# What data do we need for training an AV motion planner?

Long Chen∗, Lukas Platinsky∗, Stefanie Speichert∗, Błażej Osiński,
Oliver Scheel, Yawei Ye, Hugo Grimmett, Luca del Pero, and Peter Ondruska
∗Equal contributionAuthors are with Lyft Level 5 self-driving
division.Stefanie Speichert is also affiliated with the University of
Edinburgh.Project website: https://planning.l5kit.org

###### Abstract

We investigate what grade of sensor data is required for training an
imitation-learning-based AV planner on human expert demonstration. Machine-
learned planners [1] are very hungry for training data, which is usually
collected using vehicles equipped with the same sensors used for autonomous
operation [1]. This is costly and non-scalable. If cheaper sensors could be
used for collection instead, data availability would go up, which is crucial
in a field where data volume requirements are large and availability is small.
We present experiments using up to 1000 hours worth of expert demonstration
and find that training with 10x lower-quality data outperforms 1x AV-grade
data in terms of planner performance (see Fig. 1). The important implication
of this is that cheaper sensors can indeed be used. This serves to improve
data access and democratize the field of imitation-based motion planning.
Alongside this, we perform a sensitivity analysis of planner performance as a
function of perception range, field-of-view, accuracy, and data volume, and
reason about why lower-quality data still provide good planning results.

##  I INTRODUCTION

While human drivers can skillfully navigate complex and varied scenarios
involving multiple traffic agents safely, motion planning remains one of the
hardest problems in autonomous driving. This has motivated the recent interest
in planning approaches for Autonomous Vehicles (AVs) based on imitation-
learning [1, 2]. These methods learn to mimic human behaviour from real-world
human driving examples.

These approaches are extremely data-hungry, due to the complexity of the
traffic scenarios and their variety: the so-called _long tail_ of rare events.
Some methods attempt to reduce the need for large volumes of human examples by
employing data augmentation techniques, or by synthesizing rare cases in
simulation [3]. However, just as the advancements in object detection and
classification required large, real-world data sets (e.g., [4, 5]), we propose
that motion planning also requires _a large corpus of real human driving
data_. The problem is that these data are not readily available, and capturing
it typically requires a fleet of vehicles equipped with expensive AV-grade
sensors, the same that the AV needs to operate in autonomous mode (so that the
planner can use the same perception system both at train and test time [1]).
This significant barrier to entry stifles progress in the field.

We ask ourselves: do we really need a fleet with AV-grade sensors to collect
training data for an AV motion planner? If vehicles equipped with commodity
sensors were sufficient, data would be more readily available and this
research problem could be democratized.

Figure 1: Large data sets of expert driving demonstrations for training motion
planners need not come from expensive fleets of AVs. We evaluate motion
planner error rate (lower is better), and observe that a planner trained on
expert demonstrations collected using AVs (AV-grade, left) has inferior
performance to one trained on larger volumes of lower-quality data (middle).
Combining the two data sources further improves performance (right), as this
bridges the domain gap between the sensors used for training and those used by
the AV at test time. This opens the door to crowd-sourced data collection
using more affordable sensor configurations than those on AVs.

In this paper, we investigate the data properties we need to train an AV
planner for urban operation, by comparing the performance of a state-of-the-
art machine-learned planner as we train it on expert demonstration data with
varying levels of quality. Modern planning approaches ([1, 6]) take as input
the output of a vehicle’s perception system (_perception output_) containing
the 3D positions of other traffic agents. The supervision is provided by the
human driving the vehicle. By taking AV-grade perception output and
progressively limiting its key functional properties, like range and field-of-
view (FoV), we simulate the lower-quality data that we could get from a
variety of commodity sensors. Our experiments with this data inform what
quality we need (and hence which sensors we need), and help us understand the
trade-offs between data quantity and quality.

Our key contributions are as follows:

  * •

We show that training on 10x more data with lower-quality improves on 1x AV-
grade data (see Fig.1). This has an important implication: we can collect
expert demonstration data for training using vehicles equipped with sensors
that are much cheaper than the AV-grade. This opens the door to scalable,
cost-effective ways to collect large volumes of human driving examples for
teaching AV systems how to drive.

  * •

We present a sensitivity analysis showing which aspects of data quality are
most important for planner performance and why quantity is more important than
quality. We derive this from an extensive quantitative evaluation supported by
attention-based analysis.

##  II RELATED WORK

In this paper we build on recent advances in modular machine-learned planning
systems.

End-to-end vs modular approaches. Current motion planning methods can be
broadly categorized into end-to-end and modular (also known as mid-to-mid).
End-to-end methods consume raw sensor data and output steering commands. A
notable example is imitation learning for lane following [2], and learning
end-to-end driving from simulation [7]. We refer to [8] for a broad review.
Modular methods (e.g. [1]) subdivide the problem into sub-tasks (typically,
perception, prediction, planning and control), each feeding on the output of
the previous one. Sub-tasks are naturally more contained and easier to solve,
and having intermediate outputs makes it easier to interpret the final output.
More recently, Zeng et al. [9] proposed to bridge the two approaches with an
end-to-end architecture producing (and trained on) intermediate outputs. Here,
we use modular approach, where the planning module can transfer across
different sensor configurations as it takes as input a shared representation,
i.e. the output of the perception system.

ML planners. An overview of classical planning approaches, including for
example expert systems, can be found in [10]. Recently, there has been
interest in machine-learning (ML) approaches trained on expert demonstration
[1, 6], which have the potential to scale with the data.

Relationship between planning and perception. Other work has studied the
impact of perception quality on planning, either using involved hand-crafted
features (e.g. the nuScenes Detection Score in [11]), or as a function of the
performance of the planner [12]. While the latter work is relevant, our goal
is not to propose a metric to evaluate perception performance, but rather to
find what sensors we need to collect training data for planning. Wong et. al
[3] simulate perception output for testing the AV planning system. Their focus
is on synthesising realistic perception output as if it were captured by AV
sensors, and not different levels of perception quality for simulating
capturing data from cheaper sensors like we do here.

Training with different levels of supervisions. In our work, we train on a
huge amount of lower-quality data and fine-tune on a small amount of high-
quality data. This relates to self-supervised learning, which has recently
revolutionized natural language processing [13, 14] and is in the process of
radically changing computer vision [15, 16]. These approaches employ a pre-
training phase, in which a neural network is trained on a large collection of
unlabeled text/images, and then fine-tuned for the target task on a magnitude
smaller collection. Our work is also related to weakly supervised learning,
where training happens on data with noisy or incomplete supervision, for
example training object detectors on entire images rather than manually
annotate 2D bounding boxes [17]). In our case, what changes is the quality of
the sensors used to collect the training data.

There are several transfer learning techniques to tackle domain shift between
training and testing (e.g. [18]). One option is fine-tuning, which is widely
used in literature, e.g. [19, 20]. Another is domain adaptation, i.e. aiming
to explicitly reduce the domain gap between domains, often used in computer
vision domains, for example for synthesizing examples in new domains or style
transfer [21, 22, 23]. However, little work has been done to study transfer
learning for motion planning.

Data availability. Much previous work relied on proprietary data [1].
Available datasets for planning are few and of moderate size, e.g. [24, 11],
pointing to the data availability problem mentioned in earlier sections. In
our experiments, we use the recent Lyft dataset [25], containing 1000 hours of
expert demonstrations collected by AVs in urban settings.

Figure 2: An overview of our methodology (sec. III). Firstly, we collect
expert demonstrations using AV-grade sensors, and extract perception
information (which contains agent tracks). Secondly, we simulate what this
data would look like if collected with lower-quality sensors by reducing the
maximum range and field-of-view of the perception information. We then compare
the performances of ML planners trained on these training sets with varying
quality. This lets us do a comprehensive evaluation without having to build or
deploy a wide variety of sensor configurations. To address domain shift, we
fine-tune the planner on small amounts of data collected with AV-grade
sensors, which is what the planner uses at test time.

##  III Methodology

To understand what type of data we can use to train an AV motion planner, we
start by asking two questions: what type of data do we need (_quality_) and
how much (_quantity_)?

By data quality we mean the accuracy and robustness of the perception system
that the planner uses as input both at test and train time [1]. The perception
sytem outputs the 3D positions of other traffic agents like cars and
pedestrians (traffic agents), which provides crucial context, as these
positions influence the trajectory followed by the human driver on which the
planner is trained. Quality is driven by the sensors installed on the vehicle,
for example, an expensive AV-grade LIDAR can estimate depth much more
accurately than a commodity LIDAR, while radars have longer range than a
commodity camera system.

To understand data quality requirements, we propose to compare the performance
at test time of a state-of-the-art ML-planner as we train it on input data
collected with different sensors. However, this would require building and
deploying a variety of different sensor configurations, and collecting enough
training data with each of them. Instead, we propose to take a dataset of
expert driving examples with corresponding AV-grade perception output (Fig.
2), and then simulate what this data would look like if it were collected by a
wide variety of cheaper sensors. We do this by altering three key dimensions
of the perception data:

  1. 1.

Range: The maximum distance that sensors can see objects, e.g. 40m, 30m, etc.

  2. 2.

Field-of-view (FoV): How wide sensors can see, e.g. 90°, 180°)

  3. 3.

Geometric accuracy: How accurate the perception of nearby vehicles is, e.g.
the positional and rotational error of detected agents

We therefore start from a dataset of expert-driven examples with AV-grade
quality perception, then alter the perception quality along one or more of
these three dimensions, and finally train a planner using these data. We
repeat this process with varying degrees of quality alteration (see Fig. 2). A
nice property of using the same training samples for all experiments is that
it allows us a fair comparison: the difference in performance is only due to
difference in quality, and not, for example, due to one dataset containing
more diverse examples than the other.

At test time we always use AV-grade input since, while training data can come
from many sources, the goal is to deploy the planner on an actual AV. This
introduces domain shift, since the input representation has a different
distribution at test time (AV-grade) compared to training (lower-quality).
Addressing this is an important part of our methodology. Having investigated
data quality with this procedure, we conclude with an analysis on the
relationship between data quantity and quality requirements (Sec. IV).

Figure 3: We use a state-of-the-art ML planner (Sec. III-A) which takes as
input a mid-level representation constructed by rasterizing the agent tracks
detected from the perception system on top of an HD semantic map. The network
predicts a trajectory for the AV to follow, minimising the distance from the
trajectory followed by the expert at training time.

In what follows, we discuss the ML planner we use in our experiments (Sec.
III-A), how we alter data quality (Sec. III-B), and how we address domain
shift (Sec. III-C).

###  III-A ML Planner

For our experiments we use a state-of-the art ML motion planner trained via
imitation learning, similar to e.g. ChaufferNet [1]. The input representation
to the planner is a birds-eye-view rasterization of the current driving scene
centered around the _ego vehicle_ (the vehicle carrying the sensors). Agent
tracks are rendered as 2D bounding boxes on top of the semantic map of the
area, e.g. lane geometry, crosswalks, etc. (see Fig. 3). The network predicts
the trajectory to follow for the next $T$ time steps, which we denote by
$p=(p_{1},\ldots,p_{T})$. The training loss minimises the L2 distance between
$p$ and the trajectory $\hat{p}$ followed by the human expert:

$l=\sum_{t=1}^{T}|p-\hat{p}|_{2}$ (1)

We use a ResNet-50 backbone. The network outputs a trajectory for a 1.2 second
($T=12$ steps) prediction horizon. During training we add synthetic
perturbations by alternating the ground-truth trajectories to the left or
right side using Ackerman steering [26] for realistic kinematics. This was
shown to achieve better generalisation, which is needed for closed-loop
testing [1].

###  III-B Altering data quality

Range and field-of-view: We reduce the maximum range of the sensors by
removing portions of the agent tracks that are beyond a given distance in the
AV-grade data (e.g. 20m, 30m, etc.). We reduce the field-of-view (e.g. 90∘,
180∘) by removing agent tracks outside of it (compare Fig. 2). The field-of-
view is centered along the forward direction of the ego vehicle, e.g. 90∘
degrees denoting 45∘ on each side.

(a) Collision rate

(b) ADE

(c) ADE (fine-tuned)

Figure 4: (a)-(b) Altering range and FoV of the training data (Sec. IV-C)
impacts performance at test time, with similar trends on both collision rate
(a) and ADE (b). There is a significant gap for data with 20m range, while for
larger ranges the gap is much smaller. Performance changes smoothly as we
alter FoV. (c) If we fine-tune (FT) on a small quantity of AV-grade data (10
hours), the gap between training entirely on AV-grade is now much smaller,
showing that we can better leverage low quality data if we resolve domain
shift (Sec. III-C). Compare this to (b), where we run the same experiment
without fine-tuning.  Figure 5: Quantity vs quality. Training on 100 hours of
low-quality data performs better than training on 10 hours of AV-grade for a
wide variety of range/field-of-view combinations. If we fine-tune on 10 hours
AV-grade, training on 100 hours of low-quality data approaches 100 hours of
AV-grade (similarly, training on 1000h of low-quality and fine-tuning on 100h
of AV-grade approaches 1000h of AV-grade).

Geometric accuracy: We add noise to the position and rotation of the AV-grade
perception tracks using noise models approximating the accuracy we would
expect from cheaper sensors.

Specifically, we add positional noise to entire agent tracks by applying
random offsets to (x,y) agent coordinates in an agent-centered coordinate
system. We measure the extent of the positional noise in terms of Intersection
over Union (IoU), which is a standard metric for evaluating the accuracy of
agent tracks (e.g. [27]). We specify an IoU noise level that we would like to
alter the agent accuracy to. We then add random positional noise such that the
noisy position will have the given IoU overlapping with the true position of
the agent in the AV-grade data. More specifically, we apply random noise along
the longitudinal axis using a uniform distribution, and add noise along the
horizontal axis so that the specified IoU level is met. We sample this value
once per agent track, and apply it to all agent positions in the track (this
is equivalent to shifting the entire track). This is more realistic than, say,
applying i.i.d noise to consecutive agent positions within the same track, as
tracks from any sensor are typically processed with some form of smoothing.

For rotational noise, we add a random offset to the angle between the agent
direction relative to ego, which we sample from $r\cdot\mathcal{N}(0,\,1)$,
where $r$ is the maximum rotational noise we use in the experiments.

###  III-C Addressing domain shift

Using an input representation with different properties at training and test
time introduces domain shift. While our representation is the same in both
cases (rasterised BEV, Fig. 2), the input agent tracks follow a different
distribution and have different levels of noise.

Here, we fine-tune the planner on a relatively small amount of AV-grade data.
Intuitively, we first learn how to plan from a large corpus of training data
collected with low-quality sensors. Fine-tuning allows us to transfer this
wealth of knowledge for use on the AV, which uses different, better sensors at
test time. This is an integral part of our proposal of leveraging commodity
sensors to collect human driving data. In our experiments we only fine-tune
for 2 epochs and use 1/10th of the original base learning rate to avoid
overfitting to the small AV-grade dataset.

(a)

(b) Figure 6: (a)-(b) The proportion of influential agents (sec. IV-B) out of
all agents observed in the validation set as a function of their distance to
ego (a) and position in the vehicle’s field-of-view (b). Most influential
agents are within the 40m range and in front of ego (as expected). (c) ADE as
we add positional and rotational error to the agent tracks in the training
data (Sec. III-B). While test time performance degrades, the impact is
negligible compared to altering range and FoV (Fig. 4).

(c)

##  IV RESULTS

In this section we investigate the impact of training data quality for an ML
planner (see Sec. IV-C), and the relationship between quantity and quality
(see Sec. IV-E). We train our ML planner on either AV-grade data or lower-
quality data, and always test on AV-grade data (unless specified otherwise),
as our goal is ultimately to deploy the ML planner on a modern AV irrespective
of the data it is trained on. We use standard metrics for this domain (Sec.
IV-B).

###  IV-A Dataset

Our experiments are conducted on the Lyft Level 5 Prediction
Dataset111https://self-driving.lyft.com/level5/data/ [25] which contains
$>1000$h of expert driving demonstrations with corresponding AV-grade
perception output. It was collected in the Palo Alto area by a fleet of 20
AVs, each having 7 cameras, 3 LiDARs, and 5 radars. The data is provided in
independent chunks of 25s called _evaluation scenes_ , and perception output
is refreshed every 10 Hz. In all our experiments, we use the first 10h, 100h
and 1000h data from the L5 training dataset for training, and the original
validation dataset for testing, and let our planner predict at a rate of 10Hz,
i.e. every time a new perception observation is available (we call these
_steps_).

###  IV-B Metrics

####  IV-B1 Collision rate (closed-loop evaluation)

In closed-loop evaluation we allow the ML planner to control the ego vehicle
for an extended period of time. Specifically, at time $t=0$ we use the planner
to predict the ego location at timesteps $t_{1},t_{2}...t_{n}$ (sec. III-A),
and advance the ego to the location at $t_{1}$. We then feed the planner a new
BEV generated at this location to obtain a new prediction. We continue doing
this until we reach the end of the evaluation scene, unless the ego collides
with an agent or the planner prediction deviates from the ground truth expert
trajectory too much where the perception is not reliable anymore. In these
last two cases, we stop evaluating the scene. The metric we report for closed-
loop is the _collision rate_ , i.e. the total number of collisions divided by
the total number of steps in the dataset.

####  IV-B2 ADE (Open-loop error)

Unlike in closed loop evaluation, here the planner makes a prediction for each
step in a scene independently, and always starts from the ground-truth expert
position (no unrolling). We measure the difference between the predicted
trajectory and the expert trajectory over the 1.2s prediction horizon using
(1), and average over all steps - this is called Average Distance Error
(_ADE_). Such open-loop error is commonly used in the field ([1], [9]). While
it does not provide a holistic view of whether the planner generalises well to
the situation when it actually has control over the car like in closed-loop
[28], it is less noisy and much more robust to problems like non-reactivity of
the agents and deviations in position impacting perception.

####  IV-B3 Agent influence

Agent influence measures the impact of a specific agent track around the ego
vehicle on the output of the planner. It is similar to [12] and helps us
understand how this influence varies as a function of the relevant data
quality dimensions (Sec. III-B). For example, the distance of influential
agents from the ego vehicle impacts the range we need from the sensors, while
their relative position to the ego vehicle impacts FoV. We compute influence
for agent $a$ as follows: at test time, we first use the planner to predict a
trajectory $p$ as usual. We then generate a second raster by masking agent $a$
and let the planner predict a new trajectory $p_{a}$. We define agent
influence to be $\lvert\lvert p-p_{a}\rvert\rvert_{2}$, which intuitively
indicates how different the planner would behave if it could not see $a$. We
further define an agent as _very influential_ if this L2 distance is $>$0.1,
and _slightly influential_ if between 0.1 and 0.01. We compute this for all
agents and all steps in the validation dataset.

(a)

(b)

(c)

(d)

Figure 7: In scenario (a), a model trained on data with AV-grade range is
mostly influenced by the only far agent (red) in front of the ego-vehicle
(green), while a model trained on 20m range is not (b). We measure this using
our agent influence metric (Sec. IV-B), i.e. how much the agent influences the
planning decision. We observe the same phenomenon for FoV: when training on
$360^{\circ}$ (AV-grade), ego is influenced by agents behind (c), but is not
when training on $270^{\circ}$ (d).

###  IV-C Impact of data quality

To measure the impact of data quality, we alter the AV-grade perception tracks
in the training data along the three dimensions mentioned in Sec. III. For
range and FoV (Sec. III-B) we use the following buckets:

  * •

Range: 20m, 40m, 60m, full range (AV-grade)

  * •

FoV: 70°, 130°, 270°, 360° (AV-grade)

We train the planner on all possible permutations of range and FoV (16 in
total) and report collision rate and ADE in Fig. 4. All results are using the
same training set of 100 hours (we always train for 15 epochs), and are
evaluated on the full Lyft validation set. For both metrics, performance
increases significantly when the sensor range goes from 20m to 40m, and the
FoV from 70° to 130°. However, longer range or wider FoV do not significantly
impact performance, and sensors that can achieve 40m range and 130° FoV can
already provide valuable training data.

Next, we analyse geometric accuracy (Sec. III-B) by using the following
buckets:

  * •

Positional error: 0.1 IOU, 1.0 IOU (zero error)

  * •

Rotational error: 30°, 0°(zero error)

The results in Fig. 6(c) show that the planner is quite robust to this type of
orientation and positional error (1 cm difference in ADE at test time). For
this reason, we do not explore geometry accuracy any further in the next
experiments.

Our agent influence metric allows us to gain further insights in these
results. When training on AV-grade data, we note that the most influential
agents are within the 40m range from the ego vehicle (Fig. 6(a)). This is
consistent with Fig. 4, where training on more than 40m brings negligible
improvement. For FoV (Fig. 6(b)), agents in front of the ego are the most
influential, but we can see a non negligible mass of important agents also at
the back. This is aligned with our results in Fig. 4, where can see a
noticeable improvement in performance when training on $360^{\circ}$ compared
to $270^{\circ}$. A qualitative example is shown in Fig. 7.

###  IV-D Domain Shift

We repeat the experiment in Fig. 4(b) by fine-tuning each model on 10h of AV-
grade data. Results in Fig. 4(c) show how much we can mitigate domain shift by
exploiting a very limited amount of data from the target AV sensor
configuration, and even models trained with only 20m range now become much
more competitive.

###  IV-E Relationship between data quality and quantity

In this section we analyse the relationship between data quantity and quality.
In Fig. 4 we saw a performance degradation when we alter quality while
training on the same amount of data (100h). Fig. 5 clearly shows that training
on 1000 hours low-quality data (10x) performs better at test time than 100
hours AV-grade (1x) for a variety of range/FoV configurations. As we fine-tune
on additional 100h AV-grade data (Sec. III-C), we see that performance
approaches training on 1000h AV-grade.

##  V Conclusion

We have shown that it is possible to train an AV planner on human expert
demonstrations collected with sensors that are of lower quality than AV
sensors. Our results suggest that data quantity trumps quality, and that it is
better to have a lot of expert demonstrations with lower-quality, than a
smaller amount of demonstrations that are AV-grade. For example, 100 hours
with 40m range and $270^{\circ}$ field-of-view data outperforms 10 hours of
AV-grade quality (Fig. 5). Moreover, it approaches 100 hours AV-grade (the
same amount) after fine-tuning on a small amount of AV-grade data, showing we
can tackle domain shift. All in all this shows the promise of a crowd-sourcing
approach for democratising the collection of training data for training AV
motion planners.

We can use our results to inform the choice of sensors for collecting expert
demonstration data, choosing the trade-off between sensor complexity (and
cost) and expected performance in urban environments (Fig. 4(c)). The fact
that several sensor configurations are competitive shows that we could combine
data collected with a variety of heterogeneous sensors for training.
Furthermore, we can apply the methodology presented in this paper to analyse
data coming from different environments (e.g. highways) to understand the
trade-offs between the sensor configurations in this setting.

In future work, we plan to experiment with data collected using real commodity
sensors, rather than using our data-altering approach. Moreover, we will study
how these results generalise to different areas and conditions.

## ACKNOWLEDGMENTS

We would like to acknowledge and thank Sergey Zagoruyko for the valuable
suggestions for this paper.

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# Amenable wreath products with non almost finite actions of mean dimension
zero

Matthieu Joseph  Université Paris-Saclay, CNRS, Laboratoire de mathématiques
d’Orsay, 91405, Orsay<EMAIL_ADDRESS>
###### Abstract.

Almost finiteness was introduced in the seminal work of Kerr as an dynamical
analogue of $\mathcal{Z}$-stability in the Toms-Winter conjecture. In this
article, we provide the first examples of minimal, topologically free actions
of amenable groups that have mean dimension zero but are not almost finite.
More precisely, we prove that there exists an infinite family of amenable
wreath products that admit topologically free, minimal profinite actions on
the Cantor space which fail to be almost finite. Furthermore, these actions
have dynamical comparison. This intriguing new phenomenon shows that Kerr’s
dynamical analogue of Toms-Winter conjecture fails for minimal, topologically
free actions of amenable groups.

The notion of allosteric group holds a significant position in our study. A
group is allosteric if it admits a minimal action on a compact space with an
invariant ergodic probability measure that is topologically free but not
essentially free. We study allostery of wreath products and provide the first
examples of allosteric amenable groups.

###### 2020 Mathematics Subject Classification:

37A15, 37B05, 20E08, 20E26, 20E15, 46L35

###### Contents

  1. 1 Introduction
  2. 2 Almost finiteness and dynamical comparison
  3. 3 A profinite criterion for allostery
  4. 4 Allostery and wreath products
  5. 5 Proof of the main theorem

##  1\. Introduction

Topological dynamics and $\mathrm{C}^{*}$-algebra theory have a long
intertwined story with the former offering a wide source of meaningful
examples for the latter via the notion of crossed product. In the last decade,
a major breakthrough occurred in the ambitious classification program for
simple separable nuclear $\mathrm{C}^{*}$-algebras that was launched by
Elliott in the $1980$s. This program, which proposes to classify a certain
class of $\mathrm{C}^{*}$-algebras using $K$-theoretic and tracial data has
somehow come to a conclusion with the following classification theorem built
on the work of many researchers.

######  Classification Theorem ([CET+21, Cor. D]).

Unital, separable, simple, nuclear, $\mathcal{Z}$-stable
$\mathrm{C}^{*}$-algebras satisfying the Universal Coefficient Theorem are
classified by their Elliott invariant.

It has naturally become an essential task to find large classes of
$\mathrm{C}^{*}$-algebras that are covered by the classification theorem.
Crossed products arising from topological dynamics are typical candidates: if
$\Gamma\curvearrowright X$ is a continuous action on a compact metric space,
then the reduced crossed product $C(X)\rtimes_{r}\Gamma$ is unital and
separable, is simple if and only if $\Gamma\curvearrowright X$ is
topologically free and minimal [AS94], is nuclear if and only if the action
$\Gamma\curvearrowright X$ is amenable [AD87]. In the case where $\Gamma$ is
amenable, then the action $\Gamma\curvearrowright X$ is automatically
amenable. Moreover, if $\Gamma\curvearrowright X$ is amenable, then
$C(X)\rtimes_{r}\Gamma$ satisfies the Universal Coefficient Theorem [Tu99,
Prop. 10.7]. In order to understand whether such crossed products are covered
by the classification theorem, it remains to understand their
$\mathcal{Z}$-stability.

Even though $\mathcal{Z}$-stability for an amenable action of a non-amenable
group is a stimulating question which has received attention recently (see for
instance [GGK+22] and [GGKN22]) we focus in the present paper on actions of
amenable groups. In this context, several techniques with a dynamical flavor
have been developped recently to prove $\mathcal{Z}$-stability.

Mean dimension (as defined by Gromov) appears to be a meaningful invariant in
this context. This invariant, associated to any topological action, measures
how the dimension grows asymptotically and is known to be zero when the space
is finite dimensional or when the action has zero entropy. For a minimal
action of $\mathbb{Z}$ on a compact metric space $X$, mean dimension zero
implies that $C(X)\rtimes\mathbb{Z}$ is $\mathcal{Z}$-stable, as a combination
of [TW13] when $X$ is finite dimensional and [EN17] when $X$ is infinite
dimensional. Conversely, Giol and Kerr exhibit examples of minimal
$\mathbb{Z}$-actions of mean dimension nonzero whose associated crossed
product is not $\mathcal{Z}$-stable [GK10]. More recently, Niu proved that for
any free minimal action of $\mathbb{Z}^{d}$ on a compact metric space $X$,
mean dimension zero implies $\mathcal{Z}$-stability of
$C(X)\rtimes\mathbb{Z}^{d}$ [Niu19]. For minimal, topologically free actions
of amenable groups, mean dimension zero is conjectured to be the suitable
setting in which $\mathcal{Z}$-stability holds.

Another fruitful technique for $\mathcal{Z}$-stability is the recent dynamical
notion of _almost finiteness_ due to Kerr [Ker20]. Almost finiteness is
defined for group actions on compact metric spaces as a sort of topological
version of the Ornstein-Weiss tower theorem in measurable dynamics. It was
first used in [CJK+18] to prove that the crossed product associated with any
almost finite free minimal action on the Cantor space is $\mathcal{Z}$-stable.
Kerr was then able to remove the zero-dimensional assumption on the space: he
proved that crossed products associated with free minimal almost finite
actions of amenable groups are $\mathcal{Z}$-stable [Ker20, Thm. 12.4] and
therefore are covered by the classification theorem. A large class of group
actions are covered by this theorem: by a recent result of Kerr and Naryshkin,
any free minimal action of an elementary amenable group on a finite-
dimensional compact metric space is almost finite [KN21], see also [Nar23] for
an even larger class of group actions. The main result of this article is to
provide the first examples of minimal, topologically free actions of amenable
groups on zero dimensional spaces (and therefore of mean dimension zero) that
are not almost finite.

######  Theorem 1.1.

Let $\Lambda$ be a finitely generated torsion free nilpotent group and let
$d\geq 1$. Then $\mathbb{Z}^{d}\wr\Lambda$ admits minimal, topologically free,
profinite actions on the Cantor space that are not almost finite.

Here an action $\Gamma\curvearrowright X$ on a zero-dimensional compact metric
space $X$ is profinite if the orbits of the action of $\Gamma$ on the set
$\mathrm{Clo}(X)$ of clopen subsets of $X$ are all finite. We refer to Section
3 for a precise definition of profinite actions.

The wreath products covered by Theorem 1.1 are all elementary amenable, so by
contrast, all their free minimal actions on finite-dimensional compact metric
spaces are almost finite by Kerr and Naryshkin’s result [KN21].

As a consequence, the hope to use almost finiteness in order to prove
$\mathcal{Z}$-stability of the crossed products associated with the actions of
Theorem 1.1 is vain.

In his seminal work [Ker20], Kerr developped a dynamical counterpart of the so
called Toms-Winter conjecture for simple separable nuclear
$\mathrm{C}^{*}$-algebras. With a view toward finding dynamical analogues of
finite nuclear dimension, $\mathcal{Z}$-stability and strict comparison in the
Toms-Winter conjecture, Kerr introduced the following properties for actions
of amenable groups on compact spaces:

  1. (i)

finite tower dimension,

  2. (ii)

almost finiteness,

  3. (iii)

dynamical comparison,

and proved that for a free minimal action of an amenable group $\Gamma$ on a
finite dimensional compact metric space $X$, with finitely many invariant
ergodic measures, (i)$\Rightarrow$(ii)$\Leftrightarrow$(iii), see [Ker20, Thm.
9.3]. The actions that we construct in Theorem 1.1 are profinite. Since
profinite minimal actions are uniquely ergodic and have comparison (see Lemma
2.4), this shows that Kerr’s dynamical version of Toms-Winter conjecture fails
for topologically free actions of amenable groups (see Corollary 2.5).
Therefore, a dynamical analogue of Toms-Winter conjecture remains to be
understood for topologically free minimal actions.

The notion of allostery, which has its roots in subgroup dynamics, will be the
main tool in the proof of Theorem 1.1. A minimal ergodic action
$\Gamma\curvearrowright(X,\mu)$ is an action by homeomorphisms of $\Gamma$ on
a compact metric space $X$, which is minimal (every orbit is dense), with an
ergodic $\Gamma$-invariant Borel probability measure $\mu$. We say that a
minimal ergodic action is:

  * •

topologically free if the set of points with trivial stabilizer is comeager,
that is contains a dense countable intersection of open sets.

  * •

essentially free if the set of points with trivial stabilizer has full
measure.

It is a classical result that essential freeness implies topological freeness
for a minimal ergodic action, see for instance [Jos21, Lem. 2.2]. The study of
the converse, which is false in general, will be the main tool in this article
and led the author to introduce the following denomination in [Jos21].

######  Definition 1.2.

A minimal ergodic action $\Gamma\curvearrowright(X,\mu)$ is allosteric if it
is topologically free but not essentially free.

In general, the $\mathcal{Z}$-stability of $C(X)\rtimes\Gamma$, and therefore
its classifiability, where $\Gamma\curvearrowright(X,\mu)$ is an allosteric
action of an amenable group $\Gamma$, remains an open question.

######  Question 1.3.

Let $\Gamma$ be an amenable group. Let $\Gamma\curvearrowright(X,\mu)$ be a
minimal ergodic action of mean dimension zero. If
$\Gamma\curvearrowright(X,\mu)$ is allosteric, is the crossed product
$C(X)\rtimes\Gamma$ classifiable by its Elliott invariant?

A countable group $\Gamma$ is allosteric if it admits an allosteric action.
The existence of allosteric groups, and more precisely groups which admit
allosteric profinite actions, was asked by Grigorchuk, Nekrashevich and
Sushchanskii in [GNS00, Prob. 7.3.3]. The first examples of allosteric groups
were provided by Bergeron and Gaboriau in [BG04]. They proved that any non-
amenable free product of two nontrivial residually finite groups is
allosteric. An independent proof of this result for free groups of finite rank
was obtained by Abért and Elek in the unpublished paper [AE07]. In [AE12],
Abért and Elek proved that the free product of four copies of the cyclic group
$C_{2}$ admits an allosteric action whose orbit equivalence relation is
measure hyperfinite. In [Jos21], the author proved that the fundamental group
of any non-amenable surface group is allosteric, providing the first examples
of allosteric groups with one end. In [BG04], [AE07], [AE12], and [Jos21], the
allosteric actions obtained are all profinite, which answers positively the
question [GNS00, Prob. 7.3.3].

As of now, examples of allosteric groups are rare. By contrast, there are
plenty of groups that are known to be non-allosteric. This is the case for
countable groups $\Gamma$ such that the space $\mathrm{Sub}(\Gamma)$ of
subgroups of $\Gamma$ is at most countable [Jos21, Cor. 2.4]. Examples of
groups with only countably many subgroups are finitely generated nilpotent
groups, more generally polycyclic groups, extensions of Noetherian groups by
groups with only countably many subgroups (e.g. solvable Baumslag-Solitar
groups $\mathrm{BS}(1,n)$) as observed in [BLT19, Cor. 8.4]. Another class of
non-allosteric groups is given that the groups whose ergodic invariant random
subgroups (that is, the ergodic probability measures on $\mathrm{Sub}(\Gamma)$
that are invariant by conjugation) are all atomic. Among such groups, the most
emblematic ones are lattices in simple higher rank Lie groups [SZ94], but
plenty of other classes of groups are known to admit only atomic ergodic
invariant random subgroups, see for instance [Bek20], [Cre17], [CP17], [DM14]
or [PT16]. More surprisingly, there exist non-allosteric groups with plenty of
ergodic invariant random subgroups. This is the case for instance for the
group $\mathrm{FSym}(\mathbb{N})$ of finitely supported permutations on
$\mathbb{N}$. Indeed, one can check using [Ver12] that any ergodic invariant
random subgroup of $\mathrm{FSym}(\mathbb{N})$ contains the normal subgroup
$\mathrm{Alt}(\mathbb{N})$ in its support. This fact implies that
$\mathrm{FSym}(\mathbb{N})$ is not allosteric. A similar kind of argument
applies for weakly branch groups: by the double commutator lemma [Zhe19], any
ergodic invariant random subgroup of a weakly branch group contains a non-
trivial uniformly recurrent subgroup in its support, and thus weakly branch
groups are not allosteric.

With a view toward proving Theorem 1.1, we study in Section 4 allostery for
wreath products. Given two countable groups $\Gamma,\Lambda$, the wreath
product $\Gamma\wr\Lambda$ is the group

$\Gamma\wr\Lambda\coloneqq\Big{(}\bigoplus_{\Lambda}\Gamma\Big{)}\rtimes\Lambda,$

where $\Lambda$ acts on the direct sum $\bigoplus_{\Lambda}\Gamma$ by shifting
the copies of $\Gamma$. Given a group $\Gamma$ and a prime number $p$, we say
that $\Gamma$ is residually $p$-finite if for every nontrivial element
$\gamma\in\Gamma$, there exists a normal subgroup $N\trianglelefteq\Gamma$
such that $\Gamma/N$ is a finite $p$-group and $\gamma\notin N$.

We prove that wreath products with an abundance of finite $p$-quotients are
allosteric.

######  Theorem 1.4.

Let $\Lambda$ be a countable group and let $d\in\mathbb{N}^{*}$. Assume that
there exist infinitely many prime numbers $p$ such that $\Lambda$ is a
residually $p$-finite group. Then $\mathbb{Z}^{d}\wr\Lambda$ admits profinite
allosteric actions.

To prove this theorem, we develop a profinite criterion that implies
allostery. This criterion, which is explained in Section 3, uses a profinite
construction and rely on the existence of a sequence of finite index subgroups
which mimic allostery at finite stages. In Section 4, we prove Theorem 1.4 by
showing that $\mathbb{Z}^{d}\wr\Lambda$ satisfies this criterion. In
particular, the allosteric actions that we obtain are all profinite.

As a corollary of Theorem 1.4, we provide the first examples of _amenable_
allosteric groups, which answers a question of Ortega and Scarparo [OS22, Rem.
2.6].

######  Corollary 1.5.

Let $\Lambda$ be a finitely generated torsion-free nilpotent group and let
$d\in\mathbb{N}^{*}$. Then $\mathbb{Z}^{d}\wr\Lambda$ is an amenable
allosteric group.

###### Proof of Corollary 1.5.

If $\Lambda$ is finitely generated torsion-free and nilpotent, then $\Lambda$
is residually $p$-finite for every prime $p$ by a result of Gruenberg [Gru57].
Therefore $\mathbb{Z}^{d}\wr\Lambda$ is allosteric and amenable. ∎

Again, the allosteric actions that we obtain in Corollary 1.5 are profinite.
The groups under consideration in this corollary all have infinite asymptotic
dimension, but, after a first version of this paper appeared, Hirshberg and Wu
proved – using the same ideas and methods developed in the present paper – the
following general result: the wreath product $\Gamma\wr\Lambda$ of any
nontrivial residually finite abelian group $\Gamma$ by any countably infinite
residually finite group $\Lambda$ admits profinite allosteric actions, see
[HW23, Cor. 11.3].

The proof of Theorem 1.1 will be provided in Section 5 at the end of this
paper. Let us sketch its proof here. The key observation in order to prove
this result is that allosteric actions of amenable groups cannot be almost
finite, as almost finite actions are always essentially free for any invariant
probability measure (see Lemma 2.2 for of proof of this fact for zero-
dimensional compact metric spaces). Since the allosteric actions that we
construct in Corollary 1.5 are profinite, this provides minimal, topologically
free profinite actions on the Cantor space that are not almost finite.

##  2\. Almost finiteness and dynamical comparison

In this section we discuss the notions of almost finiteness and dynamical
comparison for zero-dimensional compact metric spaces.

A compact metric space $X$ is zero-dimensional if it admits a basis of clopen
sets. Almost finiteness was defined by Kerr for group actions on compact
metric spaces [Ker20, Def. 8.2] but we will restrict in this paper to group
actions on zero-dimensional compact metric spaces as the definition is easier
to state in this case.

######  Definition 2.1.

An action $\Gamma\curvearrowright X$ on a compact metric zero-dimensional
space $X$ is almost finite if for all finite $K\Subset\Gamma$ (here and
thereafter, $\Subset$ stands for “is a finite subset of”) and $\varepsilon>0$,
there exists $V_{1},\dots,V_{n}\subseteq X$ clopen and
$S_{1},\dots,S_{n}\Subset\Gamma$ such that

  * •

the sets $sV_{i}$ for $s\in S_{i}$ and $i\in\\{1,\dots,n\\}$ are pairwise
disjoint,

  * •

for all $i\in\\{1,\dots,n\\}$, $\lvert KS_{i}\triangle
S_{i}\rvert<\varepsilon\lvert S_{i}\rvert$,

  * •

$X=\displaystyle\bigsqcup_{i=1}^{n}S_{i}V_{i}$.

Observe that the notion of almost finiteness makes sense for action of
_amenable_ groups, as the finite sets $S_{i}$ are $(K,\varepsilon)$-Følner
sets of $\Gamma$. If $\Gamma\curvearrowright X$ is an action on a compact
space, we denote by $\mathrm{Prob}_{\Gamma}(X)$ the space of
$\Gamma$-invariant Borel probability measures on $X$.

######  Lemma 2.2.

Let $\Gamma\curvearrowright X$ be an action on a zero-dimensional compact
metric space, which is almost finite. Then for any
$\mu\in\mathrm{Prob}_{\Gamma}(X)$, the action $\Gamma\curvearrowright(X,\mu)$
is essentially free.

###### Proof.

Fix $\mu\in\mathrm{Prob}_{\Gamma}(X)$. Let
$\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}$, let $K\coloneqq\\{\gamma^{-1}\\}$
and fix $\varepsilon>0$. Let $V_{1},\dots,V_{n}\subseteq X$ clopen and
$S_{1},\dots,S_{n}\Subset\Gamma$ finite that witness almost finiteness for the
pair $(K,\varepsilon)$. If $\\{x\in X\colon\gamma x=x\\}$ intersects $sV_{i}$,
then $\gamma sV_{i}$ and $sV_{i}$ are not disjoint, so $s\in S_{i}\setminus
KS_{i}$. Thus

$\displaystyle\mu(\\{x\in X\colon\gamma x=x\\})$
$\displaystyle\leq\sum_{i=1}^{n}\lvert KS_{i}\triangle S_{i}\rvert\mu(V_{i})$
$\displaystyle\leq\varepsilon\sum_{i=1}^{n}\lvert
S_{i}\rvert\mu(V_{i})=\varepsilon.$

This implies that $\Gamma\curvearrowright(X,\mu)$ is essentially free. ∎

######  Definition 2.3.

An action $\Gamma\curvearrowright X$ on a compact metric zero-dimensional
space has comparison if for every nonempty clopen sets $A,B\subseteq X$
satisfying $\mu(A)<\mu(B)$ for all $\mu\in\mathrm{Prob}_{\Gamma}(X)$, there
exists a partition $A=\bigsqcup_{i=1}^{n}C_{i}$ into clopen sets and elements
$\gamma_{1},\dots,\gamma_{n}\in\Gamma$ such that $(\gamma_{i}C_{i})_{1\leq
i\leq n}$ are pairwise disjoint subsets of $B$.

The rest of this section is devoted to the proof that profinite minimal
actions have dynamical comparison. Recall that an action
$\Gamma\curvearrowright X$ on a zero-dimensional compact metric space $X$ is
profinite if the orbits of the $\Gamma$-action on the set $\mathrm{Clo}(X)$ of
clopen subsets of $X$ are all finite. In the proof of the following lemma, we
will need the fact that profinite minimal actions are uniquely ergodic (see
Lemma 3.1 for a more precise statement). Let us also provide some
preliminaries on Boolean algebras. The powerset $2^{X}$ together with the
operations of union, intersection and complement, is a Boolean algebra and
$\mathrm{Clo}(X)$ is a Boolean subalgebra of it. Given a Boolean algebra
$\mathcal{B}$ and a subset $E\subseteq\mathcal{B}$, the intersection of all
Boolean subalgebras of $\mathcal{B}$ that contains $E$ is called the Boolean
algebra generated by $E$. A Boolean algebra generated by a finite set is
necessarily finite ([GH09, Ch. 11, Cor. 2]). Let $\mathcal{B}$ be a Boolean
subalgebra of the powerset $2^{X}$. An atom of $\mathcal{B}$ is any non-empty
element $A\in\mathcal{B}$ such that the only elements of $\mathcal{B}$
contained in $A$ are $\emptyset$ and $A$. If $\mathcal{B}$ is finite, then any
element of $\mathcal{B}$ can be written uniquely as a union of atoms of
$\mathcal{B}$ ([GH09, Ch. 11, Thm. 2]). We are now ready to state and prove
the following.

######  Lemma 2.4.

Any profinite minimal action $\Gamma\curvearrowright X$ has comparison.

###### Proof.

Let $\mu$ be the unique $\Gamma$-invariant probability measure on $X$. Fix
$A,B$ clopen such that $\mu(A)<\mu(B)$. Since $\Gamma\curvearrowright X$ is
profinite, the set $E$ of all $\Gamma$-translates of $A$ and of $B$ is finite.
Therefore, the Boolean algebra $\mathcal{B}$ generated by $E$ is finite. Let
$\mathcal{A}\subseteq\mathcal{B}$ denotes the set of atoms of $\mathcal{B}$.
By definition of an atom, the set $\mathcal{A}$ is $\Gamma$-invariant and two
atoms are either disjoint of equal. For all $C\in\mathcal{A}$, we have
$\bigcup_{\gamma\in\Gamma}\gamma C=X$ by minimality. This shows that
$\Gamma\curvearrowright\mathcal{A}$ is transitive and that $\mathcal{A}$ forms
a partition of $X$ whose pieces all have the same $\mu$-measure. Since $A$ and
$B$ are in $\mathcal{B}$, they can be expressed uniquely as a disjoint union
of atoms. So there exists
$C_{1},\dots,C_{n},C_{1}^{\prime},\dots,C_{m}^{\prime}\in\mathcal{A}$ such
that $A=\bigsqcup_{i=1}^{n}C_{i}$ and $B=\bigsqcup_{j=1}^{m}C_{j}^{\prime}$.
Since $\mu(A)<\mu(B)$, then $n<m$ and by transitivity of
$\Gamma\curvearrowright\mathcal{A}$, there exist
$\gamma_{1},\dots,\gamma_{n}\in\Gamma$ such that $(\gamma_{i}C_{i})_{1\leq
i\leq n}$ are pairwise disjoint subsets of $B$. ∎

We therefore obtain the following result as a Corollary of Theorem 1.4.

######  Corollary 2.5.

Let $\Lambda$ be a finitely generated torsion free nilpotent group and let
$d\in\mathbb{N}^{*}$. Then $\mathbb{Z}^{d}\wr\Lambda$ admits a minimal,
topologically free action $\Gamma\curvearrowright X$ on a finite dimensional
compact metric space with finitely many invariant ergodic measures, which is
not almost finite but have comparison.

By contrast, Kerr proved that for any _free_ minimal action of an amenable
group on a finite-dimensional compact metric space with finitely many
invariant ergodic measures, almost finiteness is equivalent to comparison
[Ker20, Thm. 9.3].

##  3\. A profinite criterion for allostery

In this section we provide a profinite criterion which implies allostery. This
criterion was used without being made explicit in the proof that the
fundamental group of any hyperbolic surface group is allosteric [Jos21]. We
provide here a precise and explicit criterion. Let us first recall some basic
notions on profinite actions. In Section 2, we defined profinite actions as
actions $\Gamma\curvearrowright X$ on zero-dimensional compact metric spaces
$X$ such that the $\Gamma$-orbit of every clopen set is finite. We provide
here an equivalent definition with inverse limit of actions on finite sets.

Let $(I,\leq)$ be a directed _countable_ poset. For all $i\in I$, let
$\Gamma\curvearrowright X_{i}$ be an action on a finite set. Assume that for
all $i\leq j$, we have a $\Gamma$-equivariant surjective map $f_{ij}:X_{j}\to
X_{i}$, such that

  * •

$f_{ii}$ is the identity on $X_{i}$,

  * •

$f_{ik}=f_{ij}\circ f_{jk}$ for all $i\leq j\leq k$.

The inverse limit of the finite spaces $X_{i}$ is the space

$\underset{i\in I}{\varprojlim}\
\\!X_{i}\coloneqq\left\\{(x_{i})\in\prod_{i\in I}X_{i}\colon
x_{i}=f_{ij}(x_{j})\text{ for all }i\leq j\right\\}.$

This space is closed, thus compact metrizable, and totally disconnected in the
product topology. The diagonal action of $\Gamma$ on $\prod_{i\in I}X_{i}$
restricts to an action by homeomorphisms of $\Gamma$ on $\varprojlim X_{i}$.

If each $X_{i}$ is endowed with a $\Gamma$-invariant probability measure
$\mu_{i}$ such that $(f_{ij})_{*}\mu_{j}=\mu_{i}$ for all $i\leq j$, we let
$\mu$ be the unique Borel probability measure on $\varprojlim X_{i}$ that
projects for every $j\in I$ onto $\mu_{j}$ via the canonical projection
$\pi_{j}:\varprojlim X_{i}\to X_{j}$. The $\Gamma$-action on $\varprojlim
X_{i}$ preserves $\mu$ and is called the inverse limit of the p.m.p. actions
$\Gamma\curvearrowright(X_{i},\mu_{i})$. A p.m.p. action of $\Gamma$ is
profinite if it is measurably isomorphic to an inverse limit of p.m.p.
$\Gamma$-actions on finite sets. The following lemma is well-known, see
[Gri11, Prop. 4.1] for a proof.

######  Lemma 3.1.

Let $\Gamma\curvearrowright\varprojlim(X_{i},\mu_{i})$ be the inverse limit of
the p.m.p. finite actions $\Gamma\curvearrowright(X_{i},\mu_{i})$ and let
$\mu$ denotes the inverse limit of the $\mu_{i}$. Then the following are
equivalent:

  1. (1)

For every $i\in I$, $\Gamma\curvearrowright X_{i}$ is transitive and $\mu_{i}$
is the uniform probability measure on $X_{i}$.

  2. (2)

The action $\Gamma\curvearrowright\varprojlim X_{i}$ is minimal.

  3. (3)

The p.m.p. action $\Gamma\curvearrowright(\varprojlim X_{i},\mu)$ is ergodic.

  4. (4)

The action $\Gamma\curvearrowright\varprojlim X_{i}$ is uniquely ergodic.

We are now ready to state the allosteric criterion.

######  Theorem 3.2 (Allosteric criterion).

Let $\Gamma$ be a countable group and let $S\leq\Gamma$ be a nontrivial
subgroup. Fix a family
$(\varepsilon_{\gamma})_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}$ of real
numbers in $]0,1[$ such that
$\prod_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}(1-\varepsilon_{\gamma})>0$
and a family $(n_{\gamma})_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}$ of
pairwise coprime integers. Assume that for all
$\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}$, there exists a finite index
subgroup $\Gamma_{\gamma}\leq\Gamma$, whose index is a power of $n_{\gamma}$,
such that

  1. (1)

$\gamma\notin\Gamma_{\gamma}$,

  2. (2)

$\lvert\\{q\in\Gamma/\Gamma_{\gamma}\colon\forall s\in
S,sq=q\\}\rvert\geq(1-\varepsilon_{\gamma})[\Gamma:\Gamma_{\gamma}]$.

Then the profinite action

$\Gamma\curvearrowright\underset{F\Subset\Gamma\setminus\\{e_{\Gamma}\\}}{\varprojlim}\Big{(}\Gamma/\bigcap_{\gamma\in
F}\Gamma_{\gamma},\mu_{F}\Big{)},$

where $\mu_{F}$ is the uniform probability measure on
$\Gamma/\bigcap_{\gamma\in F}\Gamma_{\gamma}$, is allosteric.

Before we give the proof of this criterion, let us make some remarks. First,
if $\Gamma$ satisfies this criterion, then $\Gamma$ is residually finite. In
fact, we don’t know any example of a non-residually finite allosteric group.
In the criterion, the assumption that the
$(n_{\gamma})_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}$ are pairwise coprime
is used to obtain a minimal ergodic profinite action. Item 1 is used to get
topological freeness of the profinite action, whereas Item 2 is used to get
non-essential freeness of the profinite action.

###### Proof.

For all $F\Subset\Gamma\setminus\\{e_{\Gamma}\\}$, we denote by $X_{F}$ the
finite set $\Gamma/\bigcap_{\gamma\in F}\Gamma_{\gamma}$. We denote by $X$ the
inverse limit $\varprojlim X_{F}$, by $\mu$ the probability measure on $X$
that projects onto $\mu_{F}$ and by $\alpha$ the profinite action
$\Gamma\curvearrowright(X,\mu)$. For all
$F\Subset\Gamma\setminus\\{e_{\Gamma}\\}$, the action $\Gamma\curvearrowright
X_{F}$ is transitive, therefore we get by Lemma 3.1 that $\alpha$ is a minimal
ergodic action. Let us prove that $\alpha$ is topologically free but not
essentially free. For $x\in X$, we denote by
$\mathrm{Stab}_{\alpha}(x)\coloneqq\\{\gamma\in\Gamma\colon\gamma x=x\\}$ the
stabilizer of $x$ for the action $\alpha$.

We start with topological freeness. For all
$F\Subset\Gamma\setminus\\{e_{\Gamma}\\}$, let
$y_{F}\coloneqq\bigcap_{\gamma\in F}\Gamma_{\gamma}\in X_{F}$ and let
$y\coloneqq(y_{F})\in X$. By assumption, $\gamma\notin\Gamma_{\gamma}$ for all
$\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}$ and thus
$\mathrm{Stab}_{\alpha}(y)=\\{e_{\Gamma}\\}$. Since $\alpha$ is minimal, this
implies that a dense set of points have trivial stabilizer. Moreover, the set
$\mathrm{Stab}_{\alpha}^{-1}(\\{e_{\Gamma}\\})=\bigcap_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}\\{x\in
X\colon\gamma x\neq x\\}$ is always $G_{\delta}$. Therefore, it is comeager
and this shows that $\alpha$ is topologically free.

We now prove that $\alpha$ is not essentially free. Since the index of
$[\Gamma:\Gamma_{\gamma}]$ is a power of $n_{\gamma}$ for all
$\gamma\in\Gamma$ and the $n_{\gamma}$’s are pairwise coprime, the action
$\Gamma\curvearrowright X_{F}$ is isomorphic to the diagonal action
$\Gamma\curvearrowright\prod_{\gamma\in F}\Gamma/\Gamma_{\gamma}$ of the left
coset actions. Therefore for all $F\Subset\Gamma\setminus\\{e_{\Gamma}\\}$,

$\displaystyle\frac{\lvert\\{x\in X_{F}\colon\forall s\in
S,sx=x\\}\rvert}{\lvert X_{F}\rvert}$ $\displaystyle=\prod_{\gamma\in
F}\frac{\lvert\\{q\in\Gamma/\Gamma_{\gamma}\colon\forall s\in
S,sq=q\\}\rvert}{[\Gamma:\Gamma_{\gamma}]}$ $\displaystyle\geq\prod_{\gamma\in
F}(1-\varepsilon_{\gamma}).$

By definition of $\mu$, this implies that

$\mu(\\{x\in X\colon
S\leq\mathrm{Stab}_{\alpha}(x)\\})\geq\prod_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}(1-\varepsilon_{\gamma})>0$

which shows that $\alpha$ is not essentially free. ∎

##  4\. Allostery and wreath products

In this section, we apply the allosteric criterion of Theorem 3.2 to prove
that some wreath products are allosteric. In the sequel, we will only work
with wreath products of the form $A\wr\Lambda$ with $A$ abelian and we recall
the definitions in this case. Let $A$ and $\Lambda$ be two countable groups
with $A$ abelian. The support of a function $f:\Lambda\to A$ is defined by
$\mathrm{supp}(f)\coloneqq\\{\lambda\in\Lambda\colon f(\lambda)\neq 0\\}$.
Pointwise addition defines a group operation on $A^{\Lambda}$. The subgroup of
$A^{\Lambda}$ consisting of functions whose support is finite is denoted by
$\bigoplus_{\lambda\in\Lambda}A$. The wreath product of $A$ by $\Lambda$,
denoted by $A\wr\Lambda$, is the semi-direct product

$A\wr\Lambda\coloneqq\textstyle\big{(}\bigoplus_{\lambda\in\Lambda}A\big{)}\rtimes\Lambda$

where $\Lambda$ acts by shifting the copies of $A$ in
$\bigoplus_{\lambda\in\Lambda}A$. In other words, the group multiplication is
given by
$(f,\gamma)(f^{\prime},\gamma^{\prime})\coloneqq(f+f^{\prime}(\gamma^{-1}\cdot),\gamma\gamma^{\prime})$.

The following theorem uses the allosteric criterion described in Theorem 3.2
to prove that wreath products with an abundance of $p$-finite quotients are
allosteric.

######  Theorem 4.1.

Let $\Lambda$ be a countable group. Assume that there exist infinitely many
prime numbers $p$ such that $\Lambda$ is a residually $p$-finite group. Then
for all $d\geq 1$, the wreath product $\mathbb{Z}^{d}\wr\Lambda$ is
allosteric.

###### Proof.

Let $\Gamma\coloneqq\mathbb{Z}^{d}\wr\Lambda$. Fix
$(p_{\gamma})_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}$ a sequence of
pairwise distinct prime numbers, such that for all nontrivial element
$\gamma=(g,\delta)$ in $\Gamma$, the group $\Lambda$ is a residually
$p_{\gamma}$-finite group and
$g(\mathrm{supp}(g))\cap(p_{\gamma}\mathbb{Z})^{d}=\varnothing$. Fix also a
sequence $(\varepsilon_{\gamma})_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}$
of real numbers in $]0,1[$ such that
$\prod_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}(1-\varepsilon_{\gamma})>0$.

In the remainder of the proof, we fix a nontrivial element
$\gamma\coloneqq(g,\delta)$ in $\Gamma$ and we will construct a finite index
subgroup $\Gamma_{\gamma}$ satisfying the assumptions of the allosteric
criterion (Theorem 3.2). Fix $l\in\mathbb{N}$ such that
$l>\lvert\mathrm{supp}(g)\rvert$. Let us fix a finite index normal subgroup
$\Lambda_{\gamma}\trianglelefteq\Lambda$ by considering the following two
cases.

  * •

If $\delta=e_{\Lambda}$, then fix $\Lambda_{\gamma}\trianglelefteq\Lambda$ of
finite index such that:

    * –

$[\Lambda:\Lambda_{\gamma}]$ is a power of $p_{\gamma}$ that satisfies
$l<\varepsilon_{\gamma}[\Lambda:\Lambda_{\gamma}]$,

    * –

for all distinct $\lambda,\lambda^{\prime}\in\mathrm{supp}(g)$,
$\lambda^{-1}\lambda^{\prime}\notin\Lambda_{\gamma}$.

  * •

If $\delta\neq e_{\Lambda}$, then fix $\Lambda_{\gamma}\trianglelefteq\Lambda$
of finite index such that:

    * –

$[\Lambda:\Lambda_{\gamma}]$ is a power of $p_{\gamma}$ that satisfies
$l<\varepsilon_{\gamma}[\Lambda:\Lambda_{\gamma}]$,

    * –

for all distinct $\lambda,\lambda^{\prime}\in\mathrm{supp}(g)$,
$\lambda^{-1}\lambda^{\prime}\notin\Lambda_{\gamma}$,

    * –

$\delta\notin\Lambda_{\gamma}$.

Such a subgroup $\Lambda_{\gamma}$ exists because $\Lambda$ is a residually
$p_{\gamma}$-finite group. Observe that the second condition implies that any
two distinct elements in $\mathrm{supp}(g)$ belong to distinct left cosets of
$\Lambda_{\gamma}$. Fix a subset $E\subseteq\Lambda/\Lambda_{\gamma}$ of
cardinality $l$ such that all the left cosets of $\Lambda_{\gamma}$ that
intersect $\mathrm{supp}(g)$ belong to $E$. This is possible since
$l>\lvert\mathrm{supp}(g)\rvert$. Let $A_{\gamma}$ be the subgroup of
$\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d}$ given by

$A_{\gamma}\coloneqq\Big{\\{}f\in\textstyle\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d}\colon\sum_{\lambda\in
q}f(\lambda)\in(p_{\gamma}\mathbb{Z})^{d}\text{ for all }q\in E\Big{\\}}.$

######  Claim 1.

$A_{\gamma}$ is $\Lambda_{\gamma}$-invariant.

###### Proof of the claim.

Let $f\in A_{\gamma}$ and $\lambda\in\Lambda_{\gamma}$. Since
$\Lambda_{\gamma}$ is normal in $\Lambda$, then for all $q\in E$, we have
$\lambda^{-1}q=q$. Therefore,

$\sum_{\lambda^{\prime}\in
q}f(\lambda^{-1}\lambda^{\prime})=\sum_{\lambda^{\prime}\in
q}f(\lambda^{\prime})\in(p_{\gamma}\mathbb{Z})^{d}.$

This shows that $A_{\gamma}$ is $\Lambda_{\gamma}$-invariant. ∎

Let us define $\Gamma_{\gamma}\coloneqq A_{\gamma}\rtimes\Lambda_{\gamma}$.
This is a finite index subgroup of $\Gamma$.

######  Claim 2.

$[\Gamma:\Gamma_{\gamma}]=[\Lambda:\Lambda_{\gamma}][\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d}:A_{\gamma}]=[\Lambda:\Lambda_{\gamma}](p_{\gamma})^{ld}$,
which is a power of $p_{\gamma}$.

###### Proof of the claim.

Let us denote by $\boldsymbol{0}$ the neutral element of
$\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d}$. The proof is in two steps.
Firstly, the map given by

$\begin{array}[]{l|rcl}\phi:&\Lambda/\Lambda_{\gamma}&\longrightarrow&\Gamma/(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}\\\
&\xi\Lambda_{\gamma}&\longmapsto&(\boldsymbol{0},\xi)(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}\end{array}$

is well defined, as if $\xi_{1}\Lambda_{\gamma}=\xi_{2}\Lambda_{\gamma}$, then
$(\boldsymbol{0},\xi_{1})^{-1}(\boldsymbol{0},\xi_{2})=(\boldsymbol{0},\xi_{1}^{-1}\xi_{2})$
belongs to
$(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}$. It is
moreover straightfoward to check that $\phi$ is one-to-one and onto. Therefore
$[\Gamma:(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}]=[\Lambda:\Lambda_{\gamma}]$.
Secondly, the map

$\begin{array}[]{l|rcl}\psi:&(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})/A_{\gamma}&\longrightarrow&(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}/\Gamma_{\gamma}\\\
&f+A_{\gamma}&\longmapsto&(f,e_{\Lambda})\Gamma_{\gamma}\end{array}$

is well defined, as if $f_{1}+A_{\gamma}=f_{2}+A_{\gamma}$, then
$(f_{1},e_{\Lambda})^{-1}(f_{2},e_{\Lambda})=(f_{2}-f_{1},e_{\Lambda})$
belongs to $\Gamma_{\gamma}$. Once again, a straightforward computation shows
that $\psi$ is one-to-one and onto. Therefore,
$[(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}:\Gamma_{\gamma}]=[\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d}:A_{\gamma}]=(p_{\gamma})^{ld}$
and this finishes the proof. ∎

######  Claim 3.

The element $\gamma=(g,\delta)$ doesn’t belong to $\Gamma_{\gamma}$.

###### Proof of the claim.

If $\delta\neq e_{\Lambda}$, then $\delta\notin\Lambda_{\gamma}$ by
construction. Therefore $\gamma\notin\Gamma_{\gamma}$. If
$\delta=e_{\Lambda}$, then $\mathrm{supp}(g)$ is not empty since
$(g,\delta)\neq e_{\Gamma}$. Let $\lambda\in\mathrm{supp}(g)$ and let
$q\in\Lambda/\Lambda_{\gamma}$ be such that $\lambda\in q$. By construction of
$E$, we have $q\in E$. Since any two distinct elements in $\mathrm{supp}(g)$
belong to different left cosets of $\Lambda_{\gamma}$, we get that

$\sum_{\lambda^{\prime}\in q}g(\lambda^{\prime})=g(\lambda)$

which does not belong to $(p_{\gamma}\mathbb{Z})^{d}$ because
$g(\mathrm{supp}(g))\cap(p_{\gamma}\mathbb{Z})^{d}=\varnothing$. Therefore
$\gamma\notin\Gamma_{\gamma}$. ∎

Let $S$ be the subgroup of $\mathbb{Z}^{d}\wr\Lambda$ defined by

$S\coloneqq\big{\\{}(f,e_{\Lambda})\in\mathbb{Z}^{d}\wr\Lambda\colon\mathrm{supp}(f)\subseteq\\{e_{\Lambda}\\}\big{\\}}.$

######  Claim 4.

$\lvert\\{q\in\Gamma/\Gamma_{\gamma}\colon\forall s\in
S,sq=q\\}\rvert\geq(1-\varepsilon_{\gamma})[\Gamma:\Gamma_{\gamma}]$.

###### Proof of the claim.

Let
$\pi:\Gamma/\Gamma_{\gamma}\twoheadrightarrow\Gamma/(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}$
be the quotient map. This is a $\Gamma$-invariant map whose fibers are all of
cardinality $[\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d}:A_{\gamma}]$ by
Claim 2. Moreover, remark that for all $(f,\lambda)\in\Gamma$, we have
$\pi((f,\lambda)\Gamma_{\gamma})=(\boldsymbol{0},\lambda)(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}$.
For all $i\in\\{1,\dots,d\\}$, we denote by $t_{i}$ the element of
$\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d}$ such that
$\mathrm{supp}(t_{i})\subseteq\\{e_{\Lambda}\\}$ and $t_{i}(e_{\Lambda})$ is
the $i$-th element of the canonical basis of $\mathbb{Z}^{d}$. Let us also
define $s_{i}\coloneqq(t_{i},e_{\Lambda})\in S$. Then the group $S$ is
generated by $s_{1},\dots,s_{d}$. Therefore, a coset
$q\in\Gamma/\Gamma_{\gamma}$ is fixed by every element of $S$ if and only if
it is fixed by all the elements $s_{1},\dots,s_{d}$. So without loss of
generality, it is sufficient to prove that the number of
$q\in\Gamma/\Gamma_{\gamma}$ that are fixed by $s_{1}$ is
$\geq(1-\varepsilon_{\gamma})[\Gamma:\Gamma_{\gamma}]$. For all
$q=(f,\lambda)\Gamma_{\gamma}\in\Gamma/\Gamma_{\gamma}$, we have

$\displaystyle s_{1}q=q$
$\displaystyle\Leftrightarrow(f,\lambda)^{-1}(t_{1},e_{\Lambda})(f,\lambda)\in\Gamma_{\gamma}$
$\displaystyle\Leftrightarrow(t_{1}(\lambda\cdot),e_{\Lambda})\in\Gamma_{\gamma}$
(1) $\displaystyle\Leftrightarrow\forall q^{\prime}\in
E,\sum_{\lambda^{\prime}\in
q^{\prime}}t_{1}(\lambda\lambda^{\prime})\in(p_{\gamma}\mathbb{Z})^{d}.$

This equivalence shows that for all $q\in\Gamma/\Gamma_{\gamma}$, we have
$s_{1}q=q$ if and only if $s_{1}\pi(q)=\pi(q)$. Thus, we get that

$\textstyle\\{q\in\Gamma/\Gamma_{\gamma}\colon
s_{1}q=q\\}=[\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d}:A_{\gamma}]\\{q\in\Gamma/(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}\colon
s_{1}q=q\\}.$

Using the bijection $\phi$ defined in the proof of Claim 2, the equivalence
(4) implies that there is exactly $[\Lambda:\Lambda_{\gamma}]-l$ elements
$q\in\Gamma/(\bigoplus_{\lambda\in\Lambda}\mathbb{Z}^{d})\rtimes\Lambda_{\gamma}$
such that $s_{1}q=q$. We therefore obtain

$\displaystyle\frac{\lvert\\{q\in\Gamma/\Gamma_{\gamma}\colon
s_{1}q=q\\}\rvert}{[\Gamma:\Gamma_{\gamma}]}$
$\displaystyle=\frac{([\Lambda:\Lambda_{\gamma}]-l)[\bigoplus_{\Lambda}\mathbb{Z}^{d}:A_{\gamma}]}{[\Lambda:\Lambda_{\gamma}][\bigoplus_{\Lambda}\mathbb{Z}^{d}:A_{\gamma}]}$
$\displaystyle>1-\varepsilon_{\gamma},$

which finishes the proof of the claim. ∎

The sequence of finite-index subgroups
$(\Gamma_{\gamma})_{\gamma\in\Gamma\setminus\\{e_{\Gamma}\\}}$ satisfies all
the assumptions of Theorem 3.2, therefore $\Gamma=\mathbb{Z}^{d}\wr\Lambda$ is
allosteric. ∎

##  5\. Proof of the main theorem

We are now ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Let $\Lambda$ be a finitely generated torsion free nilpotent group and let
$d\in\mathbb{N}^{*}$. Let $\Gamma\coloneqq\mathbb{Z}^{d}\wr\Lambda$. By a
result of Gruenberg [Gru57], $\Lambda$ is residually $p$-finite for every
prime $p$. By Theorem 1.4, $\Gamma$ admits profinite allosteric actions. These
actions are therefore topologically free, minimal actions on the Cantor space.
Since they are not essentially free, they cannot be almost finite by Lemma
2.2. This proves the theorem. ∎

## Acknowledgment

I thank David Kerr, Petr Naryshkin and Owen Tanner for a stimulating
discussion on allostery and $\mathrm{C}^{*}$-algebras, as well as for their
comments on a preliminary version of this work. I also thank the organizers of
the conference _“Group Actions: Dynamics, Measure, Topology”_ in Münster
during which this interaction took place. I thank Damien Gaboriau for very
helpful comments on a first version of this paper. Finally, I thank the
reviewer for many remarks and corrections and for pointing out a mistake in an
earlier version of the proof of Claim 2.

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