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# High-altitude polar NM with the new DAQ system as a tool to study details of
the cosmic-ray induced nucleonic cascade

###### Abstract

A neutron monitor (NM) is, since the 1950s, a standard ground-based detector
whose count rate reflects cosmic-ray variability. The worldwide network of NMs
forms a rough spectrometer for cosmic rays. Recently, a brand-new data
acquisition (DAQ) system has been installed on the DOMC and DOMB NMs, located
at the Concordia research station on the Central Antarctic plateau. The new
DAQ system digitizes, at a 2-MHz sampling rate, and records all individual
pulses corresponding to secondary particles in the detector. An analysis of
the pulse characteristics (viz. shape, magnitude, duration, waiting time) has
been performed, and several clearly distinguishable branches were identified:
(A) corresponding to signal from individual secondary neutrons; (B)
representing the detector’s noise; (C) double pulses corresponding to the
shortly separated nucleons of the same atmospheric cascades; (D) very-high
multiple pulses which are likely caused by atmospheric muons; and (E) double
pulses potentially caused by contamination of the neighbouring detector. An
analysis of the waiting time distributions has revealed two clearly
distinguishable peaks: peak (I) at about 1 msec being related to the intra-
cascade diffusion and thermalisation of secondary atmospheric neutrons; and
peak (II) at 30 – 1000 msec corresponding to individual atmospheric cascades.
This opens a new possibility to study spectra of cosmic-ray particles in a
single location as well as details of the cosmic-ray induced atmospheric
cascades, using the same dataset.

JGR: Space Physics

Space Physics and Astronomy Research Unit, University of Oulu, Finland
Sodankylä Geophysical Observatory, University of Oulu, Finland Ioffe Physical-
Technical Institute RAS, St. Petersburg, Russia Center for Space Research,
North-West University, Potchefstroom, South Africa

Ilya<EMAIL_ADDRESS>
New data-acquisition (DAQ) systems, digitizing all pulses at a 2-MHz sampling
rate, is installed and tested at DOMC/DOMB neutron monitors.

Several branches are identified in the pulse parameters, corresponding to
different processes.

The new DAQ system allows studying inter-cascade, intra-cascade and
instrumental response separately from the same dataset.

##  1 Introduction

A neutron monitor (NM) is a standard ground-based detector to monitor cosmic-
ray variability in the near-Earth environment [Shea Smart (2000), Vainio .
(2009), Usoskin . (2017)]. The design of the NM was developed in 1957 (called
IGY — International Geophysical Year) and improved in 1964 (called NM64), and
since then it is used as a standard detector [Simpson (1958), Simpson (2000),
Stoker (2009)]. NMs record primarily the secondary nucleonic component (mostly
neutrons) of the cosmic-ray induced atmospheric cascade with a small fraction
of counts caused by muons. Its count rate is defined by the flux of primary
(impinging on the top of the atmosphere) cosmic rays, as a combination of the
cosmic-ray energy spectrum, detector’s yield function and geomagnetic rigidity
cutoff [Clem Dorman (2000), Mishev . (2020)]. The NM is an energy-integrating
detector, with the effective energy ranging from about 12 GeV for polar NMs to
35 GeV for equatorial ones [Asvestari . (2017)]. The sensitivity of NMs to
low-energy cosmic rays is highest in polar regions (low or no geomagnetic
shielding) and high altitudes (lower atmospheric shielding) and decreases
towards equatorial latitudes. The worldwide network of NMs can act as a giant
spectrometer able to roughly estimate the spectrum of both galactic cosmic
rays [<]e.g.,¿[]dorman04 and relativistic solar protons
[<]e.g.,¿[]duggal79,mishev14. Along with the count rate, the multiplicity of
NM counts (the average number of pulses within a short time interval) is
sometimes studied [Dorman (2004), Balabin . (2011)] as a rough index of the
spectral hardness of cosmic rays. Sometimes NM are accompanied by separate
muon detectors to measure high-energy cosmic rays.

This work is focused at two mini-NMs, DOMC and DOMB, located at the Concordia
Antarctic Research station on top of Dome C, Central Antarctic plateau
(75∘06’S, 123∘23’E, 3233 m above sea level) [Poluianov . (2015)]. They are
ones of the most sensitive NMs to lower energy cosmic rays (including solar
energetic particles) thanks to the highly elevated polar location. Each NM has
one BF3-filled detector surrounded by reflecting and moderating layers of
polyethylene. In addition, DOMC has a layer of lead serving as a neutron
producer to increase the detector efficiency. DOMB has no lead layer and
therefore has lower efficiency than DOMC, but is more sensitive to low-energy
secondary cosmic-ray particles [Vashenyuk . (2007)]. Recently, those
instruments got a major upgrade of the data-acquisition system (DAQ).
Traditionally, a standard NM records only the count rate of cosmic rays, while
the new electronics of DOMC and DOMB digitizes individual detector pulses with
a sub-microsecond precision [<]2-MHz sampling rate, see¿[]strauss20. In this
work, we study new opportunities provided by the DAQ upgrade, by using the
statistic of recorded pulses and show that it allows one to study details of
the cosmic-ray induced atmospheric cascade with instruments like DOMB and
DOMC.

Figure 1:  Upper panel: 2D histogram of the magnitude vs. length of pulses
recorded by DOMC NM (about $3\cdot$$10^{8}$ pulses) during the analyzed period
(no additional threshold applied). Letters denote different branches as
discussed in the text and summarized in Section 3. Separation between the
branches is shown by solid lines. Bottom panel: Typical profiles of pulses
from different branches shown in the upper panel. Panel D depicts an extremely
long multiple pulse, while a typical pulse shape of branch D is shown in
Figure 2D.

##  2 DOMC/DOMB DAQ electronic system

The new DAQ system is built with a single-board computer Raspberry Pi 3B and
easily replaceable modules responsible for the operation of different
subsystems (high voltage power supply, pre-amplifier and detector signal
processor, temperature-pressure-humidity sensors and others) – see full
details in strauss20.

Pulse signals coming from the detector are amplified and digitized by a signal
registration board. It has a dedicated microcontroller PIC32 and built-in
10-bit analogue-to-digital converter (ADC) with the reference voltage set at
3.3 V. Each pulse is sampled at the frequency of 2 MHz (viz. 0.5 $\mu$sec),
and the information about its magnitude time profile is stored in a buffer of
the board. The central computer has software-defined discriminators in pulse’s
magnitude and length. If a pulse in the buffer matches both criteria, it gets
recorded in a data file by the central computer. The files are compressed and
sent to the data server of Oulu cosmic ray station for further import to
databases cosmicrays.oulu.fi and nmdb.eu.

Figure 2:  The same as in Figure 1 but for the DOMB NM (about $3\cdot$$10^{8}$
pulses).

DOMC and DOMB have the following default thresholds for registered pulses: the
magnitude discriminator at 0.2 V and the minimum pulse length of five sample
points (2.5 $\mu$sec). All pulses failing to meet these criteria are ignored
by the DAQ system. Since each pulse with the magnitude exceeding 0.2 V and
longer than 2 $\mu$sec is digitized, a higher-value thresholds can be applied
electronically in the off-line analysis.

Analyses of the pulse shapes and statistics are presented in the subsequent
sections.

##  3 Analysis of pulse shapes

Here, we used data from DOMC for 01-Jan through 31-May-2020 (about $3\cdot
10^{8}$ pulses recorded), and from DOMB for 13-Aug through 31-Oct-2019
($4\cdot 10^{7}$ individual pulses recorded). These periods correspond to very
quiet solar conditions (solar cycle minimum) with relatively low heliospheric
modulation and high intensity of galactic cosmic rays (GCR). There were no
solar particle events or other notable transients during the studied period.
The period of January through May 2020 corresponded to a transition from
polar-day (around-a-clock insolation) to polar-night (no sunlight) conditions
and formation of the polar vortex. The period of August – October 2019 was
characterized by very stable and cold weather with a stable polar vortex. An
example of temporal variability of the pressure-corrected count rates in
different branches (see below) are shown in Figure 3 for the first 42 days of
each analyzed period. For correction, the following barometric coefficients
were used: -0.769 %/hPa for DOMC and -0.754 %/hPa for DOMB (the reference
pressure level 650 hPa) as defined during the operation of the detectors (see,
e.g., metadata in http://cosmicrays.oulu.fi). One can see that the overall
pressure-corrected count-rate vary within $\pm 1$% only, but the variability
in other branches is greater as described below.

Figure 3:  Pressure-corrected and normalized per unity count rates of NMs in
different branches (see text) for the first 42 days of each NM dataset. Upper
panel: Hourly values for DOMC for the period 01-Jan through 11-Feb-2020- The
thick black line depicts the anomaly, from the mean level 15.6 km, of the
geopotential 100 hPa height $h_{100}$ (right-hand Y-axis) over the Concordia
station (ERA-5 data from ECMWF
https://www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era5). Lower
panel: Daily means for DOMB for the period 13-Aug through 25-Sep-2019.

For the analysis, we used the following pulse parameters: magnitude $A$ (the
maximum voltage); length $t$ (the duration, in $\mu$sec of the signal being
above the selected threshold); the $e$-folding decay time $\tau$ of the
exponential decline of the pulse voltage after the maximum, and the waiting
time $\Delta T$ between onsets of subsequent pulses (in $\mu$sec).

First, we analyzed the relation between the magnitude $A$ and the length $t$
of all individual pulses. These distributions are presented in Figures 1 and
2, respectively for the DOMC and DOMB NMs [<]cf. Figure 4 in¿[]strauss20.
Several clearly distinguished branches can be identified (see statistic in
Table 1):

###  3.1 Branch A: Normal pulses

The main branch contains the majority ($>$90 %) of the pulses. It consists of
single, well-defined pulses with a fast rise (a few $\mu$sec) followed by an
exponential decline with the $e$-folding decay time $\tau\approx 8$ $\mu$sec,
defined by the pre-amplifier’s circuit (relaxation of a capacitor). Samples of
typical pulses in this branch can be observed in Figures 1A and 2A for DOMC
and DOMB NMs, respectively. The magnitude takes the entire range up to 3 V,
and duration 5 – 20 $\mu$sec, with a tendency that higher pulses are slightly
longer, as they decay to the detection threshold level longer. However, since
low pulses ($A$$<$0.5 V) are mixed with the branch B (noise), we consider
normal pulses as those with $A$$>$0.5 V. The time variability of the count
rate in this branch is perfectly corrected for pressure (the formal Pearson
correlation between the daily count rate and the pressure is $r=$-0.02) and
remains stable within $\pm 1$% in both DOMC and DOMB. This is the clear signal
part which forms the main fraction ($\approx 91$%) of the count rate, while
the remaining 8 – 9 % of pulses need more discussion.

###  3.2 Branch B: Noise

This branch contains very low ($A<0.6$ V) pulses without any clear shape (see
Figures 1B and 2B). The positive correlation between $A$ and $t$ is expected
for the same reason as for branch A. These pulses are likely related to
electronic noise. This component comprises 5 – 7 % of the total number of
pulses but can be reduced to $<$1%, thus improving the signal-to-noise ratio
from $\approx$15 to $>$90 by increasing the magnitude threshold level up to
0.5 – 0.6 V. Since the short-length ($<$12 $\mu$sec) pulses are contributed
also from branch A (normal pulses), the percentage above is a conservative
upper limit. This branch depicts hardly any time variability as expected for
the noise, but after nominal pressure correction (Figure 3) it appears ’over-
corrected’ and depicts slight variability in phase with the pressure
(r=0.29$\pm$0.15, $p-$value =$0.03$).

###  3.3 Branch C: Contribution of atmospheric cascade

This branch consists of moderately high ($A>1.5$ V) and longer (20 – 35
$\mu$sec) pulses. They are typically double peaks (Figure 1C) where the second
peak starts before the first one drops below the detection level. The negative
correlation between the magnitude and the length is understandable, as the
second pulse starts over non-zero background. The separation between the
pulses is from 5 – 20 $\mu$sec. The sub-pulses are totally consistent with the
pulses in branch A in duration and decay time ($\tau\approx 8\,\mu$sec). This
branch is clear in DOMC data, comprising about 3% of all pulses, but is hardly
visible (only 0.24 %) in the DOMB dataset, implying that is caused by a
nucleonic component. The time separation between the subpulses ($<20\,\mu$sec)
is much longer than the characteristic time (expected to be of the order of
several nanoseconds) of a cascade within the detector itself, including lead
producer, but shorter than the full development of the atmospheric cascades
(see Section 4). It is likely related to a tail of the WTD for the atmospheric
cascade development (Section 4), when the time separation between secondary
nucleons of the same cascade appear shorter than the single pulse length of
$\approx$ 20 $\mu$sec. Such pulses are registered as long ones composed of two
partly overlapping pulses and form branch C. Contributions of nucleons with
longer time separation make single pulses associated to branch A. The count-
rate variability in this branch (Figure 3) still depicts dependence (under-
correction) on pressure after the nominal pressure correction
(r=-0.26$\pm$0.16, $p$=$0.05$), implying a stronger dependence on pressure
than that for branch A. We note that the standard NM64 electronic setup with
the dead-time (20 $\mu$sec) makes this branch indistinguishable from branch A.

###  3.4 Branch D: Possible contribution of muons

This branch is characterized by very high magnitude ($A>3$ V, viz. near the
upper bound of ADC) and long duration of pulses. Most numerous here are
saturated pulses (Figure 2D) but there are also very long pulses, up to 110
$\mu$sec in length, that include a sequence of short but very high sub-pulses
following each other by several $\mu$sec (Figure 1D). The saturated pulses
would require, if fitted with a ‘standard’ pulse shape (Fig. 1A), a very high
magnitude of up to 30 V, viz. a factor of 10 greater than normal pulses.
Multiple pulses may contain up to 10 short sub-pulses, also implying an
enhanced yield. This points to a different type of process producing pulses in
branch D. The most plausible candidate is the process of direct multiple-ion
production by a cascade muon in BF3 gas inside the NM proportional counter
[Knoll (2010), Siciliano Kouzes (2012)]. This can ignite multiple, nearly
simultaneous electromagnetic avalanches inside the counter, leading to more
‘energetic’ recorded pulses (voltage exceeding the upper limit and/or multiple
overlapping pusles – see Figures 2D and 1D, respectively). This process has
not been considered nor properly modelled for NMs, where its contribution is
small, in contrast to the usually considered muon contribution to NM counts,
viz. muon-induced production of neutrons in the lead producer [Clem Dorman
(2000)] with subsequent detection in the counter [Maurin . (2015), Mangeard .
(2016)]. Such muon-induced neutrons are detected in a usual way and cannot be
distinguished from the signal of the hadronic component. Accordingly, they
appear in branch A and cannot contribute to branch D. Therefore, we can
speculate that the branch D is likely related to non-hadronic particles
producing abnormally energetic pulses via direct ionization of the filling
gas. A more detailed study of this process is planned for the future.

Count-rate in this branch (Figure 3) depicts strong variability: $\pm$25% for
DOMC and $\pm$5% for DOMB, which can be explained by a strong change in the
atmospheric density profile and by a stable vortex conditions, during the two
analyzed periods, respectively. For comparison, we shown in Figure 3A the
temporal variability, for the same period, of the anomaly of the geopotential
height of the $\Delta h_{100}$ (100 hPa) atmospheric level, which roughly
corresponds to the mean height of the muon production and affects the muon
flux near ground. The branch-D count rate co-varies in sync with $\Delta
h_{100}$ ($r$=-0.84, $p$-value $<10^{-6}$), and the magnitude of the muon-flux
variability is consistent with the $\Delta h_{100}=0.5$ km using the measured
spectrum of muons [Boezio . (2003)] at a 100-hPa level and relativistic time
dilation. This confirms the muon origin of this branch since no other source
can reliably explain it. We emphasize that this effect is strong for the high
altitude of DOMC location but fades towards lower heights (because of the
higher energy of muons that can reach lower altitudes) and accounts for only
5% at sea level of 1013 hPa. Thus, the NM with a high magnitude threshold
($V_{0}>3.1$ V) can operate as a low-efficiency muon detector.

###  3.5 Branch E: Possible contamination from DOMC

There is a branch with pulses of a normal magnitude ($A$ between 1 – 2.5 V)
and double length (35 – 40 $\mu$sec) clearly visible for DOMB NM (Figure 2).
This branch contains 0.28 % of all pulses and is composed of double pulses
(Figure 2E) separated by 10 – 20 $\mu$sec. Interestingly, this branch is not
distinguishable in DOMC data (Figure 1). We do not have a clear understanding
of the origin of this branch but may speculate that it is possible
contamination from multiple neutrons produced by the lead producer of the
neighbouring DOMC detector. Because of the diffusive propagation, the neutrons
may arrive to DOMB at slightly different times. The detectors are located
about 1 meter apart of each other, leading to a 20-$\mu$sec traversing time
for one meter of air and several g/cm2 of the moderator/reflector layers. We
note that if the two pulses are separated by more than 25 $\mu$sec so that the
voltage drops below the discriminator’s level, the pulses are counted as two
separate ones. An insignificant hint on triple pulses can be observed in
Figure 2 at 50 $\mu$sec. We note that 20 $\mu$sec is known as electronic
‘dead-time’ of a standard NM [Hatton Carimichael (1964)]. Nothing conclusive
can be seen in the time variability of this branch, because of the low
statistics (about 1 count per minute).

This branch could be potentially caused also by noise in the pre-amplifier’s
electric circuit, by producing an ‘echo’ of the signal with a delay time of
$\approx$20 $\mu$sec, viz. double – triple characteristic time of the circuit
(8 $\mu$sec). However, this is unlikely since such echoing was not observed
during electronic tests of the DAQ board.

Table 1: Different branches of pulses (magnitude-vs-length) of the DOMC and
DOMB detector (see Figures 1 and 2, respectively). Percentages for the
branches were computed using the boundaries indicated in Figures 1 and 2.

Branch | percentage |  Possible origin
---|---|---
| DOMC | DOMB |
A | 90.98 | 91.49 |  Main branch: single pulses
B | 5.20† | 7.84† |  Noise
C | 3.15 | 0.23 |  Multiple pulses from the same atmospheric cascade
D | 0.68 | 0.16 |  Possible muon contribution
E | – | 0.28 |  Possible contamination from the neighbouring detector

† the percentage is a conservative upper bound as this branch includes also
pulses from branch A.

##  4 Waiting time distribution

Figure 4:  Logarithmically binned histograms of the waiting-time ($\Delta T$)
distribution of pulses recorded by DOMC NM for different discriminator’s
magnitude threshold levels $V_{0}$, as indicated (in volts) in the legends.
Red curves represent the theoretically expected distribution for randomly
occurring independent events, corresponding to the count rate. The blue dashed
curve in panel a) represents a simulated delay of the arrival time of $\leq$10
keV neutrons with respect to the cascade-front arrival for a 10 GeV cosmic-ray
proton.

Next, we have analyzed the waiting-time distribution (WTD) of recorded pulses.
We define the waiting time $\Delta T$ as the time interval between the onsets
of consecutive pulses recorded by the DAQ system of a NM. Since the length of
individual pulses can reach 100 $\mu$sec (Figures 1 and 2), we analyse the WTD
for $\Delta T>100\,\mu$sec. The observed WTDs are plotted, as logarithmically-
binned histograms in Figures 4 and 5, respectively for DOMC and DOMB NMs, for
different values of the magnitude threshold $V_{0}$ so that $A\geq V_{0}$. The
distributions for different values of $V_{0}$ have a similar shape with two
clearly separated peaks: one (called peak $I$) at $\Delta T\approx 1$ msec,
the other (peak $II$) located between 30 msec and 10 sec depending on the
value of $V_{0}$ and the detector type.

###  4.1 Peak I: In the atmospheric cascade

The first WTD peak (peak $I$) is located at $\Delta T\approx 1$ msec. Its
location is very stable and does not depend on the $V_{0}$ values, nor on the
detector type. On the other hand, the peak broadens for higher threshold
values extending its tail to longer waiting times, up to several milliseconds.
The contribution of this peak to the total count rate varies from about 35%
(no additional threshold) down to 2.5% ($V_{0}=2.5$ V) for DOMC NM, and from
7% down to 0.1% for the DOMB NM, respectively. We note that, while the bulk of
secondary protons, muons, and electromagnetic components of the cascade arrive
to a detector as a relatively thin front, secondary neutrons diffuse in the
atmosphere, leading to a wide spread in time and lateral distribution compared
to other secondaries [Grieder (2011)].

Figure 5:  The same as Figure 4 but for DOMB NM.

In order to check that, we performed a numerical simulation of the development
of the atmospheric cascade using the Monte-Carlo simulation toolbox Geant4
v.10.6.0 [Agostinelli . (2003), Allison . (2006)] with the physics list
QGSP_BIC_HP (Quark-Gluon String model, Geant4 Binary Cascade model, High-
Precision neutron package) [Geant4 collaboration (2020)]. We simulated
$10^{6}$ atmospheric cascades initiated by incident protons with the kinetic
energy of 10 GeV impinging vertically on the top of the atmosphere. We note
that this energy corresponds to the effective energy of a polar NM to GCR
[Kudela . (2000), Asvestari . (2017)] and thus roughly represents the relation
between the NM count rate and CR variability. During the simulations, we
traced secondary neutrons and recorded their crossing of the reference level
of 650 g/cm2 where the DOMC/DOMB NMs are located. For each neutron crossing,
the neutron’s energy, location with respect to the cascade axis and the time
since the first interaction were recorded for further analysis. Neutrons were
found to spread as far as 6 km from the cascade axis [<]see also¿[]paschalis11
with the delay of up to 80 milliseconds. Next, we built a logarithmically-
binned histogram of the WTD between arrivals of epithermal neutrons ($\leq 10$
keV) with spatial separation less than 1 meter, from the same atmospheric
cascade, as shown by the blue dashed curve in Figures 4a and 5a. The
distribution reasonably well matches both the location and width of peak I.
The height of the distribution was scaled up to match the observed WTD, while
keeping the shape and location of the peak. Similar results can be obtained
for other energies of the primary particle since the 1-msec time is caused by
diffusion and thermalization of secondary neutrons from 1 MeV (evaporation
peak) to 10 keV energy, and not by the development of a cascade per se.

Accordingly, WTD peak I can be reliably associated with the neutron diffusion
and thermalization within atmospheric cascades, and the time of about one
millisecond is a typical time-scale for such a process. These pulses generally
contribute to the well-known multiplicity of NM counts [Debrunner Walther
(1968), Balabin . (2011), Mangeard . (2016), Ruffolo . (2016)], viz. multiple
correlated pulses within the NM count rate. The mean multiplicity of the NM
count rate for different threshold $V_{0}$ values is shown in Figure 6. It is
calculated as the ratio of the total number of counts for each NM to that in
peak II, viz. the number of individual cascades. The obtained values for the
multiplicity (1.5 and 1.06 for $V_{0}=0$ for DOMC and DOMB, respectively) are
slightly higher than those measured at sea level [Hatton Carimichael (1964)]
but lower than that for an air-borne NM [Kent . (1968)].

Figure 6:  Mean multiplicity of the DOMC (black squares) and DOMB (blue open
circles) NMs as a function of the threshold $V_{0}$.

###  4.2 Peak II: Individual atmospheric cascades

The other peak (called peak II) is quite broad and corresponds to the waiting
times from about 20 msec to several seconds. The distribution is slightly
distorted between 10 and 40 msec, probably due to interference with the power-
line frequency (50 Hz). The peak has a well-defined smooth shape, and its
location depends on the detector (DOMB and DOMC) and the value of the
threshold $V_{0}$ (Figures 4 and 5). This peak in WTD corresponds to
individual atmospheric cascades caused by the primary cosmic-ray particles. In
order to illustrate this, we have also plotted (as red curves) the
analytically expected WTD for randomly occurring independent pulses with the
occurrence probability defined by the observed count rate $\nu$ of a given NM
(DOMC or DOMB) for a given value $V_{0}$. WTD of independently occurring
events with the occurrence probability $\nu$ is expected to be exponential
with the characteristic decay time $1/\nu$, which takes for a logarithmically
binned histogram the shape shown by the red curves. One can see that the
analytical WTD perfectly describes the observed peak II for different values
of $V_{0}$ and for both DOMC and DOMB, confirming its relation to the
individual atmospheric cascades.

The observed WTD is consistent with the recently introduced delay
time–geometry correction factor for the NM yield function, details are given
elsewhere [Mishev . (2013), Mishev . (2020)].

##  5 Discussion and conclusions

The new DAQ system of neutron monitors makes it possible to study different
processes induced by cosmic rays in the atmosphere and the detector itself. In
Sections 3 and 4 we have analyzed different pulse parameters and waiting
times. Somewhat similar analyses were made also earlier [Hatton Tomlinson
(1968)] using oscilloscopes and were based on small statistic. The new DAQ
system allows continuous analysis of the pulses, e.g., the statistic shown
here includes $3\cdot 10^{8}$ and $4\cdot 10^{7}$ individual pulses for the
DOMC and DOMB, respectively, for illustration, but it can be much greater.
Thanks to the fully digitized pulses, the same dataset recorded by the new DAQ
system allows studying different processes separately.

Individual atmospheric cascades can be studied as pulses separated by more
than two milliseconds, which is close to the standard NM detection mode with
the dead-time of 1.2 msec [Hatton Carimichael (1964)]. It is important that,
in contrast to the standard NM64 DAQ system with the fixed “dead-time”,
waiting time between pulses can be now performed by software data-processing.

Development of the atmospheric cascade can be studied using pulses with
waiting times between 0.3 and 3 msec. This is not usually done in the NM data
analysis directly but rather via the multiplicity (or ‘leader-fraction’)
analyses. With the new DAQ system, one can study details of the cascade
development for each cascade individually, including also the multiplicity.

Development of the atmospheric cascade in the vicinity of the detector can be
studied using the pulse-shape analysis, in particular, in branches C and D in
Figures 1 and 2. In particular, with the high magnitude threshold, the NM can
operate as a low-efficiency muon detector. We are not aware of any similar
analysis done previously. Moreover, a detailed analysis of branches C and D
allows one to discriminate hadrons from muons, and hence to use an NM in a
regime of muon detector, using the same dataset.

The noise is shown to form a well-separated branch, being characterized by the
low pulse magnitude. Accordingly, by selecting the discriminator threshold to
the value of $V_{0}=$ 0.5 – 0.6 V (the default is 0.2 V) one can cut off the
noise effectively without a significant reduction of regular pulses. Placing
the threshold to 0.5 V eliminates about 90% of the noise and only a few
percent of normal pulses, thus increasing the signal-to-noise ratio by an
order of magnitude.

The standard NM is an energy-integrating detector and cannot measure the
energy spectrum of cosmic rays. Combining NM with different geomagnetic cutoff
rigidities to the worldwide network makes them a very rough spectrometer
[Moraal . (2000)]. It works reasonably well for GCR [Caballero-Lopez Moraal
(2012)] but cannot be directly applied to a study of solar energetic particles
because of the possible anisotropy which can be large during impulsive (phase
of) events [<]e.g.,¿[]mishev14. Data from different NMs cannot be compared
directly without a complicated analysis of magnetospheric transport of charged
particles [<]e.g.,¿[]smart00. The use of a pair of standard and bare NMs in
the same location (e.g. at the South Pole, SANAE or Dome C stations) provides
a rough measure of the hardness of the cosmic-ray spectrum [Caballero-Lopez
Moraal (2016), Nuntiyakul . (2018)], based on the ratio of their count rates.
Here we propose that the use of different $A$-values in the new DAQ-system
dataset may provide an estimate of the cosmic-ray spectrum. Specifically, a
muon-related branch of the pulses can be separated, providing muon counts in
the same detector. Fine analysis of relatively weak ground-level solar
particle events is often very limited due to the shortage of information about
the energy spectrum since such events are seen only by a few high-altitude
polar NMs with no significant response from sea-level and non-polar
instruments [Mishev . (2017)]. In such a case, any spectral information is
very crucial and any new addition significantly increases the quality of the
analysis. Varying the value of $A$, one can obtain several spectral points. In
particular, the measured waiting time and magnitude distributions of pulses
could serve for that purpose. In an ideal case, a strong SEP event, such as
the GLE#69 of 20-Jan-2005, could be used to ‘calibrate’ the detector, but such
events occur very seldom. Alternatively, this possibility can be explored
quantitatively with a full simulation of the detector-and-atmosphere response
to the primary particles. This is doable with the modern Monte-Carlo
simulation techniques, but is beyond the framework of this paper and left for
forthcoming work.

In summary:

  1. 1.

A new DAQ system has been installed on DOMC and DOMB NMs that has gathered a
large amount of fully digitized, at a sub-$\mu$sec sampling rate, individual
pulses.

  2. 2.

An analysis of the data has demonstrated clustering of pulses to several
branches: (A) the main branch representing secondary neutrons; (B) detector’s
electronic noise; (C) double pulses caused by shortly separated pulses of the
same atmospheric cascade; (D) multiple pulses likely related to atmospheric
muons; (E) possible contamination of DOMB detector by neutrons scattered from
the neighbouring DOMC detector.

  3. 3.

An analysis of the waiting time distributions has revealed two clearly
distinguishable peaks: peaks I at about 1 msec related to the intra-cascade
diffusion and thermalisation of secondary atmospheric neutrons; and peak II
(30 – 1000 msec) corresponding to individual atmospheric cascades.

  4. 4.

It is shown that a NM with the new DAQ system can provide also data on muon
flux, using the same dataset.

This opens a new possibility to study spectra of cosmic-ray particles in a
single location and details of cosmic-ray induced atmospheric cascades.

###### Acknowledgements.

We are grateful to the personnel of Concordia station hosting the DOMC/DOMB
instrumentation. Operation of DOMC/DOMB NMs is possible thanks to the
hospitality of the Italian polar programme PNRA (via the LTCPAA PNRA 2015/AC3
and the BSRN PNRA OSS-06 projects) and the French Polar Institute IPEV. It is
supported by the Academy of Finland (projects CRIPA-X No. 304435, ESPERA No.
321882, QUASARE 330063, HEAIM-2 330427), and Finnish Antarctic Research
Program (FINNARP). DOMC/DOMB NM data can be obtained from
http://cosmicrays.oulu.fi, courtesy of the Sodankylä Geophysical Observatory.

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# Proper proximality for groups acting on trees

Changying Ding and Srivatsav Kunnawalkam Elayavalli
<EMAIL_ADDRESS><EMAIL_ADDRESS>https://sites.google.com/view/srivatsavke Department of Mathematics
Vanderbilt University, 1326 Stevenson Center, Station B 407807, Nashville, TN
37240

###### Abstract.

In this paper, the notion of proper proximality (introduced in [BIP18]) is
studied for various families of groups that act on trees. We show that if a
group acts non-elementarily by isometries on a tree such that for any two
edges, the intersection of their edge stabilizers is finite, then $G$ is
properly proximal. We then provide a complete classification result for proper
proximality among graph products of non-trivial groups, generalizing recent
work of Duchesne, Tucker-Drob and Wesolek classifying inner amenability for
graph products. As a consequence of the above result we obtain the absence of
Cartan subalgebras and Cartan-rigidity in properly proximal graph products of
weakly amenable groups with Cowling-Haagerup constant 1.

##  1\. Introduction and statements of main results

The goal of this paper is to provide several new examples of _properly
proximal_ groups. The authors of [BIP18] who introduced this property were
motivated by the program of classifying group von Neumann algebras. Proper
proximality is a dynamical/geometric property by nature, so it is
independently of interest to group theorists and geometers. One advantage of
proper proximality is that it applies to a robust family of non-amenable
groups, including all lattices in non compact semisimple Lie groups. This, in
particular, allowed the authors of [BIP18] to demonstrate the first
$W^{*}$-strong rigidity results for compact actions of higher rank lattices:

######  Theorem 1.1 (Theorem 1.1 in [BIP18]).

For all properly proximal groups $G$, the group von Neumann algbra $L(G)$ has
no weakly compact Cartan subalgebras in the sense of Ozawa and Popa [OP10].
Moreover, for any free ergodic probability measure preserving (p.m.p.) action
$\sigma:G\curvearrowright(X,\mu)$, the crossed product
$L^{\infty}(X,\mu)\rtimes G$ admits a weakly compact Cartan subalgebra $A$ if
and only if $\sigma$ is weakly compact and, in this case, $A$ is unitary
conjugate with $L^{\infty}(X,\mu)$.

Our first main result proves proper proximality for a family of groups acting
on trees. Unless otherwise mentioned, all groups in this paper are
_countable_.

######  Theorem 1.2.

Let an infinite countable $G$ act on a countably infinite tree $T$ such that:

  1. (1)

the action is non-elementary on the Bowditch compactification $\Delta T$;

  2. (2)

for any pairs of distinct edges $e,f\in E({T})$, one has
$\mathrm{Stab}(e)\cap\mathrm{Stab}(f)$ is finite.

Then $G$ is properly proximal.

The most natural examples of groups with the above phenomenon arise from
fundamental groups of graphs of groups, via Bass-Serre theory. Particular
cases of these include amalgamated free products and HNN-extensions.

A subgroup $H$ is almost malnormal in $G$ if for any $g\in G\setminus H$ one
has $|gHg^{-1}\cap H|<\infty$.

######  Corollary 1.3.

If $G=G_{1}*_{H}G_{2}$ is a countable group such that $H$ is almost malnormal
in $G$ and $[G_{1}:H]\geq 3$, $[G_{2}:H]\geq 2$, then $G$ is properly
proximal.

######  Corollary 1.4.

If $G*_{H\sim K}$ is an HNN-extension of a countable group $G$ over almost
malnormal subgroups $H$ and $K$, and $[G:H]\geq 3$, then $G*_{H\sim K}$ is
properly proximal.

######  Remark 1.5.

A group is said to be a _convergence group_ if it admits a non-elementary
convergence action (see [Bow99]). In [Dah03] and [Osi06], many amalgamated
products and HNN-extensions are shown to be convergence groups and hence
properly proximal (see Example 4.6 in [BIP18]) . However, the above
corollaries include examples that are not non-elementary convergence groups.
For instance, consider $G=(G_{1}\wr\Gamma)*_{\Gamma}(G_{2}\wr\Gamma)$, where
$G_{1}$ and $G_{2}$ are non-trivial amenable groups and $\Gamma$ is any
infinite countable group. Suppose there exists a non-elementary convergence
action $G\curvearrowright K$ on some compact space $K$, then as
$\oplus_{\Gamma}G_{i}$ is infinite amenable for $i=1,2$, there must exist
$a_{i},b_{i}\in K$ such that $\\{a_{i},b_{i}\\}$ is fixed by
$\oplus_{\Gamma}G_{i}$ set-wise. Since the normalizer of
$\oplus_{\Gamma}G_{i}$ is $G_{i}\wr\Gamma$, we have $G_{i}\wr\Gamma$ fixes
$\\{a_{i},b_{i}\\}$ as well. Now taking any infinite sequence in $\Gamma$, we
see from north south dynamics that $\\{a_{1},b_{1}\\}=\\{a_{2},b_{2}\\}$,
hence, $G$ fixes $\\{a_{1},a_{2}\\}$ which contradicts the fact that the
action is non elementary.

######  Remark 1.6.

We would like to point out for the particular cases of groups in Corollaries
1.3 and 1.4, the consequence of Theorem 1.1 is weaker than what is already
known in the literature, due to work of Ioana in [Ioa15], and subsequently
Vaes in [Vae13].

Our second main result is a complete classification for graph products of non-
trivial groups. We provide the following algorithm to decide if a graph
product of groups is properly proximal:

######  Theorem 1.7.

Let $\mathcal{G}_{pp}$ be the collection of graph products of non-trivial
groups, defined recursively:

  1. (1)

If $\Gamma$ satisfies the condition that the radius of $\Gamma$ (see Section
2.1), is at least 2 and there does not exist a pair of vertices
$v_{1},v_{2}\in V(\Gamma)$ such that $(u,v_{i})\in E(\Gamma)$ for all $u\in
V(\Gamma)\setminus\\{v_{1},v_{2}\\}$, and $i=1,2$ then
$\Gamma(G)\in\mathcal{G}_{pp}$, for any arbitrary choice of non-trivial groups
$G_{v}$.

  2. (2)

If $r(\Gamma)=2$ and there are vertices $v_{1},v_{2}\in V(\Gamma)$ such that
$(u,v_{i})\in E(\Gamma)$ for all $i\in\\{1,2\\}$, $u\in
V(\Gamma)\setminus\\{v_{1},v_{2}\\}$, and $|G_{v_{i}}|\geq 3$ for some
$i=1,2$, then $\Gamma_{S}(G)\in\mathcal{G}_{pp}$, where $\Gamma_{S}$ is the
subgraph induced by the vertices $S=V(\Gamma)\setminus\\{v_{1},v_{2}\\}$

  3. (3)

If $r(\Gamma)=1$ with some $v\in V(\Gamma)$ such that $(v,u)\in E(\Gamma)$ for
all $u\neq v$, and $G_{v}$ is properly proximal, then
$\Gamma_{T}(G)\in\mathcal{G}_{pp}$, where $\Gamma_{T}$ is the induced subgraph
by $T=V(\Gamma)\setminus\\{v\\}$.

Then, a graph product $\Gamma(G)$ is properly proximal if and only if
$\Gamma(G)\in\mathcal{G}_{pp}$.

Note that since proper proximality implies non-inner amenability, the above
theorem generalizes Theorem 4.14 in [DTDW19] where the same classification
result is obtained in the context of inner amenability.

Note also that from Theorem 1.7, we immediately deduce:

######  Corollary 1.8.

Arbitrary finite graph products of properly proximal groups are properly
proximal.

Rigidity properties of graph products of groups has gathered considerable
interest recently. For instance, [CdSS18] shows primeness results for the
group von Neumann algebras associated to many graph products of groups. Also,
[Cas20] shows strong solidity and/or absence of Cartan for classes of Hecke
von Neumann algebras associated to some graph products. As a consequence of
Theorem 1.7, we obtain several new examples of groups satisfying the
conclusion of Theorem 1.1.

Moreover, if one has a combination of weak amenability and proper proximality,
we can deduce absence of Cartan subalgebras and $\mathcal{C}$-rigidity in the
sense of [PV14]111By $\mathcal{C}$-rigidity for $G$, we mean
$L^{\infty}(X,\mu)\rtimes G$ admits a unique Cartan subalgebra up to unitary
conjugation, for any free ergodic pmp action
$\sigma:G\curvearrowright(X,\mu)$. See [PV14] for more details. (see [BIP18,
Theorem 1.5]). In [Rec17] Reckwerdt shows that graph products of weakly
amenable groups with Cowling-Haagerup constant $1$ continue to be weakly
amenable. Therefore we obtain:

######  Corollary 1.9.

Let $\Gamma(G)\in\mathcal{G}_{pp}$ be as in Theorem 1.7 and further assume
that $G_{v}$ is weakly amenable for all $v\in V(\Gamma)$ with Cowling-Haagerup
constant $1$. Then $L(\Gamma(G))$ admits no Cartan subalgebras. Moreover,
$\Gamma(G)$ is $\mathcal{C}$-rigid.

We ask the following general question in the context of our result above:

######  Question 1.10.

Which graph products of groups do not admit Cartan subalgebras in their group
von Neumann algebras?

### Acknowledgements:

The authors are very grateful to their advisor J. Peterson his encouragement
and several helpful comments. They also thank D. Osin for helpful
conversations. Thanks are due to R. Boutonnet, I. Chifan and A. Ioana for
taking a look at our preprint and offering valuable advice and feedback.

##  2\. Preliminaries

###  2.1. Graph products of groups

Throughout this paper we denote by $\Gamma$ a non-empty finite simple (every
edge connects two distinct vertices) undirected graph, with vertex set
$V(\Gamma)$ and edge set $E(\Gamma)$. For $v\in V(\Gamma)$, its link is
$S_{v}=\\{w\in V(\Gamma)\mid(v,w)\in E(\Gamma)\\}$. For any two vertices
$v,w\in V(\Gamma)$, let $d(v,w)\in\mathbb{N}\cup\\{\infty\\}$ be the length of
the shortest path between $v$ and $w$. For any subset $T\subset V(\Gamma)$,
denote the induced subgraph on $T$ by $\Gamma_{T}$.

The _radius_ of a graph $\Gamma$ is given by

$r(\Gamma)=\inf_{v\in V(\Gamma)}\sup_{w\in V(\Gamma)}d(u,v).$

And a graph $\Gamma$ is called _irreducible_ if the complement graph
$\Gamma^{c}$ (i.e, the graph consisting of the same vertices, and $(u,v)\in
E(\Gamma^{c})$ if and only if $(u,v)\notin E(\Gamma)$) is connected

Given a $\Gamma$ be a graph, and $\\{G_{v}\\}_{v\in V(\Gamma)}$ a family of
countable groups labeled by the vertex set of $\Gamma$, the _graph product_
denoted by $G=\Gamma(G)$ is the quotient group of the free product $*_{v\in
V(\Gamma)}G_{v}$, with relations $[g,h]=1$ for all $g\in G_{u}$ and $h\in
G_{v}$ with $(u,v)\in E(\Gamma)$.

######  Definition 2.1.

Consider an amalgamated free product group $G_{1}*_{H}G_{2}$. Choose $T_{i}$ a
transversal for the cosets $\\{Hx:x\in G_{i}\\}$. A normal word is a word
$g=ht_{1}\ldots t_{k}$ where $h\in H$, $k\geq 0$ and $t_{j}\in
T_{i_{j}}\setminus\\{1\\}$ for some $i_{j}\in\\{1,2\\}$ and $i_{j}\neq
i_{j+1}$ for $1\leq j\leq k-1$.

In amalgamated free products, it is well-known that every element can also be
represented by a unique normal word (see [BH99, Theorem 3.7]). As is the case
for free products, elements in graph products also admit normal forms.

######  Definition 2.2 (Definition 3.5 in [Gre90]).

Let $G=\Gamma\\{G_{v}\\}_{v\in V(\Gamma)}$ be a graph product of groups. A
word $g_{1}g_{2},\ldots g_{n}\in G$ is said to be _reduced_ if the following
hold:

  1. (1)

$g_{i}\in G_{v_{i}}$ for all $i\in\\{1,2,\ldots n\\}$, where $v_{i}\in
V(\Gamma)$.

  2. (2)

$g_{i}\neq 1$ for all $i\in\\{1,2,\ldots,n\\}$.

  3. (3)

$\forall i\leq k<j$ such that

$[g_{i},g_{i+1}]=[g_{i},g_{i+2}]=\cdots=[g_{i},g_{k}]=1,$

and

$[g_{k+1},g_{j}]=[g_{k+2},g_{j}]=\cdots=[g_{j-1},g_{j}]=1,$

then $v_{i}\neq v_{j}$.

######  Theorem 2.3 (Theorem 3.9 in [Gre90]).

Let $G=\Gamma\\{G_{v}\\}_{v\in V(\Gamma)}$ be a graph products of groups.
Then, each non-trivial element $g\in G$ can be uniquely (up to commuting
segments) expressed as a product:

$g=g_{1}\ldots g_{n}$

where $g_{1}\ldots g_{n}$ is a reduced word.

Graph products decompose naturally as amalgamated free products. We record
this below in the following well-known lemma:

######  Lemma 2.4.

Let $\Gamma(G)$ be a graph product and let $v\in V(\Gamma)$. Then, $G$ is
cannonically isomorphic to $G_{1}*_{H}G_{2}$ where
$G_{1}=\Gamma_{S_{v}\cup\\{v\\}}(G)$,
$G_{2}=\Gamma_{V(\Gamma)\setminus\\{v\\}}(G)$ and $H=\Gamma_{S_{v}}(G)$.

###  2.2. Proper proximality

As stated in the introduction, the family of properly proximal groups is a
robust family including the following classes of groups:

  1. (1)

Non-amenable bi-exact groups.

  2. (2)

Non-elementary convergence groups.

  3. (3)

Lattices in non-compact semi-simple Lie groups.

  4. (4)

Groups admitting a proper 1-cocycle into a non-amenable representation.

  5. (5)

Groups acting properly non-elementarily by isometries on proper CAT(0) spaces.

  6. (6)

Non-elementary mapping class groups.

  7. (7)

Groups measure equivalent to any of the above.

Items (1) to (4) are results of [BIP18], items (5) and (6) are results of
[HHL20], and item (7) is due to [IPR19].

We say a sequence $\\{g_{n}\\}_{n\in\mathbb{N}}\in G$ goes to infinity
_relative_ to a countable family of subgroups $\\{H_{i}\\}_{i\in I}$, denoted
by $g_{n}\to\infty/\\{H_{i}\\}_{i\in I}$, if for any $t_{1},\ t_{2}\in G$ and
any $i\in I$, there exists $N\in\mathbb{N}$ such that for all $n\geq N$,
$g_{n}\notin t_{1}H_{i}t_{2}$.

Consider the $C^{*}$-subalgebra $c_{0}(G,\mathcal{S})\subset\ell^{\infty}(G)$
consisting of functions $f$ such that for all $g_{n}\to\infty/\mathcal{S}$,
$f(g_{n})\to 0$, where $\mathcal{S}=\\{H_{i}\\}_{i\in I}$. It contains the
ideal $c_{0}(G)$ consisting of functions vanish at infinity. Observe also that
$c_{0}(G,\mathcal{S})$ is globally left and right $G$ invariant. Therefore
$\ell^{\infty}(G)/c_{0}(G,\mathcal{S})$ is isomorphic to $C(X_{\mathcal{S}})$,
where $X_{\mathcal{S}}$ is a closed left and right invariant subset of the
Stone-Cech boundary $\Delta G\setminus G$ and is called the _boundary piece_
associated to a collection of subgroups $\mathcal{S}$.

######  Definition 2.5 (Theorem 4.3, [BIP18]).

Given a group $G$ and a family of subgroups $\mathcal{S}$, we say $G$ is
properly proximal relative to $\mathcal{S}$ if one of the following equivalent
conditions holds:

  1. (1)

There is an action $G\curvearrowright K$ by homeomorphisms on a compact
Hausdorff space $K$ such that there is no $G$-invariant probability measure on
$K$ and there exists a probability measure $\eta\in\mathrm{Prob}(K)$ with
$X_{\mathcal{S}}=\partial_{\eta}G:=\\{\omega\in\Delta G\setminus
G\mid\lim_{g\to\omega}^{wk^{*}}g\cdot\eta-(gh)\cdot\eta=0\mathrm{\ for\ any\
}h\in G\\}$.

  2. (2)

There are actions by homeomorphisms $G\curvearrowright K_{i}$, $i=1,\cdots,k$
on compact Hausdorff spaces $K_{i}$ such that there is no $G$-invariant
measure on any $K_{i}$ and there exists a probability measure
$\eta_{i}\in\mathrm{Prob}(K_{i})$ with
$X_{\mathcal{S}}\subset\cup_{i=1}^{k}\partial_{\eta_{i}}G$.

  3. (3)

There is no left-$G$ invariant state on
$\left(\ell^{\infty}(G)/c_{0}(G,\mathcal{S})\right)^{G_{r}}$ (i.e, the right
$G$ invariant subspace of $\ell^{\infty}(G)/c_{0}(G,\mathcal{S})$).

  4. (4)

There is no left-$G$ invariant state on
$\left(\left(\ell^{\infty}(G)/c_{0}(G,\mathcal{S})\right)^{**}\right)^{G_{r}}$.

The group $G$ is properly proximal if it is properly proximal relative to the
trivial subgroup.

Note that, from the above definition/theorem, if one has a finite collection
of subgroups $\\{H_{i}\\}_{i=1}^{n}$ of $G$, such that $G$ is properly
proximal relative to $\\{H_{i}\\}$ for each $i=1,\cdots,n$ where
$\cup_{i=1}^{n}{X_{H_{i}}}=\Delta G\setminus G$, then $G$ is properly
proximal.

###  2.3. Groups acting on trees

We say that an action of $G$ on a set $X$ is non-elementary if it doesn’t
preserve any set of cardinality at most 2.

Given a simplicial tree $T$, let $\Delta T$ denote the compactification
introduced in [Bow12] (see [BO08, $\mathsection\mathsection$ 5.2] for
details). The compactification $\Delta T$ is defined to be $V(T)\sqcup\partial
T$ as a set, where $\partial T$ is the Gromov boundary. For each $x\in\Delta
T$ and each finite set $F\subset V(T)$, set $U(x,F)=\\{y\in\Delta
T\mid[x,y]\cap F=\emptyset\\}\cup\\{x\\}$ where $[x,y]$ denotes the unique
geodesic path between $x$ and $y$. Then $\\{U(x,F)\mid F\subset V(T)\
\mathrm{finite}\\}_{x\in\Delta T}$ forms a basis for the _Bowditch topology_
on $\Delta T$. It is well known that $\Delta T$ equipped with this topology is
compact and Hausdorff and any isometric action $G\curvearrowright T$ extends
to an action by homeomorphisms on $\Delta T$.

If $G$ is an amalgamated free product or HNN-extension, then we have an action
by homeomorphisms of $G\curvearrowright\Delta T$, where $T$ is the Bass-Serre
tree associated to $G$ (see [Ser80]). Also note that if $T$ is countable, then
the topology on $\Delta T$ is second countable hence metrizable as the basis
may be taken as $\\{U(x,F)\mid F\subset V(T)\ \mathrm{finite}\\}_{x\in V(T)}$.

##  3\. Proofs of main theorems

###  3.1. Proof of Theorem 1.2

The proof of Theorem 1.2, splits into two steps. The first step is to
establish proper proximality relative to the edge stabilizers, and the second
step is to upgrade this to proper proximality using the malnormality
condition. The way we accomplish step 1 is by obtaining a relative north-south
dynamics for the action on the Bowditch compactification of the tree. For step
2, we use the double dual proximal space
$\left(\left(\ell^{\infty}(G)/c_{0}(G,\mathcal{S})\right)^{**}\right)^{G_{r}}$
as a filler space to create several right invariant projections that are
indexed by left cosets of the edge stabilizers in $\mathcal{S}$. Under the
malnormality assumption, we see that these are orthogonal projections. From
step 1, we see that any $G$-invariant state vanishes on the complement of the
sum of these projections. To conclude, we $G$-equivariantly embed $C(\Delta
T)$ into
$\left(\left(\ell^{\infty}(G)/c_{0}(G,\mathcal{S})\right)^{**}\right)^{G_{r}}$
along these projections. This contradicts the fact that
$G\curvearrowright\Delta T$ is non-elementary.

We begin by showing the relative north-south dynamics result described above.
We remark that a result in this flavor in the setting of graphs of convergence
groups was recently obtained by R. Tomar in Section 4 of [Tom21].

######  Proposition 3.1.

Let $G$ be a group acting on a tree $T$ with $|V(T)|=\infty$ by isometries.
Set $\mathcal{S}=\\{\mathrm{Stab}(e)\mid e\in E(T)\\}$ and suppose
$g_{n}\to\infty/\mathcal{S}$. Then there exists a subsequence $(g_{n_{k}})$
and points $a$, $b$ in $\Delta T$ such that for any $x\in\Delta
T\setminus\\{b\\}$, we have $lim_{k\to\infty}g_{n_{k}}x=a$. Furthermore, if
the action $G\curvearrowright\Delta T$ is non-elementary, then $G$ is properly
proximal relative to $\mathcal{S}$.

###### Proof.

For any such a sequence $\\{g_{n}\\}$, one may pick a vertex $o\in V(T)$ such
that $|\\{g_{n}o\\}|$ and $|\\{g_{n}^{-1}o\\}|$ are infinite. Indeed, for any
two distinct vertices $u$ and $v$, the map $n\mapsto\\{g_{n}\cdot[u,v]\\}$
must be finite-to-one, since $n\mapsto g_{n}e$ is finite-to-one for any edge
$e$ between $u$ and $v$. Thus there exists a subsequence $\\{g_{n_{k}}\\}_{k}$
and distinct vertices $u$ and $v$ such that $k\mapsto g_{n_{k}}u$ and
$k\mapsto g_{n_{k}}v$ are both one-to-one. Since we also have
$g^{-1}_{n_{k}}\to\infty/\mathcal{S}$, the same argument shows that either
$|\\{g_{n_{k}}^{-1}u\\}_{k}|=\infty$ or $|\\{g_{n_{k}}^{-1}v\\}_{k}|=\infty$.

Now pick a subsquence $\\{g_{n_{k}}\\}$ such that $\lim g_{n_{k}}o=a$ and
$\lim g_{n_{k}}^{-1}o=b$.

Case 1: $a\in\Delta T\setminus T$.

Since $d(o,g_{n_{k}}o)=d(g_{n_{k}}^{-1}o,o)$, $b$ is also in $\Delta
T\setminus T$. The argument in this case proceeds similarly to the argument in
[Tuk94]. Let $U_{a}=U(a,\\{e\\})$ and $U_{b}=U(b,\\{f\\})$ be basic open sets
in $\Delta T$, where $e$ and $f$ are edges with endpoints $e_{1},e_{2}$ and
$f_{1},f_{2}$ respectively. For $v\in\Delta T$ denote by $\beta_{v}$ the
geodesic from $o$ to $v$. Denote by $\langle x,y\rangle_{o}$ the Gromov
product, given by the distance between $o$ and the center of the unique
geoedesic tripod formed by $x,y$ and $o$. For all $c\notin U_{b}$, i.e,
$[b,c]\cap[f_{1},f_{2}]\neq\emptyset$, we have

$\liminf_{m,n\to\infty}\langle\beta_{b}(m),\beta_{c}(n)\rangle_{o}\leq\max\\{d(o,f_{1}),d(o,f_{2})\\}=:d_{1}$

Define $V(b;2d_{1})=\\{x\in\Delta T\ |\
\liminf_{m,n\to\infty}\langle\beta_{x}(m),\beta_{b}(n)\rangle_{o}\geq
2d_{1}\\}$. From the above, we have $U_{b}^{c}\subset V(b,2d_{1})^{c}$. Now
let $c\in V(b,2d_{1})$. We have:

$\liminf_{m,n\to\infty}\langle\beta_{a}(m),g_{n_{k}}\beta_{c}(n)\rangle_{o}\geq\min\\{\liminf_{m\to\infty}\langle\beta_{a}(m),g_{n_{k}}o\rangle_{o},\liminf_{n\to\infty}\langle
g_{n_{k}}o,g_{n_{k}}\beta_{c}(n)\rangle_{o}\\}.$

Note that
$\lim_{m\to\infty}\langle\beta_{a}(m),g_{n_{k}}0\rangle_{o}\to\infty$ as
$k\to\infty$; also,

$\langle g_{n_{k}}o,g_{n_{k}}\beta_{c}(n)\rangle_{o}=\langle
o,\beta_{c}(n)\rangle_{g_{n_{k}}^{-1}o}\to\infty$

as $k\to\infty$ since $c\notin V(b,2d_{1})$. Thus $\forall R>0$, there exists
$K_{0}(R)$ such that for all $k\geq K_{0}(R)$, we have
$g_{n_{k}}(V(b,2d_{1}))\subset V(a,R)$.

Now, suppose for all $k>0$, there exists $k_{0}>k$ and $c\in U_{b}^{c}$ such
that $g_{n_{k}}(c)\notin U(a,\\{e\\})$, that is,

$\liminf_{m,n\to\infty}\langle\beta_{a}(n),g_{n_{k}}\beta_{c}(m)\rangle_{o}\leq\min\\{d(o,e_{1}),d(o,e_{2})\\}=:d_{2}.$

Take $R=2d_{2}$. Then $g_{n_{k}}(c)\notin U(a,\\{f\\})$ but $g_{n_{k}}(c)\in
V(a,2d_{2})$ for all $k>K(R)$, which is a contradiction. Hence, we have the
conclusion in the case that $a\in\Delta T\setminus T$.

Case 2: $a\in T$. Let $x\in\Delta T$. We claim that
$\lim_{k\to\infty}g_{n_{k}}(x)=a$.

First suppose $x\in T$. For any open set $U(a,\\{e\\})$, we have the
following:

$|\\{n\in\mathbb{N}:g_{n}[o,x]\ni e\\}|<\infty.$

Indeed, if the above set was infinite, then there would be an infinite set
$I\subset\mathbb{N}$ with $g_{n_{i}}^{-1}e\in[o,x]$ for $i\in I$. Since
$[o,x]$ is a fixed finite length path, this contradicts the fact that
$g_{n_{i}}$ escapes all edge stabilizers. Now, since $g_{n}(o)\to a$, we have
that $\exists N>0$ such that for all $n\geq N$, $g_{n}o\in U(a,\\{e\\})$.
Hence for $n$ sufficiently large, we have that $\\{e\\}$ is not in the segment
$[g_{n}o,g_{n}x]$ and $\\{e\\}$ is not in the segment $[g_{n}o,a]$, hence
$\\{e\\}$ is not in the segment $[g_{n}x,a]$ which gives us what is required.

Suppose $x\in\partial T$. We have that $\\{g_{n}o\\}$ is bounded, and hence
let $d_{0}=\sup_{n}d(a,g_{n}o)$. Given an open set $U(a,\\{e\\})$, let
$d_{1}=d(a,e_{1})+1$, where $d$ is the combinatorial metric on $T$. Let
$N=d_{0}+d_{1}+1$. For any $n\geq N$, we apply the previous argument to the
point $\beta_{x}(n)$ to obtain that $e\notin g_{m}[\alpha(0),\beta_{x}(n)]$
for large $m$. Moreover, due to the bound on $d(a,e)$, we have that
$e\notin[\beta_{x}(n),x]$.

To show relative proper proximality, consider a sequence $\\{g_{n}\\}$ with
$g_{n}\to\infty/\mathcal{S}$. As mentioned in [Bow99, Section 2], if
$G\curvearrowright\Delta T$ is non-elementary then the limit set is perfect,
i.e., there exists a compact Hausdorff perfect set inside $\Delta T$, which
implies the existence of some diffuse measure. Fix a diffuse probability
measure $\eta$ on $\Delta T$, and let $a,b\in\Delta T$ and $\\{g_{n_{k}}\\}$
be as in the above arguments. By [FLM18, Lemma 8.3], for any $h\in G$,
$\lim_{k\to\infty}g_{n_{k}}h\eta=\delta_{a}$ and thus
$X_{\mathcal{S}}=\partial_{\eta}G$. Moreover, there is no $G$-invariant
probability measure on $\Delta T$. Indeed, if there were a $G$-invariant
probability measure $\mu$, then we see from the north-south dynamics that
$\mu(\\{a,b\\})=1$ where $a$ and $b$ are some north and south poles. This
contradicts the fact that the action is non-elementary. ∎

Applying the above proposition to the Bass-Serre tree associated with an
amalgamated free product, we obtain:

######  Corollary 3.2.

Given $G_{1},G_{2}$ and $H$ with $H\leq G_{i}$, $i=1,2$. Then
$G=G_{1}\ast_{H}G_{2}$ is properly proximal relative to $H$ if $[G_{i}:H]\geq
3$ and $[G_{j}:H]\geq 2$, for $\\{i,j\\}=\\{1,2\\}$.

###### Proof.

This follows by considering the standard action on the compactification of the
Bass-Serre tree associated to the amalgamated free product. This action is
non-elementary on $\Delta T$ because the boundary is infinite and orbits of
the action on the boundary are infinite. Moreover, the edge stabilizers are
precisely conjugates of $H$. ∎

The following is a direct proof of the above corollary without involving Bass-
Serre theory. We would like to point out that the technical difference between
non-inner amenability and proper proximality is clearly seen in this proof.

###### Alternative proof of Corollary 3.2.

Fix a choice of transversals $T=\\{e\\}\cup\\{t_{i}\\}_{i\in I}$ for cosets
$\\{Hx:x\in G_{1}\\}$ and $S=\\{e\\}\cup\\{s_{j}\\}_{j\in J}$ for cosets
$\\{Hx:x\in G_{2}\\}$. Denote by $p_{1}$ the characteristic function on the
set of elements $g$ whose normal form begins with $ht_{i}$, where $h\in H$,
$i\in I$; similarly, let $p_{2}$ be characteristic function on the set of
elements $g$ whose normal form begins with $hs_{j}$, where $h\in H$, $j\in J$.

Claim: $p_{i}\in(\ell^{\infty}(G)/c_{0}(G,H))^{G_{r}}$ for $i=1,2$.

Suppose there exists some $g\in G$ and a sequence $g_{n}\to\infty/H$ such that
$\lim_{n\to\infty}p_{1}(g_{n})-p_{1}(g_{n}g)\neq 0$, then we may extract a
subsequence, still denoted by $(g_{n})$, with
$\lim_{n\to\infty}p_{1}(g_{n})-p_{1}(g_{n}g)=1$. Let
$g_{n}=h_{n}t_{1,n}s_{1,n}\dots t_{k_{n},n}s_{k_{n},n}$ be the normal form of
each $g_{n}$, where $h_{n}\in H$, $t_{i,n}\in T\setminus\\{e\\}$ and
$S_{i,n}\in S\setminus\\{e\\}$, with $t_{1,n}$ and $s_{k_{n},n}$ possibly
being $e$. Note that since $g_{n}\to\infty/H$, $\\{h_{n}^{-1}g_{n}\mid
n\in\mathbb{N}\\}$ is a infinite set. Set $g=ht_{1}s_{1}\dots t_{m}s_{m}$ to
be its normal form. As $\lim_{n\to\infty}p_{1}(g_{n})-p_{1}(g_{n}g)=1$, for
large enough $n$ we have $t_{1,n}\in T\setminus\\{e\\}$ while
$g_{n}g\notin\operatorname{supp}(p_{1})$. Note that

$\displaystyle g_{n}g$ $\displaystyle=h_{n}t_{1,n}s_{1,n}\dots
t_{k_{n},n}s_{k_{n},n}ht_{1}s_{1}\dots t_{m}s_{m}$
$\displaystyle=h_{n}h^{\prime}t_{1,n}^{\prime}s_{1,n}^{\prime}\dots
t_{k_{n},n}^{\prime}s_{k_{n},n}^{\prime}t_{1}s_{1}\dots t_{m}s_{m},$

where $t_{i,n}^{\prime}\in T\setminus\\{e\\}$, $s_{i,n}^{\prime}\in S$ are
obtained as we move $h$ to the front. By our assumption,
$g_{n}g\notin\operatorname{supp}(p_{1})$, i.e., $t_{1,n}^{\prime}$ would
disappear after possible cancelations for all large enough $n$; however $g$ is
given while $\\{h_{n}^{-1}g_{n}\mid n\in\mathbb{N}\\}$ is a infinite set, we
arrive at a contradiction. This proves the claim.

Now take $g_{1},g_{2}\in G_{i}\setminus H$ and $g_{3}\in G_{j}\setminus H$,
which is allowed by the assumption that $[G_{i}:H]\geq 3$ and $[G_{j}:H]\geq
2$. Notice that $g_{1}p_{1}+g_{2}p_{1}\leq p_{2}$ while $g_{3}p_{2}\leq p_{1}$
and $p_{1}+p_{2}=1$. If there exists a left $G$-invariant state $\varphi$ on
$(\ell^{\infty}(G)/c_{0}(G,H))^{G_{r}}$, then
$2\varphi(p_{1})\leq\varphi(p_{2})\leq\varphi(p_{1})$, i.e.,
$\varphi(p_{1})=\varphi(p_{2})=0$ while $1=\varphi(p_{1}+p_{2})$, which is a
contradiction. ∎

###### Proof of Theorem 1.2.

Suppose $G$ is not properly proximal. Then there exists a left $G$-invariant
state $\varphi$ on the space
$\left((\ell^{\infty}(G)/c_{0}(G))^{**}\right)^{G_{r}}$. Let
$\\{H_{i}\\}_{i\in I}$ be a family of subgroups of $G$ such that
$\bigsqcup_{i\in I}\\{gH_{i}g^{-1}\mid g\in G\\}=\\{{\rm stab}(e)\mid e\in
E(t)\\}$. Set $p_{i}=\bigvee_{t,s\in G}1_{tH_{i}s}$ to be a projection in
$\left((\ell^{\infty}(G)/c_{0}(G))^{**}\right)^{G_{r}}$. From Proposition 3.1,
we obtain that $\varphi(p)=1$, where $p=\bigvee_{i\in I}p_{i}$. Indeed, we
have that $G$ is properly proximal relative to $\mathcal{S}=\\{{\rm
stab}(e)\mid e\in E(t)\\}$ and so there is no left invariant state on
$\left(\left(\ell^{\infty}(G)/c_{0}(G,\mathcal{S})\right)^{**}\right)^{G_{r}}$.
But observe that
$\left(\left(\ell^{\infty}(G)/c_{0}(G,\mathcal{S})\right)^{**}\right)^{G_{r}}=(1-p)\left(\ell^{\infty}(G)/c_{0}(G)\right)^{G_{r}}$,
and hence $\varphi(1-p)=0$. For $t\in G$ and $i\in I$, put
$p_{t,i}=\bigvee_{s\in
G}1_{tH_{i}s}\in\left((\ell^{\infty}(G)/c_{0}(G))^{**}\right)^{G_{r}}$ and we
have $\bigvee_{t\in G}p_{t,i}=p_{i}$. Let $\\{F_{n}\\}$ be a sequence of
increasing finite subsets such that $\cup_{n}F_{n}=G$. For each $t\in G$,
consider projections
$p_{t,i,n}=1_{tH_{i}F_{n}}\in(\ell^{\infty}(G)/c_{0}(G))^{**}$ and then
$\lim_{n\to\infty}p_{t,i,n}=p_{t,i}$ in SOT.

Note that $|tH_{i}s\cap H_{j}|<\infty$ for any $i,j\in I$ (not necessarily
distinct), $s\in G$ and $t\in G$ such that $tH_{i}t^{-1}\neq H_{j}$; indeed,
suppose there exists some $ths\in tH_{i}s\cap H_{j}$. Then for any
$th^{\prime}s\in tH_{i}s\cap H_{j}$, we have
$ths(th^{\prime}s)^{-1}=thh^{\prime-1}t^{-1}\in tH_{i}t^{-1}\cap H_{j}$, which
is finite as for any pair of distinct edges $e,f\in E(T)$,
$\mathrm{Stab}(e)\cap\mathrm{Stab}(f)$ is finite. Therefore we have
$p_{t,i,n}p_{s,j,n}=0$ for all $t,s\in G$ such that $tH_{i}t^{-1}\neq
sH_{j}s^{-1}$. Since the $(p_{t,i,n})$ and $(p_{s,j,n})$ are bounded,
$\lim_{n\to\infty}p_{t,i,n}p_{s,j,n}=p_{t,i}p_{s,j}$, i.e.,
$p_{t,i}p_{s,j}=0$.

Consider a $G$-equivariant unital embedding
$\psi:\ell^{\infty}(V(T))\ni\delta_{v}\mapsto\sum_{e\in
E_{v}}\delta_{e}/2\in\ell^{\infty}(E(T))$, where $E_{v}=\\{e\in E(T)\mid v\in
e\\}$. Given $e\in E(T)$, there exists a unique $i\in I$ and some $t\in G$
such that ${\rm stab}(e)=tH_{i}t^{-1}$. Thus $p_{t,i}$ is a unique projection
corresponding to $e$ and we denote it by $p(e)$. Therefore we obtain an
embedding
$\iota:\ell^{\infty}(E(T))\to\left((\ell^{\infty}(G)/c_{0}(G))^{**}\right)^{G_{r}}$
given by $\iota(\delta_{e})=p(e)$ and it is easy to check this embedding is
unital and $G$-equivariant. Finally, as $C(\Delta
T)\subset\ell^{\infty}(V(T))$, we have a $G$-invariant state
$\varphi\circ\iota\circ\psi$ on $C(\Delta T)$, which contradicts the
assumption that $G\curvearrowright\Delta T$ is non-elementary. ∎

###  3.2. Proof of Theorem 1.7

The main idea in proving Theorem 1.7 is to consider various amalgamated
product decompositions in a graph product, and obtain relative proper
proximality relative to each of these amalgams. Then, we show that the Stone-
Cech boundary of the graph product is filled by each of these boundary pieces
coming from the amalgams, thereby showing proper proximality. The other key
step is to show that if a product of groups is properly proximal, then each of
the groups has to be properly proximal. This is obtained by establishing a
natural isomorphism of the proximal spaces, at the level of the double dual.
The classification result is then obtained by a careful analysis of some
cases, given by the radius of the graphs.

Given two subgraphs $\Gamma_{1},\Gamma_{2}$ of a graph $\Gamma$, denote by
$\Gamma_{1}\cap\Gamma_{2}$ the subgraph of $\Gamma$ generated by the vertex
set $V(\Gamma_{1})\cap V(\Gamma_{2})$. As a convention, $\Gamma(G)$ is set to
be the trivial group if $V(\Gamma)=\emptyset$.

######  Lemma 3.3.

Let $\Gamma(G)$ be a graph product group with graph $\Gamma$ and generating
groups $\\{G_{v}\mid v\in V(\Gamma)\\}$. For any subgraphs $\Gamma_{1}$,
$\Gamma_{2}$ of $\Gamma$ and $g,h\in\Gamma(G)$, we have $\Gamma_{1}(G)\cap
g\Gamma_{2}(G)h\subset\cup_{i=1}^{n}c_{i}(\Gamma_{1}\cap\Gamma_{2})(G)d_{i}$,
for some finite subset $F=\\{c_{i},d_{i}\mid 1\leq i\leq
n\\}\subset\Gamma(G)$.

###### Proof.

Given $g,h\in\Gamma(G)$ with $g=g_{1}\cdots g_{k}$ and $h=h_{1}\cdots h_{m}$
as their reduced words respectively, let $F=\\{c_{i},d_{i}\\}$ be the finite
set consisting of words of the form $c_{i}=\prod_{j}g_{i_{j}}$ and
$d_{i}=\prod_{j}h_{i_{k}}$, where $(i_{j})$ and $(i_{k})$ are increasing. We
claim that $F$ is the required set. It suffices to show every $i$, we have
$\Gamma_{1}(G)\cap
c_{i}\Gamma_{2}(G)d_{i}\in\cup_{j=1}^{n}c_{j}(\Gamma_{1}\cap\Gamma_{2})(G)d_{j}$.
We show this by induction on $n(t)=|\\{t_{j}\mid 1\leq j\leq l,\ t_{j}\in
G_{v}\mathrm{\ for\ some\ }v\notin\Gamma_{1}\cap\Gamma_{2}\\}|$, where
$t=t_{1}\cdots t_{l}$ is a reduced word in $\Gamma_{2}(G)$ such that
$gth\in\Gamma_{1}(G)$. The claim is true for $n=0$, as $c_{i}td_{i}\in
c_{i}(\Gamma_{1}\cap\Gamma_{2})(G)d_{i}$.

Suppose the claim is true for $n\leq K$ and let $n=K+1$. As
$n=|\bigcup_{v\notin V(\Gamma_{1}\cap\Gamma_{2})}\\{t_{j}\mid 1\leq j\leq
l,t_{j}\in G_{v}\\}\big{|}$, we may assume $|\\{t_{j}\mid 1\leq j\leq
l,t_{j}\in G_{v_{0}}\\}|\geq 1$ for some $v_{0}\notin
V(\Gamma_{1}\cap\Gamma_{2})$. For a word $w$ define $\hat{v}_{0}(w)$ to be the
ordered set of letters in $w$ (ordered by the left to right order in the word
$w$) that do not belong to $G_{v_{0}}$. For an ordered set of letters $u$
denote by $u_{\Gamma(G)}$ the product of letters of $u$ in $\Gamma(G)$. Denote
by $w^{*}$ the reduced word of $c_{i}td_{i}$ and note
$\hat{v}_{0}(w^{*})_{\Gamma(G)}=w^{*}$ as $c_{i}td_{i}\in\Gamma_{1}(G)$.

Claim: $\hat{v}_{0}(c_{i}td_{i})_{\Gamma(G)}={c_{i}td_{i}}_{\Gamma(G)}=w^{*}$.

Denote the word $c_{i}td_{i}$ by $p_{1}p_{2}\cdots p_{n}$ and suppose
$p_{i_{1}},\dots,p_{i_{d}}$ are letters that belong to $G_{v_{0}}$. Define
words $w_{1}=p_{1}\cdots p_{i_{1}-1}$, $w_{j}=p_{i_{j-1}+1}\ldots p_{i_{j}-1}$
for $1<j\leq d$ and $w_{d+1}=p_{i_{d}+1}\ldots p_{n}$. Upon individually
reducing these words, we obtain the words $w_{i}^{*}$. Now, we have
${c_{i}td_{i}}_{\Gamma(G)}={w^{*}_{1}p_{i_{1}}w^{*}_{2}p_{i_{2}}\ldots
p_{i_{d}}w^{*}_{d+1}}_{\Gamma(G)}$. We may assume that
${w^{*}_{j}}_{\Gamma(G)}$ does not commute with $G_{v_{0}}$ for some $2\leq
j\leq d$. Then, ${w^{*}_{j}}_{\Gamma(G)}$ neither commute with
$a={w^{*}_{1}p_{i_{1}}\ldots p_{i_{j-1}}}_{\Gamma(G)}$ nor with
$b={p_{i_{j}}w^{*}_{j+1}\ldots w^{*}_{d+1}}_{\Gamma(G)}$. It follows that
$p_{j}p_{j+1}\ldots p_{d}=1$ and $p_{1}\ldots p_{j-1}=1$. Suppose otherwise
and then ${c_{i}td_{i}}_{\Gamma(G)}={aw^{*}_{j}b}_{\Gamma(G)}$ while
$a,b\notin\ker(\pi_{v_{0}})$, where $\pi_{v_{0}}$ is the canonical surjection
from $\Gamma(G)\to G_{v_{0}}$ . Since $w^{*}_{j}$ doesn’t commute with
$G_{v_{0}}$ we see ${c_{i}td_{i}}_{\Gamma(G)}\notin\Gamma_{1}(G)$ which is a
contradiction. We proceed recursively to conclude the claim.

Now it suffices to show
$\hat{v}_{0}(c_{i}td_{i})\in\cup_{j=1}^{n}c_{j}(\Gamma_{1}\cap\Gamma_{2})(G)d_{j}$,
and this follows from the inductive hypothesis as
$\hat{v_{0}}(c_{i}td_{i})=c_{i^{\prime}}\hat{v_{0}}(t)d_{i^{\prime}}$ for some
other $i^{\prime}$, and $n(\hat{v_{0}}(t))<n(t)=K+1$. ∎

######  Lemma 3.4.

Let $\Gamma$ be a graph. Then,

$\bigcup_{v\in V(\Gamma)}X_{\Gamma_{S_{v}}(G)}=\Delta\Gamma(G).$

###### Proof.

Set $V(\Gamma)=\\{v_{i}\mid i=1,\dots,n\\}$ and
$\Gamma_{i}=\Gamma_{S_{V_{i}}}$ for each $i$. Note that this statement is
equivalent to

$\bigcap_{i=1}^{n}c_{0}(\Gamma(G),\Gamma_{i}(G))=c_{0}(\Gamma(G))$

and thus it suffices to show
$|\bigcap_{i=1}^{n}s_{i}\Gamma_{i}(G)t_{i}|<\infty$ for any
$s_{i},t_{i}\in\Gamma(G)$.

By Lemma 3.3,
$\bigcap_{i=1}^{n}s_{i}\Gamma_{i}(G)t_{i}\subset\bigcup_{j=1}^{m}g_{j}(\bigcap_{i=1}^{n}\Gamma_{i})(G)h_{j}$
for some finite subset $\\{g_{j},h_{j}\mid 1\leq j\leq m\\}$ of $\Gamma(G)$.
Note that $\cap_{i=1}^{n}S_{V_{i}}=\emptyset$ and thus
$(\bigcap_{i=1}^{n}\Gamma_{i})(G)=\\{e\\}$, i.e.,
$\bigcap_{i=1}^{n}s_{i}\Gamma_{i}(G)t_{i}$ is finite. ∎

######  Proposition 3.5.

Let $\Gamma$ be a finite graph. The following are equivalent:

  1. (1)

$\Gamma$ satisfies the following condition: $r(\Gamma)$ is at least 2 and
there does not exist a pair of vertices $v_{1},v_{2}\in V(\Gamma)$ such that
$(u,v_{i})\in E(\Gamma)$ for all $u\in V(\Gamma)\setminus\\{v_{1},v_{2}\\}$,
and $i=1,2$.

  2. (2)

For all choices of non-trivial groups $G_{v}$, $v\in V(\Gamma)$, the graph
product $\Gamma(G)$ is properly proximal.

###### Proof of Proposition 3.5.

First, we show $(2)\implies(1)$. Suppose there is a radius 2 graph such that
there is a pair of vertices $v_{1},v_{2}$ satisfying $(u,v_{i})\in E(\Gamma)$
for all $u\in V(\Gamma)\setminus\\{v_{1},v_{2}\\}$, and $i=1,2$, then one can
simply consider $G_{v_{1}}=\mathbb{Z}/2\mathbb{Z}=G_{v_{2}}$. Since
$(v_{1},v_{2})\notin E(\Gamma)$, we see that
$\Gamma(G)=\mathbb{Z}_{2}*\mathbb{Z}_{2}\times\Gamma_{V(\Gamma)\setminus\\{v_{1},v_{2}\\}}(G)$
is inner amenable, and therefore not properly proximal by Proposition 4.11 in
[BIP18].

For $(1)\implies(2)$, observe that by Lemma 2.4, we have a natural amalgamated
free product decomposition: $\Gamma(G)\cong G_{1}*_{H}G_{2}$, using the same
notation. Since $\Gamma$ satisfies (1), we have
$\text{max}\\{[G_{1}:H],[G_{2}:H]\\}=\infty$ and $H$ is neither $G_{1}$ nor
$G_{2}$. Then it follows immediately from Lemma 3.4 and Corollary 3.2. ∎

It follows directly that for an arbitrary choice of non-trivial groups indexed
by vertices, $\Gamma(G)$ is properly proximal, provided that $\Gamma$ is an
irreducible graph or a graph satisfying $r(\Gamma)\geq 3$.

######  Proposition 3.6.

Let $\Gamma$ be such that $r(\Gamma)\geq 2$. Then $\Gamma(G)$ is either
properly proximal or contains an infinite amenable summand.

###### Proof.

The only case to check is if $\Gamma$ has radius 2 and it doesn’t satisfy
condition (1) of Proposition 3.5. That is, there exists vertices $v_{1},v_{2}$
such that $S=V(\Gamma)\setminus\\{v_{1},v_{2}\\}$ is connected to $v_{1}$ and
$v_{2}$. Then since $v_{1}$ is not connected to $v_{2}$ (else it violates the
radius 2 condition), we have
$\Gamma(G)\cong\left(G_{v_{1}}*G_{v_{2}}\right)\times\Gamma_{S}(G)$. The left
summand is either infinite amenable ($\mathbb{Z}_{2}*\mathbb{Z}_{2}$) or
properly proximal (since they are non-trivial free products). The above
decomposition can be iterated on $S$ and further on, using the hypothesis. The
result follows from the fact that properly proximal groups cannot have
infinite amenable summands ([BIP18, Proposition 4.11]). ∎

We thank J. Peterson for suggesting the proof of the following proposition. A
generalization of the result below will appear in [DKEP21]. The converse
already appears as Proposition 4.10 as in [BIP18].

######  Proposition 3.7.

If $G=G_{1}\times G_{2}$ is properly proximal, then both $G_{1}$ and $G_{2}$
are properly proximal.

###### Proof.

Define the projection $p=\bigvee_{g\in
G_{2}}1_{G_{1}\times\\{g\\}}\in\left(\ell^{\infty}(G)/c_{0}(G)\right)^{**}$.
We claim the following isomorphism holds:

$\bigoplus_{G_{2}}^{\infty}(\ell^{\infty}G_{1}/c_{0}(G_{1}))^{**}\cong
p(\ell^{\infty}G/c_{0}(G))^{**},$

where the direct sum is equipped with $\ell^{\infty}$-norm. Indeed, for any
finite subset $F\subset G_{2}$, consider the isomorphism

$\Theta_{F}:\bigoplus_{F}\ell^{\infty}G_{1}\ni(f_{h})_{h\in
F}\mapsto\tilde{f}_{F}\in 1_{G_{1}\times F}\ell^{\infty}G,$

where $\tilde{f}_{F}(g,h)=f_{h}(g)$ for $g\in G_{1},$ $h\in F$. Composing with
the quotient map, one has an isomorphism
$\Theta_{F}:\bigoplus_{F}(\ell^{\infty}G_{1}/c_{0}(G_{1}))\to 1_{G_{1}\times
F}(\ell^{\infty}G/c_{0}(G))$. Furthermore, we may extend this map to a weak∗
continuous isomorphism
$\tilde{\Theta}_{F}:\bigoplus_{F}(\ell^{\infty}G_{1}/c_{0}(G_{1}))^{**}\to
1_{G_{1}\times F}(\ell^{\infty}G/c_{0}(G))^{**}$.

For each $(f_{h})_{h\in
G_{2}}\in\bigoplus_{G_{2}}(\ell^{\infty}G_{1}/c_{0}(G_{1}))^{**}$, define
$\tilde{\Theta}((f_{h}))$ to be the SOT limit point of the net
$\\{\tilde{\Theta}_{F}((f_{h})_{h\in F})\\}_{F}$ in
$p(\ell^{\infty}G/c_{0}(G))^{**}$, where $F$ ranges over all finite subsets of
$G_{2}$; conversely, for $pg\in p(\ell^{\infty}G/c_{0}(G))^{**}$, set
$\tilde{\Theta}^{-1}(pg)$ to be the weak∗ limit of
$\\{\tilde{\Theta}_{F}^{-1}(1_{G_{1}\times F}g)\\}_{F}$ in
$\bigoplus_{G_{2}}(\ell^{\infty}G_{1}/c_{0}(G_{1}))^{**}$. Thus
$\tilde{\Theta}$ implements the required isomorphism, which is also
$G$-equivariant by construction.

Now by taking the $G$-right invariant subspaces in both sides, we have

$\big{(}(\ell^{\infty}G_{1}/c_{0}(G_{1}))^{**}\big{)}^{{G_{1}}_{r}}=\left(\bigoplus_{G_{2}}^{\infty}(\ell^{\infty}G_{1}/c_{0}(G))^{**}\right)^{G_{r}}\cong
p\left((\ell^{\infty}G/c_{0}(G))^{**}\right)^{G_{r}}.$

Now suppose there is a left $G_{1}$-invariant state $\varphi$ on
$\big{(}(\ell^{\infty}G_{1}/c_{0}(G_{1}))^{**}\big{)}^{{G_{1}}_{r}}$, then
$\varphi\circ\tilde{\Theta}^{-1}$ is a left $G$-invariant state on
$p\left((\ell^{\infty}G/c_{0}(G))^{**}\right)^{G_{r}}$. Set
$\psi(f)=\varphi\circ\tilde{\Theta}^{-1}(pf)$ for
$f\in\left((\ell^{\infty}G/c_{0}(G))^{**}\right)^{G_{r}}$ and we obtain a
$G$-invariant state, which contradicts the proper proximality of $G$. ∎

###### Proof of Theorem 1.7.

Suppose $\Gamma$ satisfies condition (1) in Proposition 3.5, then clearly
$\Gamma(G)\in\mathcal{G}_{pp}$ if and only if $\Gamma(G)$ is properly
proximal. If $\Gamma$ is of radius 2, and it doesn’t satisfy condition (1) of
Theorem 3.5, then $\Gamma(G)$ can be decomposed as
$\Gamma_{\\{v_{1},v_{2}\\}}(G)\times\Gamma_{\\{v_{1},v_{2}\\}^{c}}(G)$. Hence
by Proposition 3.7, it follows that $\Gamma(G)$ is properly proximal if and
only if $|G_{v_{i}}|\geq 3$ for some $i\in\\{1,2\\}$, and
$\Gamma_{\\{v_{1},v_{2}\\}^{c}}(G)\in\mathcal{G}_{pp}$. Finally suppose
$\Gamma$ has radius 1. Then the group decomposes as
$\Gamma(G)=\Gamma_{v}(G)\times\Gamma_{\\{v\\}^{c}}(G)$, whence we apply the
same argument as before. ∎

######  Remark 3.8.

In [HHL20] it is shown that proper CAT(0) cubical complex groups are properly
proximal. Graph products admit natural actions on CAT(0) cube complexes due to
[Mei96] and [Dav98], however these actions are in general, not proper. Thus
Theorem 1.7 cannot be deduced directly from [HHL20].

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