action is sufficiently collapsed, then also the group $\Gamma$ virtually
splits a non-trivial free abelian subgroup. Unfortunately, while this is
always the case for non-positively curved Riemannian manifolds, by the work of
Eberlein [Ebe83] (indeed, in that case it is always possible to split
virtually a non-trivial free abelian subgroup whose rank coincides exactly
with the rank of the maximal Euclidean factor of the space), this is no longer
true for discrete, cocompact groups of general, geodesically complete
CAT$(0)$-spaces, as [CM19, Example 1] shows. More examples of this phenomenon
can be found in [LM21].
In this section, we present a basic example, constructed in [LM21], showing
that, no matter how the action is collapsed, the virtual splitting cannot be
proved under our hypotheses. Then we will prove a splitting theorem under the
additional assumption that the isometry group of $X$ is a Lie group; this
result holds in particular, by Proposition 2.3, for all CAT$(0)$-homology
manifolds.

###  6.1. Group splitting: a counterexample

######  Example 6.1.

We construct a sequence of proper, geodesically complete, CAT$(0)$-spaces
$X_{j}$, all $(P_{0},r_{0})$-packed for the same constants $P_{0},r_{0}>0$,
and a sequence of discrete, nonsingular groups
$\Gamma_{j}<\textup{Isom}(X_{j})$ which are all $D_{0}$-cocompact for some
$D_{0}>0$ such that:

  * (a)

each $\Gamma_{j}$ has no non-trivial, virtually normal, abelian subgroups;

  * (b)

$\text{sys}^{\diamond}(\Gamma_{j},X_{j})\to 0$ as $j\to+\infty$.

Recall that a subgroup $A$ of a group $\Gamma$ is virtually normal if there
exists a finite index subgroup $\Gamma^{\prime}$ such that
$A\cap\Gamma^{\prime}$ is normal in $\Gamma^{\prime}$.
Observe that (b) gives strong restrictions to the spaces $X_{j}$ and the
groups $\Gamma_{j}$ in virtue of Remark 5.2; indeed, the spaces $X_{j}$ will
split a Euclidean factor and the groups $\Gamma_{j}$ will have a commensurated
non-trivial, abelian subgroup.
On the other hand, the condition that $\Gamma$ virtually splits a free abelian
factor of positive rank is equivalent to have a virtually normal, non trivial,
free abelian subgroup, by [CM19, Theorem 2]; so, these groups $\Gamma_{j}$ do
not split.
The idea is very simple. We start as in [LM21] from the lattices
${\mathcal{L}}_{+},{\mathcal{L}}_{-}$ of $\mathbb{Z}^{2}=\langle
a,b\,\,|\,\,[a,b]=1\rangle$ respectively generated by
${\mathcal{B}}_{+}=\\{a^{2}b^{-1},ab^{2}\\}$ and by
${\mathcal{B}}_{-}=\\{a^{2}b,a^{-1}b^{2}\\}$, and define the (torsion-free)
HNN-extension

$\Gamma=\langle
a,b,t\,\,|\,\,[a,b]=1,\,\,ta^{2}b^{-1}t^{-1}=a^{2}b,\,\,tab^{2}t^{-1}=a^{-1}b^{2}\rangle$

that is $\Gamma=\mathbb{Z}^{2}\ast\varphi$, where
$\varphi:{\mathcal{L}}_{+}\rightarrow{\mathcal{L}}_{-}$ is given by
$\varphi(a^{2}b^{-1})=a^{2}b$ and $\varphi(ab^{2})=a^{-1}b^{2}$. We denote by
$T$ its Bass-Serre tree: notice this is a locally finite tree, since
$[\mathbb{Z}^{2}:{\mathcal{L}}_{\pm}]=5$. The natural action of $\Gamma$ on
$T$ is not discrete (since the stabilizers of the vertices are infinite), non-
elementary and not quasi-parabolic (that is without a global fixed point in
$\partial T$, since $\Gamma$ is not an ascending HNN-extension, cp. [MO15]).
The same is true for any finite index subgroup $\Gamma^{\prime}$ of $\Gamma$,
in particular $L\Gamma^{\prime}$ is the smallest closed
$\Gamma^{\prime}$-invariant subset of $\partial T$. Then, the group $\Gamma$
does not virtually normalize any non-trivial, abelian subgroup $A$ (actually,
if an abelian group $A$ was normal in $\Gamma^{\prime}$ then $\Gamma^{\prime}$
would preserve the limit set $LA$, hence $LA=L\Gamma^{\prime}=L\Gamma$, by
minimality; but this would imply that the action is elementary, since $A$ is
abelian).
We also consider the action of $\Gamma$ on $\mathbb{R}^{2}$, where $a$ and $b$
act as orthogonal translations of length $1$ and $t$ as a rotation around the
origin of angle $\arccos(\frac{3}{5})$. Also this action is not discrete.
However, the diagonal action of $\Gamma$ on $X=\mathbb{R}^{2}\times T$ is
discrete and $1$-cocompact. Observe that $X$ is proper, geodesically complete,
CAT$(0)$ and $(P_{0},1)$-packed for some $P_{0}>0$.
By definition, we can see $\Gamma$ as a subgroup of
$\text{Isom}(\mathbb{R}^{2})\times\text{Isom}(T)$.
We now take $X_{j}=\frac{1}{j}\cdot\mathbb{R}^{2}\times T$ and define
$\Gamma_{j}:=\Gamma$ with its natural action on $X_{j}$ induced by the action
of $\Gamma$ on $X$. The sequence of groups $\Gamma_{j}$ acting on $X_{j}$
clearly satisfies the conditions (a) and (b).

The behaviour of this sequence will be further studied in Section 7.11,
Example 7.14.

###  6.2. Group splitting: when the isometry group is Lie

The purpose of this section is to prove that a discrete isometry group
$\Gamma$ acting on a $\textup{CAT}(0)$-space $X$ with a nontrivial Euclidean
factor ${\mathbb{R}}^{k}$ virtually splits a free abelian subgroup of rank
$k$, provided that Isom$(X)$ is a Lie group. This fact stems almost
immediately from [CM19]. However, we need a more precise result, that is to
bound the index in $\Gamma$ of the subgroup $\check{\Gamma}$ which splits
${\mathbb{Z}}^{k}$; this will be crucial for our convergence results.

######  Theorem 6.2.

Given $P_{0},r_{0},D_{0}$, there exist
$\sigma_{0}\\!=\sigma_{0}(P_{0},r_{0},D_{0})\\!>0$ (the same constant of
Theorem D) and $I_{0}=I_{0}(P_{0},r_{0},D_{0})$ such that the following holds.
Let $X$ be any proper, geodesically complete, $(P_{0},r_{0})$-packed,
$\textup{CAT}(0)$-space whose isometry group is a Lie group, and let
$X=Y\times\mathbb{R}^{n}$ be the splitting of the maximal Euclidean factor of
$X$. Then, for every discrete, $D_{0}$-cocompact group
$\Gamma<\textup{Isom}(X)$ we have:

  * (i)

$\Gamma$ has a normal, finite index subgroup $\check{\Gamma}$ which splits as
$\check{\Gamma}=\check{\Gamma}_{Y}\times\mathbb{Z}^{n}$;

  * (ii)

the projections of $\Gamma$ on $\textup{Isom}(Y)$ and
$\textup{Isom}(\mathbb{R}^{n})$ are discrete, as well as the factors
$\check{\Gamma}_{Y}$ and $\mathbb{Z}^{n}$ of $\check{\Gamma}$;

  * (iii)

$\textup{sys}^{\diamond}(\Gamma_{Y},Y)\geq\sigma_{0}$.

Moreover, if $\Gamma$ is nonsingular then $\check{\Gamma}$ can be chosen of
index $[\Gamma:\check{\Gamma}]\leq I_{0}$ and $\check{\Gamma}_{Y}$ is
nonsingular. As a result, if $\Gamma$ is nonsingular, $M=\Gamma\backslash X$
is the quotient by a finite group of isometries
$\Lambda\cong\Gamma/\check{\Gamma}$ with $|\Lambda|\leq I_{0}$ of a space
$\check{M}$ which splits isometrically as $\check{N}\times\mathbb{T}^{n}$,
where $\check{N}=\check{\Gamma}_{Y}\backslash Y$ and $\mathbb{T}^{n}$ is a
flat $n$-dimensional torus.

We also say that $\Gamma$ and $M$ virtually split as
$\check{\Gamma}_{Y}\times\mathbb{Z}^{n}$ and $\check{N}\times\mathbb{T}^{n}$,
respectively.

###### Proof.

By Proposition 2.2, the space $X$ splits isometrically as $M\times
N\times\mathbb{R}^{n}$ and
$\text{Isom}(X)\cong{\mathcal{S}}\times{\mathcal{D}}\times{\mathcal{E}}_{n}$
where ${\mathcal{S}}\cong\text{Isom}(M)$ is a semi-simple Lie group with
trivial center and no compact factors, ${\mathcal{D}}\cong\text{Isom}(N)$ is
totally disconnected and ${\mathcal{E}}_{n}\cong\text{Isom}(\mathbb{R}^{n})$.
Since $\text{Isom}(X)$ is assumed to be a Lie group then ${\mathcal{D}}$ must
be discrete. Let $\Gamma_{{\mathcal{S}}},\Gamma_{{\mathcal{D}}}$ and
$\Gamma_{{\mathcal{E}}_{n}}$ be the projections of $\Gamma$ on
${\mathcal{S}},{\mathcal{D}}$ and ${\mathcal{E}}_{n}$ respectively. By [CM19,
Theorem 2(i)] there exists a free abelian subgroup $A$ of $\Gamma$ of rank $n$
which is commensurated in $\Gamma$, and its projection $A_{{\mathcal{E}}_{n}}$
on ${\mathcal{E}}_{n}$ is a crystallographic group. Arguing as in the proof of
that theorem, we also deduce that the image $A_{{\mathcal{S}}}$ of $A$ on
${\mathcal{S}}$ is finite by Borel density. On the other hand, the projection
$A_{\mathcal{D}}$ of $A$ on ${\mathcal{D}}$ is an abelian, commensurated
subgroup of $\Gamma_{\mathcal{D}}$. Since $\Gamma_{\mathcal{D}}$ acts
cocompactly, hence minimally, on the factor $N$, and every irreducible factor
of $N$ is not Euclidean, Proposition 4 of [CM19] implies that
$A_{\mathcal{D}}$ either fixes a point or acts minimally on ${\mathcal{D}}$;
but an abelian group cannot act minimally on a proper, non-trivial,
CAT$(0)$-space without Euclidean factors, therefore $A_{\mathcal{D}}$ fixes a
point in $N$. This implies that $A_{\mathcal{D}}$ is precompact, therefore
finite, since ${\mathcal{D}}$ is discrete.
Let now $\\{a_{1},\ldots,a_{n}\\}$ be a generating set of $A$. For every
$i=1,\ldots,n$ we write
$a_{i}=(s_{i},d_{i},\tau_{i})\in{\mathcal{S}}\times{\mathcal{D}}\times{\mathcal{E}}_{n}$.
The discussion above implies that there exists $K\in\mathbb{N}^{*}$ such that
$a_{i}^{K}=(\text{id},\text{id},\tau_{i}^{K})$ for all $i$, where $\tau_{i}$
is a translation of $\mathbb{R}^{n}$. Therefore the group

$B=\\{(\text{id},\text{id},\tau)\in\Gamma\text{ s.t.
}\tau\in\text{Transl}(\mathbb{R}^{n})\\}$

is a normal, free abelian subgroup of rank $n$ of $\Gamma$. It then follows by
[CM19, Theorem 2.(ii)] that $\Gamma$ has a normal, finite index subgroup
$\check{\Gamma}$ which splits as
$\check{\Gamma}=\check{\Gamma}_{Y}\times\mathbb{Z}^{n}$, where
$\check{\Gamma}_{Y}$ is a discrete group of $Y:=M\times N$, and
$\mathbb{Z}^{n}$ acts as a lattice on $\mathbb{R}^{n}$. It is then immediate
that also the projections $\Gamma_{Y}$ and $\Gamma_{\mathbb{R}^{n}}$ of
$\Gamma$ act discretely on the respective factors. This proves (i) and (ii).
Moreover, $Y$ is again a proper, geodesically complete,
$(P_{0},r_{0})$-packed, CAT$(0)$-space and $\Gamma_{Y}$ acts
$D_{0}$-cocompactly on Y. Since $Y$ has no Euclidean factors, Theorem 4.1
implies that
$\textup{sys}^{\diamond}(\Gamma_{Y},Y)\geq\sigma_{0}:=\sigma_{P_{0},r_{0},D_{0}}(\varepsilon_{0})$,
where $\varepsilon_{0}$ is the Margulis constant, which proves (iii).
Finally, by Theorem A the possible isomorphism types of nonsingular groups
$\Gamma$ satisfying the assumptions of Theorem 6.2 are finite, so the index of
$\check{\Gamma}$ in $\Gamma$ can be supposed uniformly bounded by a constant
$I_{0}$ only depending on $P_{0},r_{0}$ and $D_{0}$. The last assertion is
consequence of (i) and of this bound. The fact that $\check{\Gamma}_{Y}$ is
nonsingular if $\Gamma$ is nonsingular is obvious. ∎

We remark that we do not know examples of proper, geodesically complete,
$\textup{CAT}(0)$-space $X$ whose isometry group is Lie, and possessing a
singular, discrete group $\Gamma<\textup{Isom}(X)$.

##  7\. Convergence and collapsing

The main purpose of this last section is to investigate the possible limits of
sequences of CAT$(0)$-groups and spaces. First of all we briefly recall the
notion of equivariant Gromov-Hausdorff convergence and ultralimits.

###  7.1. Equivariant Gromov-Hausdorff convergence and ultralimits

An isometric action on a pointed space is a triple $(X,x,\Gamma)$ where $X$ is
a proper metric space, $x\in X$ is a basepoint and $\Gamma$ is a closed (but
not necessarily discrete, nor cocompact) subgroup of the isometry group
$\text{Isom}(X)$.
An equivariant isometry between isometric actions of pointed spaces
$(X,x,\Gamma)$ and $(Y,y,\Lambda)$ is an isometry $F\colon X\to Y$ such that

  * –

$F(x)=y$;

  * –

$F_{*}\colon\textup{Isom}(X)\to\textup{Isom}(Y)$ defined by $F_{*}(g)=F\circ
g\circ F^{-1}$ is an isomorphism between $\Gamma$ and $\Lambda$.

The best known notion of convergence for isometric actions of pointed spaces
is the _equivariant pointed Gromov-Hausdorff_ convergence, as defined by K.
Fukaya in [Fuk86]: we will write
$(X_{j},x_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},x_{\infty},\Gamma_{\infty})$ for a sequence
$(X_{j},x_{j},\Gamma_{j})$ of isometric actions converging in the equivariant
pointed Gromov-Hausdorff to an isometric action
$(X_{\infty},x_{\infty},\Gamma_{\infty})$. Forgetting about the group actions
and considering just pointed metric spaces $(X_{j},x_{j})$, this convergence
reduces to the usual _pointed Gromov-Hausdorff_ convergence: we will write
$(X_{j},x_{j})\underset{\textup{pGH}}{\longrightarrow}(X_{\infty},x_{\infty})$
for a sequence of pointed metric spaces $(X_{j},x_{j})$ converging to the
pointed metric space $(X_{\infty},x_{\infty})$ in this sense. If moreover all
the spaces $X_{j}$ under consideration are compact we can even drop the
basepoints, and we will simply write
$X_{j}\underset{\textup{GH}}{\longrightarrow}X_{\infty}$ when $X_{j}$
converges to the compact metric space $X_{\infty}$.
We do not recall here the definition of equivariant, pointed Gromov-Hausdorff
convergence, since we will follow an equivalent approach which uses
ultralimits, which is more adapted to our purposes. We will present this
approach in this section, and then recall the relations with the usual Gromov-
Hausdorff convergence, referring to [Jan17] and [Cav22a]; see also [DK18] and
[CS21] for more details on ultralimits. On the other hand, we will state our
results in terms of Gromov-Hausdorff convergence, which is more commonly used
in literature.

A non-principal ultrafilter $\omega$ is a finitely additive measure on
$\mathbb{N}$ such that $\omega(A)\in\\{0,1\\}$ for every
$A\subseteq\mathbb{N}$ and $\omega(A)=0$ for every finite subset of
$\mathbb{N}$. Accordingly, we will write $\omega$-a.s. or for
$\omega$-a.e.$(j)$ in the usual measure theoretic sense. Given a bounded
sequence $(a_{j})$ of real numbers and a non-principal ultrafilter $\omega$
there exists a unique $a\in\mathbb{R}$ such that for every $\varepsilon>0$ the
set $\\{j\in\mathbb{N}\text{ s.t. }|a_{j}-a|<\varepsilon\\}$ has
$\omega$-measure $1$ (cp. [DK18, Lemma 10.25]). The real number $a$ is then
called the ultralimit of the sequence $a_{j}$ and it is denoted by
$\omega$-$\lim a_{j}$.

A non-principal ultrafilter $\omega$ being given, one can define the
ultralimit pointed metric space
$(X_{\omega},x_{\omega})=\omega$-$\lim(X_{j},x_{j})$ of any sequence of
pointed metric spaces $(X_{j},x_{j})$:
– first, one says that a sequence $(y_{j})$, where $y_{j}\in X_{j}$ for every
$j$, is admissible if there exists $M$ such that $d(x_{j},y_{j})\leq M$ for
$\omega$-a.e.$(j)$;
– then, one defines $(X_{\omega},x_{\omega})$ as set of admissible sequences
$(y_{j})$ modulo the relation $(y_{j})\sim(y_{j}^{\prime})$ if and only if
$\omega$-$\lim d(y_{j},y_{j}^{\prime})=0$.
The point of $X_{\omega}$ defined by the class of the sequence $(y_{j})$ is
denoted by $y_{\omega}=\omega$-$\lim y_{j}$. Finally, the formula
$d(\omega$-$\lim y_{j},\omega$-$\lim y_{j}^{\prime})=\omega$-$\lim
d(y_{j},y_{j}^{\prime})$ defines a metric on $X_{\omega}$ which is called the
ultralimit distance on $X_{\omega}$.

Using a non-principal ultrafilter $\omega$, one can also talk of limits of
isometries and of isometry groups of pointed metric spaces. A sequence of
isometries $g_{j}$ of pointed metric spaces $(X_{j},x_{j})$ is admissible if
there exists $M\geq 0$ such that $d(g_{j}x_{j},x_{j})\leq M$ $\omega$-a.s. Any
such sequence defines a limit isometry $g_{\omega}=\omega$-$\lim g_{j}$ of
$X_{\omega}=\omega$-$\lim(X_{j},x_{j})$ by the formula:
$g_{\omega}y_{\omega}=\omega$-$\lim g_{j}y_{j}$ ([DK18, Lemma 10.48]). Given a
sequence of groups $\Gamma_{j}<\text{Isom}(X_{j})$ we set

$\Gamma_{\omega}=\\{\omega\text{-}\lim g_{j}\text{ s.t.
}g_{j}\in\Gamma_{j}\text{ for }\omega\text{-a.e.}(j)\\}.$

In particular the elements of $\Gamma_{\omega}$ are ultralimits of admissible
sequences.

######  Lemma 7.1 ([Cav22a], Lemma 3.7).

The composition of admissible sequences of isometries is an admissible
sequence of isometries and the limit of the composition is the composition of
the limits.

Therefore, one has a well-defined composition law on $\Gamma_{\omega}$: if
$g_{\omega}=\omega$-$\lim g_{j}$ and $h_{\omega}=\omega$-$\lim h_{j}$ we set
$g_{\omega}\,\circ\,h_{\omega}:=\omega\text{-}\lim(g_{j}\circ h_{j}).$ With
this operation $\Gamma_{\omega}$ becomes a group of isometries of
$X_{\omega}$, which we call the ultralimit group of the sequence of groups
$\Gamma_{j}$. Notice that if $X_{\omega}$ is proper then $\Gamma_{\omega}$ is
always a closed subgroup of isometries of $X_{\omega}$ [Cav22a, Proposition
3.8].

In conclusion, a non-principal ultrafilter $\omega$ being given, for any
sequence of isometric actions on pointed spaces $(X_{j},x_{j},\Gamma_{j})$
there exists an ultralimit isometric action on a pointed space

$(X_{\omega},x_{\omega},\Gamma_{\omega})=\omega\\!-\\!\lim(X_{j},x_{j},\Gamma_{j}).$

We now turn to the equivalence between the ultralimit approach and the Gromov-
Hausdorff convergence, which is resumed by the following result:

######  Proposition 7.2 ([Cav22a], Proposition 3.13 & Corollary 3.14).

Let $(X_{j},x_{j},\Gamma_{j})$ be a sequence of isometric actions of pointed
spaces:

  * (i)

if $(X_{j},x_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},x_{\infty},\Gamma_{\infty})$, then
$(X_{\omega},x_{\omega},\Gamma_{\omega})\cong(X_{\infty},x_{\infty},\Gamma_{\infty})$
for every non-principal ultrafilter $\omega$;

  * (ii)

reciprocally, if $\omega$ is a non-principal ultrafilter and
$(X_{\omega},x_{\omega})$ is proper, then
$(X_{j_{k}},x_{j_{k}},\Gamma_{j_{k}})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\omega},x_{\omega},\Gamma_{\omega})$ for some
subsequence $\\{{j_{k}}\\}$.
Moreover, if for every non-principal ultrafilter $\omega$ the ultralimit
$(X_{\omega},x_{\omega},\Gamma_{\omega})$ is equivariantly isometric to the
same isometric action $(X,x,\Gamma)$, with $X$ proper, then
$(X_{j},x_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X,x,\Gamma)$.

Notice that this statement reduces to the standard relations between pointed
Gromov-Hausdorff convergence (resp. simple Gromov-Hausdorff convergence) and
ultralimits as studied for instance in [Jan17], when $\Gamma_{j}=1$ for all
$j$ (resp. when the spaces $X_{j}$ are compact).

The sequence $(X_{j},x_{j},\Gamma_{j})$ is called $D$-cocompact if each
$\Gamma_{j}$ is $D$-cocompact. It is not difficult to show that the ultralimit
of a sequence of isometric actions on pointed spaces does not depend on the
choice of the basepoints, provided that the actions have uniformly bounded
codiameter:

######  Lemma 7.3.

Let $(X_{j},x_{j},\Gamma_{j})$ be a sequence of isometric actions on pointed
spaces, which are all $D$-cocompact and let $x_{j}^{\prime}\in X_{j}$ be
different basepoints. Then, for any non-principal ultrafilter $\omega$, the
ultralimit of $(X_{j},x_{j},\Gamma_{j})$ is equivariantly isometric to the
ultralimit of $(X_{j},x_{j}^{\prime},\Gamma_{j})$.

Therefore, when considering the convergence of uniformily cocompact isometric
actions, we will often omit the basepoint $x$, if unnecessary for our
arguments.
Finally, we remark that the equivariant pointed Gromov-Hausdorff convergence
of a sequence of isometric actions with uniformly bounded codiameter implies
the pointed Gromov-Hausdorff convergence of the quotients.

######  Lemma 7.4 ([Fuk86], Theorem 2.1).

Let $(X_{j},\Gamma_{j})$ be a given sequence of $D$-cocompact isometric
actions.
If $(X_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},\Gamma_{\infty})$ then $\Gamma_{j}\backslash
X_{j}=:M_{j}\underset{\textup{GH}}{\longrightarrow}M_{\infty}:=\Gamma_{\infty}\backslash
X_{\infty}$.

Of course, even if the groups $\Gamma_{j}$ are all discrete, the group
$\Gamma_{\infty}$ may be not discrete and the structure of the quotient
$\Gamma_{\infty}\backslash X_{\infty}$ is not clear at all. Our next goal is
to clarify what happens in our specific setting.

With this purpose, we now precisely define the following classes of isometric
actions of CAT$(0)$-groups we are interested in:

  * -

$\textup{CAT}_{0}(D_{0})$: this is the set of isometric actions $(X,\Gamma)$
where $X$ is a proper, geodesically complete, CAT$(0)$-space, and $\Gamma$ is
a $D_{0}$-cocompact, discrete subgroup of $\text{Isom}(X)$;

  * -

$\textup{CAT}_{0}(P_{0},r_{0},D_{0})$: the subset of $\textup{CAT}_{0}(D_{0})$
made of the actions $(X,\Gamma)$ such that $X$ is, moreover,
$(P_{0},r_{0})$-packed.

We will denote by ${\mathcal{O}}\text{-CAT}_{0}(D_{0})$,
${\mathcal{O}}\text{-CAT}_{0}(P_{0},r_{0},D_{0})$ the respective classes of
quotients $M=\Gamma\backslash X$: these are compact CAT$(0)$-orbispaces, with
$(X,\Gamma)$ respectively in $\text{CAT}_{0}(D_{0})$ and
$\text{CAT}_{0}(P_{0},r_{0},D_{0})$.

We say that a sequence of spaces $X_{j}$ (or a sequence of actions on $X_{j}$)
is uniformly packed if there exists $(P_{0},r_{0})$ such that every $X_{j}$ is
$(P_{0},r_{0})$-packed. By [CS21, Theorem 6.1], the class of proper,
geodesically complete, pointed $\textup{CAT}(0)$-spaces $(X_{j},x_{j})$ which
are $(P_{0},r_{0})$-packed is closed under ultralimits and compact with
respect to the pointed Gromov-Hausdorff convergence. Moreover, the proof of
[Cav22a, Lemma 5.4] implies that

$\textup{CAT}_{0}(D_{0})=\bigcup_{P_{0},r_{0}}\textup{CAT}_{0}(P_{0},r_{0},D_{0}).$

More precisely, we have:

######  Proposition 7.5.

A subset $K\subseteq\textup{CAT}_{0}(D_{0})$ is precompact (with respect to
the equivariant pointed Gromov-Hausdorff convergence) if and only if there
exist $P_{0},r_{0}>0$ such that
$K\subseteq\textup{CAT}_{0}(P_{0},r_{0},D_{0})$.

###### Proof.

One direction is provided by [Cav22a, Lemma 5.8], which shows that if
$(X_{j},x_{j},\Gamma_{j})$ is a sequence of $D_{0}$-cocompact isometric
actions on pointed spaces which converges in the equivariant pointed Gromov-
Hausdorff sense to some space $(X_{\infty},x_{\infty},\Gamma_{\infty})$, then
the sequence $(X_{j},x_{j},\Gamma_{j})$ is uniformly packed. The other
implication follows by [CS21, Theorem 6.4] and by the assumption of
$D_{0}$-cocompactness. ∎

Let us consider a sequence of isometric actions $(X_{j},\Gamma_{j})$ in
$\textup{CAT}_{0}(D_{0})$, $(X_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},\Gamma_{\infty})$. Our goal is to describe
the limit group $\Gamma_{\infty}$ acting on $\Gamma_{\infty}$ and the quotient
space $\Gamma_{\infty}\backslash X_{\infty}$.

######  Definition 7.6 (Standard setting of convergence).

We say that we are in the _standard setting of convergence_ when we have a
sequence $(X_{j},\Gamma_{j})$ in $\textup{CAT}_{0}(P_{0},r_{0},D_{0})$ such
that $(X_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},\Gamma_{\infty})$.
We say that we are in the _nonsingular_ , standard setting of convergence if
we are in the standard setting of convergence and each group $\Gamma_{j}$ is
nonsingular.
We will denote by $M_{j}=\Gamma_{j}\backslash X_{j}$ and
$M_{\infty}=\Gamma_{\infty}\backslash X_{\infty}$ the quotient spaces. The
standard setting of convergence (or, equivalently, the convergence of the
isometric actions $(X_{j},\Gamma_{j})$ and of the orbispaces $M_{j}$) will be
called:

  * -

_without collapsing_ if
$\limsup_{j\to+\infty}\textup{sys}^{\diamond}(\Gamma_{j},X_{j})>0$;

  * -

_with collapsing_ if
$\liminf_{j\to+\infty}\textup{sys}^{\diamond}(\Gamma_{j},X_{j})=0$.

These cases are not apparently mutually exclusive, but indeed they are.
Actually, the two cases are respectively equivalent to the the conditions that
the dimension of the limit $M_{\infty}=\Gamma_{\infty}\backslash X_{\infty}$
equals the dimension of the quotients $M_{j}$ or decreases, as we will prove
in Subsection 7.5 (Theorem 7.15). This is a very intuitive but not trivial
result, and will follow by a careful analysis of both the collapsed and non-
collapsed case.

###  7.2. Almost stabilizers

Recall the function
$\sigma_{P_{0},r_{0},D_{0}}:(0,\varepsilon_{0}]\rightarrow(0,\varepsilon_{0}]$
given by Theorem 4.1, where $\varepsilon_{0}$ is the Margulis constant, and
the constant $\sigma_{0}=\sigma_{P_{0},r_{0},D_{0}}(\varepsilon_{0})$ of
Theorem 4.1 (only depending on the packing constants $(P_{0},r_{0})$).
Remark that taking $\varepsilon=\varepsilon_{0}$, Theorem 4.1.(iv) yields, for
any $x\in X$, a slice $\\{y\\}\times\mathbb{R}^{k}$ which is preserved by
$\overline{\Gamma}_{\sigma_{0}}(x)$ (as
$\sigma_{0}<\varepsilon_{0}^{\ast}<\varepsilon_{0}$), provided that the free
systole of $\Gamma$ is sufficently small so that $X$ splits as
$Y\times\mathbb{R}^{k}$. In the study of converging sequences it will be
useful to know that, for at least a specific point $x$, the slice can be
chosen to be the one passing through $x$. This is proved by the following:

######  Proposition 7.7.

Let $X$ be a proper, geodesically complete, $(P_{0},r_{0})$-packed
$\textup{CAT}(0)$-space, and let $\Gamma$ be any discrete, $D_{0}$-cocompact
subgroup of $\textup{Isom}(X)$. Assume that
$\textup{sys}^{\diamond}(\Gamma,X)<\sigma_{0}$, so that $X$ splits as
$Y\times\mathbb{R}^{k}$ by Theorem 4.1. Then there exists $x_{0}=(y_{0},{\bf
v})\in X$ such that $\overline{\Gamma}_{\sigma_{0}}(x_{0})$ preserves
$\\{y_{0}\\}\times\mathbb{R}^{k}$.

The proof is based on a maximality argument, similar to the proof of [CS22,
Theorem 3.1]. We just need a very basic fact:

######  Lemma 7.8 ([CS22, Lemma 3.3]).

For every $x\in X$ and $r>0$ there exists an open set $U\ni x$ such that
$\overline{\Sigma}_{r}(y)\subseteq\overline{\Sigma}_{r}(x)$ for all $y\in U$.

###### Proof of Proposition 7.7.

We introduce a partial order on $X$ defined by

$x\preceq x^{\prime}\mbox{ if and only if
}\overline{\Sigma}_{\sigma_{0}}(x)\subseteq\overline{\Sigma}_{\sigma_{0}}(x^{\prime}).$

We can show that there is a maximal element as in the proof of [CS22, Theorem
3.1, Step 3], which we report here for the reader’s convenience:
– Lemma 7.8 implies that the function
$x\mapsto\\#\overline{\Sigma}_{\sigma_{0}}(x)$ is upper semicontinuous and
clearly $\Gamma$-invariant, so (by cocompactness) it has a maximum, say $L$;
– now, given any chain $x_{1}\preceq x_{2}\preceq\ldots$, that is
$\overline{\Sigma}_{\sigma_{0}}(x_{1})\subseteq\overline{\Sigma}_{\sigma_{0}}(x_{2})\subseteq\ldots$,
the sequence must stabilize since the cardinality of each set is at most $L$,
thus giving an upper bound for the chain; then by Zorn’s lemma there exists a
maximal point $x\in X$ for $\preceq$.
Apply Theorem 4.1.(iii) to the point $x$: there exists $y_{0}\in Y$ such that
$\overline{\Gamma}_{\sigma_{0}}(x)$ preserves
$\\{y_{0}\\}\\!\times\mathbb{R}^{k}$. If
$x\in\\{y_{0}\\}\\!\times\mathbb{R}^{k}$, then we set $x_{0}=x$ and there is
nothing more to prove. Otherwise call $x_{0}$ the projection of $x$ on the
closed, convex, $\overline{\Gamma}_{\sigma_{0}}(x)$-invariant subset
$\\{y_{0}\\}\\!\times\mathbb{R}^{k}$. Let $c\colon[0,d(x,x_{0})]\to X$ be the
geodesic $[x,x_{0}]$ and set

$T=\sup\\{t\in[0,d(x,x_{0})]\text{ s.t.
}\overline{\Gamma}_{\sigma_{0}}(c(t))=\overline{\Gamma}_{\sigma_{0}}(x)\\}.$

By definition $T\geq 0$ and we claim that $T=d(x,x_{0})$ and it is a maximum.
In fact, assume
$\overline{\Gamma}_{\sigma_{0}}(c(t))=\overline{\Gamma}_{\sigma_{0}}(x)$.
Then,
$\overline{\Gamma}_{\sigma_{0}}(c(t+t^{\prime}))\subseteq\overline{\Gamma}_{\sigma_{0}}(c(t))=\overline{\Gamma}_{\sigma_{0}}(x)$
for all $t^{\prime}>0$ small enough, by Lemma 7.8. On the other hand, for
every $g\in\overline{\Gamma}_{\sigma_{0}}(x)$ we have $d(x_{0},gx_{0})\leq
d(x,gx)$ (since $g$ acts on $Y$ fixing $y_{0}$). Therefore, by the convexity
of the displacement functions we deduce that
$\overline{\Gamma}_{\sigma_{0}}(x)\subseteq\overline{\Gamma}_{\sigma}(c(t+t^{\prime}))$
too. This shows that the supremum defining $T$ is not realized, unless
$T=d(x,x_{0})$. Let now $0\leq t_{n}<T$ such that $t_{n}\to T$, so the points
$c(t_{n})$ converge to $c(T)$. By Lemma 7.8 we get
$\overline{\Sigma}_{\sigma_{0}}(c(t_{n}))\subseteq\overline{\Sigma}_{\sigma_{0}}(c(T))$
for all $n$ big enough. But
$\overline{\Sigma}_{\sigma_{0}}(c(t_{n}))=\overline{\Sigma}_{\sigma_{0}}(x)$
is maximal for $\preceq$, therefore
$\overline{\Sigma}_{\sigma_{0}}(x)=\overline{\Sigma}_{\sigma_{0}}(c(t_{n}))=\overline{\Sigma}_{\sigma}(c(T))$
for all $n$ big enough. This implies that $T=d(x,x_{0})$ is actually a
maximum. Hence,
$\overline{\Gamma}_{\sigma_{0}}(x_{0})=\overline{\Gamma}_{\sigma_{0}}(x)$, and
$x_{0}$ is the point we are looking for. ∎

Another consequence of the work that we developed in Section 4 is the
following control of the almost stabilizers, which will be useful in studying
converging sequences. Recall the constant $J_{0}$ introduced at the beginning
of Section 4. Then:

######  Corollary 7.9.

Let $X$ be any proper, geodesically complete, $(P_{0},r_{0})$-packed
$\textup{CAT}(0)$-space, and let $\Gamma$ be any discrete, $D_{0}$-cocompact
subgroup of $\textup{Isom}(X)$. Let
$\sigma(\Gamma,X)\\!:\\!=\\!\sigma_{P_{0},r_{0},D_{0}}(\varepsilon^{\diamond})$
be the constant obtained for
$\varepsilon^{\diamond}\\!=\\!\min\left\\{\varepsilon_{0},\frac{\textup{sys}^{\diamond}(\Gamma,X)}{4J_{0}}\right\\}$.
Then

  * (i)

the almost stabilizers $\overline{\Gamma}_{\sigma(\Gamma,X)}(x)$ are finite,
for all $x\in X$;

  * (ii)

there exists $x_{0}\in X$ such that
$\overline{\Gamma}_{\sigma(\Gamma,X)}(x_{0})$ fixes $x_{0}$.

Observe that, since by (i) the group $\overline{\Gamma}_{\sigma(\Gamma,X)}(x)$
is finite, then it has a fixed point (since it acts on a CAT$(0)$-space, cp.
[BH13, Chapter II, Corollary 2.8]). Here we are saying that for at least one
specific point $x_{0}$ we have that
$\overline{\Gamma}_{\sigma(\Gamma,X)}(x_{0})$ fixes exactly $x_{0}$.

###### Proof of Corollary 7.9.

We first prove part (i), and set, for short, $\sigma=\sigma(\Gamma,X)$.
Suppose that there exists $x\in X$ such that $\overline{\Gamma}_{\sigma}(x)$
is not finite: since $\overline{\Gamma}_{\sigma}(x)$ is finitely generated and
virtually abelian, then it contains a hyperbolic isometry, so
$\text{rk}(\overline{\Gamma}_{\sigma}(x))\geq 1$. Then, we can repeat all the
arguments in the proof of Theorem 4.1 (namely, the construction of the
sequence $\varepsilon_{i}$, and Propositions 4.3 & 4.5) for
$\varepsilon=\varepsilon^{\diamond}$ to show that $X$ splits isometrically and
$\Gamma$-invariantly as $Y\times\mathbb{R}^{k}$, with $k\geq 1$, and that
there exists $\varepsilon^{\ast}\in\left(\sigma,\varepsilon^{\diamond}\right)$
such that $\overline{\Gamma}_{\varepsilon^{\ast}}(x)$ projects on
$\text{Isom}(\mathbb{R}^{k})$ as a crystallographic group. Moreover, we can
find elements of $\overline{\Sigma}_{4J_{0}\cdot\varepsilon^{\ast}}(x)$ that
project to translations on $\text{Isom}(\mathbb{R}^{k})$. But these elements
are hyperbolic isometries of $\Gamma$ with translation length at most
$4J_{0}\cdot\varepsilon^{\ast}<\textup{sys}^{\diamond}(\Gamma,X)$, a
contradiction.
Assertion (ii) follows from an argument similar to that of Proposition 7.7: we
consider the same partial order $x\preceq
x^{\prime}\Leftrightarrow\overline{\Sigma}_{\sigma}(x)\subseteq\overline{\Sigma}_{\sigma}(x^{\prime})$,
and find a maximal element $x$. The group $\overline{\Gamma}_{\sigma}(x)$
being finite, the closed, convex set
$\text{Fix}(\overline{\Gamma}_{\sigma}(x))$ is not empty. If
$x\in\text{Fix}(\overline{\Gamma}_{\sigma}(x))$, then $x_{0}=x$ and there is
nothing more to prove. Otherwise we call $x_{0}$ the projection of $x$ on
$\text{Fix}(\overline{\Gamma}_{\sigma}(x))$, consider again the geodesic
$c\colon[0,d(x,x_{0})]\to[x,x_{0}]\subset X$ and we show as in Proposition 7.7
that

$T:=\sup\\{t\in[0,d(x,x_{0})]\text{ s.t.
}\overline{\Gamma}_{\sigma}(c(t))=\overline{\Gamma}_{\sigma}(x)\\}=d(x,x_{0})$

by Lemma 7.8 and the convexity of the displacement function. Moreover, again
by Lemma 7.8 and by the maximality of $x$ it follows that $T=d(x,x_{0})$ is a
maximum, hence
$\overline{\Gamma}_{\sigma}(x_{0})=\overline{\Gamma}_{\sigma}(x)$, and $x_{0}$
is the announced fixed point. ∎

###  7.3. Convergence without collapsing

######  Theorem 7.10.

Assume that we are in the standard setting of convergence of
$(X_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},\Gamma_{\infty})$, without collapsing. Then:

  * (i)

the limit group $\Gamma_{\infty}$ is closed and totally disconnected;

  * (ii)

moreover, if all the the groups $\Gamma_{j}$ are nonsingular, then
$\Gamma_{\infty}$ is discrete and isomorphic to $\Gamma_{j}$ for $j$ big
enough.

Furthermore, in case (ii), the limit of the orbispaces
$M_{j}=\Gamma_{j}\backslash X_{j}$ is isometric to the orbispace
$M_{\infty}:=\Gamma_{\infty}\backslash X_{\infty}$.

###### Proof of Theorem 7.10.

We choose a subsequence $\\{j_{h}\\}$ be for which the limit
$\lim_{h\to+\infty}\text{sys}^{\diamond}(\Gamma_{j_{h}},X_{j_{h}})>0$ exists,
and we fix a non-principal ultrafilter $\omega$ with $\omega(\\{j_{h}\\})=1$
([Jan17, Lemma 3.2]). Therefore we assume that

$\textup{sys}^{\diamond}(\Gamma_{j},X_{j})\geq\varepsilon>0\mbox{ for
}\omega\mbox{-a.e.}(j).$

By Proposition 7.2, it is enough to show the thesis for the ultralimit group
$\Gamma_{\omega}$. We may assume that $\varepsilon\leq\varepsilon_{0}$, so let
$\sigma=\sigma_{P_{0},r_{0},D_{0}}(\frac{\varepsilon}{4J_{0}})$ be the
constant provided by Theorem 4.1. By Lemma 7.3 the limit triple does not
depend on the choice of the basepoints, so we may assume that the basepoints
$x_{j}$ are those provided by Corollary 7.9(ii), that is the almost
stabilizers $\overline{\Gamma}_{\frac{\sigma}{2}}(x_{j})<\Gamma_{j}$ fix the
basepoint $x_{j}$ for $\omega$-a.e.$(j)$. We have to show that
$\Gamma_{\omega}^{\circ}=\\{\text{id}\\}$.
It is clear that if $g_{\omega}=\omega$-$\lim
g_{j}\in\overline{\Sigma}_{\frac{\sigma}{2}}(x_{\omega})$ then
$g_{j}\in\overline{\Sigma}_{\sigma}(x_{j})$ for $\omega$-a.e.$(j)$. Therefore,
$g_{j}x_{j}=x_{j}$ by our choice of $x_{j}$, so
$g_{\omega}x_{\omega}=x_{\omega}$. Notice that the subset
$\overline{\Sigma}_{\frac{\sigma}{2}}(x_{\omega})\subset\Gamma_{\omega}$ has
non-empty interior: in fact, calling
$\varphi_{x_{\omega}}:\Gamma_{\omega}\rightarrow X_{\omega}$ the continuous
map $\varphi_{x_{\omega}}(g_{\omega})=g_{\omega}x_{\omega}$, the open set
$U=\varphi_{x_{\omega}}^{-1}(B(x_{\omega},{\frac{\sigma}{2}}))$ is contained
in $\overline{\Sigma}_{\frac{\sigma}{2}}(x_{\omega})$. Therefore
$U\cap\Gamma^{\circ}_{\omega}$ is an open neighbourhood of the identity in
$\Gamma^{\circ}_{\omega}$, and it generates $\Gamma^{\circ}_{\omega}$; it then
follows that
$\langle\Sigma_{\frac{\sigma}{2}}(x_{\omega})\rangle\cap\Gamma_{\omega}^{\circ}=\Gamma_{\omega}^{\circ}$.
In particular every element of $\Gamma_{\omega}^{\circ}$ fixes $x_{\omega}$.
The set $\text{Fix}(\Gamma_{\omega}^{\circ})$ is convex, closed, non-empty and
$\Gamma_{\omega}$-invariant because $\Gamma_{\omega}^{\circ}$ is a normal
subgroup of $\Gamma_{\omega}$. The action of $\Gamma_{\omega}$ on $X_{\omega}$
is clearly $D_{0}$-cocompact, hence minimal, so
$\text{Fix}(\Gamma_{\omega}^{\circ})=X_{\omega}$ and
$\Gamma_{\omega}^{\circ}=\\{\text{id}\\}$. The fact that $\Gamma_{\omega}$ is
closed follows by [Cav22a, Proposition 3.8].
To prove part (ii) first recall that by Theorem 3.1, up to replacing
$\varepsilon$ with a smaller number (and possibly changing again the basepoint
by Lemma 7.3), we also have $\text{sys}(\Gamma_{j},x_{j})\geq\varepsilon>0$
for $\omega$-a.e.$(j)$. It is then easy to deduce that
$\textup{sys}(\Gamma_{\omega},x_{\omega})\geq\varepsilon$ too (actually, if
$g_{\omega}=\omega$-$\lim g_{j}$ is any element of $\Gamma_{\omega}$, then
$d(x_{j},g_{j}x_{j})\geq\varepsilon$ for $\omega$-a.e.$(j)$, hence
$d(x_{\omega},g_{\omega}x_{\omega})\geq\varepsilon$). Since $X_{\omega}$ is
proper, this is enough to show that $\Gamma_{\omega}$ is discrete, because the
orbit $\Gamma_{\omega}x_{\omega}$ is discrete and the stabilizer of
$x_{\omega}$ is trivial.
Finally, choose $D_{0}<D<D_{0}+1$ such that
$d(x_{\omega},g_{\omega}x_{\omega})\neq 2D$ for all
$g_{\omega}\in\Gamma_{\omega}$, and consider the generating sets
$\Sigma_{j}:=\overline{\Sigma}_{2D}(x_{j})$ and
$\Sigma_{\omega}:=\overline{\Sigma}_{2D}(x_{\omega})$ for $\Gamma_{j}$ and
$\Gamma_{\omega}$, respectively. It is classical that the spaces $X_{j}$ and
$X_{\omega}$ are respectively quasi-isometric to the Cayley graphs of
$\text{Cay}(\Gamma_{j},\Sigma_{j})$ and
$\text{Cay}(\Gamma_{\omega},\Sigma_{\omega})$ with the respective word metrics
$d_{\Sigma_{j}}$ and $d_{\Sigma_{\omega}}$, namely

$\frac{1}{2D}\cdot d(gx_{j},g^{\prime}x_{j})\leq
d_{\Sigma_{j}}(g,g^{\prime})\leq\frac{1}{2(D-D_{0})}d(gx_{j},g^{\prime}x_{j})+1$

and analogously for $d_{\Sigma_{\omega}}$ and $X_{\omega}$. From this, one can
prove exactly as in [Cav22a, Proposition 7.6] that

$\big{(}\text{Cay}(\Gamma_{\omega},\Sigma_{\omega}),\\{\text{id}\\}\big{)}=\omega\text{-}\lim\big{(}\text{Cay}(\Gamma_{j},\Sigma_{j}),\\{\text{id}\\}\big{)}.$

But, since by Proposition 5.1 we only have finitely many marked groups
$(\Gamma_{j},\Sigma_{j})$, this clearly implies that the
$(\Gamma_{j},\Sigma_{j})$’s must be all isomorphic for $j$ big enough, in
particular isomorphic to $(\Gamma_{\omega},\Sigma_{\omega})$.
The last assertion follows from the fact that the class of proper,
geodesically complete, $(P_{0},r_{0})$-packed, pointed
$\textup{CAT}(0)$-spaces $(X_{j},x_{j})$ is closed under ultralimits (cp.
[CS21, Theorem 6.1]) and from Lemma 7.4; clearly, the diameter of the limit
$\Gamma_{\omega}\backslash X_{\omega}$ is still bounded by $D_{0}$. ∎

###  7.4. Convergence with collapsing

######  Theorem 7.11.

Assume that we are in the standard setting of convergence of
$(X_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},\Gamma_{\infty})$ with collapsing, and let
$\Gamma_{\infty}^{\circ}$ be the identity component of $\Gamma_{\infty}$.
Then:

  * (i)

$X_{\infty}$ splits isometrically and $\Gamma_{\infty}$-invariantly as
$X_{\infty}^{\prime}\times\mathbb{R}^{\ell}$, for some $\ell\geq 1$;

  * (ii)

$\Gamma_{\infty}^{\circ}=\\{\textup{id}\\}\times\textup{Transl}(\mathbb{R}^{\ell})$,
and $X_{\infty}^{\prime}=\Gamma_{\infty}^{\circ}\backslash X_{\infty}$.

In particular, $X_{\infty}^{\prime}$ is proper, geodesically complete,
$(P_{0},r_{0})$-packed and $\textup{CAT}(0)$. Moreover the quotient spaces
$M_{j}=\Gamma_{j}\backslash X_{j}$ converge to
$M_{\infty}=\Gamma_{\infty}\backslash X_{\infty}$, which is isometric to the
quotient of $X_{\infty}^{\prime}$ by the closed, totally disconnected group
$\Gamma_{\infty}^{\prime}:=\Gamma_{\infty}/\Gamma_{\infty}^{\circ}$.

Recall that all the spaces under consideration, being $(P_{0},r_{0})$ packed,
have dimension $\leq n_{0}=P_{0}/2$, by Proposition 2.4(ii). Also, recall the
constant $J_{0}$ introduced at the beginning of Section 4, which bounds the
index of the canonical lattice ${\mathcal{L}}(G)$ in any crystallographic
group $G$ of $\mathbb{R}^{k}$, for $k\leq n_{0}$.
To prove Theorem 7.11, we will use the following general fact, whose proof is
straightforward and left to the reader:

######  Lemma 7.12.

Let
$(X_{j},x_{j},\Gamma_{j}),(X_{j}^{\prime},x_{j}^{\prime},\Gamma_{j}^{\prime})$
be two sequences as above, $\omega$ be a non-principal ultrafilter. Then the
ultralimit of $(X_{j}\times
X_{j}^{\prime},(x_{j},x_{j}^{\prime}),\Gamma_{j}\times\Gamma_{j}^{\prime})$ is
equivariantly isometric to $(X_{\omega}\times
X_{\omega}^{\prime},(x_{\omega},x_{\omega}^{\prime}),\Gamma_{\omega}\times\Gamma_{\omega}^{\prime})$.

We will also need the following, whose proof is included for completeness:

######  Lemma 7.13.

Let $A$ be a group of translations of $\mathbb{R}^{k}$. Then
$A\cong\mathbb{R}^{\ell}\times\mathbb{Z}^{d}$ with $\ell+d\leq k$. Moreover,
there is a corresponding $A$-invariant metric factorization of
$\mathbb{R}^{k}$ as $\mathbb{R}^{\ell}\times\mathbb{R}^{k-\ell}$, such that
the connected component $A^{\circ}\cong\mathbb{R}^{\ell}$ can be identified
with $\textup{Transl}(\mathbb{R}^{\ell})$.

###### Proof.

Let $\tau_{\bf v}$ denote the translation of $\mathbb{R}^{k}$ by a vector $\bf
v$. Consider the connected component $A^{\circ}$ of the identity of $A$: it is
an abelian, connected group of translations. The group $A^{\circ}$ acts
transitively on the the vector subspace $V=\\{{\bf v}\in\mathbb{R}^{k}\text{
s.t. }\tau_{\bf v}\in A^{\circ}\\}$; moreover, as $A^{\circ}$ acts by
translations, the stabilizer of any point ${\bf v}$ is trivial, therefore
$A^{\circ}$ can be identified with $V\cong\mathbb{R}^{\ell}$. We can split
$\mathbb{R}^{k}$ isometrically as $V\oplus V^{\perp}$, where $V^{\perp}$ is
the orthogonal complement of $V$. Since $A$ acts by translations, each
isometry $a\in A$ can be decomposed as $(a_{1},a_{2})$, with $a_{1}\in V$ and
$a_{2}\in V^{\perp}$. Notice that the isometry $(\text{id},a_{2})\in A$
because the action of $A^{\circ}$ on $V$ is transitive; so, if
$(a_{1},a_{2})\in A$, also the isometry $(a_{1},\text{id})$ belongs to $A$. In
other words, $A=A^{\circ}\times A^{\perp}$, where $A^{\perp}$ is the
projection of $A$ on $\text{Isom}(V^{\perp})$, Notice that
$A^{\perp}=A/A^{\circ}$ is totally disconnected, hence discrete and free
abelian, so it is isomorphic to some $\mathbb{Z}^{d}$ with $d\leq k-\ell$,
acting on the factor $\mathbb{R}^{k-\ell}$. ∎

###### Proof of Theorem 7.11.

As we did in the proof of Theorem 7.10 we fix a non-principal ultrafilter
$\omega$ such that $\omega$-$\lim\textup{sys}^{\diamond}(\Gamma_{j},X_{j})=0$,
and by Proposition 7.2 it is enough to show the thesis for the ultralimit
$(X_{\omega},\Gamma_{\omega})$.
By our choice of the ultrafilter $\omega$, we have
$\textup{sys}^{\diamond}(\Gamma_{j},X_{j})\leq\sigma_{0}$ for
$\omega$-a.e.$(j)$, where
$\sigma_{0}=\sigma_{P_{0},r_{0},D_{0}}(\varepsilon_{0})$ is the constant
provided by Theorem 4.1 for $\varepsilon=\varepsilon_{0}$, the Margulis
constant; so $X_{j}$ splits as $Y_{j}\times\mathbb{R}^{k_{j}}$ with $k_{j}\geq
1$ for $\omega$-a.e.$(j)$. By Lemma 7.3 the limit does not depend on the
choice of the basepoints, so we can assume that the basepoints are the points
$x_{j}=(y_{j},{\bf v_{j}})$ provided by Proposition 7.7, that is such that
$\overline{\Gamma}_{\sigma_{0}}(x_{j})$ preserves the slice
$\\{y_{j}\\}\times\mathbb{R}^{k_{j}}$. Moreover, if $k_{\omega}=\omega$-$\lim
k_{j}$, we have $k_{j}=k_{\omega}$ for $\omega$-a.e.$(j)$ and $k_{\omega}\leq
n_{0}$.
Now, choose a positive
$\varepsilon\leq\frac{1}{2}\min\left\\{\sigma_{0},\sqrt{2\sin\left(\frac{\pi}{J(k)}\right)}\right\\}$
(where $J_{0}$ is the constant introduced at the beginning of Section 4,
bounding the index of the canonical lattice ${\mathcal{L}}(G)$ in any
crystallographic group $G$ of $\mathbb{R}^{k}$, for $k\leq n_{0}$). Then, by
Lemma 2.7, we deduce that every element $g$ of a crystallographic group of
$\mathbb{R}^{k}$, for $k\leq n_{0}$, moving every point of
$B_{\mathbb{R}^{k}}({\bf O},\frac{1}{2\varepsilon})$ less than $2\varepsilon$
is a translation. We consider the open subset of $\Gamma_{\omega}$ defined by

$U(x_{\omega},\varepsilon):=\big{\\{}g_{\omega}\in\Gamma_{\omega}\text{ s.t.
}d(y_{\omega},g_{\omega}y_{\omega})<\varepsilon\text{ for all }y_{\omega}\in
B_{X_{\omega}}(x_{\omega},1/\varepsilon)\big{\\}}.$

Since $\langle U(x_{\omega},\varepsilon)\rangle$ is an open subgroup of
$\Gamma_{\omega}$ containing the identity, we have $\langle
U(x_{\omega},\varepsilon)\rangle\cap\Gamma_{\omega}^{\circ}=\Gamma_{\omega}^{\circ}$,
for every $\varepsilon>0$. Remark that if $g_{\omega}=\omega$-$\lim g_{j}$ is
in $U(x_{\omega},\varepsilon)$, then for $\omega$-a.e.$(j)$ the isometry
$g_{j}$ belongs to

$U(x_{j},2\varepsilon):=\big{\\{}g_{j}\in\Gamma_{j}\text{ s.t.
}d(y_{j},g_{j}y_{j})<2\varepsilon\text{ for all }y_{j}\in
B_{X_{j}}\left(x_{j},1/(2\varepsilon)\right)\big{\\}}.$

Every element $g_{j}\in U(x_{j},2\varepsilon)$ belongs to
$\overline{\Gamma}_{\sigma_{0}}(x_{j})$ since $2\varepsilon\leq\sigma_{0}$,
therefore it acts on $X_{j}=Y_{j}\times\mathbb{R}^{k_{\omega}}$ as
$g_{j}=(g_{j}^{\prime},g_{j}^{\prime\prime})$, where $g^{\prime}_{j}$ fixes
$y_{j}$ (because of our choice of basepoints); moreover,
$g_{j}^{\prime\prime}$ is a global translation of $\mathbb{R}^{k_{\omega}}$,
since it moves the points of $B_{\mathbb{R}^{k_{\omega}}}({\bf
O},\frac{1}{2\varepsilon})$ less than $2\varepsilon$. An application of Lemma
7.12 says that also $X_{\omega}$ splits isometrically as
$Y_{\omega}\times\mathbb{R}^{k_{\omega}}$, where $Y_{\omega}$ is the
ultralimit of the $(Y_{j},y_{j})$’s. Moreover clearly $\Gamma_{\omega}$
preserves the product decomposition. In particular every element of
$g_{\omega}\in\Gamma_{\omega}^{\circ}$ can be written as a product
$u_{\omega}(1)\cdots u_{\omega}(n)$, with each $u_{\omega}(k)$ in
$U(x_{\omega},\varepsilon)$; as $u_{\omega}(k)=\omega$-$\lim u_{j}(k)$ with
$u_{j}(k)=(u_{j}(k)^{\prime},u_{j}(k)^{\prime\prime})\in
U(x_{j},2\varepsilon)$, where $u_{j}(k)^{\prime}$ fixes $y_{j}$ and
$u_{j}(k)^{\prime\prime}$ is a translation, it follows that also $g_{\omega}$
can be written as
$(g_{\omega}^{\prime},g_{\omega}^{\prime\prime})\in\text{Isom}(Y_{\omega})\times\text{Isom}(\mathbb{R}^{k_{\omega}})$
where $g_{\omega}^{\prime}y_{\omega}=y_{\omega}$ and
$g_{\omega}^{\prime\prime}$ is a global translation (being the ultralimit of
Euclidean translations).
Let us call $p\colon\Gamma_{\omega}\to\text{Isom}(Y_{\omega})$ the projection
map. The group $p(\Gamma_{\omega}^{\circ})$ is normal in $p(\Gamma_{\omega})$.
The set of fixed points $\text{Fix}(p(\Gamma_{\omega}^{\circ}))$ is closed,
convex, non-empty and $p(\Gamma_{\omega})$-invariant (since
$p(\Gamma_{\omega}^{\circ})$ is normal in $p(\Gamma_{\omega}))$. Since
$\Gamma_{\omega}$ is clearly $D_{0}$-cocompact, so it is $p(\Gamma_{\omega})$.
Then the action of $p(\Gamma_{\omega})$ on $Y_{\omega}$ is minimal (cp.
[CM09b, Lemma 3.13]) which implies that
$\text{Fix}(p(\Gamma_{\omega}^{\circ}))=Y_{\omega}$, that is
$p(\Gamma_{\omega}^{\circ})=\\{\text{id}\\}$. We conclude that
$\Gamma_{\omega}^{\circ}$ is a connected subgroup of
$\\{\text{id}\\}\times\text{Transl}(\mathbb{R}^{k_{\omega}})$.
By Lemma 7.13, $\mathbb{R}^{k_{\omega}}$ splits isometrically as
$\mathbb{R}^{\ell}\times\mathbb{R}^{k_{\omega}-\ell}$ and
$\Gamma_{\omega}^{\circ}$ can be identified with the subgroup of translations
of the factor $\mathbb{R}^{\ell}$, for some $\ell\leq k_{\omega}$. Setting
$X_{\omega}^{\prime}:=(Y_{\omega}\times\mathbb{R}^{k_{\omega}-\ell})$, this is
still a proper, geodesically complete, $(P_{0},r_{0})$-packed,
$\text{CAT}(0)$-space, and clearly
$X_{\omega}^{\prime}=\Gamma_{\omega}^{\circ}\backslash X_{\omega}$. Notice
that, since $\Gamma_{\omega}^{\circ}$ is normal in $\Gamma_{\omega}$, then the
splitting $X_{\omega}=X_{\omega}^{\prime}\times\mathbb{R}^{\ell}$ is
$\Gamma_{\omega}$-invariant.
Let us now show that $\Gamma_{\omega}^{\circ}$ is non-trivial, hence $\ell\geq
1$. Actually, if $\Gamma_{\omega}^{\circ}$ was trivial then $\Gamma_{\omega}$
would be totally disconnected, and by [Cap09, Corollary 3.3], we would have
$\text{sys}^{\diamond}(\Gamma_{\omega},X_{\omega})>0$. However, we are able to
contruct hyperbolic isometries of $\Gamma_{\omega}$ with arbitrarily small
translation length, which will prove that $\Gamma_{\omega}^{\circ}$ is non-
trivial. Indeed, fix any $\lambda>0$ and an error $\delta>0$. By the
collapsing assumption we can find hyperbolic isometries $g_{j}\in\Gamma_{j}$
with $\ell(g_{j})\leq\delta$, for $\omega$-a.e.$(j)$. By
$D_{0}$-cocompactness, up to conjugating $g_{j}$ we can suppose $g_{j}$ has an
axis at distance at most $D_{0}$ from $x_{j}$. Take a power $m_{j}$ of $g_{j}$
such that $\lambda<\ell(g_{j}^{m_{j}})\leq\lambda+\delta$. Then, the sequence
$(g_{j}^{m_{j}})$ is admissible and defines a hyperbolic element of
$\Gamma_{\omega}$ whose translation length is between $\lambda$ and
$\lambda+\delta$. By the arbitrariness of $\lambda$ and $\delta$ we conclude.
Finally, the quotients $M_{j}=\Gamma_{j}\backslash X_{j}$ converge to
$M_{\infty}=\Gamma_{\infty}\backslash X_{\infty}=\Gamma_{\omega}\backslash
X_{\omega}$, by Lemma 7.4, where $\Gamma_{\omega}\backslash
X_{\omega}=(\Gamma_{\omega}/\Gamma_{\omega}^{\circ})\backslash(\Gamma_{\omega}^{\circ}\backslash
X_{\omega})=\Gamma_{\omega}^{\prime}\backslash X_{\omega}^{\prime}$.
It is easy to see that the totally disconnected group
$\Gamma_{\omega}/\Gamma_{\omega}^{\circ}$ is closed as a subgroup of
Isom$(X_{\omega}^{\prime})$. ∎

######  Remark 7.14.

In full generality, the conclusion of Theorem 7.11 cannot be improved saying
that the totally disconnected group $\Gamma_{\infty}/\Gamma_{\infty}^{\circ}$
is discrete.
Indeed, let $(X_{j},\Gamma_{j})$ be the sequence considered in the Example
6.1, where none of the groups $\Gamma_{j}$ have non-trivial, abelian,
virtually normal subgroups. The sequence converges in the pointed equivariant
Gromov-Hausdorff sense to a limit isometric action
$(X_{\infty},\Gamma_{\infty})$, where $X_{\infty}=\mathbb{R}^{2}\times T$ and
the connected component of $\Gamma_{\infty}$ is
$\Gamma_{\infty}^{\circ}=\text{Transl}(\mathbb{R}^{2})\times\\{\text{id}\\}$,
so $X_{\infty}^{\prime}=T$. The group
$\Gamma_{\infty}/\Gamma_{\infty}^{\circ}$, acting on $T$, contains the
infinite order, elliptic isometries induced by $a$ and $b$, so it is not
discrete. In particular $\Gamma_{\infty}$ is not a Lie group.
On the other hand observe that, in general, if $\Gamma_{\infty}$ is a Lie
group, then one can use the same proof of [SRZ22, Lemma 58] to produce a non-
trivial, nilpotent, normal subgroup of $\Gamma_{j}$, for $j$ big enough. This
means that the existence of non-trivial, nilpotent, normal subgroups in
$\Gamma_{j}$ for $j$ big enough, and the corresponding splitting of the groups
$\Gamma_{j}$, is strictly related to the structure of the limit group.

In Section 7.6 we will see a more precise relation between the metric limit
$M_{\infty}$ and the approximants $M_{j}$ in the standard, nonsingular setting
of convergence in the collapsing case, provided that the isometry groups
Isom$(X_{j})$ are Lie.

###  7.5. Limit dimension and the Euclidean factor

If $X\in\textup{CAT}_{0}(P_{0},r_{0},D_{0})$ then dim$(X)\leq n_{0}=P_{0}/2$,
by Proposition 2.4(ii). Moreover, if
$(X_{j},x_{j})\underset{\textup{pGH}}{\longrightarrow}(X_{\infty},x_{\infty})$
then, by [CS21, Theorem 6.5]

$\text{dim}(X_{\infty})\leq\liminf_{j\rightarrow+\infty}\text{dim}(X_{j}).$

The following theorem precisely relates the collapsing (as defined in 7.6) in
the standard setting of convergence to the dimension of the limit quotients,
and is a direct consequence of the convergence Theorems 7.10 & 7.11.

######  Theorem 7.15 (Characterization of collapsing).

In the standard setting of convergence of
$(X_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},\Gamma_{\infty})$, let
$M_{j}=\Gamma_{j}\backslash X_{j}$ and $M_{\infty}=\Gamma_{\infty}\backslash
X_{\infty}$ be the quotient spaces. Then:

  * (i)

the sequence is non-collapsing if and only if
$\textup{TD}(M_{\infty})=\lim_{j\to+\infty}\textup{TD}(M_{j})$;

  * (ii)

the sequence is collapsing if and only if
$\textup{TD}(M_{\infty})<\lim_{j\to+\infty}\textup{TD}(M_{j})$.

Moreover, in the above characterization, the topological dimension TD can be
replaced by the Hausdorff dimension HD.

We just need the following additional fact in order to establish the
persistence of the dimension.

######  Lemma 7.16.

Let $X$ be a proper, geodesically complete, $\textup{CAT}(0)$-space and
$\Gamma<\textup{Isom}(X)$ be closed, cocompact and totally disconnected. Then
the quotient metric space $M=\Gamma\backslash X$ satisfies

$\textup{HD}(M)=\textup{HD}(X)=\textup{TD}(X)=\textup{TD}(M).$

###### Proof.

The quotient map $p\colon X\to M$ is open because it is the quotient under a
group action. Moreover the fiber $p^{-1}([y])$ is discrete for all $[y]\in
M$.Indeed, suppose there are points $g_{j}y$ accumulating to $x\in X$, with
$g_{j}\in\Gamma$. By Ascoli-Arzelà we can extract a subsequence, which we
denote again $(g_{j})$, converging to $g_{\infty}\in\Gamma$ with respect to
the compact-open topology. Since $X$ is geodesically complete, then the
stabilizer $\text{Stab}_{\Gamma}(y)$ is open (see [CM09b, Theorem 6.1]). In
particular, $g_{\infty}^{-1}g_{j}y=y$ and we have $g_{j}y=g_{\infty}y$ for
$j\gg 0$. This shows that the fibers are discrete. Then, by [Eng78, Theorem
1.12.7], we deduce that $\text{TD}(X)=\text{TD}(M)$. On the other hand, since
the projection $p$ is $1$-Lipschitz, we have
$\textup{HD}(M)\leq\textup{HD}(X)$. Moreover, $\textup{HD}(X)=\textup{TD}(X)$
because of [LN19, Theorem 1.1]. As $\textup{TD}(M)\leq\textup{HD}(M)$ always
holds, we get

$\textup{TD}(M)\leq\textup{HD}(M)\leq\textup{HD}(X)=\textup{TD}(X)=\textup{TD}(M).\qed$

###### Proof of Theorem 7.15.

Since the groups are all $D_{0}$-cocompact, by [CS21, Proposition 6.5] we have

$\text{TD}(X_{\infty})=\text{HD}(X_{\infty})=\lim_{j\to+\infty}\text{HD}(X_{j})=\lim_{j\to+\infty}\text{TD}(X_{j}),$

so the limit exists. Moreover,
$\text{TD}(M_{j})=\text{HD}(M_{j})=\text{TD}(X_{j})=\text{HD}(X_{j})$ for
every $j$ by Lemma 7.16. Therefore, we also have

$\text{TD}(X_{\infty})=\text{HD}(X_{\infty})=\lim_{j\to+\infty}\text{HD}(M_{j})=\lim_{j\to+\infty}\text{TD}(M_{j}).$

Suppose first that the sequence is non-collapsing. In this case $M_{\infty}$
is isometric to the quotient of $X_{\infty}$ by a closed, cocompact, totally
disconnected group by Theorem 7.10. Hence
$\text{TD}(M_{\infty})=\text{TD}(X_{\infty})=\lim_{j\to+\infty}\text{TD}(M_{j})$
by Lemma 7.16, and similarly for the Hausdorff dimension.
Suppose now that the sequence is collapsing. Theorem 7.11 says that
$M_{\infty}$ is the quotient of a proper, geodesically complete, CAT0-space
$X^{\prime}_{\infty}$ of dimension strictly smaller than the dimension of
$X_{\infty}$, by a closed, cocompact, totally disconnected group. Again Lemma
7.16 then implies that

$\text{TD}(M_{\infty})=\text{HD}(M_{\infty})<\text{HD}(X_{\infty})=\text{TD}(X_{\infty})=\lim_{j\to+\infty}\text{TD}(M_{j})$

and similarly for the Hausdorff dimension.∎

Notice that since we proved that $\lim_{j\to+\infty}\text{TD}(M_{j})$ exists,
Theorem 7.15 excludes to have sequences $(X_{j},x_{j},\Gamma_{j})$ converging
with mixed behaviour (that is, such that along some subsequence the
convergence is collapsed, and along other subsequences it is non-collapsed).

Another consequence of the analysis of convergence in the standard,
nonsingular setting, combined with the renormalization Theorem E, is the
persistence of the Euclidean factor at the limit:

######  Corollary 7.17 (Euclidean factor of limits).

In the standard, nonsingular setting of convergence of
$(X_{j},\Gamma_{j})\underset{\textup{eq-
pGH}}{\longrightarrow}(X_{\infty},\Gamma_{\infty})$, let $k_{j}$ be the
dimension of the Euclidean factor of $X_{j}$. Then
$k_{\infty}:=\lim_{j\rightarrow+\infty}k_{j}$ exists and equals the dimension
of the Euclidean factor of $X_{\infty}$.

###### Proof.

By Proposition 7.2 it is sufficient to prove that for any ultrafilter $\omega$
the ultralimit $k_{\omega}=\omega$-$\lim k_{j}$ is the same and equals the
dimension of the Euclidean factor of $X_{\omega}$. By the renormalization
Theorem E and by Theorem 3.1, up to passing to an action of the groups
$\Gamma_{j}$ on new spaces $X_{j}^{\prime}$ we may assume that the sequence is
non-collapsing. Notice that this does not change the isometry type of the
limit space $X_{\omega}$ because the spaces $X_{j}^{\prime}$ provided by the
renormalization Theorem E are isometric to $X_{j}$. Then, we can apply the
convergence Theorem 7.10 and deduce that the limit group $\Gamma_{\omega}$ is
discrete (and cocompact). Now, by [CM19, Theorem 2], for any CAT$(0)$-space
$X$ possessing a discrete, cocompact group $\Gamma$, the Euclidean rank
coincides with the maximum rank of a free abelian commensurated subgroup
$A<\Gamma$. But this maximum rank is stable under ultralimits in the standard,
nonsingular setting of convergence without collapsing, since by Theorem 7.10
the groups $\Gamma_{j}$ are all isomorphic to $\Gamma_{\omega}$ for $j\gg 0$.
∎

In contrast, if the quotient spaces $M_{j}=\Gamma_{j}\backslash X_{j}$,
$M_{\infty}=\Gamma_{\infty}\backslash X_{\infty}$ split (or virtually split)
metrically maximal dimensional tori $\mathbb{T}^{k_{j}}$ and
$\mathbb{T}^{k_{\infty}}$, the dimension $k_{\infty}$ of the limit tori in the
collapsing case is generally different to the limit of the $k_{j}$’s.

###  7.6. Collapsing when the isometry groups are Lie

We prove here the convergence Theorem H stated in the introduction, where we
suppose to have a converging sequence of nonsingular CAT$(0)$-orbifolds
$M_{j}=\Gamma\backslash X_{j}$ where all the groups Isom$(X_{j})$ are Lie.
This includes, for instance, all collapsing sequences of nonpositively curved
Riemannian manifolds or, more generally, of CAT$(0)$-homology manifolds.

For the proof, we need the following

######  Lemma 7.18.

Let $(X_{j},x_{j},\Gamma_{j})$ be a sequence as above, and $\omega$ a non-
principal ultrafilter. Let $\check{\Gamma}_{j}$ be subgroups of $\Gamma_{j}$
of index $[\Gamma_{j}:\check{\Gamma}_{j}]\leq I$ for $\omega$-a.e.$(j)$. Then
$[\Gamma_{\omega}:\check{\Gamma}_{\omega}]\leq I$.

###### Proof.

Suppose to find $g_{\omega,1},\ldots,g_{\omega,{I+1}}\in\Gamma_{\omega}$ such
that $g_{\omega,m}\Gamma_{\omega}^{\prime}\neq
g_{\omega,n}\Gamma_{\omega}^{\prime}$ for all different
$m,n\in\\{1,\ldots,I+1\\}$. We write $g_{\omega,m}=\omega$-$\lim g_{j,m}$. We
can find different $m,n\in\\{1,\ldots,I+1\\}$ such that
$g_{j,m}^{-1}g_{j,n}\in\Gamma_{j}^{\prime}$ for $\omega$-a.e.$(j)$. By
definition this implies that
$g_{\omega,m}^{-1}g_{\omega,n}\in\Gamma_{\omega}^{\prime}$, a contradiction. ∎

###### Proof of Theorem H.

Let $\omega$ be any non-principal ultrafilter, and let
$(X_{\omega},\Gamma_{\omega})$ be the ultralimit of the
$(X_{j},\Gamma_{j})$’s. We know that
$\omega$-$\lim\textup{sys}^{\diamond}(\Gamma_{j},X_{j})=0$. Therefore, the
first assertion about $M_{j}$ follows from Theorem 6.2: for $j\gg 0$, the
space $X_{j}$ splits as $Y_{j}\times{\mathbb{R}}^{k_{j}}$, where
${\mathbb{R}}^{k_{j}}$ is the Euclidean factor, and there is a normal subgroup
$\check{\Gamma}_{j}=\check{\Gamma}_{Y_{j}}\times{\mathbb{Z}}^{k_{j}}\triangleleft\Gamma_{j}$
of index $[\Gamma_{j}:\check{\Gamma}_{j}]\leq I_{0}$ such that
$\check{M}_{j}=\check{\Gamma}_{j}\backslash
X_{j}=\left(\check{\Gamma}_{Y_{j}}\backslash
Y_{j}\right)\times\left({\mathbb{Z}}^{k_{j}}\backslash{\mathbb{R}}^{k_{j}}\right)=\check{N}_{j}\times{\mathbb{T}}^{k_{j}}$
and $\text{sys}^{\diamond}(\check{\Gamma}_{Y_{j}},Y_{j})\geq\sigma_{0}$.
Let $(\check{\Gamma}_{\omega},X_{\omega})$,
$(\check{\Gamma}_{Y_{\omega}},Y_{\omega})$ and
$(A_{\omega},\mathbb{R}^{k_{\omega}})$ be the ultralimits of the groups
$\check{\Gamma}_{j}$, $\check{\Gamma}_{Y_{j}}$ and $\mathbb{Z}^{k_{j}}$ acting
respectively on $X_{j}$, $Y_{j}$ and $\mathbb{R}^{k_{j}}$. By Lemma 7.12 we
have corresponding splittings
$X_{\omega}=Y_{\omega}\times{\mathbb{R}}^{k_{\omega}}$ and
$\check{\Gamma}_{\omega}=\check{\Gamma}_{Y_{\omega}}\times A_{\omega}$, with
$k_{\omega}=\omega$-$\lim k_{j}$; moreover, by Corollary 7.17,
$k_{\omega}=k_{\infty}$ is the dimension of the Euclidean factor of
$X_{\omega}$.
The groups $\check{\Gamma}_{Y_{j}}$ act nonsingularly without collapsing on
$Y_{j}$ by Theorem 6.2, which implies that $\check{\Gamma}_{Y_{\omega}}$ acts
discretely on $Y_{\omega}$, by Theorem 7.10(ii). Therefore, the
$\check{N}_{j}$’s converge without collapsing to an orbispace
$\check{N}_{\omega}=\check{\Gamma}_{Y_{\omega}}\backslash Y_{\omega}$ which is
still in ${\mathcal{NS}}$-CAT${}_{0}(P_{0},r_{0},D_{0})$. This proves (i).
Moreover, since $[\Gamma_{j}:\check{\Gamma}_{j}]\leq I_{0}$, Lemma 7.18
implies that $[\Gamma_{\omega}:\check{\Gamma}_{\omega}]\leq I_{0}$ too.
Therefore the subgroup $\check{\Gamma}_{\omega}$ is open in $\Gamma_{\omega}$
and
$\check{\Gamma}_{\omega}^{\circ}=\Gamma_{\omega}^{\circ}=\mathbb{R}^{\ell}$,
where $\ell\geq 1$ is provided by Theorem 7.11. On the other hand,
$A_{\omega}$ acts cocompactly by translations on $\mathbb{R}^{k_{\omega}}$, so
Lemma 7.13 implies that $A_{\omega}$ is isomorphic to the group
$\mathbb{Z}^{k_{\omega}-\ell^{\prime}}\times\mathbb{R}^{\ell^{\prime}}$ acting
on
$\mathbb{R}^{k_{\omega}}\cong\mathbb{R}^{k_{\omega}-\ell^{\prime}}\times\mathbb{R}^{\ell^{\prime}}$,
for some $\ell^{\prime}\geq 0$.As $\check{\Gamma}_{Y_{\omega}}$ acts
discretely on $Y_{\omega}$, it follows that
$\check{\Gamma}_{\omega}^{\circ}=A_{\omega}^{\circ}=\mathbb{R}^{\ell^{\prime}}$,
therefore $\ell=\ell^{\prime}$.
Moreover, since the groups ${\mathbb{Z}}^{k_{j}}$ converge to
$A_{\omega}=\mathbb{Z}^{k_{\omega}-\ell}\times\mathbb{R}^{\ell}$, and the
factors act separately by translations on the factors
$\mathbb{R}^{k_{\omega}-\ell}$ and $\mathbb{R}^{\ell}$ of
$\mathbb{R}^{k_{\omega}}$ , it is clear from Lemma 7.4 that the tori
${\mathbb{T}}^{k_{j}}={\mathbb{Z}}^{k_{j}}\backslash{\mathbb{R}}^{k_{j}}$
converge to the limit torus
$\mathbb{Z}^{k_{\omega}-\ell}\backslash\mathbb{R}^{k_{\omega}-\ell}={\mathbb{T}}^{k_{\omega}-\ell}$.
This proves (ii).
Now, the group
$\check{\Gamma}_{\omega}^{\prime}:=\check{\Gamma}_{\omega}/\check{\Gamma}_{\omega}^{\circ}=\check{\Gamma}_{Y_{\omega}}\times\mathbb{Z}^{k_{\omega}-\ell}$
has index $\leq I_{0}$ in
$\Gamma_{\omega}^{\prime}=\Gamma_{\omega}/\Gamma_{\omega}^{\circ}$; and, since
the first one is discrete, it follows that also $\Gamma_{\omega}^{\prime}$ is
discrete. By Theorem 7.11 it follows that the spaces $M_{j}$ converge to
$M_{\omega}=\Gamma_{\omega}^{\prime}\backslash X_{\omega}^{\prime}$, where
$X_{\omega}^{\prime}=\Gamma_{\omega}^{\circ}\backslash X_{\omega}$ hence
$M_{\omega}\in{\mathcal{NS}}$-CAT${}_{0}(P_{0},r_{0},D_{0})$, proving (iii).
Finally, remark that since $\check{\Gamma}_{j}$ is normal in $\Gamma_{j}$ for
all $j$, then $\check{\Gamma}_{\omega}$ is normal in $\Gamma_{\omega}$ (cp.
[CS20, Lemma 6.14.(iii)]), and in turns this implies that
$\check{\Gamma}_{\omega}^{\prime}\triangleleft\Gamma_{\omega}^{\prime}$.
Therefore, this space $M_{\omega}$ is the quotient of
$\check{M}_{\omega}:=\check{\Gamma}_{\omega}^{\prime}\backslash
X_{\omega}^{\prime}$ by the group
$\Lambda_{\omega}=\Gamma_{\omega}^{\prime}/\check{\Gamma}_{\omega}^{\prime}$
which has cardinality $\leq I_{0}$. This proves (iv). ∎

As a result of the convergence theorems proved in this section, we obtain the
compactness Corollary I for CAT$(0)$-homology orbifolds. Recall the classes
${\mathcal{H}}{\mathcal{O}}\text{-CAT}_{0}(D_{0})$ and
${\mathcal{H}}{\mathcal{O}}\text{-CAT}_{0}(P_{0},r_{0},D_{0})$ defined in the
introduction, consisting of all the quotients $M=\Gamma\backslash X$ with
$(X,\Gamma)$ belonging respectively to $\text{CAT}_{0}(D_{0})$ and
$\text{CAT}_{0}(P_{0},r_{0},D_{0})$, where $X$ is supposed, in addition, to be
a homology manifold.

###### Proof of Corollary I.

Let $M_{j}=\Gamma_{j}\backslash X_{j}$ be a sequence in
${\mathcal{H}}{\mathcal{O}}\text{-CAT}_{0}(P_{0},r_{0},D_{0})$. Since the
$X_{j}$ are homology manifolds, then $\Gamma_{j}$ is nonsingular and
Isom$(X_{j})$ is a Lie group for every $j$, as recalled in Section 3 and by
Proposition 2.3.
By [CS21, Theorem 6.1], the class $\text{CAT}_{0}(P_{0},r_{0},D_{0})$ is
compact with respect to the Gromov-Hausdorff convergence. Moreover, if
$(X_{j},x_{j})\underset{\textup{pGH}}{\longrightarrow}(X_{\infty},x_{\infty})$,
then $X_{\infty}$ is again a homology manifold, by [LN18, Lemma 3.3].
Now, by Theorem 7.15, the sequence $(X_{j},\Gamma_{j})$ is either non-
collapsing, or collapsing. In the first case, by Theorem 7.10 the quotients
$M_{j}$ converge to some space $M_{\infty}=\Gamma_{\infty}\backslash
X_{\infty}$, for a discrete subgroup $\Gamma_{\infty}$ of Isom$(X_{\infty})$,
hence it belongs to
${\mathcal{H}}{\mathcal{O}}\text{-CAT}_{0}(P_{0},r_{0},D_{0})$. In the second
case, the $M_{j}$’s converge to a lower dimensional limit space $M_{\infty}$,
which by Theorem H is isometric to the quotient of the space
$X_{\infty}^{\prime}$ (a proper, geodesically complete, $(P_{0},r_{0})$-packed
CAT$(0)$-space, by Theorem 7.11) by the discrete, $D_{0}$-cocompact group
$\Gamma_{\infty}^{\prime}$. Moreover, if
$X_{\infty}=Y_{\infty}\times{\mathbb{R}}^{k_{\infty}}$ is the splitting of the
Euclidean factor, we have
$X_{\infty}^{\prime}=Y_{\infty}\times{\mathbb{R}}^{k_{\infty}-\ell}$ by the
proof of Theorem 7.11 and by Corollary 7.17. By the Kunneth’s formula it then
follows that $Y_{\infty}$, and in turn $X_{\infty}^{\prime}$, are homology
manifolds; therefore, $M_{\infty}$ still belongs to
${\mathcal{H}}{\mathcal{O}}\text{-CAT}_{0}(P_{0},r_{0},D_{0})$.
Finally, the fact that ${\mathcal{H}}{\mathcal{O}}\text{-CAT}_{0}(D_{0})$ is
closed with respect to the Gromov-Hausdorff convergence follows from
Proposition 7.5 and the compactness of
${\mathcal{H}}{\mathcal{O}}\text{-CAT}_{0}(P_{0},r_{0},D_{0})$. ∎

######  Remark 7.19.

_The result above can be strengthened in the Riemannian setting. Suppose
$M_{j}=\Gamma_{j}\backslash X_{j}$ is a sequence in
$\mathcal{O}\textup{-CAT}_{0}(P_{0},r_{0},D_{0})$ converging to $M_{\infty}$,
where each $X_{j}$ is a Riemannian manifold. Then $M_{\infty}$ is isometric to
the quotient $\Gamma_{\infty}^{\prime}\backslash X_{\infty}^{\prime}$, where
$X_{\infty}^{\prime}$ is a _topological_ manifold. This follows by the same
argument of the proof of Corollary I and [LN18, Theorem 1.3]._

We finally prove the isolation results of flats in
${\mathcal{H}}{\mathcal{O}}\textup{-CAT}_{0}(P_{0},r_{0},D_{0})$.

###### Proof of Corollary J.

By assumption, the homology orbifolds $M_{j}=\Gamma_{j}\backslash X_{j}$
converge in ${\mathcal{H}}{\mathcal{O}}$-CAT${}_{0}(P_{0},r_{0},D_{0})$ to a
flat orbifold $M_{\infty}=\Gamma\backslash\mathbb{R}^{k}$, where $\Gamma$ is a
crystallographic group of $\mathbb{R}^{k}$. We fix a non-principal ultrafilter
$\omega$, and consider the ultralimit $(X_{\omega},\Gamma_{\omega})$ of the
sequence $(X_{j},\Gamma_{j})$ and the quotient orbispace
$M_{\omega}=\Gamma_{\omega}\backslash X_{\omega}$. We will treat separately
the cases where the $M_{j}$ converge collapsing or not.
Since the isometric actions $(\Gamma_{j},X_{j})$ belong to
CAT${}_{0}(P_{0},r_{0},D_{0})$, if the sequence is non-collapsing, then the
group $\Gamma_{\omega}$ is discrete by Theorem 7.10(ii). Moreover, the
ultralimit $X_{\omega}$ is again a homology manifold, so $M_{\omega}$ is a
rigid orbispace in the sense of [DLHG90]. Then, as $M_{\omega}$ is isometric
to the (rigid) orbispace $M_{\infty}$, their orbifold fundamental group are
isomorphic, hence $\Gamma_{\omega}\cong\Gamma$. Theorem 7.10 then ensures that
$\Gamma_{j}$ is isomorphic to $\Gamma_{\omega}$, and in turn to $\Gamma$, for
$j\gg 0$; in particular, $\Gamma_{j}$ is virtually abelian. Then [AB98,
Corollary C] says that $X_{j}$ is an Euclidean space, actually isometric to
$\mathbb{R}^{k}$.
On the other hand, if the sequence is collapsing, then by Theorem H we know
that the limit space $M_{\infty}$ is a finite quotient of a CAT$(0)$-orbispace
$\check{M}_{\infty}=\check{N}_{\infty}\times\mathbb{T}^{k_{\infty}}$, and that
the $M_{j}$’s virtually split as
$\check{M}_{j}\\!=\check{N}_{j}\times\mathbb{T}^{k_{j}}_{j}$ for a sequence
$\check{N}_{j}$ which converges without collapsing to $\check{N}_{\infty}$.
Notice that the orbispaces $\check{N}_{j}$ belong again to
${\mathcal{H}}{\mathcal{O}}$-CAT${}_{0}(P_{0},r_{0},D_{0})$ (since, with the
notation used in Theorem H, $X_{j}=Y_{j}\times{\mathbb{R}}^{k_{j}}$ and
$\check{N}_{j}$ is quotient of the factor $Y_{j}$, which is a homology
manifold by the Kunneth formula). Now, since $M_{\infty}$ is flat, also
$\check{N}_{\infty}$ is flat. By the proof in the non-collapsing case, we
conclude that the orbispaces $\check{N}_{j}$ are flat for $j\gg 0$, and in
turn $\check{M}_{j}$ and $M_{j}$ too.
This shows that the subset Flat$(P_{0},r_{0},D_{0})$ of
${\mathcal{H}}{\mathcal{O}}$-CAT${}_{0}(P_{0},r_{0},D_{0})$ is open.
The fact that Flat$(P_{0},r_{0},D_{0})$ is closed follows from Theorem
7.10(ii) and again [AB98, Corollary C] for a sequence of flat orbifolds
$M_{j}$ which converges without collapsing, and from Theorem H(ii) when the
$M_{j}$’s collapse.
Finally, the subset Flat$(P_{0},r_{0},D_{0})$ is clearly connected (since
every flat $M$ can be connected to a point by rescaling it by
$\lambda\rightarrow 0$, remaining in the class
$\texttt{Flat}(P_{0},r_{0},D_{0})$). Then, being a clopen subset, it is
precisely the connected component of $M_{0}=\\{\ast\\}$. ∎

Remark that, if ${\mathcal{H}}{\mathcal{O}}$-CAT${}_{0}(P_{0},r_{0})$ denotes
the class of compact CAT$(0)$-homology orbifolds $M=\Gamma\backslash X$ such
that $X$ is $(P_{0},r_{0})$-packed, since any convergent sequence has diameter
definitely bounded, the same arguments of Corollary J show that the connected
component of ${\mathcal{H}}{\mathcal{O}}$-CAT${}_{0}(P_{0},r_{0})$ is again
the subset Flat$(P_{0},r_{0})$ of its flats.

In contrast, remark that any flat $M$ can be arbitrarily well approximated in
the Gromov-Hausdorff topology by non-flat CAT$(0)$-orbispaces
$M_{j}=\Gamma_{j}\backslash X_{j}$ using non-uniformily packed $X_{j}$ (just
glue to $M$ arbitrarily small circles).

Finally, Corollary J is false in the full class of orbispaces
$\mathcal{O}$-$\textup{CAT}_{0}(P_{0},r_{0},D_{0})$. For instance the sequence
of non-flat orbispaces of Remark 7.14 converge to the flat space
$\mathbb{S}^{1}$. This sequence is even contained in
$\mathcal{L}$-$\textup{CAT}_{0}(P_{0},r_{0},D_{0})$, i.e. each space of the
sequence is locally CAT$(0$).

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