# Improved Flexible Coaxial Ribbon Cable for High-Density Superconducting
Arrays

Jennifer Pearl Smith *  —1, Benjamin A. Mazin *  —1, Alirio Boaventura2, Kyle
J. Thompson2, Miguel Daal *  —1  1University of California Santa Barbara,
Department of Physics and Astronomy, Santa Barbara, CA 93106 USA (email:
<EMAIL_ADDRESS> 2Maybell Quantum Industries, Denver, CO 80221 USA

###### Abstract

Superconducting arrays often require specialized, high-density cryogenic
cabling capable of transporting electrical signals across temperature stages
with minimal loss, crosstalk, and thermal conductivity. We report improvements
to the design and fabrication of previously published superconducting 53 wt%
Nb-47 wt% Ti (Nb47Ti) FLexible coAXial ribbon cables (FLAX). We used 3D
electromagnetic simulations to inform design changes to improve the
characteristic impedance of the cable and the connector transition. We
increased the center conductor diameter from 0.003 inches to 0.005 inches
which lowered the cable characteristic impedance from $\sim$60 $\bm{\Omega}$
to $\sim$53 $\bm{\Omega}$. This change had a negligible impact on the computed
heat load which we estimate to be 5 nW per trace from 1 K to 100 mK with a
1-ft cable. This is approximately half the calculated heat load for the
smallest commercially available superconducting coax. We also modified the
transition board to include a capacitive coupling between the upper ground
plane and signal traces that mitigates the inductive transition. We tested
these changes in a 5-trace, 1-ft long cable at 4 K and found the microwave
transmission improved from 6 dB to 1.5 dB of attenuation at 8 GHz. This loss
is comparable to commercial superconducting coax and 3$\times$ lower than
commercial NbTi-on-polyimide flex cables at 8 GHz. The nearest-neighbor
forward crosstalk remained less than -40 dB at 8 GHz. We compare key
performance metrics with commercially available superconducting coax and NbTi-
on-polyimide flex cables and we share initial progress on commercialization of
this technology by Maybell Quantum Industries.

###### Index Terms:

Superconducting cables, Superconducting microwave devices, Superconducting
filaments and wires, Superconducting resonators.

##  I Introduction

Superconducting devices are driving technology advancements in quantum
computing[1, 2], single-photon-counting sensors[3, 4], low-noise
amplifiers[5], and more. These technologies all require specialized cryogenic
wiring capable of transporting electrical signals across temperature stages
with minimal loss, crosstalk, and thermal conductivity. The development of
high-density cryogenic wiring has proved to be a bottleneck in scaling up
superconducting systems that need potentially thousands of cryogenic I/O
channels to meaningfully operate [6].

Progress has been slowed in part by stringent performance requirements that
limit wiring solutions to materials that are difficult to work with. Niobium-
titanium (NbTi) is one such material that has very low thermal conductivity
and is superconducting under 9 K, providing excellent cryogenic transmission
and isolation with a fraction of the heat load afforded by other metals[7].
This unique combination of electrical and thermal properties has led NbTi to
become the conductor of choice for cryogenic wiring despite the fact it cannot
be soldered and is otherwise difficult to work with and connectorize [8].
Similarly, Teflon-type materials make ideal dielectric mediums because they
are flexible with low thermal conductivity and dielectric loss. Unfortunately,
these materials come with a host of manufacturing challenges including
extremely high melting temperatures, non-trivial phase changes, potential to
produce highly-toxic byproducts, and inability to adhere well to other
materials [9, 10].

Despite the fabrication challenges, vendors have managed to fabricate
individual cryogenic coax lines by extruding NbTi tubes threaded with Teflon-
coated NbTi wire [11, 12, 13]. These cables boast excellent transmission and
isolation but are semi-rigid and cumbersome to use in small cryogenic volumes.
Furthermore, the extrusion process restricts manufacturing to relatively large
cross sections resulting excessive heat loads. More recently, processes have
been developed that involve sputtering NbTi onto a polyimide material such as
Kapton to form stripline traces[14, 15, 16]. NbTi-on-polyimide cables are more
flexible and have lower heat load than the aforementioned semi-rigid cables
but have higher dielectric loss that scales with frequency. This loss can be
prohibitive to applications carrying RF or Microwave signals from cryogenic
devices before amplification where the signals are small and vulnerable to
noise. Neither of these leading commercial options met the transmission,
crosstalk, and heat load requirements for our application delivering 4-8 GHz
signals to an array of 20,000+ microwave kinetic inductance detectors (MKIDs)
at 10 mK [17, 3].

Unable to buy suitable cryogenic wiring, we previously developed our own in-
house cable manufacturing process at the University of California, Santa
Barbara that involves spot welding NbTi foil around PFA-coated NbTi wires to
form a flexible coaxial ribbon cable (FLAX) [18]. This technology aims to
replicate the near-perfect isolation and transmission offered by semi-rigid,
superconducting coax but in a smaller, more flexible format that reduces heat
load and allows for increased I/O density. In [18], we demonstrated
competitive heat load and isolation but the transmission loss could prove to
be a dominant noise source for highly-sensitive experiments like those
requiring quantum-noise-limited amplifiers. Another drawback was the highly
variable, manual, and labor intensive manufacturing process [18].

In this paper, we present improvements to the cable design and show 3x
improved cable transmission without negatively impacting the isolation or heat
load. We share initial progress by Maybell Quantum Industries on a roll-to-
roll manufacturing system to streamline and standardize cable fabrication in
preparation for commercialization. Lastly, we compare the latest performance
data with the smallest commercially available semi-rigid superconducting coax
from CryoCoax111CryoCoax - Intelliconnect, 448 Old Lantana Road, Crossville,
TN, USA. P/N: BCB012[11] and the NbTi-on-polyimide Cri/oFlex®3 from Delft
Circuits222Delft Circuits, 2627 AN, Delft, Netherlands. P/N: Cri/oFlex®3[16].

##  II Manufacturing Improvements

The previous iteration of the FLAX cable was manufactured using individual,
hand-placed spot welds to act as vias between the cable traces [18]. While
effective, this process was slow and error-prone because any welds placed too
close to the center wire result in an electrical short, rendering the cable
useless. Maybell has developed a laser welding cable manufacturing system and
is developing a roll-to-roll apparatus that will greatly simplify and speed up
speed up cable manufacturing as well as increase overall uniformity. A
prototype sample laser-welded using a linearly actuated mechanical stage is
shown in Fig. 1. A close up of the continuous laser welds surrounding two
wires is show in Fig. 2. We expect cables manufactured with this system to
show enhanced durability and better impedance control due to the precise,
continuous laser welds. We also note the proposed roll-to-roll process allows
cables to be made with arbitrary length, giving more flexibility to cryogenic
routing and design.

Figure 1: Eight-trace Maybell cable prototype manufactured using linearly
actuated mechanical stage with laser welding. Two sheets of 1 mil thick NbTi
foil are welded around PFA-coated NbTi wires with a continuous laser weld.

##  III Design Improvements

Two main design improvements were made to correct for impedance mismatches in
the cable and improve the microwave transmission. First, the characteristic
impedance of the cable was adjusted closer to 50 $\Omega$. Second, the
connectorization was modified to reduce reactive impedance reflections off the
connector ends.

###  III-A Characteristic Impedance

The characteristic impedance of the cable was originally designed in [18]
assuming a perfect circular cross section for the inner and outer conductor
geometry. In reality, the manufacturing process produces an outer conductor
profile more parabolic than semicircular (see Fig. 2). This results in the
cable cross section appearing more elliptical (see Fig. 3) which causes the
outer conductor to be effectively further from the inner-conductor, elevating
the cable characteristic impedance.

We modeled the actual coaxial shape as a spline in Ansys HFSS and determined
increasing the center conductor diameter from 0.003 inches to 0.005 inches
(without changing the dielectric thickness) would effectively bring the inner
and outer conductors closer together and bring the impedance back down to the
target 50 $\Omega$.

Figure 2: Manufacturing data obtained from JT Automation during a prototype
cable manufacturing run. A.) Angle view showing a single sheet of NbTi foil
with two crimps made to house wires with continuous laser welds in-between.
B.) Aerial view of the NbTi foil showing the same welds and crimps pictured to
the left. C.) Cable profile measured in inches. In all views the cable profile
is highlighted in yellow.

###  III-B End Inductance

Figure 3: Exploded view of cable-end assembly with key dimensions shown in
inches. Drawing is not to scale. From top/back to bottom/front: G3PO half-
shell connectors are soldered to the transition board. The transition board
features 50 $\Omega$ grounded coplanar waveguide structures with the upper
ground planes connected in front to create capacitance between the upper
ground planes and signal traces. The FLAX cable center conductors are crimped
into stainless steel capillary tubes and soldered to the center traces. The
FLAX ground shield is spot welded to the upper ground planes. The cable cross
section shows the PFA (blue) insulated NbTi (grey) wire set in parabolic
crimps made in the shared NbTi foil ground shield. The two sides of the shield
are mechanically and electrically bonded with micro spot welds less than
$\lambda/16\simeq$ 2 mm (at 8 GHz) apart that run in-between the traces down
the length of the cable. Figure is modified from [18].

When traveling out of the cable and into the connector, the electric field
transitions from a coaxial mode in the cable to a coplanar waveguide mode in
the transition board, then back to a coaxial mode in the connector. The center
conductors, which behave inductively, are relatively continuous in the
transition; however, the surrounding ground planes, which provide a balancing
capacitance, undergo a large geometry shift from 3D to 2D and back. The
authors theorize that the disruption to the 3D ground plane reduces
capacitance in the transition and causes the connector ends to behave
inductively. This can be seen in the transmission spectrum from [18], which
shows ripples in the transmission consistent with the fundamental mode of the
cable growing with increasing frequency. This is consistent with reflections
off inductive features in the cable ends because the impedance of a perfect
inductor grows linearly with frequency, i.e., $Z_{L}=j\omega L$.

The authors used Ansys HFSS to simulate the connector transition and
determined the inductance in the cable ends could be mitigated by including a
capacitive feature in the transition board. This was accomplished by modifying
the upper coplanar waveguide ground planes on the transition board, bringing
them together in front of the signal trace with a 0.006 inch gap (see Fig. 3
and Fig. 4 ).

Figure 4: A.) Box end housing and grounded coplanar waveguide transition board
design used in [18] that yielded excess end inductance. B.) Box end and
modified grounded coplanar waveguide design with capacitive coupling used in
this work. C.) Connectorization scheme showing how cable is connectorized in
this work. The cable ground is spot welded to the transition board ground. The
cable center wire is exposed by cutting the foil ground and then crimping a
stainless steel tube onto the wire. The tube is soldered to the transition
board. The transition board and G3PO connectors are soldered into the box
housing.

##  IV Performance Characterization

TABLE I:  Summary of thermal, mechanical, and microwave properties of
superconducting coaxial ribbon cable, commercial semi-rigid superconducting
coax, and commercial superconducting polyimide flex cables

Thermal Load333 Mechanical Microwave444 Cable per trace [nW] All Dimensions
[inches] Values at 8 GHz 100mK to Connector OD Min. Inside Conductor
Dielectric Crosstalk Attenuation 1 K 4 K Pitch (${\diameter}$) Bend Radius
Material Material [dB] [dB] FLAX [This Work] $5$ $245$ $0.140$ $0.02$ $0.08$
NbTi PFA $-40$ $1.5$ FLAX [18] $5$ $238$ $0.140$ $0.02$ $0.08$ NbTi PFA $-40$
$7$ Semi-Rigid555 $7$ $370$ $>0.5$ $0.90$0 $0.13$ NbTi PTFE $<-60$ $0.5$
Polyimide-Flex666 TBD TBD $0.3$ $0.01$777 $0.04$ NbTi Polyimide $-60$ $7$

3 Computed using a cable length of 1 foot. 4 Values reported for $\sim$1-ft
long cables. 5 CryoCoax 0.034 inch NbTi/NbTi, P/N: BCB012 [11]. 6 Delft
Circuits Cri/oFlex®3 with NbTi conductor [16]. 7 For the microstrip geometry
this is the total cable thickness.

Design modifications were tested using a 5-trace, 1-ft long cable manufactured
in-house at the University of California, Santa Barbara using the same process
described in [18]. We compare our results with the previous iteration of the
cable as well as the smallest commercially available superconducting coax from
CryoCoax (P/N: BCB102)[11] and the only commercially available NbTi-on-
polyimide flex cable from Delft Circuits (P/N: Cri/oFlex®3) [16]. Key thermal,
mechanical, and electrical properties are summarized in Table I.

Transmission loss ($S_{21}$), crosstalk ($S_{41}$), and time domain
reflectometry were measured at 4 K with a network analyzer. The device under
test circuit consisted of the assembled FLAX cable with 3 dB cryo-attenuators
obtained from XMA888XMA Corporation-Omni Spectra, 7 Perimeter Road,
Manchester, NH. P/N: 2082-6040-03-CRYO and 25 cm nonmagnetic SMA-to-G3PO
adapter coaxial cables obtained from Koaxis999Koaxis RF Cable Assemblies, 2081
Lucon Road, Schwenksville, PA. P/N: AO10-CC047C-YO18 on either end (see Fig.
5). A short commercial coax through line was used as a calibration reference
for fridge wiring as in [18].

Figure 5: Schematic diagram depicting the device under test circuit at 4 K.
Figure is modified from [18]

###  IV-A Transmission

Figure 6: Top: $S_{21}$ (transmission) measurement of sample traces from [18]
(dashed) and this work (solid) at 4 K. Bottom: $S_{41}$ (nearest neighbor far-
end crosstalk) measurement of sample FLAX traces from the same cables at 4 K.

Transmission measurements ($S_{21}$) at 4 K for two traces from the same test
cable are plotted in Fig. 6 along with sample traces from the previous
iteration of the cable first published in [18]. The transmission is improved
from at most 7 dB of loss in [18] to 1.5 dB of loss at 8 GHz in this work. The
loss is now on-par with commercial semi-rigid superconducting coax cables and
3x better than commercial NbTi-on-polyimide cables (see Table I).

Previously, the authors had hypothesized ripples in the transmission spectrum
were caused by inductance in the connector transition that produced
reflections off the cable ends [18]. The transmission curves from this work do
not show the same ripples increasing with frequency as the previous iteration,
suggesting the capacitive tuning in the transition board successfully
mitigated the inductive connectorization. As expected, we find that
connectorization plays a critical role in the overall cable performance.

An updated time domain reflectometry measurement is shown in Fig. 7 and
confirms the larger center conductor wire lowered the impedance from $\sim$60
$\Omega$ to $\sim$53 $\Omega$. The impedance being closer to 50 $\Omega$ is
likely contributing to the improved transmission at lower frequencies where
the insertion loss dominates over the inductive ripple. The improved
characteristic impedance and transmission spectra are consistent with 3D
electromagnetic simulations the authors used to model design modifications to
the center wire and transition board.

Figure 7: Time Domain Reflectometry (TDR) measurement of the cryogenic signal
path showing the characteristic impedance at lengths along the signal path at
4 K. Traces are shown from [18] (light blue) and this work (dark blue).
Commercially available 50 $\Omega$ coaxial cables border the FLAX cables
highlighted by the double arrow. Note the 4 K TDR measurement is accurate to
$\pm$ 3 $\Omega$.

###  IV-B Crosstalk

The authors measured both the near-end crosstalk ($S_{31}$) and the far-end
crosstalk ($S_{41}$) and found the crosstalk levels to be the same. This
suggests all crosstalk is happening in the transition board and connector
ends. This is consistent with the flexible coaxial ribbon cable providing a 3D
superconducting ground around each wire making crosstalk in the cable itself
virtually impossible. Only ($S_{41}$) is shown in Fig. 6 for comparison with
[18]. We find the crosstalk remained less than -40 dB which is competitive
with commercial semi-rigid and Kapton-flex cables. We hypothesize the
crosstalk may be further reduced by adding 3D shielding on the surface of the
transition board.

###  IV-C Thermal Conductivity

Figure 8: We computed a cable thermal conductivity $G(T)$ in units of
$\mu\mathrm{WcmK}^{-1}$ by summing the thermal conductivity of each
constituent material weighted by the cross section[19, 14, 18]. The smallest
commercially available superconducting coax from CryoCoax (P/N: BCB012)
labeled “semi-rigid” (pink) is compared with this work (salmon) and [18]
(gold). The authors were unable to obtain cross-sectional dimensions for the
Delft Cri/oFlex®3 cable and so instead display a fictitious NbTi/Kapton cable,
labeled “Kapton-flex” (purple), where the Kapton and NbTi materials have been
assigned the same cross-sectional areas as the dielectric and conductor in
this work, respectively. The differences in heat load between the semi-rigid
(pink), [18] (gold), and this work (salmon) are due purely to geometrical
differences whereas the difference between the Kapton-flex (purple) and this
work (salmon) is due purely to difference in dielectric materials. Solid lines
are computed using literature values for Nb47Ti[7], PTFE[19], and Kapton [14,
20]. PTFE values were used to estimate the PFA dielectric in the FLAX cable.
Dashed lines indicate extrapolation. Figure is modified from [18].

Following previous convention, we compute $G(T)$, a length-dependent cable
thermal conductivity, by summing literature values of cable materials weighted
by their cross sections (see Fig. 8) [19, 18, 14]. The heat load between
temperature stages can be computed by integrating values in Fig. 8 from
$T_{1}$ to $T_{2}$ ($T_{1}<T_{2}$) and dividing by the cable length. Results
of this calculation are presented for select temperatures and cables in Table
I. $G(T)$ is plotted for the superconducting coaxial ribbon cable in this
work, the previous iteration [18], and the smallest commercially available
superconducting coax from CryoCoax (P/N: BCB012)[11] in Fig. 8. Delft Circuits
declined request to share cross-sectional area information for the
superconducting polyimide Cri/oFlex®3 cable (Delft Circuits, personal
communication, May 2023) and so we instead show a hypothetical NbTi-on-
polyimide cable where the polyimide101010approximated as the Nikaflex
DuPont/Nikkan Kapton film[21] used in [14]. cable dielectric and conductor
have the same cross sectional areas as the dielectric and conductor in this
work, respectively. In this way, the differences in heat load between the
three superconducting coaxes in Fig. 8 are purely geometrical whereas the
difference between this work and the polyimide/Kapton is due solely to
materials differences in the dielectric.

We find the heat load increased slightly between [18] and this work due to the
increased center conductor diameter (see Table I). Despite the increase, the
cable presented in this work remains roughly half as thermally conductive as
the smallest commercially available superconducting coax. While the heat load
of the Cri/oFlex®3 cables with NbTi conductor has not been directly measured,
Delft Circuits suggests it may be $\sim$10% of the heat load of the same
polyimide-flex cables with Ag conductor, i.e. 6x more thermally conductive
from 1 K to 100 mK than the cable presented in this work (Delft Circuits,
personal communication, May 2023)[16]. This may be because the polyimide
material makes up the bulk of the cross section and is approximately 5 times
more thermally conductive than NbTi below 1 K [7]. In this work, we
hypothesize the cable heat load is dominated by the NbTi conductor because [7]
suggests NbTi is approximately ten-times more thermally conductive than PTFE
below 1 K. To minimize heat load in cryogenic cables, we suggest NbTi/PTFE/PFA
solutions minimize conductor cross section whereas NbTi/Kapton/polyimide
technologies minimize dielectric cross section.

##  V Conclusion

We have improved the transmission 3x without negatively impacting the heat
load or isolation and made progress on commercial manufacturing of the
superconducting flexible coaxial ribbon cable presented in [18]. We adjusted
the center conductor diameter and re-designed the transition board to improve
the transmission loss while retaining strong signal isolation and low heat
load.

The cable technology presented in this work has been deployed in two
superconducting-array-based instruments: the MKID Exoplanet Camera (MEC)[3]
and the MagAO-X MKID Instrument (XKID)[22]. In both cases, FLAX cables
replaced our group’s previously developed laminated NbTi-on-Kapton microstrip
cables [14]. In MEC, the improvement in signal integrity increased the amount
of usable superconducting detectors in the array by $\sim$20%. In XKID, the
average MKID energy resolution— a key superconducting detector performance
metric, increased by a factor of 4.

This cable technology has already demonstrated promising performance gains in
superconducting detector array instruments. With the modifications in this
work along with future improvements and commercialization by Maybell Quantum
Industries, we see the potential for this technology to be widely applicable
to superconducting device systems. Future improvements may include adding 3D
shielding in the transition board which would allow for higher connector pitch
density, lowering the heat load per trace and potentially improving the
crosstalk.

Overall, we find this cable technology to be superior to commercial options
for our applications building high-density superconducting detector arrays.
The high electrical and thermal isolation coupled with low loss and
connectorized with high-density, coaxial push-on connectors makes for an easy-
to-use cryogenic wiring solution. We are looking forward to future
improvements led by Maybell Quantum Industries and we expect these results
will be especially promising for groups looking to operate high-density,
superconducting microwave device arrays.

## Acknowledgment

J. P. Smith is supported by a NASA Space Technology Research Fellowship under
grant number 80NSSC19K1126. This work was supported by a DARPA SBIR Grant
under grant number W912CG23C0002.

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# CAT(0) spaces of higher rank I

Stephan Stadler

###### Abstract.

A CAT(0) space has rank at least $n$ if every geodesic lies in an $n$-flat.
Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of
rank at least $2$ with a geometric group action is rigid – isometric to a
Riemannian symmetric space, a Euclidean building, or splits as a metric
product. This paper is the first in a series motivated by Ballmann’s
conjecture. Here we prove that a CAT(0) space of rank at least $n\geq 2$ is
rigid if it contains a periodic $n$-flat and its Tits boundary has dimension
$(n-1)$. This does not require a geometric group action. The result relies
essentially on the study of flats which do not bound flat half-spaces – so-
called Morse flats. We show that the Tits boundary $\partial_{T}F$ of a
periodic Morse $n$-flat $F$ contains a regular point – a point with a Tits-
neighborhood entirely contained in $\partial_{T}F$. More precisely, we show
that the set of singular points in $\partial_{T}F$ can be covered by finitely
many round spheres of positive codimension.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Acknowledgments
  2. 2 Preliminaries
    1. 2.1 Metric spaces
    2. 2.2 Spaces with an upper curvature bound
    3. 2.3 CAT(1) spaces
    4. 2.4 CAT(0) spaces
  3. 3 Recognizing a building at infinity
  4. 4 Rank $n$ spaces with periodic $n$-flats
    1. 4.1 From branch points to orthogonal half-planes
    2. 4.2 Orthogonal half-spaces from a sequence of lower dimensional ones
    3. 4.3 Tits boundary of a periodic Morse flat

##  1\. Introduction

###  1.1. Main results

A Hadamard manifold is a complete simply connected Riemannian manifold of non-
positive sectional curvature. It has higher rank if every complete geodesic
line is contained in an isometrically embedded Euclidean space of dimension at
least two, a so-called flat. The structure of manifolds of non-positive
curvature and higher rank was clarified in the eighties, in the series of
papers [BBE85, BBS85], culminating in Ballmann’s celebrated Higher Rank
Rigidity Theorem [Bal85, BS87, EH90]: If a higher rank Hadamard manifold
admits quotients of finite volume, then it is either a Riemannian symmetric
space or splits as a metric product. So in the presence of enough symmetry,
higher rank implies special geometry. In particular, the theorem identifies
higher rank as the exceptional case.

The modern theory of non-positive curvature focuses on metric spaces instead
of manifolds, where the curvature condition is expressed by triangle
comparison. The definition is due to Alexandrov, already in the early fifties,
and later gained considerable momentum from Gromov’s seminal paper [Gro87].
Nowadays, CAT(0) spaces – the synthetic versions of Hadamard manifolds – play
an important role in mathematics beyond geometry and topology. They appear in
many different fields such as geometric group theory, representation theory,
arithmetic and optimization. For an account of the huge amount of literature
on metric spaces with synthetic curvature bounds, see the bibliographies in
[AKP19, Bal04, BH99]. The following conjecture, formulated by Ballmann and
Buyalo in [BB08], predicts a generalization of the Higher Rank Rigidity
Theorem from Hadamard manifolds to CAT(0) spaces. It is the main motivation
for the present paper.

######  Conjecture (Higher Rank Rigidity).

Let $X$ be a locally compact geodesically complete CAT(0) space. Suppose that
$\Gamma$ is a group of isometries of $X$ with limit set
$\Lambda(\Gamma)=\partial_{\infty}X$. Then
$\operatorname{diam}(\partial_{T}X)=\pi$ implies that $X$ is a Riemannian
symmetric space or a Euclidean building of rank at least 2, or that $X$ non-
trivially splits as a metric product.

The condition on the diameter expresses higher rank, as it is equivalent to
saying that any complete geodesic bounds a flat half-plane, cf. Section 2.4.
If a group $\Gamma$ acts on a CAT(0) space $X$, then its limit set
$\Lambda(\Gamma)$ is the set of accumulation points of a $\Gamma$-orbit in the
ideal boundary $\partial_{\infty}X$. The main example of group actions on
CAT(0) spaces with full limit set are geometric actions, meaning cocompact and
properly discontinuous actions by isometries.

For singular spaces, Higher Rank Rigidity is wide open. It has only been
established in the following short list of special cases:

  * •

for a 2-dimensional simplicial complex with a piecewise smooth CAT(0) metric
and a geometric group action [BB95];

  * •

for a 3-dimensional simplicial complex with a piecewise Euclidean CAT(0)
metric and a geometric group action [BB00];

  * •

for a finite dimensional CAT(0) cube complex with a geometric group action
[CS11]111For cube complexes, the statement of Higher Rank Rigidity takes the
form of a splitting result, since neither irreducible symmetric spaces nor
irreducible Euclidean buildings are cube complexes. They actually do not carry
a piecewise cubical CAT(0) metric at all [Lee00].;

  * •

for a locally compact and geodesically complete CAT(0) space such that its
full isometry group does not fix a point at infinity and which supports a
cocompact isometric action by an amenable locally compact group [CM15];

  * •

for a locally compact and geodesically complete CAT(0) space with
1-dimensional Tits boundary which supports a geometric group action
$\Gamma\curvearrowright X$ such that the induced action on the ideal boundary
is not minimal, i.e. $\partial_{\infty}X$ contains a proper closed
$\Gamma$-invariant subset [Ric19].

Note that the interesting case in the next to last item concerns non-discrete
groups since the only locally compact geodesically complete CAT(0) space which
admits a cocompact isometric action by an amenable discrete group is the flat
Euclidean space [AB98, Corollary C].

######  Theorem A.

Let $X$ be a locally compact CAT(0) space whose Tits boundary has dimension
$n-1\geq 1$. Suppose that every geodesic in $X$ lies in an $n$-flat. If $X$
contains a periodic $n$-flat, then $X$ is a Riemannian symmetric space or a
Euclidean building, or $X$ splits as a metric product.

If the isometry group of $X$ acts cocompactly, then the condition on the
dimension of the Tits boundary simply means that $X$ does not contain an
$(n+1)$-flat [Kle99, Theorem C]. Thus, we obtain:

######  Corollary B.

Let $X$ be a locally compact CAT(0) space whose isometry group acts
cocompactly. Suppose that every geodesic in $X$ lies in an $n$-flat but $X$
does not admit $(n+1)$-flats. If $X$ contains a periodic $n$-flat, then $X$ is
a Riemannian symmetric space or a Euclidean building, or $X$ splits as a
metric product.

Another consequence of Theorem A is the following.

######  Corollary C.

Let $X$ be a locally compact CAT(0) space whose Tits boundary has dimension
$n-1\geq 1$. Suppose that every geodesic in $X$ lies in an $n$-flat. If $X$
contains a periodic $n$-flat, then $X$ has a unique decomposition into
irreducible factors

$X=\mathbb{R}^{j}\times X_{1}\times\ldots\times X_{k}\times
Y_{1}\times\ldots\times Y_{l}\times M_{1}\times\ldots\times M_{m}$

where $X_{a}$ are rank 1 spaces, $Y_{b}$ are irreducible Euclidean buildings
of higher rank and $M_{c}$ are irreducible Riemannian symmetric spaces of
higher rank.

Note that in the smooth case, the higher rank property can be expressed by
asking that every complete geodesic admits $n$ linearly independent parallel
Jacobi fields for some $n\geq 2$. In the presence of a geometric group action
these Jacobi fields can then be “integrated” to $n$-flats [Bal95, Section
IV.4]. Hence, in the context of metric spaces, the assumption on the existence
of $n$-flats in Theorem A is natural222 See [OR22] for a definition of rank
for CAT(0) spaces whose isometry groups satisfy the duality condition.. Recall
that a flat is periodic, if its stabilizer in the isometry group of the
surrounding space contains a subgroup acting geometrically on it. By [BH99,
Flat Torus Theorem 7.1], the existence of a periodic $n$-flat in $X$ is
equivalent to the presence of a subgroup
$\mathbb{Z}^{n}<\mathop{\hbox{Isom}}(X)$. We point out that Theorem A does not
assume any symmetries besides the existence of a single periodic $n$-flat.
Also, if $X$ is a rank $n$ symmetric space or Euclidean building, and $\Gamma$
acts cocompactly on $X$, then $X$ contains a periodic $n$-flat [BB95, Theorem
8.9], [Mos73, Lemma 8.3], [PR72, Theorem 2.8]. We emphasize that some symmetry
is required in order for the conclusion of Higher Rank Rigidity to hold true,
since a geodesically complete CAT(0) space whose Tits boundary $\partial_{T}X$
has diameter $\pi$ does not have to be a product or a Euclidean building
[BB00].

We now discribe the main novelty of the present paper. Recall that a point in
a metric space is called regular, if it has a neighborhood homeomorphic to an
open set in a Euclidean space. The crucial ingredient behind Theorem A is a
contribution to the following general conjecture which has been completely
open.

######  Conjecture (Lytchak’s Regular Point Conjecture).

Let $X$ be a locally compact CAT(0) space with a geometric group action. Then
$\partial_{T}X$ contains a regular point.

Gromov first raised the question which CAT(1) spaces do contain regular points
[Gro93, p.90]. A general CAT(1) space does not have to contain any regular
points. On the other hand, locally compact and geodesically complete CAT(1)
spaces always do, as shown by Lytchak-Nagano in [LN19], following ideas of
Burago-Gromov-Perelman [BGP92]. However, the Tits boundary of a locally
compact CAT(0) space with a geometric group action is typically not locally
compact.

Recall that a flat in a CAT(0) space is called Morse if it does not bound a
flat half-space. We prove:

######  Theorem D.

[Theorem 4.10] Let $X$ be a locally compact CAT(0) space. Suppose that $X$
contains a periodic Morse flat $F$. Then the complement in $\partial_{T}F$ of
the set of regular points can be covered by a finite set of round spheres of
positive codimension.

In particular, the singular set in the Tits boundary of a periodic Morse
$(n+1)$-flat is given by a union of finitely many hyperspheres and a closed
set of finite $(n-2)$-dimensional Hausdorff measure. In this sense the Tits
boundary of a periodic Morse flat resembles a Coxeter complex. This will be
used in [Sta22a] in order to control the geometry of certain submetries of the
Tits boundary. Let us emphasize again that Theorem D does not assume any
symmetry besides the periodic Morse flat, no geometric group action is
required.

The present article is the first in a series motivated by the Higher Rank
Rigidity Conjecture. The sequels all rely on the crucial new idea on how to
find regular points at infinity presented here. In [Sta22a] we prove the
following rigidity result which generalizes the Higher Rank Rigidity Theorem
for Hadamard manifolds.

######  Theorem 1.1 ([Sta22a, Main Theorem]).

Let $X$ be a locally compact CAT(0) space with a geometric group action
$\Gamma\curvearrowright X$. Suppose that there exists $n\geq 2$ such that
every geodesic in $X$ lies in an $n$-flat. If $X$ contains a periodic Morse
$n$-flat, then $X$ is a Riemannian symmetric space or a Euclidean building or
$X$ non-trivially splits as a metric product.

In [Sta22b] we confirm Higher Rank Rigidity as well as the Closing Lemma for
spaces without 3-flats. More precisely, we obtain

######  Theorem 1.2 ([Sta22b, Theorems A and B]).

Let $X$ be a locally compact geodesically complete CAT(0) space without
3-flats. Suppose $X$ admits a geometric group action $\Gamma\curvearrowright
X$. If $X$ contains a complete geodesic which does not bound a flat half-
plane, then it also contains a $\Gamma$-periodic geodesic which does not bound
a flat half-plane. If every complete geodesic in $X$ bounds a flat half-plane,
then $X$ is a Riemannian symmetric space, a Euclidean building or non-
trivially splits as a metric product.

###  1.2. Acknowledgments

I would like to use this opportunity to thank Bernhard Leeb and Alexander
Lytchak for pointing out several mistakes in a late version of this article.
Their insightful comments led to substantial improvements. I also want to
thank Bruce Kleiner for several inspiring discussions. I was supported by DFG
grant SPP 2026.

##  2\. Preliminaries

###  2.1. Metric spaces

We assume that the reader is familiar with the geometry of metric spaces with
upper curvature bound, as references we mention [AKP19, Bal04, BH99, KL97].
For the required preliminaries on buildings we refer the reader to [KL97,
Lee00].

Here we only want to agree on notation and collect some basic facts that will
be used later.

Euclidean $n$-space with its flat metric will be denoted by $\mathbb{R}^{n}$.
The unit sphere $S^{n-1}\subset\mathbb{R}^{n}$ equipped with the induced
metric will be referred to as a round sphere. Its intersection with a half-
space $\mathbb{R}^{n-1}\times[0,\infty)$ is called a round hemisphere. We
denote the distance between two points $x$ and $y$ in a metric space $X$ by
$|x,y|$. For $x\in X$ and $r>0$, we denote by $B_{r}(x)$ and $\bar{B}_{r}(x)$
the open and closed $r$-ball around $x$, respectively. Similarly, $N_{r}(A)$
and $\bar{N}_{r}(A)$ denote the open and closed $r$-neighborhood of a subset
$A\subset X$, respectively. A _geodesic_ is an isometric embedding of an
interval. It is called a geodesic segment, if it is compact. The endpoints of
a geodesic segment $c$ are denoted by $\partial c$.

A branch point is a point through which a certain geodesic can be extended in
more than one way. More precisely, a point $b$ is a branch point if there are
geodesic segments $c$, $c^{+}$ and $c^{-}$ such that $c\cup c^{\pm}$ is a
geodesic and $c^{+}\cap c^{-}=\\{b\\}$. In a _geodesic metric space_ any pair
of points is connected by a geodesic. Such a space is _geodesically complete_
if every geodesic segment is contained in a complete local geodesic.

###  2.2. Spaces with an upper curvature bound

For any CAT($\kappa$) space $X$, the angle between each pair of geodesics
starting at the same point is well defined. Let $x,y,z$ be three points at
pairwise distance less than $\frac{\pi}{\sqrt{\kappa}}$ in a CAT($\kappa$)
space $X$. Whenever $x\neq y$, the geodesic between $x$ and $y$ is unique and
will be denoted by $xy$. For $y,z\neq x$, the angle at $x$ between $xy$ and
$xz$ will be denoted by $\angle_{x}(y,z)$. The _space of directions_ or link
at a point $x\in X$ will be denoted by $(\Sigma_{x}X,\angle)$, its elements
are called directions (at $x$).

The dimension $\dim(X)$ of a CAT($\kappa$) space $X$ refers to Kleiner’s
geometric dimension [Kle99]. It vanishes precisely when the space is discrete.
In general, it is defined inductively:

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

###  2.3. CAT(1) spaces

For two CAT(1) spaces $Z_{1}$ and $Z_{2}$ we denote by $Z_{1}\circ Z_{2}$
their spherical join. It is a CAT(1) space of diameter $\pi$. A pair of points
$\xi^{-},\xi^{+}$ in a CAT(1) space $Z$ are called antipodes, if their
distance is at least $\pi$.

A subset $A\subset Z$ is called spherical, if it embeds isometrically into a
round sphere. A (spherical) $n$-lune of angle $\theta\in[0,\pi]$ in a CAT(1)
space is a closed convex subset $\lambda$ isometric to
$S^{n-1}\circ[0,\theta]$. Bigons in CAT(1) spaces lead to spherical lunes
[BB99, Lemma 2.5]. We will use the following more general result which follows
immediately from [BB99, Lemma 2.5] and [Lyt05, Lemma 4.1].

######  Lemma 2.1 (Lune Lemma).

Let $\xi$ and $\hat{\xi}$ be antipodes at distance $\pi$ in a CAT(1) space
$Z$. If $\gamma^{\pm}$ are geodesics from $\xi$ to $\hat{\xi}$, then
$\gamma^{-}\cup\gamma^{+}$ is a spherical subset in $Z$. More precisely, if
$\angle_{\xi}(\gamma^{-},\gamma^{+})\geq\pi$, then $\gamma^{-}\cup\gamma^{+}$
is a round $1$-sphere in $Z$; and if
$\angle_{\xi}(\gamma^{-},\gamma^{+})=\theta<\pi$, then
$\gamma^{-}\cup\gamma^{+}$ bounds a spherical lune of angle $\theta$.
Similarly, let $\tau^{\pm}\subset Z$ be round $n$-hemispheres with
$\partial\tau^{-}=\partial\tau^{+}$ and let $\zeta^{\pm}\in Z$ denote their
centers. If $|\zeta^{-},\zeta^{+}|\geq\pi$, then $\tau^{-}\cup\tau^{+}$ is a
round $n$-sphere in $Z$; and if $|\zeta^{-},\zeta^{+}|=\theta<\pi$, then
$\tau^{-}\cup\tau^{+}$ bounds a spherical $(n+1)$-lune of angle $\theta$.

######  Definition 2.2.

Let $Z$ be a CAT(1) space. We call a point $\xi\in Z$ $k$-regular, if it has a
neighborhood homeomorphic to an open set in $\mathbb{R}^{k}$. The point $\xi$
is called $k$-spherical, if there exists a radius $r>0$ such that
$B_{r}(\xi)\subset Z$ is isometric to an $r$-ball in $S^{k}$. In both
definitions, if it’s clear from the context, we neglect the quantification and
speak of regular respectively spherical points.

######  Lemma 2.3 ([BL06, Lemma 2.1]).

Let $Z$ be a CAT(1) space of dimension $n$. If $\sigma\subset Z$ is a round
$n$-sphere, then every point $\xi\in Z$ has an antipode in $\sigma$.

This implies the following well-known criterion for geodesic completeness.

######  Lemma 2.4.

Let $Z$ be a CAT(1) space of dimension $n$. If every point $\xi\in Z$ is
contained in a round $n$-sphere in $Z$, then $Z$ is geodesically complete.

###### Proof.

If $\xi$ is a point in $Z$ contained in a round $n$-sphere $\sigma\subset Z$,
then any local geodesic ending in the point $\xi$ can be extended beyond $\xi$
by a geodesic in $\sigma$, since any direction in $\Sigma_{\xi}Z$ has an
antipode in $\Sigma_{\xi}\sigma\cong S^{n-1}$ by Lemma 2.3. ∎

A simple but useful property in CAT(1) spaces with a geodesically complete
link is the following.

######  Lemma 2.5.

Let $Z$ be a CAT(1) space and $z\in Z$ a point where the link $\Sigma_{z}Z$ is
geodesically complete. Then for every pair of directions $v,w\in\Sigma_{z}Z$
there exists an antipode $\hat{w}\in\Sigma_{z}Z$ of $w$, such that

$\angle_{z}(\hat{w},v)=\pi-\angle_{z}(w,v).$

###### Proof.

If $\angle_{z}(v,w)=\pi$, then we can take $\hat{w}=v$. Otherwise, we extend
the geodesic from $w$ to $v$ up to distance $\pi$. The endpoint $\hat{w}$ is
then an antipode of $w$ with the required property. ∎

The proof of Theorem A relies on the following rigidity result of Lytchak.

######  Theorem 2.6 ([Lyt05, Main Theorem]).

Let $Z$ be a finite dimensional geodesically complete CAT(1) space. If $Z$ has
a proper closed subset $A$ containing with each point all of its antipodes,
then $Z$ is a spherical join or a spherical building.

###  2.4. CAT(0) spaces

The ideal boundary of a CAT(0) space $X$, equipped with the cone topology, is
denoted by $\partial_{\infty}X$. If $X$ is locally compact, then
$\partial_{\infty}X$ is compact. The Tits boundary of $X$ is denoted by
$\partial_{T}X$, it is the ideal boundary equipped with the Tits metric
$|\cdot,\cdot|_{T}$. Recall that the Tits metric is the intrinsic metric
associated to the Tits angle.

If $C$ is a closed convex subset, then it is CAT(0) with respect to the
induced metric. In this case, $C$ admits a 1-Lipschitz retraction
$\pi_{C}:X\to C$. If $C_{1}$ and $C_{2}$ are closed convex subsets, then the
distance function $d(\cdot,C_{1})|_{C_{2}}$ is convex; and constant if and
only if $\pi_{C_{1}}$ restricts to an isometric embedding on $C_{2}$. We call
$C_{1}$ and $C_{2}$ parallel, $C_{1}\|C_{2}$, if and only if
$d(\cdot,C_{1})|_{C_{2}}$ and $d(\cdot,C_{2})|_{C_{1}}$ are constant. Let
$Y\subset X$ be a geodesically complete closed convex subset. Then we define
the parallel set $P(Y)$ as the union of all closed convex subsets parallel to
$Y$. The parallel set is closed, convex and splits canonically as a metric
product

$P(Y)\cong Y\times CS(Y)$

where the cross section $CS(Y)$ is a closed convex subset.

If $X_{1}$ and $X_{2}$ are CAT(0) spaces, then their metric product
$X_{1}\times X_{2}$ is again a CAT(0) space. We have $\partial_{T}(X_{1}\times
X_{2})=\partial_{T}X_{1}\circ\partial_{T}X_{2}$ and
$\Sigma_{(x_{1},x_{2})}(X_{1}\times
X_{2})=\Sigma_{x_{1}}X_{1}\circ\Sigma_{x_{2}}X_{2}$. If $X$ is a geodesically
complete CAT(0) space, then any join decomposition of $\partial_{T}X$ is
induced by a metric product decomposition of $X$ [KL97, Proposition 2.3.7].

The Tits boundary of a closed convex subset $Y\subset X$ embeds canonically
$\partial_{T}Y\subset\partial_{T}X$. If two closed convex subsets $Y_{1}$ and
$Y_{2}$ intersect in $X$, then

$\partial_{T}(Y_{1}\cap Y_{2})=\partial_{T}Y_{1}\cap\partial_{T}Y_{2}.$

For points $\xi\in\partial_{\infty}X$ and $x\in X$ we denote the Busemann
function centered at $\xi$ based at $x$ by $b_{\xi,x}$. If $\rho:[0,\infty)\to
X$ denotes the geodesic ray asymptotic to $\xi$ and with $\rho(0)=x$, then

$b_{\xi,x}(y)=\lim\limits_{t\to\infty}(|y,\rho(t)|-t).$

It is a 1-Lipschitz convex function whose negative gradient at a point $y\in
X$ is given by $\log_{y}(\xi)$. We denote the horoball centered at a point
$\xi\in\partial_{\infty}X$ and based at the point $x\in X$ by

$HB(\xi,x):=b^{-1}_{\xi,x}((-\infty,0]).$

It is a closed convex subset with

$\partial_{T}HB(\xi,x)=\bar{B}_{\frac{\pi}{2}}(\xi)\subset\partial_{T}X.$

A $n$-flat $F$ in a CAT(0) space $X$ is a closed convex subset isometric to
$\mathbb{R}^{n}$. In particular, $\partial_{T}F\subset\partial_{T}X$ is a
round $(n-1)$-sphere. On the other hand, if $X$ is locally compact and
$\sigma\subset\partial_{T}X$ is a round $(n-1)$-sphere, then either there
exists an $n$-flat $F\subset X$ with $\partial_{T}F=\sigma$, or there exists a
round $n$-hemisphere $\tau^{+}\subset\partial_{T}X$ with
$\sigma=\partial\tau^{+}$ [Lee00, Proposition 2.1]. Consequently, if
$\partial_{T}X$ is $(n-1)$-dimensional, then any round $(n-1)$-sphere in
$\partial_{T}X$ is the Tits boundary of some $n$-flat in $X$. Moreover, a
round $n$-hemisphere $\tau^{+}\subset\partial_{T}X$ bounds a flat
$(n+1)$-half-space in $X$ if and only if its boundary $\partial\tau^{+}$
bounds an $n$-flat in $X$. A flat is called Morse, if it does not bound a flat
half-space333For a motivation of the term and further context see [HKS22a,
HKS22b]..

The following basic fact will be used repeatedly throughout the paper.

######  Lemma 2.7 ([Lee00, Sublemma 2.3]).

Let $F$ be a flat in a CAT(0) space and let $C\subset X$ be a closed convex
subset so that $\partial_{\infty}F\subset\partial_{\infty}C$. Then $C$
contains a flat $F^{\prime}$ parallel to $F$.

A flat ($n$-dimensional) half-space $H\subset X$ is a closed convex subset
isometric to a Euclidean half-space $\mathbb{R}_{+}^{n}$. Its boundary
$\partial H\subset X$ is an $(n-1)$-flat and its Tits boundary is a round
$(n-1)$-hemisphere $\partial_{T}H\subset\partial_{T}X$. Flat half-spaces will
play a central role in our arguments later, and we agree to denote them by $H$
and their boundaries by $\partial H=h$.

We define the parallel set of a round sphere $\sigma\subset\partial_{T}X$ as

$P(\sigma)=P(F)$

where $F$ is a flat in $X$ with $\partial_{T}F=\sigma$, if such a flat exists.

The proof of Theorem A relies on the following rigidity result of Leeb.

######  Theorem 2.8 ([Lee00, Main Theorem]).

Let $X$ be a locally compact, geodesically complete CAT(0) space. If
$\partial_{T}X$ is a connected thick irreducible spherical building, then $X$
is either a symmetric space or a Euclidean building.

##  3\. Recognizing a building at infinity

The aim of this section is to prove the following result.

######  Theorem 3.1.

Let $X$ be a locally compact CAT(0) space where every geodesic lies in an
$n$-flat. Suppose that $\dim(\partial_{T}X)=n-1$. If $\partial_{T}X$ contains
an open relatively compact subset $U$, then $\partial_{T}X$ is a spherical
building.

The result is similar to [BL06, Theorem 1.6] and builds on Theorem 2.6 in the
same way. Note that if we would know that any pair of antipodes in
$\partial_{T}X$ lies in a round $(n-1)$-sphere, then we could directly apply
[BL06, Theorem 1.6].

Using the structure theory of locally compact and geodesically complete spaces
with upper curvature bounds [LN19], one can deduce Theorem 3.1 form the
following similar result which will suffice for our proof of Theorem A.

######  Theorem 3.2.

Let $X$ be a locally compact CAT(0) space where every geodesic lies in an
$n$-flat. Suppose that $\dim(\partial_{T}X)=n-1$. If $\partial_{T}X$ contains
an $(n-1)$-spherical point, then $\partial_{T}X$ is a spherical building.

We need some preparation before we can provide the proof of Theorem 3.1.

Recall that a pair of antipodes in the Tits boundary of a CAT(0) space $X$
does not have to bound a complete geodesic in $X$. Nevertheless, for locally
compact geodesically complete spaces we have the following.

######  Lemma 3.3.

Let $X$ be a locally compact, geodesically complete CAT(0) space. Let $\xi$
and $\hat{\xi}$ be antipodes in $\partial_{T}X$. Then there exists a sequence
of complete geodesics $c_{k}$ in $X$ with
$\partial_{\infty}c_{k}=\\{\hat{\xi},\xi_{k}\\}$ and $\xi_{k}\to\xi$ in
$\partial_{\infty}X$.

###### Proof.

If $|\xi,\hat{\xi}|>\pi$, then there is a complete geodesic $c$ in $X$ with
$\partial_{\infty}c=\\{\xi,\hat{\xi}\\}$ [Lee00, Proposition 2.1]. Hence, we
may assume $|\xi,\hat{\xi}|=\pi$. Fix a base point $o\in X$ and let
$\rho:[0,\infty)\to X$ be a parametrization of the geodesic ray $o\xi$. Since
$X$ is locally compact and geodesically complete, all links in $X$ are
geodesically complete [LN19, Corollary 5.8, Corollary 5.9]. By Lemma 2.5, we
can choose for each $k\in\mathbb{N}$ a complete geodesic $c_{k}$ extending the
geodesic ray $\rho(k)\hat{\xi}$ and such that

$\angle_{\rho(k)}(\hat{\xi},\xi)=\pi-\angle_{\rho(k)}(\xi,\xi_{k})$

where $\xi_{k}:=c_{k}(+\infty)$. Now for every $p\in o\xi$ we have

$\angle_{p}(\xi_{k},\xi)\leq\angle_{\rho(k)}(\xi_{k},\xi)=\pi-\angle_{\rho(k)}(\xi,\hat{\xi})\to
0.$

Since $\angle_{\rho(k)}(\xi,\hat{\xi})\to|\xi,\hat{\xi}|=\pi$. As $p\in o\xi$
was arbitrary, this shows $\xi_{k}\to\xi$ in $\partial_{\infty}X$. Indeed, by
local compactness, we may assume $\xi_{k}\to\xi^{\prime}$ after passing to a
subsequence. In particular, $\angle_{x}(\xi_{k},\xi^{\prime})\to 0$ for all
$x\in X$. Now if $\xi\neq\xi^{\prime}$, then there exists $q\in o\xi$ such
that $\angle_{q}(\xi,\xi^{\prime})>0$. But

$\angle_{q}(\xi,\xi^{\prime})\leq\angle_{q}(\xi,\xi_{k})+\angle_{q}(\xi_{k},\xi^{\prime})\to
0.$

∎

######  Lemma 3.4.

Let $X$ be a locally compact CAT(0) space. Suppose that every geodesic in $X$
lies in an $n$-flat. If $\xi\in\partial_{T}X$ is an $(n-1)$-spherical point,
then for every antipode $\hat{\xi}$ of $\xi$ there exists a round
$(n-1)$-sphere $\sigma\subset\partial_{T}X$ containing $\xi$ and $\hat{\xi}$.

###### Proof.

Let $\xi\in\partial_{T}X$ be an $(n-1)$-spherical point, cf. Definition 2.2.
Then there exists $s>0$ such that $B_{s}(\xi)$ is contained in any
$(n-1)$-sphere which contains $\xi$. If $\hat{\xi}\in\partial_{T}X$ is an
antipode of $\xi$, then by Lemma 3.3, there exists a sequence of complete
geodesics $(c_{k})$ in $X$ with
$\partial_{\infty}c_{k}=\\{\xi,\hat{\xi}_{k}\\}$ and
$\hat{\xi}_{k}\to\hat{\xi}$. By assumption, there exists $n$-flats $F_{k}$
with $c_{k}\subset F_{k}$. Hence, if $\eta\in\partial_{T}X$ is a point with
$|\eta,\xi|=s$, then $|\eta,\hat{\xi}_{k}|=\pi-s$ for all $k\in\mathbb{N}$.
Thus,

$\pi=|\xi,\hat{\xi}|\leq|\xi,\eta|+|\eta,\hat{\xi}|\leq
s+\liminf\limits_{k\to\infty}|\eta,\hat{\xi}_{k}|=\pi,$

by lower semi-continuity of the Tits metric. We obtain
$|\eta,\hat{\xi}|=\pi-s$ for every point $\eta$ at distance $s$ from $\xi$. In
particular, through every such point runs a geodesic from $\xi$ to
$\hat{\xi}$. Denote by $\sigma\subset\partial_{T}X$ the union of all these
geodesics. It follows from Lemma 2.1 that $\sigma$ is a round $(n-1)$-sphere.
∎

######  Proposition 3.5.

Let $X$ be a locally compact CAT(0) space where every geodesic lies in an
$n$-flat. Suppose $\dim(\partial_{T}X)=n-1$. Then the set
$O\subset\partial_{T}X$ of $(n-1)$-spherical points is closed under taking
antipodes.

###### Proof.

Let $\xi$ be a $(n-1)$-spherical point in $\partial_{T}X$ and let $\hat{\xi}$
be an antipode. By Lemma 3.4, there exists a round $(n-1)$-sphere
$\sigma\subset\partial_{T}X$ which contains $\xi$ and $\hat{\xi}$. Since $\xi$
is spherical, there exists $s>0$ such that $B_{s}(\xi)\subset\sigma$. We claim
that $B_{s}(\hat{\xi})\subset\sigma$ holds as well. Let $\hat{\eta}$ be a
point in $B_{s}(\hat{\xi})$. Since $\dim(\partial_{T}X)=n-1$, we can extend
the geodesic $\hat{\eta}\hat{\xi}$, as in the proof of Lemma 2.4, up to an
antipode $\eta$ of $\hat{\eta}$ in $\sigma$. Then
$|\eta,\xi|=|\hat{\eta},\hat{\xi}|<s$ and therefore $\eta$ is spherical. By
Lemma 3.4, we find a round $(n-1)$-sphere
$\sigma^{\prime}\subset\partial_{T}X$ which contains $\eta$ and $\hat{\eta}$.
Because $\eta\in B_{s}(\xi)$, we must have
$B_{s}(\xi)\subset\sigma\cap\sigma^{\prime}$. By construction, $\hat{\xi}$
also lies in $\sigma\cap\sigma^{\prime}$. Now convexity of
$\sigma\cap\sigma^{\prime}$ implies $\sigma=\sigma^{\prime}$ and therefore
$\hat{\eta}\in\sigma$ as required. ∎

######  Lemma 3.6.

Let $X$ be a locally compact CAT(0) space with $\dim(\partial_{T}X)=n-1$ and
where every geodesic lies in an $n$-flat. Suppose $X$ splits as a non-trivial
metric product $X\cong X_{1}\times X_{2}$. Then we have $n=n_{1}+n_{2}$ where
$\dim(\partial_{T}X_{j})=n_{j}-1$, $j=1,2$, and every complete geodesic in
$X_{j}$ is contained in an $n_{j}$-flat.

###### Proof.

It is clear that the rank is additive. So let $c_{j}$ be a complete geodesic
in $X_{j}$ and let $\tilde{c}=(c_{1},c_{2})$ be the diagonal geodesic in
$c_{1}\times c_{2}\subset X_{1}\times X_{2}$. If $\tilde{F}$ is an $n$-flat in
$X$ which contains $\tilde{c}$, then by [KL97, Lemma 2.3.8] there are flats
$F_{j}\subset X_{j}$ such that $\tilde{F}\subset F_{1}\times F_{2}$. In
particular, $\dim(F_{j})=n_{j}$ and $c_{j}\subset F_{j}$. ∎

###### Proof of Theorem 3.2.

By assumption, $\dim(\partial_{T}X)=n-1$ and every point $\xi$ in
$\partial_{T}X$ lies in a round $(n-1)$-sphere. Hence, Lemma 2.4 implies that
$\partial_{T}X$ is geodesically complete.

We proceed by induction on the dimension of $\partial_{T}X$. If the
$\dim(\partial_{T}X)$ is 0, then $\partial_{T}X$ is discrete and there is
nothing to show. So let’s assume $\dim(\partial_{T}X)\geq 1$. Recall that two
antipodes $\xi,\hat{\xi}$ in $\partial_{T}X$ at distance $>\pi$ bound a
complete geodesic in $X$ [Lee00, Proposition 2.1]. Since every complete
geodesic in $X$ lies in an $n$-flat, we conclude that the diameter of
$\partial_{T}X$ is $\pi$.

Suppose $X$ splits as a non-trivial metric product $X\cong X_{1}\times X_{2}$.
Then by Lemma 3.6, the factors are subject to the induction hypothesis and we
may assume that $X$ is irreducible.

By assumption, the set of $(n-1)$-spherical points $O\subset\partial_{T}X$ is
non-empty. If $O=\partial_{T}X$, then $X$ is isometric to $\mathbb{R}^{n}$. So
let us assume that $O$ is a proper subset of $\partial_{T}X$. By Proposition
3.5, the set $\partial_{T}X\setminus O$ is a non-empty, closed proper subset
which contains with every point also all its Tits antipodes. Since
$\partial_{T}X$ is irreducible, Theorem 2.6 implies that it is a spherical
building. ∎

###### Proof of Theorem 3.1.

By Theorem 3.2, it is enough to find an $(n-1)$-spherical point in
$\partial_{T}X$. If $U$ is an open relatively compact subset of
$\partial_{T}X$, then $U$ contains an open subset $U^{\prime}$ homeomorphic to
an $n$-manifold [LN19, Theorem 1.2]. Hence, for a point $\xi\in U^{\prime}$
there exists an $\epsilon>0$ such that $B_{\epsilon}(\xi)\subset U^{\prime}$.
If $\sigma\subset\partial_{T}X$ denotes a round $(n-1)$-sphere with
$\xi\in\sigma$, then $\sigma\cap B_{\epsilon}(\xi)=B_{\epsilon}(\xi)$. Hence
$\xi$ is $(n-1)$-spherical. ∎

##  4\. Rank $n$ spaces with periodic $n$-flats

###  4.1. From branch points to orthogonal half-planes

######  Definition 4.1.

Let $F\subset X$ be a flat in a CAT(0) space. Then we call a flat half-space
$H\subset X$ a flat half-space orthogonal to $F$, or, if the respective flat
is clear from the context, an orthogonal flat half-space if

$H\cap F=\partial H\ \text{ and }\ \angle(H,F)\geq\frac{\pi}{2}$

where the angle is measured in the cross section of the parallel set
$P(\partial H)$. In case $H$ is 2-dimensional, we adjust to flat half-plane
orthogonal to $F$ and orthogonal flat half-plane, respectively.

######  Definition 4.2.

Let $X$ be a locally compact CAT(0) space and $F\subset X$ a flat. Let
$\mathop{\hbox{Isom}}(X)$ denote the isometry group of $X$. We say that $F$ is
periodic, if its stabilizer
$\mathop{\hbox{Stab}}(F):=\\{\gamma\in\mathop{\hbox{Isom}}(X)|\
\gamma(F)=F\\}$ contains a subgroup acting geometrically on $F$.

######  Lemma 4.3.

Let $X$ be a locally compact CAT(0) space and let $F\subset X$ be a periodic
$n$-flat with $n\geq 2$. Suppose that $F=F^{-}\times F^{+}$ is a metric
product decomposition into flats $F^{\pm}$ of dimension $n^{\pm}$. Then
$F^{\pm}$ is a periodic $n^{\pm}$-flat in the cross section $CS(F^{\mp})$.

###### Proof.

By Bieberbach’s theorem, the stabilizer of $F$ contains a free abelian
subgroup $A$ of rank $n$ which acts cocompactly on $F$ by Euclidean
translations. Since $A$ preserves $\partial_{T}F$ pointwise, it also preserves
$P(F^{\mp})$ and the splitting $P(F^{\mp})\cong F^{\mp}\times CS(F^{\mp})$.
Hence its projection to $CS(F^{\mp})$ contains a free abelian subgroup of rank
$n^{\pm}=n-n^{\mp}$ which acts geometrically by Euclidean translations on
$F^{\pm}\subset CS(F^{\mp})$. ∎

The following result implies that a branch point in the ideal boundary of a
periodic flat leads to an orthogonal flat half-plane.

######  Lemma 4.4 ([HK05, Lemma 2.3.1]).

Let $X$ be a locally compact CAT(0) space. Let $F\subset X$ be a periodic
flat. Let $\xi\in\partial_{T}F$ be a point such that there exists a sequence
$(\xi_{k})$ in $\partial_{T}X\setminus\partial_{T}F$ which converges to $\xi$
with respect to the Tits metric. Then there exists a flat half-plane $H\subset
X$ asymptotic to $\xi$ and orthogonal to $F$ (cf. Definition 4.1).

###### Proof.

Since $F$ is periodic, it is enough to find for every $R>0$ a flat strip
$S\cong\mathbb{R}\times[0,R]$ which is asymptotic to $\xi$ and orthogonal to
$F$. We fix a base point $o\in F$. For every $k\in\mathbb{N}$ we choose
$x_{k}\in o\xi_{k}$ such that $|x_{k},F|=R$. Denote by $\bar{x}_{k}\in F$ the
nearest point to $x_{k}$. Choose isometries
$\gamma_{k}\in\mathop{\hbox{Stab}}(F)$ such that all points
$\gamma_{k}(\bar{x}_{k})$ lies in a fixed compact set. After passing to a
subsequence, we obtain a limit geodesic $c_{\infty}$ which is parallel to $F$
and spans the required flats strip. More precisely, we have the convergence
$\gamma_{k}(o\xi_{k},x_{k})\to(c_{\infty},x_{\infty})$ with respect to pointed
Hausdorff topology. The complete geodesic $c_{\infty}$ is asymptotic to $\xi$
since $|\gamma_{k}\xi_{k},\xi|=|\xi_{k},\xi|\to 0$. Because
$x_{k}o\subset\bar{N}_{R}(F)$, also $c_{\infty}$ is contained in the
$R$-neighborhood of $F$ and has even distance $R$ from $F$ since
$|x_{\infty},F|=R$. Denote by $\bar{c}_{\infty}\subset F$ the complete
geodesic which contains the point $x_{\infty}$ and is asymptotic to $\xi$.
Then $c_{\infty}$ and $\bar{c}_{\infty}$ span the required flat strip by
[KL97, Lemma 2.3.5]. ∎

######  Corollary 4.5.

Let $X$ be a locally compact CAT(0) space. Let $\hat{F}\subset X$ be a
periodic $(n+1)$-flat with $\partial_{T}\hat{F}=\hat{\sigma}$. Denote by
$\lambda\subset\partial_{T}X$ a spherical $(k+1)$-lune with $k\leq n$ bounded
by round $k$-hemispheres $\tau^{\pm}$. Suppose that
$\lambda\cap\hat{\sigma}=\tau^{-}$. Then there exists a round
$(k+1)$-hemisphere $\hat{\tau}\subset\partial_{T}X$ such that
$\partial\hat{\tau}\subset\hat{\sigma}$ and
$\tau^{-}\subset\partial\hat{\tau}$. Moreover, $\hat{\tau}$ is the ideal
boundary of a flat half-space $\hat{H}$ orthogonal to $\hat{F}$.

###### Proof.

Set $\sigma=\partial\tau^{\pm}$ and write
$\hat{\sigma}=\sigma\circ\sigma^{\perp}$. Let $\hat{F}=F\times F^{\perp}$ be a
corresponding product splitting. We have $\lambda=\sigma\circ[0,\theta]$. By
Lemma 4.3, $F^{\perp}$ is a periodic flat in the cross section $CS(F)$. Denote
by $\zeta^{-}$ the center of $\tau^{-}$. Then the cross section of $\lambda$
provides a path in $\partial_{T}CS(F)$ which intersects
$\partial_{T}F^{\perp}$ only in $\zeta^{-}$. Thus, by Lemma 4.4, there exists
an orthogonal flat half-plane $H\subset CS(F)$ for $F^{\perp}$ which is
asymptotic to $\zeta^{-}$. Hence, $\hat{H}:=F\times H\subset P(F)$ and
$\hat{\tau}:=\partial_{T}\hat{H}$ define an orthogonal flat half-space and
round $(k+1)$-hemisphere, respectively, as required. ∎

###  4.2. Orthogonal half-spaces from a sequence of lower dimensional ones

In this section we prove the following key result which allows us to
understand the singular set in the Tits boundary of a periodic Morse flat. The
responsible geometric property behind the proof is flatness of intersecting
parallel sets, similar as in [Sta15, Theorem 2].

######  Proposition 4.6.

Let $X$ be a locally compact CAT(0) space. Let $\hat{F}$ be a periodic
$\hat{n}$-flat in $X$ with $\partial_{T}\hat{F}=\hat{\sigma}$. Suppose that
there is a sequence of flat half-spaces $(H_{k})$ of dimension $d\leq n$, all
orthogonal to $\hat{F}$, such that their boundary flats $h_{k}:=\partial
H_{k}$ are pairwise non-parallel. Then there exists a $(d+1)$-dimensional flat
half-space $\hat{H}$ orthogonal to $\hat{F}$. Moreover, if the sequence of
round spheres $\sigma_{k}=\partial_{\infty}h_{k}$ converges to a round sphere
$\sigma\subset\hat{\sigma}$ and for every $k\in\mathbb{N}$ the convex hull of
$\sigma_{k}$ and $\sigma$ is a round sphere of dimension $\hat{d}-1$, then we
can arrange $\hat{H}$ to have dimension $\hat{d}+1$ and such that
$\sigma\subset\partial_{\infty}\hat{H}$.

###### Proof.

Since $\hat{F}$ is periodic, we may assume that we have a convergent sequence
of pairwise distinct pointed flat half-spaces
$(H_{k},p_{k})\to(H_{\infty},p_{\infty})$, where $p_{k}\in F_{k}$. The limit
$H_{\infty}$ with boundary flat $h_{\infty}:=\partial H_{\infty}$ is then
still orthogonal to $\hat{F}$. Set $\partial_{T}H_{k}=\tau_{k}^{+}$,
$\partial_{T}H_{\infty}=\tau_{\infty}^{+}$ and denote by $\zeta_{k}$ and
$\zeta_{\infty}$ the respective centers. Because $p_{\infty}\in h_{\infty}$
and $H_{\infty}$ is orthogonal to $\hat{F}$, the Busemann function
$b_{\zeta_{\infty},p_{\infty}}$ is non-negative on $\hat{F}$. Since the
geodesic rays $p_{k}\zeta_{k}$ converge to $p_{\infty}\zeta_{\infty}$, we find
a sequence $t_{k}\to\infty$ and points $x_{k}\in p_{k}\zeta_{k}$ such that
$b_{\zeta_{\infty},p_{\infty}}(x_{k})\leq-t_{k}$. The geodesic ray
$p_{k}\zeta_{k}$ lies in $H_{k}$ and therefore $x_{k}$ lies in $P(h_{k})$.
Thus $HB(\zeta_{\infty},x_{k})\cap P(h_{k})$ is a non-empty closed convex set
and we have

$\partial_{\infty}(HB(\zeta_{\infty},x_{k})\cap
P(h_{k}))=\bar{B}_{\frac{\pi}{2}}(\zeta_{\infty})\cap\partial_{\infty}P(h_{k}).$

As $\hat{F}\subset P(h_{k})$, we see
$\partial_{\infty}h_{\infty}\subset\partial_{\infty}(HB(\zeta_{\infty},x_{k})\cap
P(h_{k}))$. From Lemma 2.7 we infer that there is a $(d-1)$-flat $E_{k}\subset
HB(\zeta_{\infty},x_{k})\cap P(h_{k})$ with
$\partial_{\infty}E_{k}=\partial_{\infty}h_{\infty}$. Since the flats $h_{k}$
are pairwise non-parallel and $E_{k}\subset P(h_{k})$, we see that there is a
flat $F_{k}$ in $P(h_{k})$ of dimension at least $d$ which contains $E_{k}$
and such that $\partial_{\infty}h_{k}\subset\partial_{\infty}F_{k}$. Note that
$b_{\zeta_{\infty},p_{\infty}}$ is bounded above on $E_{k}$ by $-t_{k}$ since
$E_{k}\subset HB(\zeta_{\infty},x_{k})$. As $b_{\zeta_{\infty},p_{\infty}}$ is
non-negative on $\hat{F}$ we conclude $E_{k}\cap
N_{t_{k}}(\hat{F})=\emptyset$. Hence if $F^{\prime}_{k}\subset\hat{F}$ denotes
the closest parallel flat to $F_{k}$, then $|F_{k},F^{\prime}_{k}|_{H}\geq
t_{k}$. In particular, $F_{k}$ and $F^{\prime}_{k}$ bound a flat strip of
dimension $\geq d+1$, orthogonal to $\hat{F}$ and of width at least $t_{k}$.
Since $\hat{F}$ is periodic and $t_{k}\to\infty$, we obtain a flat half-space
$\hat{H}$ orthogonal to $\hat{F}$ and of dimension at least $d+1$ as a limit.

To prove the finer statement, simply note that in the argument above the flat
$E_{k}\subset P(h_{k})$ spans a flat $F_{k}\subset P(h_{k})$ of dimension
$\hat{d}$ if and only if the convex hull of the spheres
$\partial_{\infty}E_{k}=\partial_{\infty}h_{\infty}$ and
$\partial_{\infty}h_{k}$ is a round sphere of dimension $\hat{d}-1$ in
$\hat{\sigma}$. The rest of the argument stays the same. ∎

###  4.3. Tits boundary of a periodic Morse flat

In this final section we provide proofs of our main results.

######  Definition 4.7.

A round sphere $\sigma\subset\hat{\sigma}$ is called singular, if there exists
a round hemisphere $\tau^{+}\subset\partial_{T}X$ with
$\partial\tau^{+}=\sigma$ and $\tau^{+}\cap\hat{\sigma}=\sigma$ such that the
union $\tau^{+}\cup\tau^{-}$ is a round sphere for every round hemisphere
$\tau^{-}\subset\hat{\sigma}$ with $\partial\tau^{-}=\sigma$. We call a
singular sphere $\sigma\subset\hat{\sigma}$ maximal, if it is not contained in
a singular sphere in $\hat{\sigma}$ of strictly larger dimension.

######  Corollary 4.8.

Let $X$ be a locally compact CAT(0) space. Let $\hat{F}$ be a periodic
$\hat{n}$-flat in $X$ with $\partial_{T}\hat{F}=\hat{\sigma}$. Then the set of
maximal singular spheres in $\hat{\sigma}$ is finite.

###### Proof.

Proposition 4.6 implies that a maximal singular sphere
$\sigma\subset\hat{\sigma}$ has to be isolated in the following sense. If
$(\sigma_{k})$ is a sequence of singular spheres which converges to a sphere
$\sigma_{\infty}\subset\sigma$, then we must have $\sigma_{k}\subset\sigma$
for almost all $k\in\mathbb{N}$. This yields the claim. ∎

Recall that a point in $\hat{\sigma}$ is regular, if it has a Tits-
neighborhood which is entirely contained in $\hat{\sigma}$. Otherwise we call
it singular.

######  Lemma 4.9.

Every singular point in $\hat{\sigma}$ is contained in a singular sphere.

###### Proof.

If $\xi\in\hat{\sigma}$ is a singular point, then there exists a sequence of
branch points $\xi_{k}\in\hat{\sigma}$ with $\xi_{k}\to\xi$. By Lemma 4.4,
every such branch point yields a flat half-plane $H_{k}$ orthogonal to
$\hat{F}$ and with $\xi_{k}\in\partial_{\infty}H_{k}$. Using the cocompact
stabilizer of $\hat{F}$ we can produce a limit flat half-plane $H_{\infty}$
which is again orthogonal to $\hat{F}$ and with
$\xi\in\partial_{\infty}H_{\infty}$. Set
$\tau^{+}_{1}:=\partial_{\infty}H_{\infty}$. If $\partial\tau^{+}_{1}$ is not
singular, then by the Lune Lemma (Lemma 2.1) we can find a half-circle
$\tau^{-}_{1}\subset\hat{\sigma}$ with
$\partial\tau^{+}_{1}=\partial\tau^{-}_{1}$ such that $\tau^{+}_{1}$ and
$\tau^{-}_{1}$ span a 2-dimensional spherical lune $\lambda_{2}$. By Corollary
4.5, there exists a 2-dimensional round hemisphere
$\tau^{+}_{2}\subset\partial_{T}X$ with
$\tau^{+}_{2}\cap\hat{\sigma}=\partial\tau^{+}_{2}$ and
$\tau^{-}_{1}\subset\partial\tau^{+}_{2}$, in particular
$\xi\in\partial\tau^{+}_{2}$. If $\partial\tau^{+}_{2}$ is not singular, we
repeat the argument to produce a 3-dimensional round hemisphere
$\tau^{+}_{3}\subset\partial_{T}X$ with $\xi\in\partial\tau^{+}_{3}$. After at
most $(n-1)$ steps we end up with a singular sphere that contains the point
$\xi$. ∎

Now we obtain the following refinement of Theorem D.

######  Theorem 4.10.

Let $X$ be a locally compact CAT(0) space. Suppose that $X$ contains a
periodic Morse flat $F$. Then the complement in $\partial_{T}F$ of the set of
regular points can be covered by a finite set of singular spheres of positive
codimension.

###### Proof.

Immediate from Corollary 4.8 and Lemma 4.9. ∎

###### Proof of Theorem A.

Let $F\subset X$ be a periodic $n$-flat. By Theorem D, $\sigma=\partial_{T}F$
contains a dense subset of spherical points. Then, by Theorem 3.2,
$\partial_{T}X$ is a spherical join or a spherical building. Since $X$ is
geodesically complete and locally compact, the claim follows from Theorem 2.8
and [KL97, Proposition 2.3.7]. ∎

###### Proof of Corollary C.

By Theorem A, Lemma 3.6 and Lemma 4.3, we can produce a product decomposition
of $X$ as claimed. Since every such decomposition induces a corresponding join
decomposition of $\partial_{T}X$. The uniqueness statement follows from the
uniqueness of such join decompositions [Lyt05, Corollary 1.2]. ∎

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| C-TLP |  | GC-TLP |  | Diff. |  | C-TLP |  | GC-TLP |  | Diff. |  | C-TLP |  | GC-TLP |  | Diff.
| Auxiliary
---
information
$175$ |  | $175$ |  | $0$ |  | $175$ |  | $175$ |  | $0$ |  | $175$ |  | $175$ |  | $0$
Modulus N | $296$ |  | $296$ |  | $0$ |  | $296$ |  | $296$ |  | $0$ |  | $296$ |  | $296$ |  | 0
Squaring |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
operations T | 56 |  | 136 |  | 80 |  | 56 |  | $28\,784$ |  | $28\,488$ |  | 56 |  | $588\,240$ |  | $587\,944$
Public key | 823 |  | 903 |  | 80 |  | 1063 |  | $29\,551$ |  | $28\,488$ |  | $3\,601\,523$ |  | $9\,213\,499$ |  | $5\,611\,976$
Random |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
generators r | 377 |  | 377 |  | 0 |  | $29\,820$ |  | $29\,820$ |  | 0 |  | $2\,975\,176$ |  | $2\,975\,176$ |  | 0
Secret |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
exponent a | 296 |  | 384 |  | 88 |  | 296 |  | $30\,520$ |  | $30\,224$ |  | 296 |  | $3\,045\,176$ |  | $3\,044\,880$
Witnesses d | 137 |  | 137 |  | 0 |  | $5820$ |  | $5820$ |  | 0 |  | $5\,700\,984$ |  | $5\,700\,984$ |  | 0
Secret key | 810 |  | 898 |  | 88 |  | $35\,936$ |  | $66\,160$ |  | $30\,224$ |  | $3\,550\,648$ |  | $6\,595\,528$ |  | $3\,044\,880$
Total | 1633 |  | 1801 |  | 168 |  | $36\,999$ |  | $95\,711$ |  | $58\,712$ |  | $3\,551\,711$ |  | $7\,184\,535$ |  | $3\,632\,824$

## Acknowledgements

Aydin Abadi was supported in part by REPHRAIN: The National Research Centre on
Privacy, Harm Reduction and Adversarial Influence Online, under UKRI grant:
EP/V011189/1. Steven J. Murdoch was supported by REPHRAIN and The Royal
Society under grant UF160505.

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##  Appendix 0.A Symmetric-key Encryption Scheme

A symmetric-key encryption scheme consists of three algorithms
$(\mathtt{SKE.keyGen},\mathtt{Enc},\mathtt{Dec})$, defined as follows. (1)
$\mathtt{SKE.keyGen}(1^{\scriptscriptstyle\lambda})\rightarrow k$ is a
probabilistic algorithm that outputs a symmetric key $k$. (2)
$\mathtt{Enc}(k,m)\rightarrow c$ takes as input $k$ and a message $m$ in some
message space and outputs a ciphertext $c$. (3) $\mathtt{Dec}(k,c)\rightarrow
m$ takes as input $k$ and a ciphertext $c$ and outputs a message $m$.

The correctness requirement is that for all messages $m$ in the message space,
it holds that

$\Pr\Big{[}\ \
\mathtt{Dec}(k,\mathtt{Enc}(k,m))=m:\mathtt{SKE.keyGen}(1^{\scriptscriptstyle\lambda})\rightarrow
k\Big{]}=1\;.$

The symmetric-key encryption scheme satisfies _indistinguishability against
chosen-plaintext attacks (IND-CPA)_ , if any PPT adversary $\mathcal{A}$ has
no more than $\frac{1}{2}+\mathsf{negl}(\lambda)$ probability in winning the
following game: the challenger generates a symmetric key
$\mathtt{SKE.keyGen}(1^{\scriptscriptstyle\lambda})\rightarrow k$. The
adversary $\mathcal{A}$ is given access to an encryption oracle
$\mathtt{Enc}(k,\cdot)$ and eventually sends to the challenger a pair of
messages $m_{0},m_{1}$ of equal length. In turn, the challenger chooses a
random bit $b$ and provides $\mathcal{A}$ with a ciphertext $\mathtt{Enc}(k,$
$m_{b})\rightarrow c_{b}$. Upon receiving $c_{b}$, $\mathcal{A}$ continues to
have access to $\mathtt{Enc}(k,\cdot)$ and wins if its guess $b^{\prime}$ is
equal to $b$.

##  Appendix 0.B Sequential and Iterated Functions

######  Definition 13 ($\Delta,\delta(\Delta))$-Sequential function)

For a function: $\delta(\Delta)$, time parameter: $\Delta$ and security
parameter: $\lambda=O(\log(|X|))$, $f:X\rightarrow Y$ is a
$(\Delta,\delta(\Delta))$-sequential function if the following conditions
hold:

  * $\bullet$

There is an algorithm that for all $x\in X$evaluates $f$ in parallel time
$\Delta$, by using $poly(\log(\Delta),\lambda)$ processors.

  * $\bullet$

For all adversaries $\mathcal{A}$ which execute in parallel time strictly less
than $\delta(\Delta)$ with $poly(\Delta,\lambda)$ processors:

$Pr\left[y_{\scriptscriptstyle A}=f(x)\middle|y_{\scriptscriptstyle
A}\stackrel{{\scriptstyle\scriptscriptstyle\$}}{{\leftarrow}}\mathcal{A}(\lambda,x),x\stackrel{{\scriptstyle\scriptscriptstyle\$}}{{\leftarrow}}X\right]\leq
negl(\lambda)$

where $\delta(\Delta)=(1-\epsilon)\Delta$ and $\epsilon<1$.

######  Definition 14 (Iterated Sequential function)

Let $\beta:X\rightarrow X$ be a $(\Delta,\delta(\Delta))$-sequential function.
A function $f:\mathbb{N}\times X\rightarrow X$ defined as
$f(k,x)=\beta^{\scriptscriptstyle(k)}(x)=\overbrace{\beta\circ\beta\circ...\circ\beta}^{\scriptscriptstyle
k\text{\ \ Times}}$ is an iterated sequential function, with round function
$\beta$, if for all $k=2^{\scriptscriptstyle o(\lambda)}$ the function
$h:X\rightarrow X$ defined by $h(x)=f(k,x)$ is
$(k\Delta,\delta(\Delta))$-sequential.

The primary property of an iterated sequential function is that the iteration
of the round function $\beta$ is the quickest way to evaluate the function.
Iterated squaring in a finite group of unknown order, is widely believed to be
a suitable candidate for an iterated sequential function. Below, we restate
its definition.

######  Assumption 1 (Iterated Squaring)

Let N be a strong RSA modulus, $r$ be a generator of
$\mathbb{Z}_{\scriptscriptstyle N}$, $\Delta$ be a time parameter, and
$T=poly(\Delta,\lambda)$. For any $\mathcal{A}$, defined above, there is a
negligible function $\mu(.)$ such that:

$Pr\left[\begin{array}[]{l}\mathcal{A}(N,r,y)\rightarrow
b\end{array}:\begin{array}[]{l}r\stackrel{{\scriptstyle\scriptscriptstyle\$}}{{\leftarrow}}\mathbb{Z}_{\scriptscriptstyle
N},b\stackrel{{\scriptstyle\scriptscriptstyle\$}}{{\leftarrow}}\\{0,1\\}\\\
\text{if}\ \ b=0,\
y\stackrel{{\scriptstyle\scriptscriptstyle\$}}{{\leftarrow}}\mathbb{Z}_{\scriptscriptstyle
N}\\\ \text{else}\ y=r^{\scriptscriptstyle 2^{\scriptscriptstyle
T}}\end{array}\right]\leq\frac{1}{2}+\mu(\lambda)$

##  Appendix 0.C Further Discussion on the Network Delay Parameter

As discussed in [16], liveness states that an honestly generated transaction
will eventually be included more than $\kappa$ blocks deep in an honest
party’s blockchain. It is parameterised by wait time: $u$ and depth: $\kappa$.
They can be fixed by setting $\kappa$ as the minimum depth of a block
considered as the blockchain’s state (i.e., a part of the blockchain that
remains unchanged with a high probability, e.g., $\kappa\geq 6$) and $u$ the
waiting time that the transaction gets $\kappa$ blocks deep. As shown in [2],
there is a slackness in honest parties’ view of the blockchain. In particular,
there is no guarantee that at any given time, all honest miners have the same
view of the blockchain or even the state. But, there is an upper bound on the
slackness, denoted by $WindowSize$, after which all honest parties would have
the same view on a certain part of the blockchain state. This means when an
honest party (e.g., the solver) propagates its transaction (containing the
proof) all honest parties will see it on their chain after at most:
$\Upsilon=WindowSize+u$ time period.