neq\emptyset$ because $g$ is
onto). Since $f$ is finite to one and $H$ is infinite, for $w\in W_{y_{1}}$
there is $y_{2}\in H$ such that $f^{-1}(w)\cap g^{-1}(y_{2})=\emptyset$,
contradicting the assumption that $W_{y_{1}}=W_{y_{2}}$. ∎

Since Presburger Arithmetic eliminates imaginaries, this gives the analogue of
Lemma 3.9(2) for stably embedded ${\mathbb{Z}}$-groups. For SW-uniformities,
the analogous statement is Lemma 2.9.

###  4.2. Vicinic dp-minimal sets

We now introduce the axiomatic setting we are interested in, but first:

######  Definition 4.4.

Let $X$ be an $A$-definable set of finite dp-rank, $a\in X$ and $B\supseteq A$
a set of parameters.

  1. (1)

The point $a$ is $B$-_generic_ in $X$ (or, generic in $X$ over $B$) if
$\mathrm{dp\text{-}rk}(a/B)=\mathrm{dp\text{-}rk}(X)$.

  2. (2)

For an $A$-generic $a\in X$, a set $U\subseteq X$ is _a $B$-generic vicinity
of $a$ in $X$_ if $a\in U$, $U$ is $B$-definable, and
$\mathrm{dp\text{-}rk}(a/B)=\mathrm{dp\text{-}rk}(X)$ (in particular,
$\mathrm{dp\text{-}rk}(U)=\mathrm{dp\text{-}rk}(X)$).

Note that if $D$ is an SW-uniformity, then a generic vicinity of a generic
point $a$ in a set $X\subseteq D^{k}$ is, in fact, a neighbourhood of $a$ in
the relative topology of $X$. The existence of generic neighbourhoods in SW-
uniformities is given by Fact 2.8.

######  Definition 4.5.

A dp-minimal set $D$ is _vicinic_ if it satisfies the following axioms:

  1. (A1)

$\dim_{\operatorname{acl}}=\mathrm{dp\text{-}rk}$; i.e. for any tuple $a\in
D^{n}$ and set $A$,
$\dim_{\operatorname{acl}}(a/A)=\mathrm{dp\text{-}rk}(a/A)$.

  2. (A2)

For any sets of parameters $A$ and $B$, for every $A$-generic elements $b\in
D^{n}$, $c\in D^{m}$ and any $B$-generic vicinity $X$ of $b$ in $D^{n}$, there
exists $C\supseteq A$ and a $C$-generic vicinity of $b$ in $X$ such that
$\mathrm{dp\text{-}rk}(b,c/A)=\mathrm{dp\text{-}rk}(b,c/C)$.

Note that in Axiom (A2) it is crucial that the parameter set $B$ need not
contain $A$. The topological intuition is that if $b$ is in the interior of a
$B$-definable set $X$ then we can find a smaller neighbourhood of $b$ defined
over a new parameter set $C$, that is generic with respect to all the initial
data.

######  Remark 4.6.

By Axiom (A1), for every parameter set $A$, every $c\in D^{m}$ is inter-
algebraic over $A$ with an $A$-generic $c^{\prime}\in D^{k}$, for
$k=\mathrm{dp\text{-}rk}(c/A)$. Thus, (A2) remains true if we drop the
requirement that $c$ is generic.

Let us note that indeed all the unstable distinguished sorts in our various
settings are vicinic:

######  Fact 4.7.

  1. (1)

Every SW-uniformity is vicinic.

  2. (2)

If ${\mathcal{K}}=(K,+,\cdot,v,\dots)$ is either V-minimal, power-bounded
T-convex, or a $p$-adically closed valued field, then all the distinguished
sorts, except k in the V-minimal setting, are vicinic.

###### Proof.

When $D$ is an SW-uniformity, Axiom (A1) holds by [34, Proposition 2.4] and
Axiom (A2) is Fact 2.8. This proves (1) and (2) follows in all cases except
when $D=K/{\mathcal{O}}$ and $D=\Gamma$ in the $p$-adic setting.

When ${\mathcal{K}}$ is $p$-adically closed, then for $K/{\mathcal{O}}$, see
Corollary 3.7 for (A1), and Proposition 3.8 for (A2). For $\Gamma$, see Fact
4.1, for (A1) and Lemma 4.2, for (A2). ∎

######  Question 4.8.

Does every dp-minimal distal structure satisfy Axiom (A2)? We cannot expect
the converse to hold, since by the above fact, the valued field sort of an
algebraically closed field of equi-characteristic $0$ satisfies Axioms (A2),
but it is not distal.

For the rest of Section 4, we assume that $D$ is a vicinic sort.

We first note an immediate implication of Axiom (A1):

######  Lemma 4.9.

Let $X$ be a definable set $a\in X$ and assume that $X$ is almost strongly
internal to $D$ over $A$. Then for any $B\supseteq A$ we have
$\mathrm{dp\text{-}rk}(a/B)=k$ if and only if there exists $a^{\prime}\in
D^{k}$ such that $\mathrm{dp\text{-}rk}(a^{\prime}/B)=k$,
$a^{\prime}\in\operatorname{dcl}(aA)$ and
$a\in\operatorname{acl}(a^{\prime}B)$.

We now start developing the technical tools needed for the construction of
infinitesimal groups in the setting of vicinic structures. We first want to
show that boxes form vicinity-bases at generic points in the following sense:

######  Lemma 4.10.

For any $A\subseteq{\mathcal{M}}$, $A$-generic $b=(b_{1},\dots,b_{n})\in
D^{n}$ and $c\in D^{m}$, and any $A$-generic vicinity $X$ of $b$ in $D^{n}$,
there exists $C\supseteq A$ and $C$-generic vicinities $I_{i}\ni b_{i}$ in
$D$, for $i=1,\dots,n$ such that $I_{1}\times\dots\times I_{n}\subseteq X$ and
$\mathrm{dp\text{-}rk}(b,c/C)=\mathrm{dp\text{-}rk}(b,c/A)=n+m$.

###### Proof.

We use induction on $n$, where the case $n=1$ is an immediate application of
(A2) to $b\in D$, and $c\in D^{m}$.

We now consider $b=(b_{1},\ldots,b_{n})$ and $X$ an $A$-vicinity of $b$ in
$D^{n}$. Let $b^{\prime}=(b_{1},\ldots,b_{n-1})$, and apply (A2) to $b_{n}\in
X_{b^{\prime}}\subseteq D$.

We then find $C^{\prime}\supseteq A$ and a $C^{\prime}$-vicinity $I\subseteq
X_{b^{\prime}}$ of $b_{n}$ in $D$, such that
$\mathrm{dp\text{-}rk}(b_{n},(b^{\prime},c)/C^{\prime})=n+m$. Let

$X_{1}:=\\{x\in D^{n-1}:(\forall y)(y\in I\to(x,y)\in X)\\}.$

This is a $C^{\prime}$-definable set, containing $b^{\prime}$. So it is a
$C^{\prime}$-vicinity of $b^{\prime}$ in $D^{n-1}$. By induction (now
replacing $c$ with $b_{n},c$), there exist $I_{i}\subseteq D$,
$i=1,\ldots,n-1$, defined over $C\supseteq C^{\prime}$, such that
$b^{\prime}\in\prod_{i=1}^{n-1}I_{i}\subseteq X_{1}$, with
$\mathrm{dp\text{-}rk}(b^{\prime},b_{n},c/C)=n+m$. The set
$\prod_{i=1}^{n}I_{i}\subseteq X$ is the desired vicinity of $b$. ∎

###  4.3. Functions with minimal fibres

The notion of definable sets with minimal fibres, with respect to some finite-
to-one projection, appeared already in Section 3. We slightly generalize.

######  Definition 4.11.

  1. (1)

For $X$ a definable set of finite dp-rank, a definable function $f:X\to Y$ has
minimal fibres if there exists some $m\in\mathbb{N}$ such that for every $y\in
Y$, we have $|f^{-1}(y)|\leq m$, and there is no definable $X_{1}\subseteq X$
(possibly over additional parameters), with
$\mathrm{dp\text{-}rk}(X_{1})=\mathrm{dp\text{-}rk}(X)$, such that for every
$y\in Y$, $|f^{-1}(y)\cap X_{1}|<m.$

  2. (2)

A set $X\subseteq D^{n}$ has minimal fibres (in $D$) if there exists a
coordinate projection $\pi:D^{n}\to D^{m}$ such that
$\mathrm{dp\text{-}rk}(X)=m$, and $\pi$ has minimal fibres.

######  Remark 4.12.

  1. (1)

Notice that an $A$-definable finite-to-one $f:X\to Y$ has minimal fibres if
and only if for every $B\supseteq A$ and every $B$-generic $a\in X$, all
elements of $[a]_{f}$ satisfy the same type over $f(a)B$.

  2. (2)

Every definable set $Z\subseteq D^{r}$ has a definable subset $Z_{1}\subseteq
Z$ (possibly over additional parameters) such that
$\mathrm{dp\text{-}rk}(Z_{1})=\mathrm{dp\text{-}rk}(Z)$ and $Z_{1}$ has
minimal fibres.

  3. (3)

If $Z\subseteq D^{r}$ has minimal fibres and $X^{\prime}\subseteq X$ with
$\mathrm{dp\text{-}rk}(X^{\prime})=\mathrm{dp\text{-}rk}(X)$ then $X^{\prime}$
has minimal fibres as well.

Before proceeding to the next lemma we recall that for any function $f:X\to
Y$, we write $[x]_{f}=f^{-1}(f(x))$.

Our next goal is to show that Axiom (A2) can be pulled back via maps with
minimal fibres, in the following sense:

######  Lemma 4.13.

Let $X$ be definable in ${\mathcal{M}}$ with $\mathrm{dp\text{-}rk}(X)=n$ and
$f:X\to D^{n}$ an $A$-definable function with minimal fibres. Let $b\in X$ be
$A$-generic and $c\in{\mathcal{M}}$ inter-algebraic over $A$ with some $d\in
D^{m}$.

Then, for every parameter set $B$ and every $B$-generic vicinity $Y\subseteq
X$ of $b$, there exists $C\supseteq A$ and a $C$-generic vicinity
$Y_{1}\subseteq Y$ of $b$, such that
$\mathrm{dp\text{-}rk}(b,c/C)=\mathrm{dp\text{-}rk}(b,c/A)$.

###### Proof.

Let $B$ be any set of parameters and $Y$ a $B$-generic vicinity of $b$ in $X$.
Since $f$ has finite fibres (because it has minimal fibres) our assumptions
imply that $f(b)$ is $A$-generic in $D^{n}$, and $f(Y)$ is a $B$-generic
vicinity of $f(b)$ in $D^{n}$. Thus, by (A2), there exists $C\supseteq A$, and
a $C$-generic vicinity $W\subseteq f(Y)$ of $f(b)$, such that
$\mathrm{dp\text{-}rk}(f(b),d/C)=\mathrm{dp\text{-}rk}(f(b),d/A)$. Hence,
$\mathrm{dp\text{-}rk}(b,c/C)=\mathrm{dp\text{-}rk}(b,c/A)$ as well.

Let $m$ be the size of maximal $f$-fibres. Since $f$ has minimal fibres, for
every $C$-generic $y\in Y$, we have $|[y]_{f}|=m$, so $[y]_{f}\subseteq Y$.
Thus, the set $W_{1}\subseteq W$, of all $w\in W$ such that $|f^{-1}(w)|=m$
satisfies $\mathrm{dp\text{-}rk}(W_{1})=\mathrm{dp\text{-}rk}(W)$, and we have
$Y_{1}:=f^{-1}(W_{1})\subseteq Y$. The set $Y_{1}$ is the desired $C$-generic
vicinity of $b$. ∎

Now we wish to pull back the conclusion of Lemma 4.10 via functions with
minimal fibres. We first note:

######  Lemma 4.14.

Assume that $f_{i}:X_{i}\to Y_{i}$, $i=1,2$, have minimal fibres. Then
$(f_{1},f_{2}):X_{1}\times X_{2}\to Y_{1}\times Y_{2}$ has minimal fibres. In
particular, if $X_{i}\subseteq D^{m_{i}}$ has minimal fibres, for $i=1,2$,
then so does $X_{1}\times X_{2}\subseteq D^{m_{1}}\times D^{m_{2}}$.

###### Proof.

Let $f=(f_{1},f_{2}):X_{1}\times X_{2}\to Y_{1}\times Y_{2}$ and assume for
simplicity of notation that it is $\emptyset$-definable. We apply Remark
4.12(1). Let $(a,b)\in X_{1}\times X_{2}$ be generic over some $B$. Then, by
sub-additivity of dp-rank, $a$ is $Bb$ generic in $X_{1}$, so all elements of
$[a]_{f_{1}}$ realize the same type over $f_{1}(a)bB$, hence also over
$f_{1}(a)f_{2}(b)bB$. It follows that all elements of
$[a]_{f_{1}}\times\\{b\\}$ satisfy the same type over $f_{1}(a)f_{2}(b)B$.
Similarly, all elements in $\\{a\\}\times[b]_{f_{2}}$ satisfy the same type
over $f_{1}(a)f_{2}(b)B$. It easily follows that all elements of $[(a,b)]_{f}$
realize the same type over $f(a,b)B$, as needed. ∎

######  Corollary 4.15.

For $i=1,2$ let $X_{i}$ be $A$-definable sets,
$\mathrm{dp\text{-}rk}(X_{i})=n_{i}$, and $f_{i}:X_{i}\to D^{n_{i}}$
$A$-definable functions with minimal fibres. Let $(d_{1},d_{2})\in X_{1}\times
X_{2}$ be $A$-generic. Then for any $A$-generic vicinity $Y\subseteq
X_{1}\times X_{2}$ of $(d_{1},d_{2})$, there exists $B\supseteq A$ and a
$B$-generic vicinity of $(d_{1},d_{2})$ of the form $I_{1}\times
I_{2}\subseteq Y$, with $I_{i}\subseteq X_{i}$.

###### Proof.

Let $f=(f_{1},f_{2}):X_{1}\times X_{2}\to D^{n_{1}}\times D^{n_{2}}$. By the
minimality assumption on $f_{1},f_{2}$, and Lemma 4.14, we have
$[(d_{1},d_{2})]_{f}\cap Y=[(d_{1},d_{2})]_{f}$. Let

$W=\\{w\in f(Y):f^{-1}(w)\cap Y=f^{-1}(w)\\}.$

Then, $(f(d_{1}),f(d_{2}))\in W$, so
$\mathrm{dp\text{-}rk}(W)=\mathrm{dp\text{-}rk}(d_{1},d_{2})=n_{1}+n_{2}$.

By Lemma 4.10, there exists $B\supseteq A$ such that $W$ contains a
$B$-generic vicinity of the form $V_{1}\times V_{2}$ of
$(f_{1}(d_{1}),f_{2}(d_{2}))$ in $f(X_{1}\times X_{2})$. By definition,
$f_{1}^{-1}(V_{1})\times f_{2}^{-1}(V_{2})$ is contained in $Y$ and it is thus
a $B$-generic vicinity of $(d_{1},d_{2})$ satisfying the requirements. ∎

###  4.4. Critical, almost critical and $D$-sets

Let ${\mathcal{M}}$ and $D$ be as before (so $D$ is still vicinic). Recall
Definition 2.6 of a set _strongly internal_ and _almost strongly internal_ to
$D$. We remind (and expand) a definition from [11]:

######  Definition 4.16.

Let $S$ be a definable set of finite dp-rank.

  1. (1)

A definable $X\subseteq S$ is $m$-internal to $D$ if there exists an $m$-to-
one $f:X\to D^{n}$,

  2. (2)

A definable set $X\subseteq S$ is $D$-critical (for $S$) if $X$ is strongly
internal to $D$, and has maximal dp-rank among all such subsets of $S$. Its
dp-rank is the $D$-critical rank of $S$.

  3. (3)

A definable set $X\subseteq S$ is _almost $D$-critical (for $S$)_ if (i) it is
almost-strongly internal to $D$ and has maximal dp-rank among all such subsets
of $S$, and (ii) it is $m$-internal for minimal $m$ among all sets satisfying
(i). We call $\mathrm{dp\text{-}rk}(X)$ the _almost $D$-critical rank of $S$_.

The set $X$ is called (almost) $D$-critical over $A$ if the corresponding map
of $X$ into $D^{n}$ is defined over $A$.

  4. (4)

A definable set $X\subseteq S$ is an (almost) $D$-set over $A$, if $X$ is an
(almost) $D$-critical set, witnessed by an $A$-definable $f:X\to D^{n}$, such
that in addition $f(X)\subseteq D^{n}$ has minimal fibres.

Notice that the $D$-critical rank of $S$ is always bounded above by the almost
$D$-critical rank of $S$, but the ranks need not be equal, as we shall now
see. Thus, a $D$-critical set is not necessarily almost $D$-critical.

######  Example 4.17.

Let ${\mathcal{K}}$ be an elementary extension of a quadratic residual
extension of $\mathbb{Q}_{p}$, as in Example 3.33. Let $C_{p}\leq
K/{\mathcal{O}}$ be a cyclic subgroup of order $p$ and
$G_{0}=(K/{\mathcal{O}})/C_{p}$ as in that example. Let
$G=K/{\mathcal{O}}\times G_{0}$. Then $G$ is almost strongly internal to
$K/{\mathcal{O}}$, since both $K/{\mathcal{O}}$ and $G_{0}$ are; so its almost
$K/{\mathcal{O}}$-critical rank is $2$.

We claim that the critical $K/{\mathcal{O}}$-rank of $G$ is $1$: The definable
set $K/{\mathcal{O}}\times\\{0\\}$ is strongly internal to $K/{\mathcal{O}}$,
so we only need to note that $G$ has no definable subset of dp-rank $2$ which
is strongly internal to $K/{\mathcal{O}}$. Indeed, if $X\subseteq G$ has dp-
rank $2$, then, just like in Lemma 3.6, it contains a definable set of the
form $Y_{1}\times Y_{2}$, with $Y_{2}\subseteq G_{0}$ of dp-rank $1$. So if
$X$ were strongly internal to $K/{\mathcal{O}}$ then $G_{0}$ would be locally
strongly internal to $K/{\mathcal{O}}$, contradicting Example 3.33.

######  Remark 4.18.

Let $X\subseteq S$ be an almost $D$-critical set, witnessed by $f:X\to D^{k}$.
Then it is not hard to see that there exists $X_{1}\subseteq X$, defined over
some $B\supseteq A$, that is a D-set for the same $S$, as witnessed by $f$. If
$f$ is injective then we can find such an $X_{1}\subseteq X$ which is a
$D$-set.

We will use the following remark implicitly throughout.

######  Remark 4.19.

  1. (1)

If $f:X\to D^{n}$ witnesses that $X$ is an (almost) $D$-set for $S$ then,
since $f(X)$ has minimal fibres, we may compose $f$ with an appropriate
coordinate projection to obtain a map with minimal fibres $\pi\circ f:X\to
D^{n}$, such that $n=\mathrm{dp\text{-}rk}(X)$. Thus, in Lemma 4.13 the
assumption that $\mathrm{dp\text{-}rk}(X)=n$ can be dropped when $X$ is an
(almost) $D$-set. Similarly, Lemma 4.15 holds when $X_{1},X_{2}$ are (almost)
$D$-sets.

  2. (2)

If $X_{1},X_{2}\subseteq S$ are (almost) $D$-sets then so is $X_{1}\times
X_{2}\subseteq S^{2}$. Indeed, by Lemma 4.14 it is sufficient to show that
$X_{1}\times X_{2}$ is (almost) $D$-critical in $S\times S$: If $Y\subseteq
S\times S$ is (almost) strongly internal to $D$ then so are the fibres $Y_{x}$
and $Y^{x}$ for every $x\in S$. In particular
$\mathrm{dp\text{-}rk}(Y_{x})\leq\mathrm{dp\text{-}rk}(X_{1})$ and
$\mathrm{dp\text{-}rk}(Y^{x})\leq\mathrm{dp\text{-}rk}(X_{2})=\mathrm{dp\text{-}rk}(X_{1})$
for any $x\in S$. By sub-additivity $\mathrm{dp\text{-}rk}(Y)\leq
2\mathrm{dp\text{-}rk}(X_{1})$, so $2\mathrm{dp\text{-}rk}(X_{1})$ is the
(almost) $D$-critical rank of $S\times S$.

  3. (3)

If $X\subseteq S$ is an (almost) $D$-set and $Y\subseteq X$ with
$\mathrm{dp\text{-}rk}(Y)=\mathrm{dp\text{-}rk}(X)$ then $Y$ is also an
(almost) $D$-set.

Below, if the ambient set is clear from the context or immaterial, we will
just refer to (almost) $D$-critical sets, without explicit mention of $S$
(though such an $S$ of finite dp-rank is always assumed to exist in the
background).

We end this section with a result on generic vicinities, which will play an
important role in the next sections.

######  Lemma 4.20.

Assume that $X$ is an (almost) $D$-set over $A$ and $d\in X$ generic over $A$.
Let $X_{i}\subseteq X$, $i=1,2$, be two $A_{i}$-generic vicinities of $d$ in
$X$ for some $A_{i}\supseteq A$.

Then there exists $C$ and a $C$-definable $U\subseteq X_{1}\cap X_{2}$ which
is a $C$-generic vicinity of $d$ in $X$.

###### Proof.

By Remark 4.19 (1) we may apply Lemma 4.13 as follows: first apply the lemma
to $X$ (viewed as $A_{2}$-definable), $d$ and the $A_{1}$-generic vicinity
$X_{1}$ (in the role of $Y$ in the lemma) to obtain $U_{1}\subseteq X_{1}$
defined over some $A_{2}^{\prime}\supseteq A_{2}$ such that $d\in U_{1}$ is
$A_{2}^{\prime}$-generic. Now $U_{1}\cap X_{2}$ is $A_{2}^{\prime}$-definable,
so an $A_{2}^{\prime}$-generic vicinity of $d$ satisfying the requirements. ∎

###  4.5. $D$-groups

We proceed with a series of technical lemmas ultimately allowing us to
construct, inside a group that is (almost) strongly internal to $D$, a
definable subset that is both (almost) strongly internal and sufficiently
closed under the group operation.

######  Lemma 4.21.

Let $G$ be an $A$-definable group. Let $X,Y\subseteq G$ be $A$-definable with
$X$ (almost) strongly internal to $D$ over $A$, and fix an $A$-generic
$(g,h)\in X\times Y$.

If $\mathrm{dp\text{-}rk}(g/A,g\cdot h)<\mathrm{dp\text{-}rk}(X)$ then there
exists a finite-to-one definable function from a subset of $X\times Y$ onto a
set $W\subseteq X\cdot Y\subseteq G$ satisfying
$\mathrm{dp\text{-}rk}(W)>\mathrm{dp\text{-}rk}(Y)$.

###### Proof.

For simplicity of notation, assume that $A=\emptyset$ and write $k=gh\in G$.
Assume that $d:=\mathrm{dp\text{-}rk}(g/k)<m:=\mathrm{dp\text{-}rk}(X)$, and
let $n=\mathrm{dp\text{-}rk}(Y)$.

Because $X$ is (almost) strongly internal to $D$ we may apply Lemma 4.9 to
obtain $a\in D^{d}$, $a\in\operatorname{dcl}(g)$ such that
$g\in\operatorname{acl}(a,k)$.

Notice that each two pairs of $g,h,k$ are interdefinable over $\emptyset$.
E.g., the map $(x,y)\mapsto(x,xy)$ sends $(g,h)$ to $(g,k)$. Thus, we have
$\mathrm{dp\text{-}rk}(a,k)=\mathrm{dp\text{-}rk}(g,k)=\mathrm{dp\text{-}rk}(g,h)=m+n$.

Since $a\in D^{d}$, we have $\mathrm{dp\text{-}rk}(a)\leq d$ so by sub-
additivity of dp-rank we have $\mathrm{dp\text{-}rk}(k/a)\geq n+m-d>n$.

Let $\varphi(x,k)$ be the formula over $a$ isolating
$\operatorname{tp}(g/a,k)$. Let $l:=|\varphi(X,k)|$ and

$Z=\\{(x,y)\in X\times Y:\varphi(x,x\cdot y)\land(\exists^{\leq
l}x^{\prime})(x^{\prime}\in X\land\varphi(x^{\prime},x\cdot y)\\}.$

This is an $a$-definable set containing $(g,h)$, thus its image under
$(x,y)\mapsto x\cdot y$, call it $W$, is also $a$-definable and contains $k$,
so $\mathrm{dp\text{-}rk}(W)\geq\mathrm{dp\text{-}rk}(k/a)>n$. Our assumption
on $\varphi$ implies that the restriction of the group multiplication to $Z$
is a finite-to-one map. ∎

The following statement is rather convoluted, although its proof is simple,
based on what we already know. It is the main technical tool towards the
definition of an (almost) $D$-group.

######  Corollary 4.22.

Let $G$ be an $A$-definable group.

(1) Assume that

  1. ($\star$)

for every definable $X,Y\subseteq G$ and a finite-to-one surjection $f:X\to
Y$, if $X$ is strongly internal to $D$ then there exists a definable subset
$Y^{\prime}\subseteq Y$ with
$\mathrm{dp\text{-}rk}(Y^{\prime})=\mathrm{dp\text{-}rk}(Y)$, such that
$Y^{\prime}$ is strongly internal to $D$.

Let $X_{1}\subseteq G$ be strongly internal to $D$ and $X_{2}\subseteq G$ be
$D$-critical, both over $A$. Then for every $A$-generic $(g,h)\in X_{1}\times
X_{2}$, we have $\mathrm{dp\text{-}rk}(g/A,g\cdot
h)=\mathrm{dp\text{-}rk}(X_{1})$.

(2) Assume that

  1. ($\star_{a}$)

for every definable $X,Y\subseteq G$ and a finite-to-one surjection $f:X\to
Y$, if $X$ is almost strongly internal to $D$ there exists a definable subset
$Y^{\prime}\subseteq Y$ with
$\mathrm{dp\text{-}rk}(Y^{\prime})=\mathrm{dp\text{-}rk}(Y)$, such that
$Y^{\prime}$ is almost strongly internal to $D$.

Let $X_{1}\subseteq G$ be almost strongly internal to $D$ and $X_{2}\subseteq
G$ be almost $D$-critical, both over $A$. Then for every $A$-generic $(g,h)\in
X_{1}\times X_{2}$, we have $\mathrm{dp\text{-}rk}(g/A,g\cdot
h)=\mathrm{dp\text{-}rk}(X_{1})$.

###### Proof.

(1) Assume $X_{1}$ is strongly internal to $D$, $X_{2}$ is $D$-critical and
assume towards a contradiction that $\mathrm{dp\text{-}rk}(g/A,g\cdot
h)<m:=\mathrm{dp\text{-}rk}(X_{1})$; then by Lemma 4.21, there exists a
finite-to-one definable function $f$ from a subset $Z$ of $X_{1}\times X_{2}$
onto a definable subset $W\subseteq X_{1}\cdot X_{2}$ with
$\mathrm{dp\text{-}rk}(W)>\mathrm{dp\text{-}rk}(X_{2})$.

Since $X_{1}$ and $X_{2}$ are both strongly internal to $D$ so is $X_{1}\times
X_{2}$, and hence also $Z$. By ($\star$), applied to $f:Z\to X_{1}\to W$,
there exists a definable subset $W_{1}\subseteq W$ with
$\mathrm{dp\text{-}rk}(W_{1})=\mathrm{dp\text{-}rk}(W)$ and an injective map
from $W_{1}$ into some $D^{p}$, namely $W_{1}$ is strongly internal to $D$ and
$\mathrm{dp\text{-}rk}(W)>\mathrm{dp\text{-}rk}(X_{2})$. This contradicts the
maximality of $\mathrm{dp\text{-}rk}(X_{2})$.

(2) This proof is almost identical, replacing everywhere “strongly internal”
with “almost strongly internal” and using $(\star_{a})$ instead of $(\star)$.
∎

The conclusions of Corollary 4.22 are important for much that follows. For the
sake of clarity of exposition, we isolate this property of groups and define:

######  Definition 4.23.

Let $D$ be a vicinic sort. An $A$-definable group $G$ is an _(almost)
$D$-group_ if its (almost) $D$-critical rank is at least $1$ and for every
$X_{1}\subseteq G$ (almost) strongly internal to $D$, every (almost)
$D$-critical set $X_{2}\subseteq G$, both over some $B\supseteq A$, and for
every $(g,h)$ generic in $X_{1}\times X_{2}$ over $B$, we have

$\mathrm{dp\text{-}rk}(g/B,g\cdot h)=\mathrm{dp\text{-}rk}(X_{1}).$

######  Remark 4.24.

By Corollary 4.22, if $D$ satisfies ($\star$) then every definable group $G$
which is locally strongly internal to $D$ is a $D$-group. If $D$ satisfies
($\star_{a}$) then every definable group $G$ which is almost locally strongly
internal to $D$ is an almost $D$-group.

To avoid any confusion we point out that though the name may suggest it, it
formally need not be the case that every $D$-group is an almost $D$-group
(Example 4.30).

We can now collect our previous results and conclude:

######  Fact 4.25.

  1. (1)

A group of finite dp-rank which is locally (almost) strongly internal to an
SW-uniformity $D$ is an (almost) $D$-group.

  2. (2)

Let ${\mathcal{K}}=(K,+,\cdot,v,\dots)$ be an expansion of a valued field, $G$
an interpretable group in ${\mathcal{K}}$ and $D$ one of the infinite
distinguished sorts.

     * $\bullet$

If ${\mathcal{K}}$ is V-minimal, $D\neq\textbf{k}$ and $G$ locally (almost)
strongly internal to $D$ then $G$ is an (almost) $D$-group.

     * $\bullet$

If ${\mathcal{K}}$ is power-bounded $T$-convex and $G$ locally (almost)
strongly internal to $D$ then $G$ is an (almost) $D$-group.

     * $\bullet$

If ${\mathcal{K}}$ is $p$-adically closed and $G$ locally almost strongly
internal to $D$ then $G$ is an almost $D$-group.

###### Proof.

By Fact 4.7, all the sorts $D$ are vicinic, so we only need to check whether
$D$ satisfies ($\star$) or ($\star_{a}$) (Remark 4.24).

For (1), When $D$ is an SW-uniformity, it satisfies both ($\star$) and
($\star_{a}$) by Lemma 2.9. For (2), if $D$ is an SW-uniformity, then by Lemma
2.9 both ($\star$) and ($\star_{a}$) hold. This covers the first two settings
and the case $D=K$ in the $p$-adically closed setting. We are left with
$\Gamma$ and $K/{\mathcal{O}}$ in the $p$-adic setting.

If $D$ is a stably embedded ${\mathbb{Z}}$-group then by elimination of
imaginaries both ($\star$) and ($\star_{a}$) hold, by Lemma 4.3. If
$D=K/{\mathcal{O}}$ then ($\star_{a}$) holds by Lemma 3.9. ∎

We also have:

######  Lemma 4.26.

Let $G$ be a definable (almost) $D$-group, and let $X_{1}\subseteq G$ be
(almost) strongly internal to $D$ and $X_{2}\subseteq G$ (almost)
$D$-critical, both over $A$.

Assume that $(g_{1},g_{2})\in X_{1}\times X_{2}$ is $A$-generic and let
$g=g_{1}g_{2}^{-1}$. Then $X_{1}\cap gX_{2}$ is an $Ag$-generic vicinity of
$g_{1}$ in $X_{1}$. In particular, if $X_{1}$ is (almost) $D$-critical, then
$X_{1}\cap gX_{2}$ is also (almost) $D$-critical.

###### Proof.

Assume for simplicity that $A=\emptyset$. As $G$ is an (almost) $D$-group, it
is easy to see that $\mathrm{dp\text{-}rk}(g_{1}/g_{1}\cdot
g_{2}^{-1})=\mathrm{dp\text{-}rk}(X_{1})$. Since $g_{1}\in(g_{1}\cdot
g_{2}^{-1}X_{2})\cap X_{1}$ and the intersection is $(g_{1}\cdot
g_{2}^{-1})$-definable, it follows that $X_{1}\cap g_{1}\cdot g_{2}^{-1}X_{2}$
is a generic vicinity of $g_{1}$ in $X_{1}$. ∎

We can now deduce the main result of this section.

######  Lemma 4.27.

Let $G$ be a definable (almost) $D$-group, $X_{1},X_{2}\subseteq G$ (almost)
$D$-sets over $A$. Assume that $(g_{1},g_{2})\in X_{1}\times X_{2}$ is
$A$-generic.

Then, there exists $B\supseteq A$ and $B$-definable subsets
$X_{i}^{\prime}\subseteq X_{i}$, such that $X_{1}^{\prime}\times
X_{2}^{\prime}$ is a $B$-generic vicinity of $(g_{1},g_{2})$ in $X_{1}\times
X_{2}$ and such that $X_{1}^{\prime}\cdot X_{2}^{\prime}\subseteq X_{1}\cdot
g_{2}$. Moreover, $X_{1}^{\prime}\cdot X_{2}^{\prime}\subseteq X_{1}\cdot g$
for every $g\in X_{2}^{\prime}$ and thus $X_{1}^{\prime}\cdot X_{2}^{\prime}$
is an (almost) $D$-set over $Bg$.

###### Proof.

We prove the lemma in case the $X_{i}$ are almost $D$-sets (and $G$ is an
almost $D$-group). The proof for the case where the $X_{i}$ are $D$-sets and
$G$ is a $D$-group is similar. For simplicity of notation, we assume
$A=\emptyset$.

Let $k=g_{1}\cdot g_{2}\in G$ and let $Y_{2}=\\{x_{2}\in X_{2}:k\in X_{1}\cdot
x_{2}\\}$. Note that $Y_{2}$ is $k$-definable and contains $g_{2}$. Since $G$
is an almost $D$-group, $\mathrm{dp\text{-}rk}(g_{2}/Ak)=n$, hence
$\mathrm{dp\text{-}rk}(Y_{2})=n$. Since $g_{1}$ is inter-algebraic over $A$
with some element in $D^{l}$ we can apply Lemma 4.13 and Remark 4.19(1) to
obtain a definable subset $Y_{2}^{\prime}\subseteq Y_{2}$ containing $g_{2}$
that is $C$-definable for some parameter set $C$ such that
$\mathrm{dp\text{-}rk}(g_{1},g_{2}/C)=\mathrm{dp\text{-}rk}(g_{1},g_{2})=2n$.
It follows that $\mathrm{dp\text{-}rk}(g_{i},k/C)=2n$, for $i=1,2$.

We let

$Z=\bigcap_{y\in Y_{2}^{\prime}}X_{1}y.$

It is $C$-definable, containing $k$, hence $\mathrm{dp\text{-}rk}(Z)=n$. Since
$g_{2}\in Y_{2}^{\prime}$, we also have $Z\subseteq X_{1}g_{2}$. Finally, we
consider

$S=\\{(x_{1},x_{2})\in X_{1}\times X_{2}:x_{1}\cdot x_{2}\in Z\\}.$

It is definable over $C$, and contains $(g_{1},g_{2})$, thus
$\mathrm{dp\text{-}rk}(S)=2n$. By Corollary 4.15, there exists
$X_{1}^{\prime}\times X_{2}^{\prime\prime}\subseteq S$, a $B$-generic vicinity
of $(g_{1},g_{2})$ in $X_{1}\times X_{2}$, for some $B\supseteq C$. Now let
$X_{2}^{\prime}=X_{2}^{\prime\prime}\cap Y_{2}^{\prime}$; it is still a
$C$-definable vicinity of $g_{2}$. Note that for every $g\in X_{2}^{\prime}$,
we have

$X_{1}^{\prime}\cdot X_{2}^{\prime}\subseteq X_{1}^{\prime}\cdot
X_{2}^{\prime\prime}\subseteq Z\subseteq X_{1}\cdot g,$

where the latter follows from the definition of $Z$ and the fact that $g\in
X_{2}^{\prime}\subseteq Y_{2}^{\prime}$. ∎

######  Remark 4.28.

A symmetric proof would give that we can find $X_{1}^{\prime}$ and
$X_{2}^{\prime}$ such that $X_{1}^{\prime}\cdot X_{2}^{\prime}\subseteq
g_{1}X_{2}$

We conclude this section with a couple of examples. One corollary of Lemma
4.27, is that given (almost) $D$-sets $X_{1},X_{2}\subseteq G$, there are
$X_{i}^{\prime}\subseteq X_{i}$, $i=1,2$, of full rank, such that
$\mathrm{dp\text{-}rk}(X_{1}^{\prime}\cdot
X_{2}^{\prime})=\mathrm{dp\text{-}rk}(X_{2})$. The first example shows that
this is the best we can hope for. Namely, that even if $X_{1},X_{2}$ are
$D$-critical, the dp-rank of $X_{1}\cdot X_{2}$ may increase:

######  Example 4.29.

Let ${\mathcal{K}}$ be a real closed valued field and $G:=K\times
K/{\mathcal{O}}$. Let $X\subseteq G$ be the graph of the quotient map
$\pi:K\to K/{\mathcal{O}}$ and $Y\subseteq G$ the graph of $-\pi$. It is clear
that $\mathrm{dp\text{-}rk}(X)=\mathrm{dp\text{-}rk}(Y)=1$ and that both are
strongly internal to $K$, and therefore $D$-critical (see Corollary 9.5). It
is easy to verify, using sub-additivity, that $X+Y=G$ so that
$\mathrm{dp\text{-}rk}(X+Y)=2$.

The next example shows that interpretable almost $D$-groups need not be
$D$-groups:

######  Example 4.30.

Let ${\mathcal{K}}$ be a sufficiently saturated elementary extension of a
quadratic residual extension of $\mathbb{Q}_{p}$ as in Example 4.17 and
$G=K/{\mathcal{O}}\times G_{0}$, where $G_{0}=(K/{\mathcal{O}})/C_{p}$ the
group from Example 4.17. Below $D=K/{\mathcal{O}}$.

As in Example 4.17, $G_{0}$ is locally almost strongly internal to $D$, thus
so is $G$. By Fact 4.25, it is an almost $D$-group, in particular,
$\mathrm{dp\text{-}rk}(g/B,g\cdot h)=\mathrm{dp\text{-}rk}(X_{2})$ for any
$B$-definable almost $D$-critical $X_{1},X_{2}$ and $B$-generic $(g,h)\in
X_{1}\times X_{2}$.

By Example 4.17, the $D$-critial rank of $G$ is $1$. To see that $G$ is not a
$D$-group we consider two $D$-sets in $G$: The subgroups
$X_{1}=K/{\mathcal{O}}\times\\{0\\}$ and $X_{2}=\\{(h,h+C_{p}):h\in
K/{\mathcal{O}}\\}$ are in bijection with $K/{\mathcal{O}}$, thus they are
$D$-sets in $G$. However, the two subgroups generate $G$, so if we pick
$(g,h)\in X_{1}\times X_{2}$ generic then $\mathrm{dp\text{-}rk}(g/g+h)=0$.

###  4.6. From almost strong internality to strong internality

In order to put the machinery for the construction of infinitesimal subgroups
in gear we need to work inside a group that is locally strongly internal to a
vicinic sort $D$. As we have seen in Example 3.33, in the case
$D=K/{\mathcal{O}}$ in the $p$-adic setting, it may happen that a group is
locally almost strongly internal to $D$, but not locally strongly internal. In
this section, we show that modding out a finite normal subgroup resolves this
problem. The proof is inspired by a result of a similar nature due to
Hrushovski and Rideau-Kikuchi, [17, Lemma 2.25].

We need (also for later use) an elementary fact from group theory:

######  Fact 4.31.

Let $G$ be a group, $A,B\subseteq G$ arbitrary subsets, $a\in A$, $b\in B$.
Assume that

$a\cdot B=A\cdot B=A\cdot b.$

Then there is a sub-monoid $S\leq G$ (i.e. $e\in S$ and $S$ closed under
multiplication) such that $A=aS$ and $B=Sb$. In particular, if $A,B$ are
finite, then $S$ is a subgroup.

###### Proof.

Note that $a^{-1}A=Bb^{-1}$. We first claim that the set $S_{1}:=\\{g\in
G:gB\subseteq B\\}$ equals $a^{-1}A$. Indeed, if $gB\subseteq B$ then $gb\in
B$, so $g\in Bb^{-1}=a^{-1}A$. Conversely, since $aB=AB$ then $a^{-1}AB=B$, so
the claim follows. Clearly, $e\in S_{1}$ and $S_{1}$ is closed under
multiplication. By symmetry, $S_{2}:=\\{g\in G:Ag^{-1}\subseteq A\\}=Bb^{-1}$,
is also a sub-monoid of $G$. As we just noted, we have $S_{1}=S_{2}$, call it
$S$, and $A=aS$, $B=Sb$.

Now, if $A$ and $B$ are finite (if one is finite then so is the other) then
$S$ is a finite sub-monoid of a group, so a subgroup. ∎

######  Lemma 4.32.

Let $G$ be a definable group of finite dp-rank and let $D$ be any definable
set. Let $X_{1},X_{2}\subseteq G$, $f_{i}:X_{i}\to D^{k}$ and $h:X_{1}\cdot
X_{2}\to D^{p}$ be generically $m$-to-one definable functions, with
$f_{1},f_{2},h$ defined over $A$. Assume, moreover, that no
$X_{i}^{\prime}\subseteq X_{i}$ of the same dp-rank is $n$-internal to $D$ for
any $n<m$.

Then for every $(a,b)\in X_{1}\times X_{2}$ generic over $A$, there is a
finite subgroup $H(a,b)\subseteq G$ such that $[a]_{f_{1}}=a\cdot H(a,b)$ and
$[b]_{f_{2}}=H(a,b)\cdot b$.

###### Proof.

By the assumption on $X_{1}$, for any $B\supseteq A$, any $B$-definable
function $F:X_{1}\to D^{q}$ (any $q$) with finite fibres is, generically, at
least $m$-to-one.

Fix $(a,b)\in X_{1}\times X_{2}$ generic over $A$. Consider the function
$F:X_{1}\to D^{k+p}$ given by $x\mapsto(f_{1}(x),h(x\cdot b))$. Notice that
$[x]_{F}=[x]_{f_{1}}\cap([x\cdot b]_{h}\cdot b^{-1})$).

Since $[a]_{F}\subseteq[a]_{f_{1}}$, the assumption that
$\mathrm{dp\text{-}rk}(a/Ab)=\mathrm{dp\text{-}rk}(X_{1})$ and the minimality
of $m$ forces $|[a]_{F}|\geq m$, so it must be that, in fact,
$[a]_{F}=[a]_{f_{1}}$. It follows that $[a]_{f_{1}}\subseteq[a\cdot
b]_{h}\cdot b^{-1}$. By assumption $|[a\cdot b]_{h}|\leq m$, so necessarily
$[a]_{f_{1}}=[a\cdot b]_{h}\cdot b^{-1}$, and so $[a]_{f_{1}}\cdot b=[a\cdot
b]_{h}$. If $a^{\prime}\in[a]_{f_{1}}$ then, as $a,a^{\prime}$ are
interalgebraic, we have
$\mathrm{dp\text{-}rk}(a^{\prime},b/A)=\mathrm{dp\text{-}rk}(a,b/A)$, so we
also get $[a^{\prime}]_{f_{1}}\cdot b=[a^{\prime}\cdot b]_{h}$, but
$[a^{\prime}]_{f_{1}}=[a]_{f_{1}}$ so $[a^{\prime}\cdot b]_{h}=[a\cdot
b]_{h}$. Since the roles of $a$ and $b$ are symmetric, we conclude also that
$[a\cdot b^{\prime}]_{h}=[a\cdot b]_{h}$ and so $[a]_{f_{1}}\cdot
b^{\prime}=[a\cdot b]_{h}$ for all $b^{\prime}\in[b]_{f_{2}}$. Therefore,

$[a]_{f_{1}}\cdot b=[a]_{f_{1}}\cdot[b]_{f_{2}}=a\cdot[b]_{f_{2}},$

so by Fact 4.31 there is a finite group as claimed. ∎

For the main result of this section, we need the following facts. The first
was proved in the context of SW-uniformities in [11, Lemma 3.14 ]. It remains
valid for vicinic structures as well:

######  Fact 4.33.

Let $\mathbb{U}\succ{\mathcal{M}}$ a monster model, $b_{1},\dots,b_{n}$ some
tuples in $\mathbb{U}$. For every ${\mathcal{M}}$-definable $X\subseteq D^{r}$
with finite fibres, there exists an $M$-generic $a\in X$ such that
$\mathrm{dp\text{-}rk}(a,b_{i}/M)=\mathrm{dp\text{-}rk}(a/M)+\mathrm{dp\text{-}rk}(b_{i}/M)$
for all $1\leq i\leq n$.

###### Proof.

This was proved in [11] under the assumption that $D$ is an SW-uniformity.
However, the proof only uses the fact that, in the notation of the above
statement, there is a finite-to-one projection $\pi:X\to M^{k}$, with
$k=\mathrm{dp\text{-}rk}(X)$, and an $M$-generic box in its image (i.e., a
product of $k$ dp-minimal subsets). In the vicinic setting, this is implied by
Axiom (A1) and Corollary 4.15. ∎

The second of the facts appearing in [11, Lemma 3.14] was proved for a single
type. The same proof works for two (or more) types:

######  Fact 4.34.

Let ${\mathcal{M}}$ be a structure of finite dp-rank and
$\mathbb{U}\succ{\mathcal{M}}$ a monster model.

For $A\subseteq\mathbb{U}$ and $a,b\in{\mathcal{M}}^{n}$, there exists a small
model ${\mathcal{N}}$, $A\subseteq{\mathcal{N}}\prec{\mathcal{M}}$, such that
$\mathrm{dp\text{-}rk}(a/A)=\mathrm{dp\text{-}rk}(a/N)$ and
$\mathrm{dp\text{-}rk}(b/A)=\mathrm{dp\text{-}rk}(b/N)$.

###### Proof.

Let $\langle I_{t}:t<k_{1}\rangle$ be mutually indiscernible sequences over