308 2

S.V. Antipin et al.

11institutetext:  Sternberg Astronomical Institute, M.V. Lomonosov Moscow
State University, Universitetskij pr., 13, Moscow, 119234, Russia, 11email:
<EMAIL_ADDRESS>22institutetext: Institute of Astronomy of the Russian
Academy of Sciences, Pyatnitskaya str., 48, Moscow, 119017, Russia
33institutetext: Department of Astronomy, University of Illinois at Urbana-
Champaign, 1002 West Green Street, Urbana, IL 61801, USA 44institutetext:
National Research University Higher School of Economics, Staraya Basmannaya
str., 21/4, Moscow, 105066, Russia

# New SU UMa-type star ZTF18abdlzhd in the Zwicky Transient Facility data

S.V. Antipin 11 A.M. Zubareva 2211 A.A. Belinski 11 M.A. Burlak 11 N.P.
Ikonnikova 11 K.L. Malanchev 3311 M.V. Kornilov 1144 E.O. Mishin 11

(2021; December 04, 2020; February 11, 2021; December 04, 2020)

###### Abstract

We carried out a search for unknown dwarf novae in a public data release of
the Zwicky Transient Facility survey and suspected that the object
ZTF18abdlzhd is a SU UMa-type star. Performed multicolor CCD observations
permit us to follow its fading from an outburst in August and an entire
superoutburst in October 2020. The duration of the superoutburst is 13 days.
We detected superhumps with period P = 0$\fd$06918(3) that are characteristic
of UGSU type stars.

###### keywords:

stars – photometry – dwarf novae – ZTF18abdlzhd

††volume: 51

##  1 Introduction

Cataclysmic variables provide opportunities to observe various accretion-
related phenomena in a setting of close binary systems. A white dwarf primary
is receiving matter via the inner Lagrangian point (L1) from a low-mass
secondary star that fills up its Roche lobe. Dwarf novae are a subclass of
cataclysmic variables where the stream of matter forms an accretion disk
around the white dwarf and the disk alters between hot (high accretion rate)
and cold (low accretion rate) states resulting in recurrent dramatic changes
in brightness.

SU UMa-type dwarf novae are known to increase their brightness by several
magnitudes for days, after which they get back to the quiescent state for a
while. Two kinds of outbursts are found in these systems, distinguished by the
outburst amplitude and duration. Short low-amplitude outbursts are called
normal, and long ones having greater amplitudes and a “plateau” phase, are
known as superoutbursts. In the course of a superoutburst, periodic brightness
variations called “superhumps” emerge. They are characterized by amplitudes up
to 0.3 mag and have periods longer than the orbital period of a system by
several percent. See more detailed information on Dwarf Novae in Warner
(1995), and specifically on UGSU-type stars in Kato et al. (2009).

We revealed the object in Zwicky Transient Facility Data Release 3 (Bellm et
al., 2019) during the dedicated search of dwarf novae. We downloaded ZTF DR3
light curves from the IRSA IPAC
server111https://irsa.ipac.caltech.edu/data/ZTF/lc_dr3/. Then we selected all
$zr$ (ZTF Sloan $r$ band, see Fig. 2 from Bellm et al. (2019)) light curves
with the following restrictions: the peak magnitude is brighter than 19.5 mag,
the amplitude is at least 1 mag, the duration is between 10 and 30 days, and
the number of observed nights is at least 10. Only good weather-condition
observations were considered. This search yielded us 15 out of 3 billion ZTF
DR3 objects.

A visual analysis of light curves was performed with the SNAD ZTF web-
viewer222https://ztf.snad.space (Malanchev et al., in
prep.333https://arxiv.org/abs/2012.01419) and the object ZTF18abdlzhd was
found as the only reliable UGSU candidate.

The outburst light curve based on ZTF data is shown in Fig. 1.

Figure 1: ZTF18abdlzhd outburst light curve from ZTF survey data. zr is for
ZTF $r$ band whose transmission curve is depicted in Fig.2 in Bellm et al.
(2019).

##  2 Observations

CCD photometry of ZTF18abdlzhd ($\alpha=23^{h}15^{m}30\fs 841,\delta=+57\degr
08\arcmin 46\farcs 98,$ J2000) carried out in August and October 2020 permit
us to follow the end of the fading stage of one outburst and a full
superoutburst of this dwarf nova. We monitored the field of the variable with
the new automated 60-cm telescope of the Caucasus Mountain Observatory of SAI
MSU. The ASA RC600 60-cm reflector was installed at the Caucasus observatory
supported by M.V. Lomonosov Moscow State University Program of Development. It
is equipped with an Andor iKon-L (DZ936N-BV) 2048 $\times$ 2048 CCD-camera and
a set of filters (see Berdnikov et al., 2020). Sloan $g$, $r$, $i$ (Fukugita
et al., 1996) and Johnson-Cusins $B$, $V$, $R_{c}$, $I_{c}$ bands were used to
explore the behaviour of ZTF18abdlzhd. Observations in $g$, $r$, $i$ were
obtained on August 19–23, 2020 (JD2459081–85, 4 nights), and CCD frames in
$B$, $V$, $R_{c}$, $I_{c}$ were taken on October 10–26, 2020 (JD2459133–149,
15 nights). The observational log is given in Table 1. The exposure times
varied from 120 seconds during the outburst to 600 seconds in minimum
brightness of the object.

Table 1: Observational log JD | Date | Number of frames (Band)
---|---|---
2459081 | August 19, 2020 | 3 ($g$), 23 ($r$), 2 ($i$)
2459082 | August 20, 2020 | 1 ($g$), 1 ($r$), 1 ($i$)
2459084 | August 22, 2020 | 1 ($r$)
2459085 | August 23, 2020 | 1 ($r$)
2459133 | October 10, 2020 | 5 ($B$), 5 ($V$), 5 ($R_{c}$), 5 ($I_{c}$)
2459134 | October 11, 2020 | 2 ($B$), 2 ($V$), 2 ($R_{c}$), 2 ($I_{c}$)
2459135 | October 12, 2020 | 2 ($B$), 2 ($V$), 2 ($R_{c}$), 2 ($I_{c}$)
2459136 | October 13, 2020 | 2 ($B$), 2 ($V$), 2 ($R_{c}$), 2 ($I_{c}$)
2459137 | October 14, 2020 | 1 ($B$), 1 ($V$), 73 ($R_{c}$), 1 ($I_{c}$)
2459138 | October 15, 2020 | 45 ($R_{c}$)
2459139 | October 16, 2020 | 43 ($R_{c}$)
2459140 | October 17, 2020 | 2 ($B$), 2 ($V$), 2 ($R_{c}$), 2 ($I_{c}$)
2459141 | October 18, 2020 | 2 ($B$), 2 ($V$), 2 ($R_{c}$), 2 ($I_{c}$)
2459144 | October 21, 2020 | 1 ($B$), 1 ($V$), 1 ($R_{c}$), 1 ($I_{c}$)
2459145 | October 22, 2020 | 1 ($B$)
2459146 | October 23, 2020 | 6 ($R_{c}$)
2459147 | October 24, 2020 | 1 ($R_{c}$)
2459148 | October 25, 2020 | 1 ($R_{c}$)
2459149 | October 26, 2020 | 1 ($R_{c}$)

To perform aperture photometry and magnitude calibration, we used
VaST444https://scan.sai.msu.ru/vast software (Sokolovsky & Lebedev, 2018). We
derived magnitudes of an ensemble of comparison stars within the field of view
from the APASS ($B$, $V$, $R_{c}$, $I_{c}$) and the PanSTARRS1 ($g$, $r$, $i$)
survey (Chambers et al., 2016).

##  3 Results

Our data cover two outbursts of ZTF18abdlzhd. Unfortunately, we have no
opportunity to classify the type of the August outburst (normal or
superoutburst, see the left panel of Fig. 2) because the observations reveal
only the final stage of the event.

Figure 2: ZTF18abdlzhd. The fading stage of the August outburst (left panel)
and the superoutburst of October 2020 (right panel).

On October 10, 2020, we found the variable in its bright state again. The
brightness of the variable star increased to the next night of observations,
so we can say that we detected the dwarf nova at the beginning of an outburst.
The total duration of this outburst is 13 days which is very similar to the
event found in ZTF data (cf. Fig. 1 and the right panel of Fig. 2).

ZTF18abdlzhd reached the maximum brightness on October, 11, $R_{c}=17.49$ mag.
Immediately after the superoutburst, the star faded to $R_{c}=21.8$ mag. In
the next few days, the dwarf nova passed beyond the detection limit (about
22.0 mag in $R_{c}$-band at 600 s exposure). Thus, the variability amplitude
is greater than 4.5 mag. We found ZTF18abdlzhd in PanSTARRS DR1 (object ID
176573488784746366) at $r$ = 22.096 $\pm$ 0.196 mag (Chambers et al., 2016)).

Color indices remain constant during the plateau phase of the outburst and are
equal to $B-V$ = 0.46 $\pm$ 0.02, $V-R_{c}$ = 0.41 $\pm$ 0.02 and $V-I_{c}$ =
0.23 $\pm$ 0.03.

During the maximum of the October 2020 outburst we found a periodic brightness
variability – superhumps – which are a distinctive feature of a superoutburst
(see Fig. 3). The most of our observations in the plateau phase of this
superoutburst were obtained in $R_{c}$ band – 161 frames for three nights
covered with photometry densely (see Table 1). For this set we removed a
linear trend from the brightness measurements before running the period search
for which WinEfk software555http://www.vgoranskij.net/software/ developed by
Dr. V.P. Goranskij was applied. The corresponding periodogram and the phased
light curve of the superhumps are given in Fig. 4. The ephemeris is as
follows:

$JD_{max}=2459137.4314+0.06918(3)\times E.$

Figure 3: Individual light curves of ZTF18abdlzhd for three nights in maximum
of the superoutburst (October 14, 15 and 16, 2020).

Figure 4: Left panel: the periodogram for three nights of observations shown
in Fig. 3. Maximum peak corresponds to the frequency 14.455(6) c/d and period
0.06918(3) days. Right panel: the corresponding phased light curve of
superhumps.

##  4 Conclusion

Based on the ZTF survey data, we suspected that object ZTF18abdlzhd is a UGSU-
type star. Our CCD observations allow us to confirm this assumption. The
duration of the October 2020 superoutburst was 13 days. We found a superhump
period P = 0$\fd$06918(3).

###### Acknowledgements.

The authors are grateful for partial support from M.V. Lomonosov Moscow State
University Program of Development. This research has been supported by the
Interdisciplinary Scientific and Educational School of Moscow University
“Fundamental and Applied Space Research”. K.L.M. and M.V.K. are supported by
RFBR grant 20-02-00779 for the preparing ZTF data and the revealing of the
object. The study was made possible through the use of the AAVSO Photometric
All-Sky Survey (APASS), funded by the Robert Martin Ayers Sciences Fund and
NSF AST-1412587.

## References

  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al., The Zwicky Transient Facility: System Overview, Performance, and First Results. 2019, PASP, 131, 018002, DOI: 10.1088/1538-3873/aaecbe
  * Berdnikov et al. (2020) Berdnikov, L. N., Belinskii, A. A., Shatskii, N. I., et al., BVIc Observations and Search for Evolutionary Variations in the Cepheid V811 Oph Period. 2020, Astronomy Reports, 64, 310, DOI: 10.1134/S1063772920040010
  * Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al., The Pan-STARRS1 Surveys. 2016, arXiv e-prints, arXiv:1612.05560
  * Fukugita et al. (1996) Fukugita, M., Ichikawa, T., Gunn, J. E., et al., The Sloan Digital Sky Survey Photometric System. 1996, AJ, 111, 1748, DOI: 10.1086/117915
  * Kato et al. (2009) Kato, T., Imada, A., Uemura, M., et al., Survey of Period Variations of Superhumps in SU UMa-Type Dwarf Novae. 2009, PASJ, 61, S395, DOI: 10.1093/pasj/61.sp2.S395
  * Sokolovsky & Lebedev (2018) Sokolovsky, K. V. & Lebedev, A. A., VaST: A variability search toolkit. 2018, Astronomy and Computing, 22, 28, DOI: 10.1016/j.ascom.2017.12.001
  * Warner (1995) Warner, B., Cataclysmic variable stars. 1995, Cambridge Astrophysics Series, 28


# Thin actions on CAT(0)-spaces

Nicola Cavallucci  and  Andrea Sambusetti  Andrea Sambusetti, Dipartimento di
Matematica “Guido Castelnuovo”, SAPIENZA Universita di Roma, Piazzale Aldo
Moro 5, I-00185 ‘ Roma.  Nicola Cavallucci, Karlsruhe Institute of Technology,
Engelstrasse 2, D-76128 Karlsruhe<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study groups of isometries of packed, geodesically complete,
CAT$(0)$-spaces for which the systole at every point is smaller than a
universal constant depending only on the packing, deducing strong rigidity
results. We show that if a space as above has some negative curvature
behaviour then it cannot support a thin action: this generalizes the classical
Margulis Lemma to a broader class of spaces.

N.Cavallucci is partially supported by the SFB/TRR 191, funded by the DFG.

A. Sambusetti is member of the Differential Geometry section of the GNSAGA-
INdAM.

##  1\. Introduction

The classical Margulis-Heintze Lemma is a cornerstone in the study of
negatively curved manifolds: it asserts that there exists some constant
$\varepsilon(n,\kappa)>0$ such that for any torsionless lattice $\Gamma$ of a
complete, Riemannian $n$-dimensional manifold $X$ with sectional curvature
$-\kappa^{2}\leq K(X)<0$ there exists a point $x$ such that

$\text{sys}(\Gamma,x):=\inf_{g\in\Gamma^{\ast}}d(x,gx)>\varepsilon(n,\kappa),$

where $\Gamma^{\ast}$ is the subset of nontrivial elements of $\Gamma$ (see
[Hei76], [BGS13], [BZ88]). Accordingly, for any discrete group $\Gamma$ acting
by isometries on a metric space $X$ we will call diastole of the action (or of
the quotient $M=\Gamma\backslash X$), the value

$\text{dias}(\Gamma,X)=\sup_{x\in X}\inf_{g\in\Gamma^{\ast}}d(x,gx).$

We say that the action of $\Gamma$ is _$\varepsilon$ -thin_ if
$\text{dias}(\Gamma,X)<\varepsilon$ (since the subset of points $x$ where
$\text{sys}(\Gamma,x)<\varepsilon$ is classically known as the
$\varepsilon$-thin subset of X).
Buyalo first, in dimension smaller than $5$ [Buy90a]-[Buy90b], and Cao-
Cheeger-Rong later, in any dimension [CCR01], studied the possibility of
extending the Margulis-Heintze Lemma to uniform lattices of non-positively
curved manifolds. They proved the following alternative: either the diastole
is bounded below by a universal positive constant $\eta(n,\kappa)>0$, or the
action of $\Gamma$ is $\eta$-thin and the manifold $\Gamma\backslash X$ admits
a so-called abelian local splitting structure. This is, roughly speaking, a
decomposition of $X$ into a union of minimal sets of hyperbolic isometries
with the additional property that if two minimal sets intersect then the
corresponding isometries commute.

######  Example 1.1 ([Gro78], Section 5 and [Buy81], Section 4).

Consider two copies $\Sigma_{1},\Sigma_{2}$ of a hyperbolic surface with
connected, geodesic boundary of length $\ell$, then take the products
$M_{i}=\Sigma_{i}\times S^{1}$ with a circle of length $\ell$, and glue
$M_{1}$ to $M_{2}$ by means of an isometry $\varphi$ of the boundary tori
$\partial M_{i}=\partial\Sigma_{i}\times S^{1}$ which interchanges the circles
$\partial\Sigma_{i}$ with $S^{1}$. This yields a nonpositively curved
$3$-manifold $M$ (a graph manifold) with sectional curvature $-1\leq k(M)\leq
0$ whose diastole can be chosen arbitrarily small, by taking $\ell\rightarrow
0$. This manifold is the prototypical example of a nonpositively curved
manifold with an abelian local splitting structure: calling $g_{1},g_{2}$ the
hyperbolic isometries of $X=\widetilde{M}$ corresponding to the boundary loops
$\partial\Sigma_{i}$, the universal cover $\widetilde{X}$ is the union of the
two closed, minimum sets
Min$(g_{i})\cong\widetilde{\Sigma}_{i}\times{\mathbb{R}}$ (where $g_{i}$ acts
as $id$ times a translation) and Min$(g_{1})\cap$Min$(g_{2})$ is the gluing
torus, on which $\langle g_{1},g_{2}\rangle$ acts on as an abelian group.

In this paper, we will generalize this result to groups acting (not
necessarily cocompactly) on CAT(0)-spaces and we draw from it a description of
spaces with thin actions. In the framework of CAT(0)-spaces, the lower
sectional curvature bound appearing in the assumptions of the classical
Margulis-Heintze Lemma can be naturally replaced by a uniform, local packing
condition: we say that a metric space $(X,d)$ satisfies the $P_{0}$-packing
condition at scale $r_{0}>0$ if all balls of radius $3r_{0}$ in $X$ contain at
most $P_{0}$ points that are $2r_{0}$-separated from each other. In [CS21] is
proved that, for complete and geodesically complete CAT(0)-spaces, a packing
condition at some scale $r_{0}$ yields an explicit, uniform control of the
packing function at any other scale $r$: therefore, for these spaces, this
condition is equivalent to similar conditions which have been considered by
other authors with different names (“uniform compactness of the family of
$r$-balls” in [Gro81]; “geometrical boundedness” in [DY05], etc.). This
packing condition should be thought as a macroscopic replacement of a lower
bound on the curvature: for Riemannian manifolds it is strictly weaker than a
lower bound of the Ricci curvature, and for general metric spaces it is weaker
than asking that $X$ is a space with curvature bounded below in the sense of
Alexandrov (see [BCGS17], Sec.3.3, for different examples and comparison
between packing and curvature conditions). Although much weaker than a
curvature bound, the packing condition implies, by Breuillard-Green-Tao’s
work, an analogue of the celebrated Margulis’ Lemma for a general metric space
$X$: there exists a constant $\varepsilon_{0}>0$, only depending on the
packing constants $P_{0},r_{0}$, such that for every discrete group of
isometries $\Gamma$ of $X$ the $\varepsilon_{0}$-almost stabilizer
$\Gamma_{\varepsilon_{0}}(x)$ of any point $x$ is virtually nilpotent (cp.
[BGT11] and Section §2.3 below for details). We call this
$\varepsilon_{0}=\varepsilon_{0}(P_{0},r_{0})$ the Margulis constant, since it
plays the role of the classical Margulis constant in this metric setting.

There are a lot of non-manifolds examples in the class of packed,
CAT(0)-spaces: limits of Hadamard manifolds with curvature bounded below;
simplicial complexes with locally constant curvature and bounded geometry
(also called $M_{\kappa}$-complexes, cp. [BH13] and [CS21]), Euclidean and
hyperbolic buildings with bounded geometry in particular; nonpositively curved
cones over compact CAT(1)-spaces, etc.

The first theorem of this paper generalizes Cao-Cheeger-Rong’s result to this
broader setting, taking into account also non-compact actions of groups,
possibly with torsion:

######  Theorem A (Extract from Theorem 3.1).

There exist two functions $\lambda_{0}=\lambda_{0}(P_{0},r_{0})$,
$\eta_{0}=(P_{0},r_{0})>0$ with the following property: for any complete,
geodesically complete, CAT$(0)$-space $X$ which is $(P_{0},r_{0})$-packed, and
any almost-cocompact discrete group of isometries $\Gamma$ of $X$, if $\Gamma$
is $\eta_{0}$-thin then $X$ can be decomposed as

(1) $X=\bigcup_{g\in\Sigma_{\lambda_{0}}}\textup{Min}(g)$

where $\Sigma_{\lambda_{0}}$ is the subset of $g\in\Gamma^{*}$ with minimal
displacement $\ell(g)\leq\lambda_{0}$.

By almost-cocompact group, we mean a group which acts cocompactly on the
$\delta$-thick subsets of $X$. The same notion was introduce in [BGS13], §8.4
with the notation InjRad$(\Gamma\backslash X)\to 0$. In particular, cocompact
groups or groups acting with quotient of finite measure are almost cocompact
(see Section §2 for the definition and the natural measure of CAT(0)-spaces
under consideration).

The proof of Theorem A is different from that for manifolds given in [CCR01]:
actually, the proof of Cao-Cheeger-Rong is based on the fact that hyperbolic
elements stabilize (i.e., for any $s>0$ and for any such $g$ there exists $n$
such that Min$(g^{n})=$Min$(g^{nk})$ for all $0<k\leq s$), a property which
does not hold in general for CAT(0)-spaces, as the following easy example
shows.

######  Example 1.2.

Let $X$ be the product of the regular tree $T$ of valency $3$ and
$\mathbb{R}$. Let $g_{T}$ be an elliptic isometry of $T$ with infinite order
and such that $\text{Fix}(g_{T}^{n})\neq\text{Fix}(g_{T}^{k})$ for all
$n,k\in\mathbb{N}$, $n\neq k$. Let $\tau$ be a translation of $\mathbb{R}$.
The isometry $g=(g_{T},\tau)$ of $X$ is hyperbolic but no power of $g$
stabilizes: indeed for every $n,k\in\mathbb{N}$, $n\neq k$ it holds
$\text{Min}(g^{n})\neq\text{Min}(g^{k})$.

A major difficulty for this is that, in this extended metric setting, there
exist different isometries which coincide on open sets. This problem is
related to the existence of non-trivial isometry whose set of fixed points has
non-empty interior. For many applications of Theorem A we will exclude this
case, restricting the attention to what we called _slim groups_ (we refer to
Section 2.2 for more details). By definition two isometries of a slim group
coinciding on an open set must coincide globally, this gives a better control
on the geometry of the group action. Every torsion-free group is trivially
slim, as well as any discrete group acting on a CAT$(0)$-homology manifold
(cp. Lemma 2.1). Moreover, as opposite to [CCR01], we also have to deal with
parabolic and elliptic isometries, as $\Gamma$ is allowed to have torsion.
On the other hand, our proof has some similarities with a minimality argument
used by Cao-Cheeger-Rong (see their proof of Theorem 0.3 in [CCR01]), and it
is inspired by statements of Ballmann-Gromov-Schroeder (Lemma 7.11 and Section
13 of [BGS13]) holding for analytic manifolds (we stress that analiticity is
crucial in [BGS13] for their argument to work); similar arguments, which use
analiticity or strictly negative curvature, can be found in [Gel11], Section 2
and [BGLS10], Section 2. The main innovation in our proof relies in the choice
of the good minimal point for the function $\Psi$ (cp. Step 3 in the proof of
Theorem 3.1), which allows us to overcome any analyticity or negative
curvature assumption.

In Theorem 3.1 we will see a stronger statement than Theorem A: the same
alternative holds for any $\lambda$ smaller than the Margulis’ constant
$\varepsilon_{0}$ and an explicit $\eta=\eta(P_{0},r_{0},\lambda)$. Here we
want just to stress that the equality (1), in particular the fact the the
union runs over minimum sets of isometries with minimal displacement less than
the Margulis constant, imposes a very strong condition on the topology of $X$.
For instance, if $X$ is a non-elementary Gromov hyperbolic space and $\Gamma$
is torsionless, then $X$ cannot satisfy (1), and the diastole is always
universally bounded away from zero. This was already proved in [BCGS17] in the
cocompact case (see proof of Proposition 5.24), and in [CS20] (Corollary 1.4)
for non-cocompact actions. The following corollary yields a simple algebraic
obstruction.

######  Corollary B.

Let $X$ be a complete, geodesically complete, CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$. Assume that $\Gamma$ is a discrete, slim,
almost-cocompact group of isometries of $X$. If
$\textup{dias}(\Gamma,X)<\eta_{0}$, then

  * (i)

either $\Gamma$ contains a subgroup isomorphic to $\mathbb{Z}\times\mathbb{Z}$

  * (ii)

or $X$ splits a non-trivial Euclidean factor.

If moreover $\Gamma$ is cocompact then (ii) can be replaced by

  * (ii’)

$X=\mathbb{R}$.

(The constant $\eta_{0}$ here and in the following is the same as in Theorem
A.)

The following theorem generalizes the lower bounds for dias$(\Gamma,X)$ of
[BCGS17] and [CS20], and completely determines the structure of packed
(complete, geodesically complete) CAT(0)-spaces with diastole small enough in
some remarkable cases:

######  Theorem C (Extract from Proposition 4.1 and Corollaries 4.2 & 4.5).

Let $X$ be a complete, geodesically complete, CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$. Assume moreover that one of the following
holds:

  * (i)

$X$ has a point of dimension $1$,

  * (ii)

or $X$ has a point with an open neighbourhood which is CAT$(-\varepsilon)$ for
some $\varepsilon>0$,

  * (iii)

or $X$ is a visibility space.

If $X$ has a discrete, slim, almost-cocompact group of isometries $\Gamma$
such that $\textup{dias}(\Gamma,X)<\eta_{0}$, then $X$ is isometric to
$\mathbb{R}$.

(See Section 2.1 for the notion of points of dimension $k$ and generalites on
the dimension for CAT(0)-spaces).

The following is another structural consequence of a small diastole for
CAT(0)-spaces in low dimension.

######  Theorem D.

Let $X$ be a complete, geodesically complete CAT$(0)$-space of dimension at
most $2$, which is $P_{0}$-packed at scale $r_{0}$, and let $\Gamma$ be a
discrete, slim, almost-cocompact group of isometries of $X$. If
$\textup{dias}(\Gamma,X)<\eta_{0}$ then $X$ is isometric to
$T\times\mathbb{R}$, where $T$ is a tree (possibily a point). If moreover
$\Gamma$ is cocompact then it virtually splits as
$\Gamma_{T}\times\mathbb{Z}$, where $\Gamma_{T}$ (resp. $\mathbb{Z}$) acts
discretely and cocompactly on $T$ (resp. $\mathbb{R}$).

This splitting result is false in higher dimensions as showed by Example 1.1.

######  Remark 1.3.

This last theorem enlightens two interesting phenomena. In dimension $\leq 2$,
if the diastole is smaller than the universal constant $\eta_{0}$ then:

  * (a)

the space is pure dimensional, that is the dimension of every point is the
same;

  * (b)

the space is collapsible, in the following sense: one can define new CAT$(0)$
metrics $d_{n}$ on $X$ in such a way that $(X,d_{n})$ is still uniformily
packed at scale $r_{0}$ and $\Gamma$ still acts by isometries with respect to
$d_{n}$, but the diastole of $\Gamma$ with respect to the metric $d_{n}$ is
smaller than $\frac{1}{n}$.
Therefore, if the dias$(\Gamma,X)$ is sufficiently small, then it can be taken
arbitrarily small by changing the metric of $X$ remaining in the class of
uniformly packed CAT$(0)$ metrics.

The result of (a) is not true in higher dimensions: it is enough to consider
the product $X=Y\times\mathbb{R}$, where $Y$ is any non-pure dimensional,
complete, geodesically complete, packed CAT$(0)$-space admitting a cocompact
group of isometries $\Gamma_{Y}$, and taking
$\Gamma=\Gamma_{Y}\times\langle\tau_{\varepsilon}\rangle$, where
$\tau_{\varepsilon}$ is a translation by $\varepsilon$.
On the other hand, we have no clue if (b) is true in higher dimensions.

We remark that Corollary B, Theorem C and Theorem D are false if the group is
not slim, as the following example due to A.Lytchak shows.

######  Example 1.4.

Let $X$ be the Cayley graph of the free group on two generators $F(a,b)$. Let
$e$ be the identity point on this graph and $\tilde{a},\tilde{b}$ the axes of
the isometries $a$ and $b$. Let $\sigma_{a},\sigma_{b}$ be the reflections
with respect to $\tilde{a}$ and $\tilde{b}$. Finally let $\Gamma$ be the group
of isometries of $X$ generated by $\\{a,b,\sigma_{a},\sigma_{b}\\}$. It is
clear that every point of $X$ is stabilized by some element of $\Gamma$, so
dias$(\Gamma,X)=0$. The only non-trivial fact is to show that $\Gamma$ is
discrete, which is equivalent to say that the stabilizer of $e$ is finite. Let
$\mathbb{S}^{1}\ast\mathbb{S}^{1}$ be the quotient of $X$ with respect to the
action of $F(a,b)$. Every element of $\Gamma$ defines an isometry of
$\mathbb{S}^{1}\ast\mathbb{S}^{1}$ fixing the gluing point. By the standard
cover theory each isometry of $\mathbb{S}^{1}\ast\mathbb{S}^{1}$ defines a
unique isometry $\tilde{f}$ of $X$ such that $\tilde{f}(e)=e$. So there exists
an injective map from Stab${}_{\Gamma}(e)$ to
Isom$(\mathbb{S}^{1}\ast\mathbb{S}^{1})$. The latter group is finite, so also
Stab${}_{\Gamma}(e)$ is finite.

In this case the decomposition given by Theorem A contains only elliptic
elements and this is why the slim assumption plays a role. The next example
shows instead that the decomposition given by Theorem A is much stronger than
a decomposition as union of minimal sets of _some_ isometries.

######  Example 1.5.

Let $Y$ be a flat torus with a loop of length $\ell$ glued at some point. Let
$X$ be its universal cover, which is a geodesically complete, CAT$(0)$ space
which is $P_{0}$-packed at scale $r_{0}$ for some $P_{0},r_{0}$. It appears as
a tree of $\mathbb{R}^{2}$. Let $\Gamma$ be the fundamental group of $Y$. It
is clear that $X$ can be written as the union of the minimal sets of the non-
trivial isometries of $\Gamma$, which is slim because it is torsion-free.
However the conclusions of Theorem C do not hold for $X$, showing that it
cannot be decomposed as in Theorem A.

We would also bring to the attention of the reader a general observation. When
$\Gamma$ is a discrete, torsionless group of isometries of our CAT(0)-space
$X$, then the quotient space $M=\Gamma\backslash X$ is a locally
CAT$(0)$-space which is endowed with a natural measure $\mu_{M}$ (defined,
locally, as explained in Section 2.1). If dias$(\Gamma,X)\geq\eta_{0}$, then
there is a ball of $M$ of radius $\frac{\eta_{0}}{2}$ which is isometric to a
ball of $X$ and in particular (using the volume estimates recalled in Section
2.3) we get

$\mu_{M}(M)\geq v_{0}=v_{0}(P_{0},r_{0})>0.$

Therefore, we can read all the above results at level of quotient spaces as a
“small volume versus rigidity” alternative. By the way of example, we just
express Theorem C(ii) in this terms.

######  Corollary E (Thm. C(ii), revisited).

Let $M$ be a compact, locally geodesically complete, locally CAT$(0)$-space
whose universal cover is $P_{0}$-packed at scale $r_{0}$. Suppose there exists
an (arbitrarily small) open set $U\subset M$ which is CAT$(-\varepsilon)$ for
some $\varepsilon>0$. If $\mu_{M}(M)<v_{0}$ then $M$, up to renormalization,
is isometric to $\mathbb{S}^{1}$.

##  2\. Preliminaries

The open (resp. closed) ball of center $x$ and radius $r$ in a metric space
$X$ will be denoted by $B(x,r)$ (resp. $\overline{B}(x,r)$). A geodesic in a
metric space $X$ is an isometry $c\colon[a,b]\to X$, where $[a,b]$ is an
interval of $\mathbb{R}$. The endpoints of the geodesic $c$ are the points
$c(a)$ and $c(b)$; a geodesic with endpoints $x,y\in X$ is also denoted by
$[x,y]$. A geodesic ray is an isometry $c\colon[0,+\infty)\to X$ and a
geodesic line is an isometry $c\colon\mathbb{R}\to X$. A metric space $X$ is
called geodesic if for every two points $x,y\in X$ there is a geodesic with
endpoints $x$ and $y$.

###  2.1. Geodesically complete CAT(0)-spaces

A metric space $X$ is CAT$(0)$ if it is geodesic and every geodesic triangle
$\Delta(x,y,z)$ is thinner than its Euclidean comparison triangle
$\overline{\Delta}(\bar{x},\bar{y},\bar{z})$, that is for any couple of points
$p\in[x,y]$ and $q\in[x,z]$ we have $d(p,q)\leq d(\bar{p},\bar{q})$ where
$\bar{p},\bar{q}$ are the comparison points of $p,q$ in
$\overline{\Delta}(\bar{x},\bar{y},\bar{z})$ (see for instance [BH13] for the
basics of CAT(0)-geometry).
As consequence of the definition every CAT$(0)$-space is uniquely geodesic:
for every two points $x,y$ there exists a unique geodesic segment with
endpoints $x$ and $y$. A CAT$(0)$-metric space is geodesically complete if any
geodesic segment $c\colon[a,b]\to X$ can be extended to a geodesic line. For
instance a complete CAT$(0)$-homology manifold is always geodesically complete
([BH13], Proposition II.5.12).
A subset $D$ of a complete, geodesically complete CAT$(0)$-space $X$ is convex
for every two points $x,y\in D$ the geodesic $[x,y]$ is contained in $D$.
The space of direction $\Sigma_{x}X$ at a point $x\in X$ is the set of
geodesic segments starting from $x$ modulo the equivalence relation $c\sim
c^{\prime}$ if and only if $\angle_{x}(c,c^{\prime})=0$. Given $x,y\in X$ we
denote by $[y]_{x}$ the element of $\Sigma_{x}X$ defined by the geodesic
segment $[x,y]$.
For a CAT$(0)$ space we denote by GD$(X)$ the _geometric dimension_ of $X$ as
defined in [Kle99]. By definition GD$(X)=0$ if $X$ is discrete and

$\text{GD}(X)=1+\sup_{x\in X}\text{GD}(\Sigma_{x}X).$

In particular a CAT$(0)$-space has geometric dimension $1$ if and only if all
its spaces of directions are discrete.

In the following, we will always assume that $X$ is a proper, geodesically
complete CAT$(0)$-space. By [LN19] we know that every point $x\in X$ has a
well defined integer dimension in the following sense: there exists
$n_{x}\in\mathbb{N}$ such that every small enough ball around $x$ has
Hausdorff dimension equal to $n_{x}$. This defines a stratification of $X$
into pieces of different integer dimensions. Namely, if $X^{k}$ denotes the
subset of points of $X$ with dimension $k$, then

$X=\bigcup_{k\in\mathbb{N}}X^{k}.$

The set $X_{k}$ contains a dense subset $M_{k}$, which is open in $X$ and
locally bi-Lipschitz homeomorphic to $\mathbb{R}^{k}$ ([LN19], Theorem 1.2).
In particular if $X_{k}\neq\emptyset$ then there exists some ball in $X$ which
is a $k$-dimensional Lipschitz manifold. The dimension of $X$ is the supremum
of the dimensions of its points: it coincides with the topological dimension
of $X$, cp. Theorem 1.1 of [LN19], and with the geometric dimension of $X$.
The formula

$\mu_{X}:=\sum_{k\in\mathbb{N}}\mathcal{H}^{k}\llcorner X^{k}$

where $\mathcal{H}^{k}$ is the $k$-dimensional Hausdorff measure, defines a
natural measure on $X$ which is locally positive and locally finite.

###  2.2. Isometries of CAT(0)-spaces

The group of isometries of $X$ will be denoted by Isom$(X)$. The translation
length of $g\in\text{Isom}(X)$ is by definition

$\ell(g):=\inf_{x\in X}d(x,gx).$

If the infimum is realized then $g$ is called elliptic if $\ell(g)=0$ and
hyperbolic otherwise; $g$ is called parabolic when the infimum is not
realized.
It is well known that the function $d_{g}\colon X\to[0,+\infty)$, $x\mapsto
d(x,gx)$ is convex. Its $\tau$-level sets, for $\tau\geq 0$, are by definition
the subsets

$M_{\tau}(g):=\\{x\in X\text{ s.t. }d(x,gx)\leq\tau\\}$

which are closed, convex subsets of $X$. We recall that the minimum set of $g$
is defined as

$\text{Min}(g):=M_{\ell(g)}(g)$

that is, $\text{Min}(g)$ is the subset of points of $X$ realizing the minimum
in the definition of the translation length of $g$. Notice that if $g$ is
parabolic then $\text{Min}(g)=\emptyset$, while if $g$ is elliptic then
$\text{Min}(g)$ is the set of points fixed by $g$, in which case it will be
denoted also by $\text{Fix}(g)$.
It is known that if $g$ in an hyperbolic isometry then $\text{Min}(g)$ splits
isometrically as a product $D(g)\times\mathbb{R}$, where $D(g)$ is a convex
subset of $X$, and $g$ acts on $D(g)\times\mathbb{R}$ respecting the product
decomposition and acting as the identity on $D(g)$ and as a translation of
length $\ell(g)$ on $\mathbb{R}$.
When dealing with torsion there is a class of elliptic isometries which
deserves a name. We say that an elliptic isometry $g$ of $X$ is _slim_ if
$\text{Fix}(g)$ has empty interior. A group of isometries $\Gamma$ is said to
be _slim_ if every non-trivial elliptic isometry of $\Gamma$ is slim. For
instance every torsion-free group is trivially slim, as well as any discrete
group of isometries of a CAT$(0)$-homology manifold.

######  Lemma 2.1.

Let $X$ be a proper CAT$(0)$-space which is a homology manifold, and let
$\Gamma$ be a discrete group of isometries of $X$. Then every elliptic
isometry of $\Gamma$ is slim.

###### Proof.

By [LN18], Theorem 1.2 there exists a locally finite subset $E$ of $X$ such
that $X\setminus E$ is a topological manifold. The set $E$ can clearly be
chosen to be invariant under the isometry group of $X$. We can assume also
that the dimension of $X$ is $\geq 3$, otherwise $X$ is already a topological
manifold. In this case the topological manifold $X\setminus E$ is connected.
Let $g$ be an elliptic isometry of $\Gamma$. Since $\Gamma$ is discrete, the
order of $g$ is finite. After taking a power of $g$, whose fixed point set
contains the fixed point set of $g$, we can suppose $g$ has prime order. By
Newman’s second theorem [New31] if $g$ fixes an open subset of $X\setminus E$
then it is the identity. This shows that Fix$(g)\cap(X\setminus E)$ has empty
interior. Also $E$ has empty interior, so Fix$(g)$ has empty interior. ∎

Let now $\Gamma$ be a subgroup of Isom$(X)$. For $x\in X$ and $\eta\geq 0$ we
set

$\Sigma_{\eta}(x):=\\{g\in\Gamma\text{ s.t. }d(x,gx)<\eta\\}$
$\overline{\Sigma}_{\eta}(x):=\\{g\in\Gamma\text{ s.t. }d(x,gx)\leq\eta\\}$

and we define $\Gamma_{\eta}(x):=\langle\Sigma_{\eta}(x)\rangle$,
$\overline{\Gamma}_{\eta}(x)=\langle\overline{\Sigma}_{\eta}(x)\rangle$. For
instance when $\eta=0$ we have
$\overline{\Sigma}_{0}(x)=\overline{\Gamma}_{0}(x)=\text{Stab}_{\Gamma}(x)$,
the stabilizer of $x$. The subgroup $\Gamma$ is discrete if for all $x\in X$
and all $\eta\geq 0$ the set $\Sigma_{\eta}(x)$ is finite.
The systole of $\Gamma$ at a point $x\in X$ is

$\text{sys}(\Gamma,x):=\inf_{g\in\Gamma^{*}}d(x,gx)$

where $\Gamma^{*}=\Gamma\setminus\\{\text{id}\\}$. Finally, the central notion
of this paper is the diastole of $\Gamma$, that is the quantity:

$\text{dias}(\Gamma,X)=\sup_{x\in X}\text{sys}(\Gamma,x).$

We end this section with some facts about finite sets of isometries we will
need later. The first one is a well-known fact about commuting isometries.

######  Proposition 2.2 ([Gel11], Lemma 2.7).

Let $X$ be a complete CAT$(0)$-space and let $g_{1},\ldots,g_{n}$ be commuting
isometries of $X$. Let $\tau_{1},\ldots,\tau_{n}\geq 0$ such that
$M_{\tau_{i}}(g_{i})\neq\emptyset$. Then
$\bigcap_{i=1}^{n}M_{\tau_{i}}(g_{i})\neq\emptyset$.

The second one is a similar statement for elliptic isometries:

######  Proposition 2.3 ([BH13], Corollary II.2.8).

Let $X$ be a complete CAT$(0)$-space and let $F$ be a finite group of
isometries of $X$. Then $\bigcap_{g\in F}\textup{Fix}(g)\neq\emptyset$.

###  2.3. Packing conditions on CAT(0)-spaces

Let $P_{0},r_{0}>0$: we say $X$ is $P_{0}$-packed at scale $r_{0}$ if for
every $x\in X$ the cardinality of every $r_{0}$-separated subset of
$\overline{B}(x,3r_{0})$ is at most $P_{0}$ (recall that a subset $Y\subseteq
X$ is $r_{0}$-separated subset if $d(y,y^{\prime})>r_{0}$ for all
$y,y^{\prime}\in Y$). We will simply say that $X$ is packed if it is
$P_{0}$-packed at scale $r_{0}$ for some $P_{0},r_{0}>0$. If $X$ is a
complete, geodesically complete CAT$(0)$-space which is $P_{0}$-packed at
scale $r_{0}$ then the dimension of $X$ is at most $\frac{P_{0}}{2}$, and for
every $R\geq 0$ and for every $x\in X$ it holds:

(2) $v(R)\leq\mu_{X}(B(x,R))\leq V(R),$

where $v(R)$ and $V(R)$ are functions only depending on the packing constants
$P_{0},r_{0}$, as showed in [CS21]. Another remarkable consequence of a
packing condition at some fixed scale is the following version of the
Margulis’ Lemma due to Breuillard-Green-Tao, which we decline for geodesically
complete CAT(0)-spaces.

######  Theorem 2.4 ([BGT11], Corollary 11.17 and Corollary 11.2).

Let $X$ be a complete, geodesically complete, CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$ and $\Gamma$ be a discrete group of isometries
of $X$. There exist $\varepsilon_{0}=\varepsilon_{0}(P_{0},r_{0})>0,$
$I_{0}=I_{0}(P_{0},r_{0})\geq 0$ and $S_{0}=S_{0}(P_{0},r_{0})\geq 0$ such
that the following holds true: for every $x\in X$ and every positive
$\eta\leq\varepsilon_{0}$ there exist normal subgroups
$\Delta_{x},\Lambda_{x}$ of $\overline{\Gamma}_{\eta}(x)$ such that

  * (i)

$[\overline{\Gamma}_{\eta}(x):\Delta_{x}]\leq I_{0}$;

  * (ii)

$\Lambda_{x}$ is a finite subgroup of $\Delta_{x}$;

  * (iii)

$\Delta_{x}/\Lambda_{x}$ is nilpotent of step $\leq S_{0}$.

In particular $\overline{\Gamma}_{\eta}(x)$ is virtually nilpotent. Moreover,
if $\Gamma$ is cocompact then $\overline{\Gamma}_{\eta}(x)$ is virtually
abelian (cp. [BH13], Theorem II.7.8).

We call this $\varepsilon_{0}(P_{0},r_{0})$ the Margulis constant, since it
plays the role of the classical Margulis constant in this metric setting. This
constant and Theorem 2.4 will play a crucial role in our main theorem.
We end this section describing more carefully the structure of the groups
$\overline{\Gamma}_{\eta}(x)$ appearing in Theorem 2.4.

######  Lemma 2.5 ([BGS13], Lemma 7.4 and Remark 7.2).

Let $X$ be a complete, CAT$(0)$-space and let $N$ be a nilpotent group of
isometries of $X$. If $N$ has a semisimple isometry then there exists a
semisimple isometry in the center of $N$. The set of semisimple isometries in
the center of $N$ forms an abelian normal subgroup of $N$.

######  Proposition 2.6.

Let $X$ be a complete, CAT$(0)$-space with $\textup{GD}(X)<+\infty$. Let
$\Gamma$ be a finitely generated, discrete, virtually nilpotent group of
isometries of $X$. Then:

  * (i)

there exists a $\Gamma$-invariant convex subset $W\times\mathbb{R}^{k}$,
$k\geq 0$, of $X$ such that each $\gamma\in\Gamma$ acts as
$(\gamma^{\prime},\gamma^{\prime\prime})$, where $\gamma^{\prime}$ is not
hyperbolic;

  * (ii)

a finite index, normal subgroup of $\Gamma$ splits as
$\Gamma_{W}\times\mathbb{Z}^{k}$, where $\Gamma_{W}$ is a finitely generated,
discrete, nilpotent group acting on $W$ without hyperbolic isometries and
$\mathbb{Z}^{k}$ acts as a lattice on $\mathbb{R}^{k}$;

  * (iii)

the set of semisimple isometries of $\Gamma_{W}\times\mathbb{Z}^{k}$ is a
virtually abelian, normal subgroup of $\Gamma$. It coincides with the set of
elements whose first component is elliptic;

  * (iv)

if $\Gamma$ does not contain parabolic elements then it is virtually abelian.

###### Proof.

We show (i) and (ii) by induction on $n:=\text{GD}(X)$, where the case $n=0$
is trivial. If $\Gamma$ does not contain hyperbolic isometries it is enough to
take $W=X$ and $k=0$. Suppose $\Gamma$ has a hyperbolic isometry and denote by
$N$ a nilpotent subgroup of finite index. Since $N$ is finitely generated and
nilpotent it is not restrictive to suppose $N$ torsion-free and normal in
$\Gamma$. Clearly also $N$ has hyperbolic isometries, so there exists a
hyperbolic isometry in the center of $N$ by Lemma 2.5. We consider the set $H$
of semisimple isometries in the center of $N$: it is an abelian subgroup of
$N$ by Lemma 2.5. Let $\gamma_{1},\ldots,\gamma_{n}\in\Gamma$ such that
$\Gamma=\bigcup_{i=1}^{n}\gamma_{i}N$. Observe that the isometries of
$\gamma_{i}H\gamma_{i}^{-1}$ are still semisimple and belong to the center of
$N$. Indeed the center of $N$ is characteristic in $N$, so it is normal in
$\Gamma$. The isometries $\\{\gamma_{i}H\gamma_{i}^{-1}\\}_{i=1,\ldots,n}$
generate an abelian group which is clearly normal in $\Gamma$ and has positive
rank. Up to passing to the characteristic subgroup of this abelian group we
can suppose to have an abelian, torsion-free, normal subgroup $A$ of $\Gamma$.
By the flat torus Theorem (cp. [BH13], Theorem II.7.1) we know that there
exists a convex subset $W^{\prime}\times\mathbb{R}^{j}$ of $X$ which is
preserved by $\Gamma$, and moreover each $\gamma\in\Gamma$ acts as
$(\gamma^{\prime},\gamma^{\prime\prime})$. Furthermore $\Gamma$ has a subgroup
of finite index which splits as $\Gamma_{W^{\prime}}\times
A\cong\Gamma_{W^{\prime}}\times\mathbb{Z}^{j}$. Let $p(\Gamma)$ be the
projection of $\Gamma$ on $\text{Isom}(W^{\prime})$. Clearly it is virtually
nilpotent and finitely generated. We need to show it is discrete. It contains
$\Gamma_{W^{\prime}}$ as finite index subgroup and it is easy to see that
$\Gamma_{W^{\prime}}$ is discrete, so also $p(\Gamma)$ is discrete. Finally
observe that since the rank of $A$ is strictly positive we have
$\text{GD}(W^{\prime})<\text{GD}(X)$, so we can apply induction on
$p(\Gamma)$. So there exists a convex subset $W\times\mathbb{R}^{k}$ which is
$p(\Gamma)$-invariant and each isometry of $p(\Gamma)$ preserves the product
decomposition, i.e. each $\gamma\in p(\Gamma)$ decomposes as
$(\gamma^{\prime},\gamma^{\prime\prime})$ with $\gamma^{\prime}$ not
hyperbolic. This means that the set
$W\times\mathbb{R}^{k}\times\mathbb{R}^{j}$ does the job for $\Gamma$.
Moreover by induction we can also assume that a finite index subgroup of
$\Gamma_{W^{\prime}}$ splits as $\Gamma_{W}\times\mathbb{Z}^{k}$, so a finite
index subgroup of $\Gamma$ splits as $\Gamma_{W}\times\mathbb{Z}^{k+j}$.
An isometry $(\gamma^{\prime},\gamma^{\prime\prime})$ of
$\Gamma_{W}\times\mathbb{Z}^{k}$ is semisimple if and only if both
$\gamma^{\prime}$ and $\gamma^{\prime\prime}$ are semisimple ([BH13],
Proposition II.6.9). Since $\gamma^{\prime\prime}$ is always semisimple
because it is an isometry of $\mathbb{R}^{k}$, we conclude that
$(\gamma^{\prime},\gamma^{\prime\prime})\in\Gamma_{W}\times\mathbb{Z}^{k}$ is
semisimple if and only if $\gamma^{\prime}$ has finite order. Since the finite
order elements of a nilpotent group form a subgroup, we see that the set of
semisimple isometries of $\Gamma_{W}\times\mathbb{Z}^{k}$ are a subgroup. Let
$(\gamma^{\prime},\gamma^{\prime\prime})$ be a semisimple isometry of
$\Gamma_{W}\times\mathbb{Z}^{k}$ and $(g^{\prime},g^{\prime\prime})$ be an
element of $\Gamma$. Then

$(g^{\prime},g^{\prime\prime})(\gamma^{\prime},\gamma^{\prime\prime})(g^{\prime-1},g^{\prime\prime-1})=(g^{\prime}\gamma^{\prime}g^{\prime-1},g^{\prime\prime}\gamma^{\prime\prime}g^{\prime\prime-1}).$

Since $\Gamma_{W}\times\mathbb{Z}^{k}$ is normal in $\Gamma$ we know that
$(g^{\prime}\gamma^{\prime}g^{\prime-1},g^{\prime\prime}\gamma^{\prime\prime}g^{\prime\prime-1})$
still belongs to $\Gamma_{W}\times\mathbb{Z}^{k}$. Moreover
$g^{\prime}\gamma^{\prime}g^{\prime-1}$ is still of finite order: this shows
that the subgroup of semisimple isometries of $\Gamma_{W}\times\mathbb{Z}^{k}$
is normal in $\Gamma$. Clearly the quotient of this group by the torsion group
of $\Gamma_{W}$, which is finite, is abelian, so this group is virtually
abelian.
Finally (iv) follows immediately: indeed if $\Gamma$ does not have parabolic
elements then also $\Gamma_{W}\times\mathbb{Z}^{k}$ does not have parabolic
elements. This means that $\Gamma_{W}$ is finite, so
$\Gamma_{W}\times\mathbb{Z}^{k}$ is virtually abelian. Therefore also $\Gamma$
is virtually abelian. ∎

###  2.4. Almost cocompact groups of CAT(0)-spaces

A group of isometries $\Gamma$ of a complete, geodesically complete
CAT$(0)$-space $X$ is called cocompact if there exists a compact subset
$K\subset X$ such that $X=\bigcup_{g\in\Gamma}gK$. The main theorem of the
paper will be proved under a weaker assumption, which we call almost-
cocompactness, which we now describe.
For $\delta>0$ we define the $\delta$-thick subset of $X$ as

$\text{Thick}_{\delta}(\Gamma,X):=\\{x\in X\text{ s.t.
sys}(\Gamma,x)\geq\delta\\}.$

This is clearly a $\Gamma$-invariant subset of $X$. We say $\Gamma$ is almost-
cocompact if the action of $\Gamma$ on $\text{Thick}_{\delta}(\Gamma,X)$ is
cocompact for all $\delta>0$. In [BGS13], Section 8, the same notion for
manifolds was introduced under the notation InjRad$(\Gamma\backslash X)\to 0$.
It is clear that if $\Gamma$ is cocompact then it is almost-cocompact.
Moreover if the global systole of the action is strictly positive then almost-
cocompactness coincides with cocompactness. Among almost-cocompact actions the
most important ones are those with finite covolume:

######  Definition 2.7.

Let $\Gamma$ be a discrete subgroup of isometries of a complete, geodesically
complete CAT$(0)$-space $X$. An open subset $\mathcal{D}\subset X$ is a
fundamental domain for $\Gamma$ if :

  * (i)

$g\mathcal{D}\cap h\mathcal{D}=\emptyset$ for all $g,h\in\Gamma$, $g\neq h$;

  * (ii)

$\bigcup_{g\in\Gamma}g\overline{\mathcal{D}}=X$.

The group $\Gamma$ is said to have finite covolume if there exists a
fundamental domain $\mathcal{D}$ for $\Gamma$ with
$\mu_{X}(\mathcal{D})<+\infty$.

######  Lemma 2.8 (Compare with [BGS13], §8.4).

Let $X$ be a complete, geodesically complete, packed, CAT$(0)$ space $X$ and
let $\Gamma$ be a discrete group of isometries of $X$ with finite covolume.
Then $\Gamma$ is almost-cocompact.

###### Proof.

Suppose $\Gamma\backslash\textup{Thick}_{\delta}(\Gamma,X)$ is not compact for
some $\delta>0$. It is complete, so it is not totally bounded. This means we
can find infinitely many disjoint balls of radius
$0<\varepsilon\leq\frac{\delta}{2}$ centered in points of
$\Gamma\backslash\textup{Thick}_{\delta}(\Gamma,X)$. By definition these balls
are isometric to the corresponding balls in $X$, in particular their volume is
bigger than a universal number $v>0$ depending on $\varepsilon$ and the
packing constants of $X$, by (2). This contradicts the finite covolume
assumption. ∎

The existence of a fundamental domain is discussed in the following result.

######  Proposition 2.9.

Let $X$ be a complete CAT$(0)$-space and let $\Gamma$ be a discrete group of
isometries. Then there exists a fundamental domain for $\Gamma$ if and only if
dias$(\Gamma,X)>0$, i.e. if there exists $x\in X$ such that
$\textup{Stab}_{\Gamma}(x)=\\{\textup{id}\\}$. In particular fundamental
domains exist when $\Gamma$ is slim, e.g. when it is torsion-free or $X$ is a
homology manifold by Lemma 2.1.

###### Proof.

Clearly dias$(\Gamma,X)>0$ if and only if there exists $x\in X$ such that
$\textup{Stab}_{\Gamma}(x)=\\{\textup{id}\\}$. If this is the case then the
set

$\mathcal{D}=\\{y\in X\text{ s.t. }d(y,x)<d(gy,x)\text{ for all
}g\in\Gamma\\}$

is a fundamental domain, since $X$ is a geodesic space. This assumption is
satisfied if $\Gamma$ is slim: indeed by Baire theorem
$\bigcup_{g\in\Gamma^{*}}\text{Fix}(g)$ has empty interior, because $\Gamma$
is countable. So there exists a point whose stabilizer is trivial.
Viceversa let us suppose that $\textup{Stab}_{\Gamma}(x)\neq\\{\textup{id}\\}$
for every $x\in X$. If $\mathcal{D}$ is any subset of $X$ and
$x\in\mathcal{D}$ then there exists a non-trivial $g\in\Gamma$ such that
$gx=x$, so $g\mathcal{D}\cap\mathcal{D}\neq\emptyset$. Therefore fundamental
domains cannot exist. ∎

Example 1.4 shows that there exist cocompact, discrete groups of isometries
$\Gamma$ of complete, geodesically complete, CAT$(0)$-spaces $X$ with
$\text{dias}(\Gamma,X)=0$.

We conclude with an important property of almost-cocompact, slim groups. A
group of isometries $\Gamma$ of a CAT$(0)$-space $X$ is said to be _minimal_
if there are no convex, closed, $\Gamma$-invariant proper subsets $C\subsetneq
X$.

######  Proposition 2.10 (Compare with [BGS13], proof of Lemma 2 at page 195).

Let $X$ be a proper, geodesically complete, CAT$(0)$-space. Let $\Gamma$ be a
discrete, slim, almost-cocompact group of isometries of $X$. Then $\Gamma$ is
minimal.

###### Proof.

Suppose to have a closed, convex, $\Gamma$-invariant subset $C\subsetneq X$.
Let $x\in X\setminus C$ and $y\in C$ be its projection on $C$. Consider the
geodesic ray $[y,x]$ and extend it to a geodesic ray $c_{x}$. By
$\Gamma$-invariance of $C$ we know that $d_{g}$ is non-decreasing along
$c_{x}$ for all $g\in\Gamma$. Suppose first there exist $\delta>0$ and a time
$t_{0}\geq 0$ such that $d(c_{x}(t_{0}),gc_{x}(t_{0}))\geq\delta>0$ for all
$g\in\Gamma^{*}$. The same holds for all $t\geq t_{0}$, so there exists a
whole geodesic ray which is contained in $\text{Thick}_{\delta}(\Gamma,X)$.
Clearly this ray does not project to a compact subset of $\Gamma\backslash X$,
a contradiction to the almost-cocompactness assumption. In the remaining case
we have that the ray $c_{x}$ is entirely contained in
$\bigcup_{g\in\Gamma^{*}}\text{Fix}(g)$, and this must happen for all possible
choices of $x\in X\setminus C$. In other words $X\setminus
C\subseteq\bigcup_{g\in\Gamma^{*}}\text{Fix}(g)$. Now, $X\setminus C$ is an
open, non-empty subset of $X$ while each $\text{Fix}(g)$ is a closed subset
with empty interior. The union is countable because $\Gamma$ is discrete, so
by Baire Theorem $\bigcup_{g\in\Gamma^{*}}\text{Fix}(g)$ has empty interior,
impossible. ∎

##  3\. Thin actions by almost-cocompact groups

From now on we fix $P_{0},r_{0}>0$ and we call $\varepsilon_{0},I_{0},S_{0}$
the constants given by Theorem 2.4. For $\lambda\geq 0$ we introduce the set

$\text{Min}_{\lambda}(\Gamma,X):=\bigcup_{g\in\Gamma^{*}\textup{ :
}\ell(g)\leq\lambda}\textup{Min}(g).$

In particular when $\lambda=0$ we denote the set
$\text{Min}_{\lambda}(\Gamma,X)$ by $\text{Fix}(\Gamma,X)$, which is the set
of points that are fixed by some non-trivial (and necessarely elliptic)
isometry. Observe that $\text{Min}_{\lambda}(\Gamma,X)$ contains
$\text{Fix}(\Gamma,X)$ for every $\lambda\geq 0$.
We state here the version of the main theorem we will prove.

######  Theorem 3.1.

Let $X$ be a complete, geodesically complete, CAT$(0)$-space that is
$P_{0}$-packed at scale $r_{0}$. Let $\Gamma$ be a discrete, almost-cocompact
group of isometries of $X$ and let $0<\lambda\leq\varepsilon_{0}$. Assume that
dias$(\Gamma,X)<\eta(P_{0},r_{0},\lambda):=\frac{\lambda}{(2I_{0}-1)(3\cdot
2^{S_{0}-1}-2)}$, then $X=\textup{Min}_{\lambda}(\Gamma,X)$.

Example 1.4 in which dias$(\Gamma,X)=0$ shows that if case (b) occurs the
elements of the decomposition can be all elliptic. Notice that if in the above
decomposition Min$(g)$ intersects Min$(g^{\prime})$ then $\\{g,g^{\prime}\\}$
generate a virtually nilpotent group (virtually abelian, if $\Gamma$ is
cocompact), by Theorem 2.4.
To prove Theorem 3.1 we need a couple of preparatory results. The first one is
a group-theoretic result for finite index subgroups:

######  Lemma 3.2 (Lemma 3.4 of [SW92]).

Let $G$ be a group generated by a finite set $S$. Let $H$ be a subgroup of $G$
with finite index $I$. Then $H$ can be generated by a set $S^{\prime}$ whose
elements have word length with respect to $S$ at most $(2I-1)$.

(We recall that, in the above setting, the word length with respect to $S$ of
an element $g\in G$ is the infimum of the natural numbers $n$ such that $g$
can be written as a product of $n$ elements of $S$.)

######  Lemma 3.3.

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

###### Proof.

By discreteness $\Sigma_{\eta}(x)=\\{g_{1},\ldots,g_{k}\\}$ and
$d(x,g_{i}x)<\eta$ for each $i=1,\ldots,k$. Hence there exists $\varepsilon>0$
such that $d(x,g_{i}x)<\eta-2\varepsilon$ for every $i=1,\ldots,k$. So we have
$\Sigma_{\eta}(y)\supseteq\Sigma_{\eta}(x)$ for every point $y\in
B(x,\varepsilon)$.
In order to prove the second part we suppose the opposite: there is a sequence
$x_{n}$ converging to $x$ such that
$\overline{\Sigma}_{\eta}(x_{n})\supset\overline{\Sigma}_{\eta}(x)$ for every
$n$: in particular there is $g_{n}\in\Gamma$ such that
$d(x_{n},g_{n}x_{n})\leq\eta$ but $d(x,g_{n}x)>\eta$. By discreteness the sets
of possible $g_{n}$’s is finite, so we can take a subsequence where $g_{n}$ is
constantly equal to a fixed $g\in\Gamma$. Therefore by continuity
$d(x,gx)=\lim_{n\to+\infty}d(x_{n},gx_{n})\leq\eta$, which is a contradiction.
∎

We are ready to prove Theorem 3.1. We present it in the cocompact case and we
will see after the proof how to do the general case.

###### Proof of Theorem 3.1.

We suppose $X\neq\text{Min}_{\lambda}(\Gamma,X)$. Our aim is to find a point
$x\in X$ such that sys$(\Gamma,x)\geq\eta=\eta(P_{0},r_{0},\lambda)$. Let
$f\colon[0,+\infty)\to[0,+\infty)$ be a continuous, decreasing function such
that $\lim_{t\to 0}f(t)=+\infty$ and $f(t)=0$ for all $t\geq\eta$. We define
the map $\Psi\colon X\setminus\text{Min}_{\lambda}(\Gamma,X)\to[0,+\infty)$ by

(3)
$\Psi(x)=\sum_{g\in\Sigma_{\eta}(x)\setminus\\{\text{id}\\}}f(d(x,gx))=\sum_{g\in\overline{\Sigma}_{\eta}(x)\setminus\\{\text{id}\\}}f(d(x,gx)),$

where the sum is equal to $0$ if $\Sigma_{\eta}(x)=\\{\text{id}\\}$. This map
is well defined on every point $x$ which does not belong to
$\text{Fix}(\Gamma,X)$, in particular it is well defined outside
$\text{Min}_{\lambda}(\Gamma,X)$. In order to get the thesis it is enough to
show that $\min\Psi=0$. Indeed if a point $x\in
X\setminus\text{Min}_{\lambda}(\Gamma,X)$ satisfies $\Psi(x)=0$ then
sys$(\Gamma,x)\geq\eta$.

Step 1: $\Psi$ is continuous and $\Gamma$-invariant.
The $\Gamma$-invariance is obvious. Let us take a sequence $x_{n}$ converging
to $x_{\infty}$. By Lemma 3.3 we have
$\Sigma_{\eta}(x_{n})\supseteq\Sigma_{\eta}(x_{\infty})$ and
$\overline{\Sigma}_{\eta}(x_{n})\subseteq\overline{\Sigma}_{\eta}(x_{\infty})$,
for $n$ big enough. Therefore, using the two expressions of $\Psi$ in (3):

$\displaystyle\liminf_{n\to+\infty}\Psi(x_{n})$
$\displaystyle=\liminf_{n\to+\infty}\sum_{g\in\Sigma_{\eta}(x_{n})\setminus\\{\text{id}\\}}f(d(x_{n},gx_{n}))$
$\displaystyle\geq\liminf_{n\to+\infty}\sum_{g\in\Sigma_{\eta}(x_{\infty})\setminus\\{\text{id}\\}}f(d(x_{n},gx_{n}))$
$\displaystyle=\sum_{g\in\Sigma_{\eta}(x_{\infty})\setminus\\{\text{id}\\}}f(d(x_{\infty},gx_{\infty}))=\Psi(x_{\infty})$

and

$\displaystyle\limsup_{n\to+\infty}\Psi(x_{n})$
$\displaystyle=\limsup_{n\to+\infty}\sum_{g\in\overline{\Sigma}_{\eta}(x_{n})\setminus\\{\text{id}\\}}f(d(x_{n},gx_{n}))$
$\displaystyle\leq\limsup_{n\to+\infty}\sum_{g\in\overline{\Sigma}_{\eta}(x_{\infty})\setminus\\{\text{id}\\}}f(d(x_{n},gx_{n}))$
$\displaystyle=\sum_{g\in\overline{\Sigma}_{\eta}(x_{\infty})\setminus\\{\text{id}\\}}f(d(x_{\infty},gx_{\infty}))=\Psi(x_{\infty}).$

Step 2: $\Psi$ has a minimum.
Let us take a sequence $x_{n}\in X\setminus\text{Min}_{\lambda}(\Gamma,X)$
such that $\Psi(x_{n})\to\inf\Psi$. By cocompactness of the action we can
suppose that $x_{n}$ converges to $x_{\infty}$ and, since $\Psi$ is
continuous, $\Psi(x_{\infty})=\inf\Psi$.

In the next step we will find a good point realizing the minimum of $\Psi$. In
order to do that we introduce the partial order $\preceq$ on $\Psi^{-1}(m)$,
$m\in[0,+\infty)$, defined by $x\preceq y$ if and only if
$\overline{\Sigma}_{\eta}(x)\subseteq\overline{\Sigma}_{\eta}(y)$.

Step 3: the mininum of $\Psi$ is realized at a point $x$ maximal for
$\preceq$.
Set $m:=\min\Psi$. We consider the set $\Psi^{-1}(m)$ with the partial order
$\preceq$. We claim that every chain has a maximal element. Let us fix a chain
$x_{1}\preceq x_{2}\preceq\ldots$. The function
$x\mapsto\\#\overline{\Sigma}_{\eta}(x)$ is upper semicontinuous by Lemma 3.3
and $\Gamma$-invariant, so it has a maximum on $X$. Moreover
$\\#\overline{\Sigma}(x_{1})\leq\\#\overline{\Sigma}(x_{2})\leq\ldots$, so
this sequence must stabilize, i.e. there exists $\bar{n}\in\mathbb{N}$ such
that $\overline{\Sigma}(x_{\bar{n}})=\overline{\Sigma}(x_{n})$ for all
$n\geq\bar{n}$. This means that $x_{\bar{n}}$ is a maximizing element of the
chain. By Zorn’s Lemma there exists a point $x$ such that $\Psi(x)=m$ and
which is maximal for $\preceq$.

Our aim is to show that $m=0$, so we suppose $m>0$ and we look for a
contradiction. Indeed we will find a point $y\in
X\setminus\text{Min}_{\lambda}(\Gamma,X)$ such that $\Psi(y)<m$. Since $m>0$
the set $\overline{\Sigma}_{\eta}(x)$ contains some non-trivial element. We
write $\overline{\Sigma}_{\eta}(x)=\\{\sigma_{1},\ldots,\sigma_{n}\\}$ and we
consider the group
$\overline{\Gamma}_{\eta}(x)=\langle\overline{\Sigma}_{\eta}(x)\rangle$.

Step 4: there exists a non-empty, closed, convex,
$\overline{\Gamma}_{\eta}(x)$-invariant subset $Y$ of $X$ not containing $x$.
By Theorem 2.4 there are groups
$\Lambda_{x},\Delta_{x}\vartriangleleft\overline{\Gamma}_{\eta}(x)$ such that:

  * (i)

$[\overline{\Gamma}_{\eta}(x):\Delta_{x}]\leq I_{0}$;

  * (ii)

$\Lambda_{x}$ is a finite subgroup of $\Delta_{x}$;

  * (iii)

$\Delta_{x}/\Lambda_{x}$ is nilpotent of step $\leq S_{0}$.

Here we divide the proof in two cases: (a) $\Lambda_{x}\neq\\{\text{id}\\}$
and (b) $\Lambda_{x}=\text{id}$ and hence $\Delta_{x}$ is nilpotent of step
$\leq S_{0}$.
In case (a) we define the set $Y=\bigcap_{g\in\Lambda_{x}}\text{Min}(g)$. It
is closed, convex and non-empty by Proposition 2.3 because $\Lambda_{x}$ is
finite. Moreover it is $\overline{\Gamma}_{\eta}(x)$-invariant since
$\Lambda_{x}$ is normal in $\overline{\Gamma}_{\eta}(x)$. Finally $x\notin Y$
because $x\notin\text{Fix}(\Gamma,X)$.
In case (b) we know that the group $\Delta_{x}$ is nilpotent of step $\leq
S_{0}$ and it has index at most $I_{0}$ in $\overline{\Gamma}_{\eta}(x)$. We
apply Lemma 3.2 to find a generating set $\Sigma^{\prime}$ of $\Delta_{x}$
such that each $\sigma^{\prime}\in\Sigma^{\prime}$ has length at most
$(2I_{0}-1)$ in the alphabet $\\{\sigma_{1},\ldots,\sigma_{n}\\}$. Since the
lower central series of $\Delta_{x}$ has length $S\leq S_{0}$ we can find
elements $\sigma_{1}^{\prime},\ldots,\sigma_{S}^{\prime}\in\Sigma^{\prime}$
such that
$[\sigma_{1}^{\prime},[\sigma_{2}^{\prime},\cdots[\sigma_{S-1}^{\prime},\sigma_{S}^{\prime}]]]$
is not trivial and belongs to the center of $\Delta_{x}$. We call $g_{0}$ such
an element and we notice that its length in the alphabet
$\\{\sigma_{1},\ldots,\sigma_{n}\\}$ is at most $(2I_{0}-1)(3\cdot
2^{S_{0}-1}-2)$.
We consider the finite set
$\text{Conj}_{\overline{\Gamma}_{\eta}(x)}(g_{0})=\\{hg_{0}h^{-1}\text{ s.t.
}h\in\overline{\Gamma}_{\eta}(x)\\}$. Since the center of $\Delta_{x}$ is
characteristic in $\Delta_{x}$ and $\Delta_{x}$ is normal in
$\overline{\Gamma}_{\eta}(x)$, then the center of $\Delta_{x}$ is normal in
$\overline{\Gamma}_{\eta}(x)$. Hence all the elements of
Conj${}_{\overline{\Gamma}_{\eta}(x)}(g_{0})$ are in the center of
$\Delta_{x}$ and commute. The isometries of
Conj${}_{\overline{\Gamma}_{\eta}(x)}(g_{0})$ are of the same type of $g_{0}$.
If $g_{0}$ is either hyperbolic or elliptic we take
$Y=\bigcap_{h\in\overline{\Gamma}_{\eta}(x)}\text{Min}(hg_{0}h^{-1})$. It is
non-empty by Proposition 2.2, convex, closed and clearly
$\overline{\Gamma}_{\eta}(x)$-invariant. Moreover by triangle inequality we
have

$d(x,g_{0}x)\leq(2I_{0}-1)(3\cdot 2^{S_{0}-1}-2)\cdot\eta=\lambda.$

This shows that $\ell(g_{0})\leq\lambda$, so by definition
$x\notin\text{Min}(g_{0})$ and in particular $x\notin Y$. If $g_{0}$ is
parabolic we take $\tau$ such that $\ell(g_{0})<\tau<d(x,g_{0}x)$ and we
consider the set
$Y=\bigcap_{h\in\overline{\Gamma}_{\eta}(x)}M_{\tau}(hg_{0}h^{-1})$. It is
again convex, closed, non-empty by Proposition 2.2 and
$\overline{\Gamma}_{\eta}(x)$-invariant. Clearly $x\notin Y$ by definition of
$\tau$.

Step 5: the minumum $m$ of $\Psi$ is 0.
Let $Y$ be the subset of the previous step. We consider the projection $y$ of
$x$ on $Y$ and we extend the geodesic segment $[y,x]$ to a geodesic ray $c$
beyond $x$. Since $Y$ is $\overline{\Gamma}_{\eta}(x)$-invariant then
$d_{\sigma_{i}}$ is a non-decreasing function along $c$ for every
$i=1,\ldots,n$. Let us suppose first they are all constant along $c$. This
means that $\sigma_{i}c$ is parallel to $c$ for every $i=1,\ldots,n$.
Therefore $gc$ is parallel to $c$ for all $g\in\overline{\Gamma}_{\eta}(x)$.
In case (a), when $\Lambda_{x}$ is not trivial, we have the stronger
conclusion $gc=c$ for all $g\in\Lambda_{x}$. In particular $x$ is fixed by all
the elements of $\Lambda_{x}$, which is in contradiction to the fact that
$x\notin\text{Min}_{\lambda}(\Gamma,X)\supseteq\text{Fix}(\Gamma,X)$.
In case (b), when $g_{0}$ is parabolic this situation cannot happen. Indeed by
our choice of $\tau$ in the definition of $Y$ we see that $g_{0}c$ cannot be
parallel to $c$. When $g_{0}$ is hyperbolic we conclude, since $Y$ is
contained in $\text{Min}(g_{0})$, that also $x$ should belong to
$\text{Min}(g_{0})$ which is impossible because $\ell(g_{0})\leq\lambda$ and
$x\notin\text{Min}_{\lambda}(\Gamma,X)$.
Therefore the quantity

$\bar{t}=\max\\{t\in[0,+\infty)\text{ s.t. }d_{\sigma_{i}}(c(t))\leq
d_{\sigma_{i}}(x)\,\,\forall i=1,\ldots,n\\}$

is finite. We denote by $\bar{x}$ the point $c(\bar{t})$. We claim that
$\overline{\Sigma}_{\eta}(\bar{x})=\overline{\Sigma}_{\eta}(x)$ and
consequently $\Psi(\bar{x})=\Psi(x)=m$. Let us define the set
$A=\\{y\in[x,\bar{x}]\text{ s.t.
}\overline{\Sigma}_{\eta}(y)=\overline{\Sigma}_{\eta}(x)\\}$. By Lemma 3.3 and
by definition of $\bar{t}$ we conclude that $A$ is open. Indeed for
$y^{\prime}$ in a open neighbourhood $U$ of $y$ we have
$\overline{\Sigma}_{\eta}(y^{\prime})\subseteq\overline{\Sigma}_{\eta}(y)=\overline{\Sigma}_{\eta}(x)$.
By definition of $\bar{t}$ we know that in $c\cap U$, which is an open set of
$[x,\bar{x}]$ the other inclusion holds. Furthermore $A$ is closed: indeed
suppose to have a sequence of points $y_{n}\in A$ converging to $y$. Therefore
$\overline{\Sigma}_{\eta}(y)\supseteq\overline{\Sigma}_{\eta}(y_{n})=\overline{\Sigma}_{\eta}(x)$
for $n$ big enough, again by Lemma 3.3. But this containement is actually an
equality, by the maximality of $x$ with respect to $\preceq$ implying $y\in
A$. Indeed observe that by definition $\Psi(y_{n})=\Psi(x)$ for every $n$ and
by continuity of $\Psi$ the same holds for $y$. Now we can use the maximality
of $x$ with respect to $\preceq$ to conclude the equality. So $A=[x,\bar{x}]$,
i.e. $\overline{\Sigma}_{\eta}(\bar{x})=\\{\sigma_{1},\ldots,\sigma_{n}\\}$.
By Lemma 3.3 it holds
$\overline{\Sigma}_{\eta}(\bar{x})\supseteq\overline{\Sigma}_{\eta}(c(\bar{t}+t))$
for every $t>0$ small enough. By definition of $\bar{x}$ for such $t$’s there
is at least one index $i\in\\{1,\ldots,n\\}$ such that
$d(\sigma_{i}c(\bar{t}+t),c(\bar{t}+t))>d(\sigma_{i}\bar{x},\bar{x})$. So

$\Psi(c(\bar{t}+t))\leq\sum_{i=1}^{n}f(d(\sigma_{i}c(\bar{t}+t),c(\bar{t}+t)))<\sum_{i=1}^{n}f(d(\sigma_{i}\bar{x},\bar{x}))=\Psi(\bar{x})=m.$

Observe that the sum is made among the same set of isometries because of the
inclusion
$\overline{\Sigma}_{\eta}(\bar{x})\supseteq\overline{\Sigma}_{\eta}(c(\bar{t}+t))$.
This contradicts the fact that $m$ is the minimum, concluding the proof. ∎

######  Remark 3.4.

The proof of Theorem 3.1 in the almost-cocompact case.
We gave the proof in the cocompact case to avoid obscuring the main arguments
with other technical details. However, the same proof works for almost
cocompact actions (and in particular for finite volume ones): the only steps
where we used cocompactness are Step 2 and Step 3. Step 2 can be replaced in
the almost-cocompact case by the following argument:
– take again a sequence $x_{n}$ such that $\Psi(x_{n})\to\inf\Psi$. The
sequence $\Psi(x_{n})$ is bounded, therefore it exists $0<\delta<\eta$ such
that $d(x_{n},gx_{n})\geq\delta$ for every $g\in\Sigma_{\eta}(x_{n})$ and for
every $n$;
– this means that sys$(\Gamma,x_{n})\geq\delta$ for every $n$, so all the
$x_{n}$’s are contained in the $\delta$-thick part
Thick${}_{\delta}(\Gamma,X)$ of $X$, which is cocompact by almost
cocompactness;
– then, we can find a subsequence, called again $x_{n}$, that converges to
$x_{\infty}$ and, since $\Psi$ is continuous, $\Psi(x_{\infty})=\inf\Psi$.
Step 3 can be replaced by a similar argument: the map
$x\mapsto\overline{\Sigma}_{\eta}(x)$ is still upper semi-continuous and
$\Gamma$-invariant. The set $\Psi^{-1}(m)$ is contained in a thick part of
$X$, as said above. By almost cocompactness there exists $L$ such that
$\overline{\Sigma}_{\eta}(x)\leq L$ for all $x\in\Psi^{-1}(m)$. The rest of
the proof do not change.

######  Remark 3.5.

An interesting, though simple, topological fact is that when
$X=\text{Min}_{\lambda}(\Gamma,X)$, since this family of closed subsets is
locally finite then we can take the union only among those which have non-
empty interior, as already observed in [CCR01], Proposition 0.2. This is of
particular interest for _slim_ actions of groups with torsion, for instance
for actions on homology manifolds.

##  4\. Splitting and rigidity of CAT$(0)$ spaces under thin, slim actions

Our aim is to find several rigidity results for spaces with small diastole.
Let $I_{0}(P_{0},r_{0})$ be the constant given by Theorem 2.4 and let $J_{0}$
be the constant given by Bieberbach’s Theorem in dimension 2: every discrete,
cocompact group of isometries of $\mathbb{R}^{2}$ contains a lattice of index
at most $J_{0}$. We set

(4)
$\lambda_{0}=\min\left\\{\varepsilon_{0},\frac{4r_{0}}{\max\\{I_{0},J_{0}\\}\cdot(P_{0}+1)}\right\\}$

and we call
$\eta_{0}=\eta_{0}(P_{0},r_{0},\lambda_{0})=\eta_{0}(P_{0},r_{0})>0$ the
associated constant given by Theorem 3.1. Let $X$ be a complete, geodesically
complete, CAT$(0)$-space which is $P_{0}$-packed at scale $r_{0}$ and let
$\Gamma$ be a discrete, almost-cocompact group of isometries of $X$. We denote
by $\Lambda$ the set of hyperbolic isometries $g$ of $\Gamma$ with
$\ell(g)\leq\lambda_{0}$ and such that $\text{Min}(g)$ has non-empty interior.
Clearly $h\Lambda h^{-1}=\Lambda$ for all $h\in\Gamma$. Theorem 3.1 and Remark
3.5 implies that if the diastole is smaller than $\eta_{0}$ and the group is
slim then $X=\bigcup_{g\in\Lambda}\text{Min}(g)$.

We start studying spaces whit points of dimension $1$:

######  Proposition 4.1.

Let $X$ be a complete, geodesically complete, CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$ and suppose that $X$ has a point of dimension
$1$. Let $\Gamma$ be a discrete, slim, almost-cocompact group of isometries of
$X$. If $\textup{dias}(\Gamma,X)<\eta_{0}$ then $X$ is isometric to
$\mathbb{R}$.

###### Proof.

We have $X=\bigcup_{g\in\Lambda}\text{Min}(g).$ Since $X$ has points of
dimension $1$ then there is an open subset $U\subseteq X$ such that every
point of $U$ has dimension $1$. In particular we can find a point $x\in X$
such that $\overline{B}(x,\varepsilon)$ is isometric to an interval for some
$\varepsilon>0$, by the discussion of Section 2.1. We take $g\in\Lambda$ such
that $x\in\text{Min}(g)$. We can identify $x$ with a point $(y,0)\in
D(g)\times\mathbb{R}$. Since $x$ is of dimension $1$ we conclude that $(y,0)$
is a point of dimension $1$, and therefore that $y$ is a point of dimension
$0$: an isolated point. Since $D(g)$ is convex, and in particular connected,
we conclude that $D(g)$ is a point. Hence Min$(g)$ is isometric to
$\mathbb{R}$. Suppose $X\neq\text{Min}(g)$. Then we can find a point
$x^{\prime}$ at distance $r_{0}$ from $\text{Min}(g)$. We consider the
projection $y$ of $x^{\prime}$ on $\text{Min}(g)$. We focus on the segment
$[y,gy]$. Clearly some element in the orbit of $\langle g\rangle x$ must lie
inside this segment. Therefore we can find a point, called again $x$, such
that $x\in[y,gy]$ and $\overline{B}(x,\varepsilon)$ is isometric to an
interval. We claim that the geodesic $[x^{\prime},gx^{\prime}]$ has length at
least $2r_{0}$. Let $c\colon[0,d(x^{\prime},gx^{\prime})]\to X$ be the
geodesic $[x^{\prime},gx^{\prime}]$. For every
$t\in[0,d(x^{\prime},gx^{\prime})]$ we consider the geodesic $[c(t),x]$. There
are only three cases: $[c(t)]_{x}=v_{-}$, $[c(t)]_{x}=v_{+}$ or $c(t)=x$,
where $v_{\pm}$ are the only two elements of $\Sigma_{x}X$. It is easy to see
that the sets $A_{\pm}=\\{t\in[0,d(x^{\prime},gx^{\prime})]\text{ s.t.
}[c(t)]_{x}=v_{\pm}\\}$ are closed. If $c(t)\neq x$ for every
$t\in[0,d(x^{\prime},gx^{\prime})]$ one concludes that
$[0,d(x^{\prime},gx^{\prime})]=A_{+}$ or
$[0,d(x^{\prime},gx^{\prime})]=A_{-}$. We claim that both these two cases are
impossible since $[x^{\prime}]_{x}\neq[gx^{\prime}]_{x}$. Indeed we consider
the convex map $d(x^{\prime},\cdot)$ on the geodesic $\text{Min}(g)$ which has
its minimum at $y$. This forces to have $[x^{\prime}]_{x}=[y]_{x}$. Indeed let
$x_{\pm\varepsilon}$ be the points along Min$(g)$ at distance $\varepsilon$
from $x$ in direction $v_{\pm}$ and suppose $[y]_{x}=v_{-}$. Then
$d(x_{-\varepsilon},x^{\prime})<d(x_{+\varepsilon},x^{\prime})$. The geodesic
$[x^{\prime},x]$ is either
$[x^{\prime},x_{-\varepsilon}]\cup[x_{-\varepsilon},x]$ or
$[x^{\prime},x_{+\varepsilon}]\cup[x_{+\varepsilon},x]$, so the inequality
above says it is the first one. In other words $[x^{\prime}]_{x}=[y]_{x}$. In
the same way $[gx^{\prime}]_{x}=[gy]_{x}$ and clearly $[y]_{x}\neq[gy]_{x}$.
Therefore there must be some $t\in[0,d(x^{\prime},gx^{\prime})]$ such that
$c(t)=x$. This forces the length of $c$ to be bigger than $2r_{0}$.
Analogously we have that $d(g^{n}x^{\prime},g^{m}x^{\prime})>2r_{0}$ for all
distincts $n,m\in\mathbb{Z}$. Moreover $d(g^{n}x^{\prime},y)\leq
r_{0}+|n|\ell(g)\leq r_{0}+|n|\cdot\frac{4r_{0}}{P_{0}+1}$ for all
$n\in\mathbb{Z}$. Therefore we can find $P_{0}+1$ distinct points of the form
$g^{n}x^{\prime}$ inside $\overline{B}(y,3r_{0})$. They form a
$2r_{0}$-separated subset of $\overline{B}(y,3r_{0})$ and this is a
contradiction to the packing assumption on $X$. ∎

Another application is a characterization of spaces with small diastole
admitting a strictly negatively curved open subset. It resembles [CCR01],
Corollary 0.7.

######  Corollary 4.2.

Let $X$ be a complete and geodesically complete CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$, and suppose that there exists some
(arbitrarily small) open set $U\subset X$ which is CAT$(-\varepsilon)$, for
some $\varepsilon>0$. Let $\Gamma$ be a discrete, slim, almost-cocompact group
of isometries of $X$: if $\textup{dias}(\Gamma,X)<\eta_{0}$ then $X$ is
isometric to $\mathbb{R}$.

###### Proof.

Again we know that $X=\bigcup_{g\in\Lambda}\text{Min}(g)$. If there exists a
point of dimension $1$ then $X$ is isometric to $\mathbb{R}$ by Proposition
4.1. Therefore we can suppose that every point of $X$ has dimension $\geq 2$.
We then take a point $x\in U$ and an isometry $g\in\Lambda$, such that
$x\in\text{Min}(g)=D(g)\times\mathbb{R}$. We identify $x$ with a point
$(y,0)\in D(g)\times\mathbb{R}$. Since every point of $X$ has dimension $\geq
2$ then $D(g)$ is not a single point, i.e. there is $y\neq y^{\prime}\in
D(g)$. By the quadrangle flat theorem ([BH13], Theorem II.2.11) we have that
the non-degenerate quadrangle $(y,0),g(y,0),g(y^{\prime},0),(y^{\prime},0)$ is
isometric to a rectangle in $\mathbb{R}^{2}$. It is then possible to find
inside $U$ a non-degenerate quadrangle isometric to a euclidean rectangle.
This contradicts the assumption on $U$. ∎

For the next applications we need the following splitting criterion.

######  Lemma 4.3.

Let $X$ be a complete, geodesically complete, CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$. Let $\Gamma$ be a discrete, slim, almost-
cocompact group of isometries of $X$. Suppose that
$X=\bigcup_{g\in\Lambda}\textup{Min}(g)$ and that if $g,h\in\Lambda$ are such
that $\textup{Min}(g)\cap\textup{Min}(h)\neq\emptyset$ then $\langle
g,h\rangle$ is virtually cyclic.
Then $X$ splits isometrically as $Y\times\mathbb{R}$. Moreover if $\Gamma$ is
finitely generated then $\Gamma$ virtually splits as
$\Gamma_{Y}\times\mathbb{Z}$, where $\Gamma_{Y}$ (resp. $\mathbb{Z}$) acts
discretely on $Y$ (resp. $\mathbb{R}$).

The first part of the proof should be compared to [BGS13], Appendix 2, Lemma 1
and Lemma 2.

###### Proof.

A first observation is that if $K$ is a compact subset of $X$ then the set
$\Lambda_{K}:=\\{g\in\Lambda\text{ s.t. }\text{Min}(g)\cap K\neq\emptyset\\}$
is finite. Indeed if $x$ is a point of $K$ then $d(x,gx)\leq
2\text{Diam}(K)+\lambda_{0}$ for all $g\in\Lambda_{K}$ and the claim follows
by discreteness of $\Gamma$.
The second observation is that the isometries of $\Lambda$ have all parallel
axes. Let $g,h\in\Lambda$ and take $x\in\text{Min}(g),y\in\text{Min}(h)$.
Applying the first claim to $K=[x,y]$ and using the connectedness of $[x,y]$
we can find a finite sequence $g=f_{0},f_{1},\ldots,f_{n}=h$ of elements of
$\Lambda$ such that $\text{Min}(f_{i})\cap\text{Min}(f_{i+1})\neq\emptyset$
for all $i=0,\ldots,n-1$. We prove that the axes of $g$ and $h$ are parallel
by induction on $n$. If $n=0$ it is trivial. If $n=1$, since $\langle
g,h\rangle$ is virtually cyclic, there exist $p,q\in\mathbb{Z}^{*}$ such that
$g^{p}=h^{q}$ and the claim follows. The induction step is similar: the axes
of $g$ are parallel to the axes of $f_{n-1}$ and there exist
$p,q\in\mathbb{Z}^{*}$ such that $f_{n-1}^{p}=h^{q}$.
Let now $c$ be one axis of some $g\in\Lambda$ and consider the subset $P_{c}$
of $X$ of all geodesic lines parallel to $c$. It is closed, convex and
$\Gamma$-invariant. Indeed if $\gamma\in\Gamma$ then $\gamma c$ is an axis of
$\gamma g\gamma^{-1}\in\Lambda$, so $\gamma c$ is parallel to $c$. By
minimality of the action, Proposition 2.10, we conclude that $X=P_{c}$ and
$P_{c}$ canonically splits isometrically as $Y\times\mathbb{R}$, cp. Theorem
II.2.14 of [BH13].

For the finitely generated case we need to show another claim: for every
compact, convex $K\subseteq X$ there exists a hyperbolic isometry
$\gamma\in\Gamma^{*}$ such that $K\subseteq\text{Min}(\gamma)$ and the axes of
$\gamma$ are parallel to $c$. We know $\Lambda_{K}$ is finite, say
$\Lambda_{K}=\\{g_{1},\ldots,g_{n}\\}$, so
$K\subseteq\bigcup_{i=1}^{n}\text{Min}(g_{i})$. By connectedness of $K$ we
know that for each $i$ there exists $j$ such that
$\text{Min}(g_{i})\cap\text{Min}(g_{j})\neq\emptyset$. An easy computation
shows that there are $p_{i}\in\mathbb{Z}^{*}$ such that
$g_{i}^{p_{i}}=g_{j}^{p_{j}}$ for all $i,j\in\\{1,\ldots,n\\}$. If we call
$\gamma$ this common power we have

$K\subseteq\bigcup_{i=1}^{n}\text{Min}(g_{i})\subseteq\bigcup_{i=1}^{n}\text{Min}(g_{i}^{p_{i}})=\text{Min}(\gamma),$

and clearly the axes of $\gamma$ are parallel to $c$.
Suppose $\Gamma$ is generated by $\\{h_{1},\ldots,h_{n}\\}$ and set
$R:=\max_{i=1,\ldots,n}d(x_{0},h_{i}x_{0})$ for some point $x_{0}\in X$. We
apply the claim above to $K=\overline{B}(x_{0},2R)$, finding a hyperbolic
isometry $\gamma\in\Gamma^{*}$ with axes parallel to $c$ such that
$\overline{B}(x_{0},2R)\subseteq\text{Min}(\gamma)$. We recall that $\Gamma$
sends geodesic lines parallel to $c$ to geodesic lines parallel to $c$. In
particular the axes of $h_{i}\gamma h_{i}^{-1}$ are parallel to the axes of
$\gamma$. By definition of $R$ and $\gamma$ we observe that
$B(h_{i}x_{0},R)\subseteq\text{Min}(\gamma)\cap\text{Min}(h_{i}\gamma
h_{i}^{-1})$. Therefore these two isometries have the same axis passing
through every point of $B(h_{i}x_{0},R)$ and they have the same translation
length. This implies that $h_{i}\gamma h_{i}^{-1}=\gamma$ on
$B(h_{i}x_{0},R)$. Hence $h_{i}\gamma h_{i}^{-1}=\gamma$ because they coincide
on an open subset. This shows that $\langle\gamma\rangle$ is normal, and even
central, in $\Gamma$. So $\Gamma$ preserves
$\text{Min}(\gamma)=X=Y\times\mathbb{R}$ because $\Gamma$ is minimal by
Proposition 2.10 and $\Gamma$ virtually splits as $\Gamma_{Y}\times\mathbb{Z}$
by Theorem II.7.1 of [BH13], with each factor acting discretely on the
corresponding space. ∎

###### Proof of Corollary B.

As usual we know that $X=\bigcup_{g\in\Lambda}\text{Min}(g)$. For every $x\in
X$ let $\overline{\Gamma}_{\lambda_{0}}(x)$ be the non-trivial group generated
by $\overline{\Sigma}_{\lambda_{0}}(x)$. By Proposition 2.6 we can find a
convex subset $W_{x}\times\mathbb{R}^{k_{x}}$ and a finite index subgroup
$\Gamma_{W_{x}}\times\mathbb{Z}^{k_{x}}$ of
$\overline{\Gamma}_{\lambda_{0}}(x)$, where $\Gamma_{W_{x}}$ acts on $W_{x}$
without hyperbolic elements and $\mathbb{Z}^{k_{x}}$ acts on
$\mathbb{R}^{k_{x}}$ as a lattice. The existence of a hyperbolic isometry
displacing $x$ by at most $\lambda_{0}$ forces to have $k_{x}\geq 1$. If
$k_{x}\geq 2$ we have the desired $\mathbb{Z}\times\mathbb{Z}$, so we can
suppose $k_{x}=1$. If $\Gamma_{W_{x}}$ contains some parabolic element we have
again $\mathbb{Z}\times\mathbb{Z}$ inside $\Gamma$. In the remaining case we
have $k_{x}=1$ and $\Gamma_{W_{x}}$ finite, so
$\overline{\Gamma}_{\lambda_{0}}(x)$ is virtually cyclic for every $x\in X$.
The assumptions of Lemma 4.3 are clearly satisfied, so $X$ splits as
$Y\times\mathbb{R}$. If moreover $\Gamma$ is cocompact (hence finitely
generated) then it has a finite index subgroup which splits as
$\Gamma_{Y}\times\mathbb{Z}$, and clearly $\Gamma_{Y}$ acts cocompactly on
$Y$. By [Swe99], Theorem 11 either $Y$ is a point or $\Gamma_{Y}$ contains an
infinite order element, because $Y$ is also geodesically complete. In the
second case we have again $\mathbb{Z}\times\mathbb{Z}$ inside $\Gamma$, in the
first one $X=\mathbb{R}$.

∎

######  Remark 4.4.

If in Corollary B we consider finitely generated almost-cocompact groups in
place of cocompact ones we can still apply Lemma 4.3 to get a finite index
subgroup of the form $\Gamma_{Y}\times\mathbb{Z}$. Moreover we could get the
same conclusion of Corollary B under these weaker assumptions if we knew that
$\Gamma_{Y}$ contains an isometry of infinite order. This last statement is
the content of the following conjecture.

######  Conjecture.

Let $X$ be a proper, CAT$(0)$-space and let $\Gamma$ be a discrete group of
isometries of $X$, not necessarely cocompact. If $\Gamma$ is torsion, i.e. it
contains only finite order elements, then $\Gamma$ is finite.
An easier version for us would be: if $\Gamma$ is almost-cocompact then it
contains an infinite order element.

By the flat torus theorem ([BH13], Theorem I.7.1), if $\Gamma$ contains
$\mathbb{Z}\times\mathbb{Z}$ then $X$ contains an isometrically embedded flat
$\mathbb{R}^{2}$. The same holds if $X$ splits as $Y\times\mathbb{R}$, with
$Y$ different from a point. This is impossible for instance if $X$ is a
visibility space (see Definition I.9.28 of [BH13]), therefore we obtain:

######  Corollary 4.5.

Let $X$ be a complete, geodesically complete, CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$, and assume that $X$ is a visibility space.
Let $\Gamma$ be a discrete, slim, almost-cocompact group of isometries of $X$:
if $\textup{dias}(\Gamma,X)<\eta_{0}$ then $X$ is isometric to $\mathbb{R}$.

The last example of characterization of spaces with small diastole is the
following, holding for spaces of dimension at most $2$.

######  Proposition 4.6.

Let $X$ be a complete, geodesically complete, CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$, and assume that the dimension of $X$ is at
most $2$. Let $\Gamma$ be a discrete, slim, almost-cocompact group of
isometries of $X$: if $\textup{dias}(\Gamma,X)<\eta_{0}$ then $X$ is isometric
to a product $T\times\mathbb{R}$, where $T$ is a geodesically complete,
simplicial tree (possibily reduced to a point). Moreover if $\Gamma$ is
cocompact then it has a finite index subgroup which splits as
$\Gamma_{T}\times\mathbb{Z}$, where $\Gamma_{T}$ (resp. $\mathbb{Z}$) acts
discretely and cocompactly on $T$ (resp. $\mathbb{R}$).

First of all we need to characterize trees as those CAT$(0)$-spaces with
geometric dimension $1$.

######  Lemma 4.7 (See also the proof of Lemma 6.3 of [OP21]).

Let $Y$ be a CAT$(0)$-space such that $\Sigma_{y}Y$ is discrete and not empty
for every $y\in Y$, in other words $\textup{GD}(Y)=1$. Then $Y$ is a real
tree, i.e. for every two points $x,z\in Y$ the image of any continuous
injective map $\alpha\colon[a,b]\to X$ such that $\alpha(a)=x$ and
$\alpha(b)=z$ coincides with the image of $[x,z]$. Moreover, if $Y$ is
geodesically complete and proper then it is a simplicial tree.

###### Proof.

Let $x,z\in Y$, $c=[x,z]$ and $\alpha$ be a continuous injective map
$\alpha\colon[a,b]\to Y$ such that $\alpha(a)=x$ and $\alpha(b)=z$. We want to
show that $\text{Im}(\alpha)=\text{Im}(c)$, where $c\colon[0,d(x,z)]\to Y$ is
the geodesic $[x,z]$. We first prove that
$\text{Im}(c)\subseteq\text{Im}(\alpha)$. Let $s\in(0,d(x,z))$. The set
$\Sigma_{c(s)}Y$ is discrete, say equal to $\\{v_{i}\\}_{i\in I}$. We consider
the sets $A_{i}=\\{t\in[a,b]\text{ s.t. }[\alpha(t)]_{c(s)}=v_{i}\\}$, for
$i\in I$. Observe that the quantity $[\alpha(t)]_{c(s)}$ is defined as soon as
$\alpha(t)\neq c(s)$. By discreteness of $\Sigma_{c(s)}Y$ it is easy to check
that each $A_{i}$ is open and closed. By connectedness of $[a,b]$ there are
two possibilities: either there exists $i\in I$ such that $A_{i}=[a,b]$ or
there is some $t\in(a,b)$ such that $\alpha(t)=c(s)$. The first possibility
has to be excluded since clearly $[\alpha(a)]_{c(s)}\neq[\alpha(b)]_{c(s)}$,
so $c(s)\in\text{Im}(\alpha)$.
It remains to prove the other inclusion. We set $A=\\{t\in[a,b]\text{ s.t.
}\alpha(t)\in\text{Im}(c)\\}$. It is a closed subset of $[a,b]$, so the
complementary is the disjoint union of countably many intervals,
$A^{c}=\bigcup_{i=1}^{N}(a_{i},b_{i})$, $N\in\mathbb{N}\cup\\{\infty\\}$. In
the interval $(a_{i},b_{i})$ we have $\text{Im}(\alpha|_{[a_{i},b_{i}]})\cap
c=\\{\alpha(a_{i}),\alpha(b_{i})\\}$. By injectivity of $\alpha$ we know that
$\alpha(a_{i})\neq\alpha(b_{i})$, so the subsegment of $c$ joining these two
points is not trivial. However applying the first part of the proof to the
continuous map $\alpha|_{[a_{i},b_{i}]}$ we obtain
$\text{Im}(\alpha|_{[a_{i},b_{i}]})\cap
c\supseteq[\alpha(a_{i}),\alpha(b_{i})]$, a contradiction. This shows that
$A=[a,b]$.
If a real tree $Y$ is locally compact and geodesically complete then it is
simplicial by the description of geodesically complete, proper,
CAT$(0)$-spaces of [LN19] recalled in Section 2.1. ∎

The second tool we need is a better description of the almost stabilizers in
dimension at most $2$.

######  Lemma 4.8.

Let $X$ be a complete, geodesically complete, CAT$(0)$-space which is
$P_{0}$-packed at scale $r_{0}$, and assume that the dimension of $X$ is at
most $2$. Let $\Gamma$ be a discrete group of isometries of $X$ and let
$\varepsilon_{0}$ be the Margulis constant given by Theorem 2.4. Then for
every $x\in X$ one of the following three mutually exclusive possibilities
occur:

  * (i)

$\overline{\Gamma}_{\varepsilon_{0}}(x)$ has no hyperbolic isometries;

  * (ii)

$\overline{\Gamma}_{\varepsilon_{0}}(x)$ has no parabolic isometries and it is
virtually $\mathbb{Z}$;

  * (iii)

$\overline{\Gamma}_{\varepsilon_{0}}(x)$ has no parabolic isometries and it
has a subgroup of index at most $J_{0}$ which is isomorphic to
$\mathbb{Z}^{2}$.

###### Proof.

As $\overline{\Gamma}_{\varepsilon_{0}}(x)$ is virtually nilpotent, by
Proposition 2.6 we can find a
$\overline{\Gamma}_{\varepsilon_{0}}(x)$-invariant convex subset
$W\times\mathbb{R}^{k}$ of $X$ and a finite index subgroup of
$\overline{\Gamma}_{\varepsilon_{0}}(x)$ of the form
$\Gamma_{W}\times\mathbb{Z}^{k}$, where $\Gamma_{W}$ acts on $W$ by parabolic
or elliptic isometries. If there is at least one hyperbolic isometry in
$\overline{\Gamma}_{\varepsilon_{0}}(x)$ then $k\geq 1$ and of course $k\leq
2$. We start with the case $k=2$. By dimensional reasons $W$ must have
dimension $0$, so it must be a point. This means that
$\overline{\Gamma}_{\varepsilon_{0}}(x)$ acts on $\mathbb{R}^{2}$ cocompactly
and discretely because it contains a lattice of $\mathbb{R}^{2}$ as finite
index subgroup. By Bieberbach’s Theorem we conclude that
$\overline{\Gamma}_{\varepsilon_{0}}(x)$ has a subgroup of index at most
$J_{0}$ which is isomorphic to $\mathbb{Z}^{2}$.
Now suppose $k=1$. Then $W$ is a CAT$(0)$-space of geometric dimension at most
$1$, so it is a real tree by Lemma 4.7 or a point. In any case $W$ has no
parabolic isometries, so $\Gamma_{W}$ contains only elliptic elements. As a
consequence $\Gamma_{W}$ is finite, so
$\overline{\Gamma}_{\varepsilon_{0}}(x)$ has no parabolic isometries and it is
virtually cyclic. ∎

###### Proof of Proposition 4.6.

We know that $X=\bigcup_{g\in\Lambda}\text{Min}(g)$. If $X$ has a point of
dimension $1$ then $X=\mathbb{R}$ by Proposition 4.1. We can therefore suppose
that every point of $X$ is of dimension $2$. The minimal set of $g\in\Lambda$
is of the form $\text{Min}(g)=D(g)\times\mathbb{R}$. Since every point of $X$
has dimension $2$ and $\text{Min}(g)$ has non-empty interior we deduce that
$D(g)$ is not a point, and by Lemma 4.7 it is a (maybe compact) real tree. If
$g,h\in\Lambda$ are such that $\text{Min}(g)\cap\text{Min}(h)\neq\emptyset$
then either $\langle g,h\rangle$ is virtually cyclic or $\langle
g^{J_{0}},h^{J_{0}}\rangle\cong\mathbb{Z}^{2}$ by Lemma 4.8. Let us study the
second case. Since $g^{J_{0}}$ and $h^{J_{0}}$ commute, the isometry
$h^{J_{0}}$ acts on $\text{Min}(g^{J_{0}})=D(g^{J_{0}})\times\mathbb{R}$
preserving the product decomposition. Let $h^{\prime}$ be the isometry of
$D(g^{J_{0}})$ induced by $h^{J_{0}}$. The isometry $h^{\prime}$ cannot have
fixed points otherwise $h^{J_{0}}$ and $g^{J_{0}}$ have a common axis, and
they would generate a virtually cyclic group. Therefore $h^{\prime}$ is a
hyperbolic isometry of $D(g^{J_{0}})$ and we call $c$ its axis. We want to
show that $D(g^{J_{0}})=c$. Suppose there exists a point $x\in D(g^{J_{0}})$
which is not in $c$. Let $y$ be its projection on $c$. As $D(g^{J_{0}})$ is a
tree, the geodesic in $D(g^{J_{0}})$, and so in $X$, between $x$ and
$h^{\prime}x$ is the concatenation
$[x,y]\cup[y,h^{\prime}y]\cup[h^{\prime}y,h^{\prime}x]$. We extend the
geodesic segment $[y,x]$ in $X$ beyond $x$, defining a geodesic ray $c_{y}$.
We do the same for the geodesic segment $[h^{\prime}y,h^{\prime}x]$ defining a
geodesic ray $c_{h^{\prime}y}$. We observe that the concatenation
$c_{y}\cup[y,h^{\prime}y]\cup c_{h^{\prime}y}$ is a geodesic line in $X$. In
particular the distance between the two points along $c_{y}$ and
$c_{h^{\prime}y}$ at distance $r_{0}$ from $y$ and $h^{\prime}y$ is at least
$2r_{0}$. We can repeat the construction for all points $(h^{\prime})^{n}x$,
extending the geodesic segments $[(h^{\prime})^{n}y,(h^{\prime})^{n}x]$ beyond
$(h^{\prime})^{n}x$ and finding points that are at distance at least $2r_{0}$
one from the other. Clearly $\ell(h^{\prime})\leq\ell(h^{J_{0}})\leq
J_{0}\cdot\lambda_{0}$, so arguing as in the proof of Proposition 4.1 we
conclude we can find more than $P_{0}$ points inside $\overline{B}(y,3r_{0})$
that are $r_{0}$-separated as soon as
$\lambda_{0}\leq\frac{4r_{0}}{J_{0}(P_{0}+1)}$ which is the case by (4). This
violates the packing assumption on $X$, so $D(g^{J_{0}})=\mathbb{R}$ and
Min$(g^{J_{0}})$ must be isometric to $\mathbb{R}^{2}$. The same conclusion
holds for Min$(h^{J_{0}})$. By the flat torus theorem we know that

$Y\times\mathbb{R}^{2}=\text{Min}(g^{J_{0}})\cap\text{Min}(h^{J_{0}})\subseteq\text{Min}(g^{J_{0}})=\mathbb{R}^{2}$

for some convex subset $Y$ of $X$. Since the dimension of $X$ is at most $2$
the set $Y$ must be a point and the inclusion above is an equality. This shows
that $\text{Min}(g^{J_{0}})=\text{Min}(h^{J_{0}})=\mathbb{R}^{2}$. The same
argument shows that also
$\text{Min}(g^{pJ_{0}})=\text{Min}(g^{J_{0}})=\mathbb{R}^{2}$ for all $p\neq
0$, because $h^{J_{0}}$ acts on $\text{Min}(g^{pJ_{0}})$. In the same way we
have $\text{Min}(h^{pJ_{0}})=\text{Min}(h^{J_{0}})=\mathbb{R}^{2}$ for all
$p\neq 0$.
We claim that $\text{Min}(f)\subseteq\mathbb{R}^{2}$ for all $f\in\Lambda$. It
is not difficult to see it by taking a path $g=f_{0},f_{1},\ldots,f_{n}=f$
such that $f_{i}\in\Lambda$ and
$\text{Min}(f_{i})\cap\text{Min}(f_{i+1})\neq\emptyset$ for all
$i=0,\ldots,n-1$. Therefore
$X=\bigcup_{g\in\Lambda}\text{Min}(g)=\mathbb{R}^{2}$ which is
$\mathbb{R}\times\mathbb{R}$. If moreover $\Gamma$ is cocompact then the
thesis is implied by Bieberbach’s Theorem in dimension $2$.
It remains the case where $\langle g,h\rangle$ is virtually cyclic for all
$g,h\in\Lambda$ such that $\text{Min}(g)\cap\text{Min}(h)\neq\emptyset$. An
application of Lemma 4.3 gives the splitting of $X$ as $T\times\mathbb{R}$.
Here $T$ is necessarely a simplicial tree because of Lemma 4.7. If $\Gamma$ is
cocompact then the thesis follows from the second part of Lemma 4.3.

∎

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