# Pyramid Point:
A Multi-Level Focusing Network for Revisiting Feature Layers

Nina Varney  Vijayan K. Asari  Quinn Graehling

###### Abstract

We present a method to learn a diverse group of object categories from an
unordered point set. We propose our Pyramid Point network, which uses a dense
pyramid structure instead of the traditional ’U’ shape, typically seen in
semantic segmentation networks. This pyramid structure gives a second look,
allowing the network to revisit different layers simultaneously, increasing
the contextual information by creating additional layers with less noise. We
introduce a Focused Kernel Point convolution (FKP Conv), which expands on the
traditional point convolutions by adding an attention mechanism to the kernel
outputs. This FKP Conv increases our feature quality and allows us to weigh
the kernel outputs dynamically. These FKP Convs are the central part of our
Recurrent FKP Bottleneck block, which makes up the backbone of our encoder.
With this distinct network, we demonstrate competitive performance on three
benchmark data sets. We also perform an ablation study to show the positive
effects of each element in our FKP Conv.

Figure 1: Example of the fundamental concept of our Pyramid Point network. The
final output is based not only on an upsampling from the previous layers; but
also on features from all network layers. By upsampling from each layer and
then concatenating the features, we can get a richer combination with less
noise.

##  1 Introduction

Recent breakthroughs in automatic driving and the decrease in LiDAR sensor
cost have increased research into 3D object detection. One of the most popular
tasks is the semantic segmentation of unorganized 3D point clouds. 3D point
clouds present a distinct type of challenge because of their sparse and
unordered nature. These point clouds exist as lists of points containing X, Y,
and Z information with other possible features like reflectively or RGB data.
The resolution and occlusions in a point cloud vary greatly, not only from
sensor-to-sensor, but also within each scene. The points are spatially
related, however the relationships are not uniform like we would find in 2D
imagery. This property makes operations like convolution difficult.

There are two main approaches to handling convolutions with unordered point
sets. The first is to transform the point clouds into ordered sets, either
through voxels or transformations, into a higher dimension. The second
approach is to redefine the convolution operation to perform on unordered
point sets. Once a convolution operation is defined, we can begin to apply
similar methods, like those used in 2D scene segmentation. Most 3D scene
segmentation methods use the traditional U-Net type structure with encoder
layers, followed by decoder layers, usually with connections between similar
sized encoder and decoder layers. However, this U structure has some
shortcomings, particularly rough results that struggle to segment fine
details. This failure is due to the max-pooling and subsampling of the feature
layers, which results in a reduced feature map resolution. This extreme
reduction in the receptive field at each encoder layer makes it very difficult
for this network structure to segment small objects with a high degree of
accuracy.

We propose our Pyramid Point network, which uses feature fusion to combat
reduced feature map resolution issues. Pyramid Point passes features in a
dense pyramid structure. This structure results in feature fusion, from the
encoding layers to the corresponding decoding layers, between encoder and
decoder units of different layers and between the decoder layers themselves.
By allowing the features to traverse between different units and layers, not
just restricting them to the defined U path, we will enable the network to
gain different receptive field views. The network output layer is in Figure 1.
This structure also provides the advantage of having several ”shallow” feature
layers, which have not gone through the entire U structure. Because the
decoding unit adds noise, features that have gone through one or two decoder
units, instead of the traditional four, are less susceptible to segmentation
errors, which is especially crucial for small objects.

In addition to the proposed pyramid structure, we also introduce the Focused
Kernel Point Convolution (FKP Conv), which adds an attention element to the
kernel outputs of a traditional kernel point convolution. This FKP Conv is the
critical element in our Recurrent FKP Bottleneck block.

In our evaluation section, we show the success of our semantic segmentation
network. We assess our performance on three different networks, DALES [32],
Paris-Lille 3D [29], and Semantic 3D [12]. We choose these three LiDAR data
sets because of their distinctly different signatures, aerial, mobile, and
terrestrial type LiDAR, respectively. This evaluation strategy allows us to
assess our network’s overall performance and robustness to different object
categories, resolutions, and occlusions. We show that our network outperforms
other networks on the DALES and Paris-Lille 3D data sets while providing
competitive performance on the Semantic 3D data set and lessening the
variation in results across object categories. Finally, we perform an ablation
study to assess the FKP Conv elements and their effect on our overall
performance. Specifically, we explore the impact of the number of hidden
layers in the recurrent FKP Conv and the implications of different pooling
types.

Our contributions are the following:

  * •

We propose our Pyramid Point network, a multi-level focusing network which
allows for dense feature fusion across all layers of the network

  * •

We introduce a Focused Kernel Point Convolution (FKP Conv) which adds spatial
and channel attention elements to the traditional kernel point convolutions

  * •

We demonstrate the effectiveness of this network on three benchmark data sets,
representing three distinct LiDAR types

##  2 Related Works

###  2.1 Semantic Segmentation of Point Clouds

Since AlexNet’s [17] debut in 2012, there has been an overwhelming amount of
research in developing deep convolutional networks for 2D imagery. As this
field continues to experience massive success, some focus has shifted into
applying deep learning to other types of data, specifically 3D imagery.
Developing deep learning methods for 3D imagery becomes increasingly difficult
when the data is unstructured, as is the case in mesh and point cloud data.

In point cloud data, many original methods focused on transforming the point
cloud into a structured data set using voxels, such as VoxNet [23], VoxelNet
[41], and Point-Voxel CNN [20]. In terms of transformation, voxels do the best
job of maintaining fidelity to the point cloud’s original shape. However,
voxels present an issue because they require that the point cloud be sampled
at a set size typically chosen by the user.

Recently, newer methods have concentrated on performing operations directly on
the points themselves. PointNet [25] was the first architecture to use point
sets as inputs, and the authors developed a method to extract global features
from these point sets. PointNet++ [26] expanded on this by adding a local
hierarchy. In addition to the local hierarchy, PointNet++ also introduced the
idea of multi-scale grouping and multi-resolution grouping to deal with the
problems of different scales in varying resolutions within a single scene.
PointNet++ popularized the idea of performing the operations in a non-
Euclidean metric space, such as using geodesic distances, to capture the
underlying points instead of the traditional metric space. Architectures such
as VoteNet [24], ShellNet [40], PointCNN [19], and SplatNet [30] expand on
using 3D points as direct inputs.

Pointwise multi-layer perceptron networks, like PointNet, were prevalent, but
several networks have recently defined specific point convolutions performed
directly on the inputs. PointwiseCNN [14], SpiderCNN [38], flex convolution
[11], and PCNN [3] all focus on defining these point convolution operations.
The most notable point convolution operation is the Kernel Point Convolution
(KP Conv) [31], which uses a linear correlation to connect the kernel points
to their closest input points. The authors also introduce the idea of a
deformable convolution, where kernel points can adapt to local geometry.

Figure 2: Example of the architecture of our Pyramid Point network. We
construct the architecture in an inverse dense pyramid shape instead of the
typical U shape. At the third layer, features are both downsampled and
upsampled, and similarly, dimensioned layers are concatenated together in the
final stages.

###  2.2 Pyramid Networks

Features used for computer vision application have long relied on scaling to
provide rich and robust features. Classical computer vision methods such as
Scale Invariant Feature Transforms (SIFT) [21] and Speeded Up Robust Features
(SURF) [5] rely heavily on these methods. This approach has also shifted into
deep learning, where architectures rely on a U-type structure to perform image
segmentation and scene understanding.

Medical image segmentation tasks have a rich history of hierarchical methods.
U-Net [27] is the prominent architecture, suggesting a hierarchical approach
to image segmentation with cross-architecture connections. From there, many
iterations such as Residual U-Net, R2 U-Net [2], and SegNet [4] made
incremental improvements. Finally, NABLA-N ($\nabla^{N}$) Net [1] proposed a
pyramid structure of hierarchical layers. While many improvements to the
typical encoder-decoder structure have been made in the 2D image space, the 3D
image space has mostly the same architectures, focusing on improvements to the
convolutional units themselves rather than the architecture.

We propose a 3D semantic segmentation network that moves away from the U
structure in favor of a pyramid network. Each layer is encoded and decoded at
multiple points within the network. This approach gives us multiple viewpoints
at different hierarchical levels and allows us to access different scales
while avoiding the noise associated with traversing through the entire
architecture.

###  2.3 Attention Modules

Attention modules have significantly impacted deep learning with their ability
to efficiently model dependencies [8, 16]. Originally developed to combat
long-range dependence in encoder-decoder-based natural language processing
methods, attention has expanded into many deep learning areas, including
computer vision. Recent proposals to the self-attention mechanism have
expanded the scope to include scene segmentation tasks [15]. Modern computer
optics allow for image captures to be represented by complex and densely
packed features. The pixels representing a specific label’s features are often
susceptible to changes in various fields, such as scaling and rotation,
leading to a lack of consistency within a semantic class. The attention module
can capture rich contextual relationships for better feature representation,
which helps alleviate intra-class conflicts and improves overall segmentation
accuracy.

Attention mechanisms have many versions, but the most popular lately are
varying channel and spatial attention combinations. RCAN [39], CBAM [37] and
ECA [34] both use a 2D convolution combined with channel attention. The
proposal of a dual attention network [10] combining the channel, and spatial
attention has gained popularity as a successful semantic segmentation method
on 2D images. This method is highly successful but can not be replicated
across many convolutions because of its high model complexity.

Some attention mechanisms are applied to the 3D point cloud space with methods
like SCA-CNN [9] and MPRM [36]. In these methods, spatial attention is
utilized to extract contextual information across the spatial domain, while
channel attention approaches are used for exploring local cross-channel
interactions. We propose spatial and channel attention combinations to a 3D
semantic segmentation network in attention-based weighting for kernel point
convolution. Before kernel summation, an attention mechanism’s introduction
will allow the network to perform kernel weighting with considerations for
factors outside of simple geometric relations.

##  3 Pyramid Point Network

###  3.1 Feature Pyramids

Research in semantic segmentation in unorganized point clouds has mainly
focused on the convolution unit of the network. Methods such as KPConv or
ConvPoint [6] have proposed new approaches to performing convolutions on 3D
points. Even though the convolution units have changed, the network remains
the same, following the basic U structure. We propose a new design of a multi-
level network, which draws from the human perception module. The U network’s
idea is that as data progresses through the network, we can gain new
information by varying the receptive fields. We look at the same scene
multiple times from different levels.

As the data progresses through the network, the encoder decreases the
network’s dimensionality while increasing the number of features; conversely,
the decoder increases the number of dimensions while reducing the number of
features. These different layers represent a mixture of high and low-level
features. The initial higher dimensional layers display low-level features
like edges, and the lower-dimensional features representing more complex and
specific features. The issue with this approach is that decoder is sensitive
to noise. Traversing the layers’ dimensionality from high to low and back to
high can contribute to a certain amount of noise, propagating with additional
feature layers.

We propose a change to combat these issues. Figure 2 shows the full
architecture of our multi-level focusing network. The encoder-decoder
structure stays the same, but the output is also upsampled in stages back to
the original layer dimensionality for each decoding stage. We take a second
look for each progressive layer, upsampling back to the actual network layers.
The second look layers are then concatenated to the original decoder layer and
processed.

This approach has two distinct advantages. The first is that it provides
multiple viewpoints at each layer, which are concatenated to make the final
prediction. These additional viewpoints increase the network results. The
other advantage is that these layers are not subject to the extra noise added
as the features traverse through the network. These layers are less processed
and, therefore, cleaner than their counterparts.

Except for the first two downsampling layers, each layer is both downsampled
and upsampled, resulting in several layers of similar features, each derived
from a different point in the network. After some experimentation, we found
the best configuration is to concatenate the similar elements at each layer of
the network. In the next sections, we discuss the adjustments we made to the
kernel point convolution and the encoder and decoder units in detail.

###  3.2 Focused Kernel Point Convolution

Figure 3: The kernel attention module. This module is applied to the kernel
outputs in order to enhance the features, before reducing the kernel
dimension.

####  3.2.1 Kernel Attention

To address the idea of scale within different objects in a scene, we introduce
the concept of kernel attention. In the original kernel point convolution
solution, the final output features are produced by summing to the outputs of
each of the $N$ number of kernels. We argue that adding attention to the
kernel modules can increase the network’s discrimination ability. A direct
linear correlation in the system may not always be the best application. By
adding an attention mechanism, we can help discriminate which of the kernel
weights has the most significant contribution.

We propose an attention module on the kernel outputs before the final
summation to produce the output features. This adjustment will allow the
network to learn discrimination between the different kernel points and allow
for a more sophisticated kernel weighting, rather than relying strictly on
geometric correlation alone. The kernel attention is as follows:

We consider the kernel outputs as $F\in\mathbb{R}^{K\times N\times D}$, where
$K$ is the number of kernel points, $N$ is the number of points and $D$ is the
feature dimension. We take the max and mean pooling of the feature $F$, with
respect to the kernel outputs, to get $F_{max}\in\mathbb{R}^{1\times N\times
D}$ and $F_{mean}\in\mathbb{R}^{1\times N\times D}$ . $F_{mean}$ and $F_{max}$
are concatenated together and run through two shared MLPs. The MLP layers are
followed by a sigmoid function, $\sigma$, which makes the kernel attention
matrix $M_{K}(F)\in\mathbb{R}^{1\times N\times D}$. In the final step, we use
element-wise multiplication to get the final feature $F^{\prime}$. The entire
process is shown below:

$\displaystyle M_{K}(F)=\sigma(MLP([F_{mean},F_{max}]))$ (1) $\displaystyle
F_{FKP}=F\otimes M_{K}(F)$

A summation across all kernel features then reduces the final enhanced kernel
outputs to make the final point features from our Focused Kernel Point
Convolution, $F_{FKP}\in\mathbb{R}^{N\times D}$. The entire kernel attention
architecture can be seen in Figure 3.

Figure 4: Example of our recurrent Focused Kernel Point Convolution bottleneck
block and our Strided Focused Kernel Point Convolution bottleneck block. The
traditional 2D convolution is replaced by a recurrent Focused Kernel Point
Convolution and Focused Kernel Point Convolution, respectively Figure 5:
Example of the recurrent Focused Kernel Point Convolution, with three hidden
layers.

###  3.3 Encoder and Decoder

Once we have established our FKP Conv, we then define our encoder’s building
blocks. We use the bottleneck structure commonly found in ResNet, with a few
adjustments. We begin with a unary convolution, with a Leaky ReLu activation.
This result is passed into a recurrent FKP Conv. We design the output of the
recurrent layer as follows:

$\displaystyle O(t)=w_{k}^{f}*x^{f}(t)+w_{k}^{r}*x^{r}(t+1)$ (2)

We can extend the equation to include any number of hidden layers. This
recurrent FKP is followed by a Leaky ReLu activation and another unary
convolution with Leaky ReLu activation. The final result is then summed with
the original input features to form the final enhanced output. Figure 5 shows
an example of the deconstructed recurrent FKP Conv layout with three hidden
layers.

Our encoder unit consists of two layers; the first is the recurrent FKP
bottleneck block followed by an FKP strided bottleneck block. This FKP strided
bottleneck uses a single strided FKP Conv in place of the recurrent FKP Conv,
as shown in Figure 4. Our decoder consists of a nearest upsampling layer,
followed by a unary convolution.

##  4 Evaluations

Method | mIoU (%) | OA (%) | ground | buildings | cars | trucks | poles | power lines | fences | veg
---|---|---|---|---|---|---|---|---|---|---
ShellNet [40] | 57.4 | 96.4 | 96.0 | 95.4 | 32.2 | 39.6 | 20.0 | 27.4 | 60.0 | 88.4
PointCNN [19] | 58.4 | 97.2 | 97.5 | 95.7 | 40.6 | 4.8 | 57.6 | 26.7 | 52.6 | 91.7
SuperPoint [18] | 60.6 | 95.5 | 94.7 | 93.4 | 62.9 | 18.7 | 28.5 | 65.2 | 33.6 | 87.9
ConvPoint [6] | 67.4 | 97.2 | 96.9 | 96.3 | 75.5 | 21.7 | 40.3 | 86.7 | 29.6 | 91.9
PointNet++ [26] | 68.3 | 95.7 | 94.1 | 89.1 | 75.4 | 30.3 | 40.0 | 79.9 | 46.2 | 91.2
Wang [35] | 72.6 | 94.6 | 95.8 | 95.2 | 86.3 | 54.2 | 43.6 | 92.4 | 58.9 | 94.1
KPConv [31] | 81.1 | 97.8 | 97.1 | 96.6 | 85.3 | 41.9 | 75.0 | 95.5 | 63.5 | 94.1
Pyramid Point | 83.6 | 98.3 | 97.8 | 97.3 | 88.4 | 47.9 | 77.6 | 96.7 | 67.5 | 95.4
Table 1: Current leaders on the DALES data set. We report the overall
accuracy, mean IoU and per class IoU, for each category. We show that our
network outperforms all other methods on the DALES data set.

Figure 6: Visual results of the Pyramid Point network on the DALES data set.
Points are colored by object category; ground (blue), buildings (red), cars
(magenta), trucks (light orange), poles (cyan), power lines (yellow), fences
(orange) and vegetation (green)

Figure 7: Visual results for the Paris-Lille 3D benchmark data set. Each point is colored by object category; ground (dark blue), buildings (light blue), pole (dark green), bollard (green), trash can (light green), barrier (yellow), pedestrian (light orange), car (dark orange) and natural (red) Method | mIoU(%) | ground | building | pole | bollard | trash can | barrier | pedestrian | car | natural
---|---|---|---|---|---|---|---|---|---|---
ConvPoint [6] | 75.9 | 99.5 | 95.1 | 71.6 | 88.7 | 46.7 | 52.9 | 53.5 | 89.4 | 85.4
RandLANet [13] | 78.5 | 99.5 | 97.0 | 71.0 | 86.7 | 50.5 | 65.5 | 49.1 | 95.3 | 91.7
MS-RRFSegNet [22] | 79.2 | 98.6 | 98.0 | 79.7 | 74.3 | 75.1 | 57.9 | 55.9 | 82.0 | 91.4
KP-FCNN [31] | 82.0 | 99.5 | 94.0 | 71.3 | 83.1 | 78.7 | 47.7 | 78.2 | 94.4 | 91.4
FKAConv [7] | 82.7 | 99.6 | 98.1 | 77.2 | 91.1 | 64.7 | 66.5 | 58.1 | 95.6 | 93.9
Pyramid Point | 82.9 | 99.6 | 97.1 | 74.6 | 84.3 | 56.0 | 65.9 | 79.1 | 95.1 | 93.9
Table 2: Benchmark results on the Paris-Lille data set. Our method comes in
first overall, with the best mean IoU. We also place first in the in the
ground, natural and pedestrian categories. Accessed November 16th 2020

We evaluate our network on three different types of LiDAR benchmark data sets;
DALES [32], Semantic3D [12], and Paris-Lille 3D [29]. We choose these
benchmarks because of their popularity and because they represent three
distinct types of LiDAR, Aerial Laser Scanners (ALS), Terrestrial Laser
Scanners (TLS), and Mobile Laser Scanners (MLS). These different sensors all
provide various object classes, resolutions, and types of occlusions and will
allow us to test our network’s robustness.

For each evaluation, we used five total layers in our architecture and began
upsampling and downsampling simultaneously at layer three. The feature
dimensions are $64/128/256/512/1024$, respectively. For our recurrent FKP Conv
bottleneck, we use three hidden layers. After several experiments, we
concluded that the best configuration for combining similarly shaped layers
was to concatenate them. We changed the subsampling parameter and neighborhood
radii based on the resolution of the point cloud. Aside from the subsampling
rate and neighborhood radius, no other parameter changes were made when
testing different data sets.

We examine our network’s performance using mean Intersection over Union (IoU)
as our primary metric. For consistency with the benchmark, we also report per
class IoU and overall accuracy, where appropriate. A discussion of the results
from each data set is below.

###  4.1 DALES

We evaluate the performance of our network on the DALES data set. DALES is a
large-scale semantic segmentation data set for aerial LiDAR, containing over
500 million points and eight object categories in rural, suburban, industrial
and commercial scenes. We used a subsampling rate of 0.25 meters and a
neighborhood radius of 15 meters.

Table 1 shows the performance of our network on DALES. We can see that the
Pyramid Point has a good performance with the highest mean IoU, of over 2.5%
more than the next closest method. One notable performance is between our
network and KPConv, whose kernel point convolution makes up the backbone of
our FKP Conv. We can see a large margin of performance improvement, especially
in some of the lower performing categories. This emphasizes our pyramid
architecture’s effectiveness and ability to exploit the features’
interconnections and bridge the performance gap between object categories. The
visual results of some sample DALES scenes are shown in Figure 6.

###  4.2 Paris-Lille 3D

We also test our method on the Paris Lille 3D data set. This data set is a
mobile LiDAR data set covering two different French cities; the training and
evaluation have ten labeled classes, covering around 2 km and 160 million
points total, providing diverse, challenging scenes. We used a subsampling
rate of 0.08 meters and a neighborhood radius of 4 meters. We compare our
performance to other methods and measure our success by comparing the mean IoU
across all different classes. Table 2 shows our results compared against
others in the benchmark. Overall, our method comes in first in this data set.
We also claim the top performance in the ground, natural and pedestrian
categories. We can also examine the qualitative results by looking at a
selection of scenes shown in Figure 7.

Figure 8: Visual results from the Semantic 3D benchmark data set. Points are colored by semantic class; man-made (orange), natural (dark green), , high veg (green), low veg (blue-green), buildings (light green), hardscape (yellow), scanning art (orange), cars (red) Method | mIoU (%) | OA (%) | man-made | natural | high veg | low veg | buildings | hardscape | scanning art | cars
---|---|---|---|---|---|---|---|---|---|---
MSDeepVoxNet [28] | 65.3 | 88.4 | 83.0 | 67.2 | 83.8 | 36.7 | 92.4 | 31.3 | 50.0 | 78.2
ShellNet [40] | 69.3 | 93.2 | 96.3 | 90.4 | 83.9 | 41.0 | 94.2 | 34.7 | 43.9 | 70.2
GACNet [33] | 70.8 | 91.9 | 86.4 | 77.7 | 88.5 | 60.6 | 94.2 | 37.3 | 43.5 | 77.8
SPG [18] | 73.2 | 94.0 | 97.4 | 92.6 | 87.9 | 44.0 | 83.2 | 31.0 | 63.5 | 76.2
KPConv [31] | 74.6 | 92.9 | 90.9 | 82.2 | 84.2 | 47.9 | 94.9 | 40.0 | 77.3 | 79.7
RandLA-Net [13] | 77.4 | 94.8 | 95.6 | 91.4 | 86.6 | 51.5 | 95.7 | 51.5 | 69.8 | 76.8
Pyramid Point | 77.3 | 94.5 | 95.7 | 90.3 | 84.4 | 50.5 | 95.4 | 45.9 | 72.7 | 83.2
Table 3:  Benchmark results for the reduced-8 Semantic 3D data set. Our method comes in second overall of the available published results. Accessed November 16th 2020 # Hidden Layers | mIoU
---|---
2 | 75.6
3 | 77.3
4 | 76.6
Table 4: Ablation study of the effect of residual layers on the Semantic3D data set  Variation | mIoU
---|---
(1) No Focused Kernel | 85.4
(2) Max Focused Kernel | 89.1
(3) Mean Focused Kernel | 88.3
(4) Max, Mean Focused Kernel | 90.2
Table 5: Ablation study on the combination of intermediate layers in the
Pyramid Point network

###  4.3 Semantic3D

The third point cloud classification evaluation is on the Semantic3D data set.
This data set is taken with a terrestrial LiDAR sensor and has over four
billion hand-labeled points with eight semantic classes. We used a subsampling
rate of 0.06 meters and a neighborhood radius of 3 meters. We use the
reduced-8 data set for our evaluations because we believe that the sampling
strategy is less biased towards the points closest to the sensor.
This data set is challenging because of its overall size, occlusion types, and
varying resolutions. The results of our network are in Table 3. Like others
[13, 31]; we only compare published methods, and we obtain the scores from
their benchmark. We can see that our method outperforms all other methods
except for RandLA Net, which performs similarly with a 0.1% difference between
the two methods. Pyramid Point also logs the highest performance in the cars
category, by 3.5%. Figure 8 shows the qualitative results from selected scenes
in Semantic3D.

###  4.4 Ablation Study

We perform two ablation studies to discover the effects of some of our
contributions to the network. We focus on the impact of the recurrent layers
and how many hidden layers contribute to positive network performances. We
also explore how the FKP Conv and variations to the pooling methods within FKP
Conv can affect the final results.

####  4.4.1 Recurrent Layers

The first set of experiments compares the full intact network and observes the
effects of different numbers of hidden layers in our recurrent structure. We
use Semantic3D data to perform this experiment. Table 4 shows the effects of
different numbers of hidden layers. Adding the recurrent layers improves the
performance, increasing when going from two to three hidden layers. This
performance drops off when increasing the number of hidden layers from three
to four.

####  4.4.2 FKP Conv

Next, we compare the FKP Conv and how it affects our network results. We first
examine the network’s effects using a kernel point convolution, without the
kernel attention elements. We explore using max and mean pooling in the
attention mechanism, and then finally, we explore the combination of all of
them to make up the final FKP Conv. For this evaluation, we use a subset of
the Paris-Lille 3D data.

Table 5 compares the results from all ablated networks. We note that adding
the kernel’s attention to the kernel point convolution gives the biggest jump
in performance; we then get a marginal improvement by considering the max and
mean pooling together to form the final FKP Conv element.

##  5 Conclusion

In this paper, we have presented a novel network for semantic segmentation in
3D point clouds. Our Pyramid Point network uses the concept of a dense pyramid
structure to increase the number of receptive fields and revisit feature
layers. This structure, which varies from the traditional U-style networks,
provides additional details and less noise in the feature channels.
We also introduced a FKP Conv, which utilizes an attention mechanism to
improve outputs from the convolutional kernel. We demonstrated the network’s
success by showing results on three prominent benchmark data sets, DALES,
Semantic3D, and Paris-Lille 3D. The Pyramid Point was the top performer in the
DALES and Paris-Lille 3D data sets and performed competitively in the
Semantic3D data set, demonstrating the network’s success in three different
types of LiDAR environments. We also performed an ablation study, which
illustrates how each element we have introduced positively contributes to our
network performance.

## Acknowledgments

This effort was supported in part by the U.S. Air Force through contract
number FA8650-20-F-1925. The views expressed in this article are those of the
authors and do not reflect on the official policy of the Air Force, Department
of Defense or the U.S. Government.

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# The Genesis of a Theorem

John Labute  Department of Mathematics and Statistics, McGill University,
Burnside Hall, 805 Sherbrooke Street West, Montreal QC H3A 2K6, Canada
<EMAIL_ADDRESS>
(Date: June 23, 2024)

###### Abstract.

In this article we trace the genesis of a theorem that gives for the first
time examples of Galois group $G_{S}$ of the maximal $p$-extension of
${\mathbb{Q}}$, unramified outside a finite set of primes not containing $p$,
that are of cohomlogical dimension $2$. The pro-$p$-group $G_{S}$ is a fab
pro-$p$-group which means that all its derived factors are finite.

This paper was written at the El Monte Eco Lodge in Mindo, Ecuador. The author
wishes to thank Mariela, Tom and their staff and guides for providing a
stimulating and restful environment

À Serre

##  1\. The Theorem

Let $p$ be a odd prime and let $S=\\{q_{1},\ldots,q_{m}\\}$ be a set of primes
$q_{i}\equiv 1$ mod $p$. Let $g_{i}$ be a primitive root mod $q_{i}$ and let
the linking number $\ell_{ij}\in{\mathbb{F}}_{p}$ be defined by

$q_{i}\equiv g_{j}^{-\ell_{ij}}\ {\rm(mod}\ q_{j}).$

If $g$ is another primitive root mod $q_{j}$ then $\ell_{ij}$ is replaced by
$c_{j}\ell_{ij}$ for some $c_{j}$. Let $G_{S}$ be the Galois group of the
maximal $p$-extension of ${\mathbb{Q}}$ which is unramified outside of the set
$S$. Not much was known about these groups; all we knew was that, by Golod-
Shafarevich, they were infinite if $m\geq 4$.

######  Theorem 1.

Suppose that $m$ is even and that

  1. (a)

$\ell_{ij}=0$ if $i,j$ are both odd,

  2. (b)

$\ell_{12}\ell_{23}\cdots\ell_{m-1,m}\ell_{m1}-\ell_{21}\ell_{32}\cdots\ell_{m,m-1}\ell_{1m}\neq
0$.

Then $G_{S}$ is of cohomological dimension 2.

######  Example 2.

For $S=\\{7,19,61,163\\}$ and $p=3$ the non-zero $\ell_{ij}$ are

$\ell_{12}=\ell_{21}=\ell_{14}=\ell_{23}=\ell{{}_{24}}=\ell_{34}=1,\
\ell_{43}=\ell_{41}=-1.$

Conditions (a) and (b) are satisfied, so ${\rm cd}(G_{S})=2$ and this gives
the first example of a fab pro-$p$-group whose cohomological dimension is $2$.
The most general statement of our criterion which covers the case $m$ is odd
is elegantly formulated by Alexander Schmidt:

######  Theorem 3.

Let $G$ be a finitely generated pro-$p$-group such that
$H^{1}(G,{\mathbb{Z}}/p{\mathbb{Z}})$ is the direct sum of non-trivial
subspaces $U,V$ and such that the cup product

$H^{1}(G,{\mathbb{Z}}/p{\mathbb{Z}})\otimes
H^{1}(G,{\mathbb{Z}}/p{\mathbb{Z}})\rightarrow
H^{2}(G,{\mathbb{Z}}/p{\mathbb{Z}})$

  1. (a)

is trivial on $U\otimes U$ and

  2. (b)

maps $U\otimes V$ surjectively onto $H^{2}(G,{\mathbb{Z}}/p{\mathbb{Z}})$,

then $G$ is of cohomological dimension $2$.

##  2\. It all began with a question of Serre

In a 1963 Séminaire Bourbaki Lecture [2] Serre proposed the following question
where $F$ is a finitely generated free pro-$p$-group and $F_{2}=F^{p}[F,F]$. :

”Soit $r\in F_{2}$, et soit $G_{r}=F/(r)$. Peut-on étendre à $G_{r}$ les
résultats démontrés par Lyndon [1] dans le cas discret? En particulier, si $r$
n’est pas une puissance $p$-ième, est-il vrai que $G_{r}$ est de dimension
cohomologique $2$?”

In the Fall of 1964 Serre introduced me to Lie algebras and the work of Lazard
on filtrations of groups in his course ”Groups and Lie Algebras” which he gave
at Harvard. He also gave me a private lecture on the Elimination Theorem for
groups and Lie algebras which as we shall see played a decisive role in the
proof of Theorem 1. My work is very much influenced by the interplay between
groups and Lie algebras. Sometimes, if one can prove the corresponding Lie
algebra question and the result is strong enough, it can be pulled back to a
proof of the original group theory question. This is the case here, at least
partially so; so let me give a sketch of my attempt to solve his question with
this in mind. I was so excited with my ”proof” that I told Serre I had
answered his question in the affirmative. It was only when I wrote down all
the details of my attempted proof that I realized that there was a natural
boundary inherent in my method. I was embarassed having to tell Serre this
especially when he said he told Tate that I had solved his question.

Let $R=(r)$ and let $M=R/[R,R]$ where $[R,R]$ is the subgroup of $R$ generated
by the commutators $[x,y]=x^{-1}y^{-1}xy$ with $x,y\in R$. Then by a result of
Brumer [3] we have ${\rm cd}(G_{r})=2$ if and only if $M$, viewed as a module
over the completed algebra ${\mathbb{Z}}_{p}[[G]]$ of $G$, is a free module of
rank $1$ generated by the image of $r$ in $M$. In [4] we showed that this was
true if $r$ was not ”too close” to a $p$-th power. More precisely, let
$(F_{n})$ be the filtration of $F$ defined by $F_{1}=F$ and
$F_{n+1}=F^{p}[F,F_{n}]$, also known as the descending $p$-central series of
$F$, and let $e$ be largest with $r\in F_{e}$; we have $e<\infty$ if $r\neq
1$, since the intersection of the subgroups $F_{n}$ is $1$. We show that ${\rm
cd}(G_{r})=2$ if $r$ is not a $p$-th power mod $F_{e+1}$.

The filtration $(F_{n})$ has two important properties:

  1. (a)

$[F_{n},F_{m}]\subset F_{n+m}$,

  2. (b)

$F_{n}^{p}\subset F_{n+1}$.

Let ${\rm gr}_{n}(G)$ be the abelian group $G_{n}/G_{n+1}$, which we denote
additively. We introduce a Lie bracket on ${\rm gr}(G)=\oplus{\rm gr}_{n}(G)$
using the commutator operation as follows: let $\xi_{n}\in{\rm
gr}_{n}(G),\eta_{m}\in{\rm gr}_{m}(G)$ be the the images of $x_{n}\in
G_{n},y_{m}\in G_{m}$ respectiively. Then, letting $[\xi_{n},\eta_{m}]$ be the
image in ${\rm gr}_{n+m}$ of $[x_{n},y_{m}]\in G_{n+m}$, we get a Lie bracket
on the graded ${\mathbb{F}}_{p}$-module ${\rm gr}(G)$. The $p$-th power
operator on $G$ induces the structure of a graded
${\mathbb{F}}_{p}[\pi]$-algebra on ${\rm gr}(G)$; namely, if $x\in F_{n}$ and
$\xi$ is its image in ${\rm gr}_{n}(G)$ then $\pi\xi$ is the image of $x^{p}$
in ${\rm gr}_{n+1}(G)$. If $F=F(x_{1},\ldots,x_{d})$ is the free group on
$x_{1},\ldots,x_{d}$ and $\xi_{i}$ is the image of $x_{i}$ in ${\rm
gr}_{1}(F)$ then $L={\rm gr}(F)$ is the free Lie algebra over
${\mathbb{F}}_{p}[\pi]$ on $\xi_{1},\ldots,\xi_{d}$.

Let $\rho$ be the image of $r$ in ${\rm gr}_{e}(F)$; this element is called
the initial form of $r$. Then $\rho$ not a multiple of $\pi$ is the same as
saying $r$ is not a $p$-th power modulo $F_{e+1}$. Let $\mathfrak{r}$ be the
ideal of $L$ generated by $\rho$ and let
$\mathfrak{m}=\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$. Let
$\mathfrak{g}=L/\mathfrak{r}$ and let $U_{\mathfrak{g}}$ be the enveloping
algebra of $\mathfrak{g}$. Then $\mathfrak{m}$ is a $U_{\mathfrak{g}}$ module
via the adjoint representation. Then the corresponding Lie algebra question
would be to show that $\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$ was a free
$U_{\mathfrak{g}}$ module of rank $1$ generated by the image of $\rho$. To
prove this we needed to prove that $\mathfrak{g}$ was torsion free as an
${\mathbb{F}}_{p}[\pi]$ module which would entail the same is true for
$U_{\mathfrak{g}}$ and hence by the Birkhoff-Witt Theorem that
$U_{\mathfrak{g}}$ has no zero divisors. It would be then a straightforward
exercise using the exact sequence

$\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]\rightarrow
I_{\mathfrak{g}}\rightarrow U_{g}\rightarrow{\mathbb{F}}_{p}[\pi]\rightarrow
0$

where $I_{\mathfrak{g}}\cong U_{\mathfrak{g}}^{d}$ is the augmentation ideal
of $U_{\mathfrak{g}}$. This yields the resolution by free $U_{\mathfrak{g}}$
modules

$0\rightarrow\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]\rightarrow
I_{\mathfrak{g}}\rightarrow U_{g}\rightarrow{\mathbb{F}}_{p}[\pi]\rightarrow
0$

showing that ${\rm cd}(\mathfrak{g})=2$.

To show that this lifts to $G=F/R$ we have to show that ${\rm gr}(G)=L/(\rho)$
or, equivalently that ${\rm gr}(R)=(\rho)$ where $R_{n}=F\cap F_{n}$. The
proof of this requires that $\mathfrak{g}$ be torsion free and utilizes
suitable Lazard filtrations of $F$ and iterative applications of Birkhoff-
Witt; it is much too lengthy to be even sketched here. The same goes for the
proof of $L/(\rho)$ being torsion free if $\rho$ is not a multiple of $\pi$.
Today I marvel that I could come up with such intricate proofs but this is
largely due to the influence of Serre at Harvard in 1964 and in Paris in
1965/67.

This proof generalizes to the case of several relators; cf. [12] but the
linear independence of the initial forms of the relators over
${\mathbb{F}}_{p}[\pi]$ is not enough to prove the result. We need to assume
that $U_{\mathfrak{g}}$ is a torsion free ${\mathbb{F}}_{p}[\pi]$ module and
that $\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$ is a free $U_{\mathfrak{g}}$
module on the images of the initial forms of the relators. We call such
relators strongly free.

As it turned out, the question of Serre had a negative answer; for example, in
the case $r=x^{p}[x^{p},y]$ as was shown by Guildenhuys [6] since $x^{p}=1$ in
$G_{r}$ and $r$ is not a $p$-th power in $F$. We still do not have a criterion
for deciding whether the group $G_{r}$ has finite or infinite cohomological
dimension.

##  3\. A fortitutitous detour to discrete groups

In this section $F$ will be a free discrete group of rank $m$. We let
$(F_{n})$ be the filtration of $F$ defined by $F_{1}=F,F_{n+1}=[F,F_{n}]$,
also known as the descending central series of $F$. The graded Lie algebra
${\rm gr}(F)$ is defined as above. It is a free Lie algebra over
${\mathbb{Z}}$. Let $r\in F$ and let $G=F/(r)$. Suppose $r\in F_{e}$, $r\notin
F_{e+1}$ with $e\geq 2$. Let $\rho$ be the image of $r$ in ${\rm gr}_{e}(F)$,
the initial form of $r$. If $\rho$ is not a proper multiple then Waldinger in
[5] showed that ${\rm gr}_{n}(G)$ is a free ${\mathbb{Z}}$ module for $e\leq
n\leq 3e$ and gave formulae for the ranks as a partial answer to a question of
Magnus who asked if ${\rm gr}(G)$ was torsion free for
$G=F/(r_{1},\ldots,\\_d)$ if the initial forms of the $r_{i}$ were linearly
independent.

In [7] it is shown that in fact ${\rm gr}(G)={\rm gr}(F)/(\rho)$ is a free
${\mathbb{Z}}$ module if $\rho$ is not a proper multiple and that the Poincaré
series of its enveloping algebra was

$\frac{1}{1-mt+t^{e}}=\prod_{n\geq 1}\frac{1}{(1-t^{n})^{g_{n}}}$

where $g_{n}$ is the rank of ${\rm gr}_{n}(G)$. A straightforward calculation
yields the formula

$ng_{n}=\sum_{d|n}\mu(n/d)[\sum_{0\leq i\leq
n/d}(-1)^{i}\frac{1}{d+i-ei}\binom{d+i-ei}{i}m^{d-ei}].$

In particular $g_{n}$ depends only on $n,e$ and $m$.

In [12] we extended these results to the case of several relators provided
that their initial forms $\rho_{1},\ldots,\rho_{d}$ satisfied the following
two conditions

  1. (a)

The enveloping algebra $U$ of ${\rm gr}(F)/(\rho_{1},\ldots,\rho_{d})$ is a
torsion free ${\mathbb{Z}}$ module;

  2. (b)

If $\mathfrak{r}=(\rho_{1},\ldots,\rho_{d})$ then
$\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$ is a free $U$ module with basis the
the images of $\rho_{1},\ldots,\rho_{d}$.

Such a sequence we also call strongly free. We also gave a method for
constructing such sequences using using the Elimination Theorem. This method
would prove decisive in the proof of Theorem 1.

######  Theorem 4 (Elimination Theorem).

Let $K$ be a commutative ring and let $L=L(X)$ be the free Lie algebra over
$K$ on the set $X$. Let $S$ be a subset of $X$ and let $\mathfrak{s}$ be the
ideal of $L(X)$ generated by $X-S$. Let $W$ be the enveloping algebra of
$L(S)$ and let $M(S)$ be the submonoid of $W$ generated by $S$.Then
$\mathfrak{s}$ is the free Lie algebra over $k$ on the elements

$T=\\{{\rm ad}(m)(x)\ |\ x\in X-S,\ m\in M(S)\\}.$

######  Corollary 5.

The $W$ module $\mathfrak{s}/[\mathfrak{s},\mathfrak{s}]$ is a free module
over $W$ with basis the image of $T$.

######  Theorem 6 (Criterion for Strong Freeness).

Let $\rho_{1},\ldots,\rho_{d}$ be elements of $\mathfrak{s}$ such that the
elements

$T_{1}=\\{{\rm ad}(m)(\rho_{j})|m\in M(S),\ 0\leq j\leq d\\}$

are part of a basis of $\mathfrak{s}$. Then $\rho_{1},\ldots\rho_{n}$ is a
strongly free sequence.

###### Proof.

The ideal $\mathfrak{r}$ of $L$ generated by $\rho_{1},\ldots,\rho_{d}$ is
generated as an ideal of $\mathfrak{s}$ by the set $T_{1}$. Since $T_{1}$ is
part of a basis of $\mathfrak{s}$ the Corollary to the Elimination Theorem
says that $\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$ is a free module over the
enveloping algebra $V$ of $\mathfrak{s}/\mathfrak{r}$ with basis the image of
$T_{1}$. The exact sequence

$0\rightarrow\mathfrak{s}/\mathfrak{r}\rightarrow U/\mathfrak{r}\rightarrow
L(S)\rightarrow 0$

splits which implies that

$U=V\otimes W=\bigoplus_{m\in M(S)}Vm.$

This implies that $U$ is a free $K$ module. To show that
$\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$ is a free modulue let
$\overline{\rho}_{i}$ be the image of $\rho_{i}$ in
$\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$ and suppose that

$\sum_{i}u_{i}\rho_{i}=0\ with\ u_{i}\in U.$

Then $u_{i}=\sum_{j}v_{ij}m_{j}$ with $u_{i}\in U$, $v_{ij}\in V$ which
implies that

$\sum_{i}u_{i}\overline{\rho}_{i}=\sum_{i,j}v_{ij}{\rm
ad}(m_{j})(\overline{\rho}_{j})=0.$

But this implies $v_{ij}=0$ since $\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$
is a free $V$ module with basis $T_{1}$.

∎

The Lie algebras $L/\mathfrak{r}$ constructed in this way are of cohomological
dimension $\leq 2$.

An important example of this is

$[\xi_{1},\xi_{2}],[\xi_{2},\xi_{3}],[\xi_{3},\xi_{4}],\dots,[\xi_{m-1},\xi_{m}],$

where $X=\\{x_{1},\ldots,x_{m}\\}$ and $S=\\{x_{1},x_{3},x_{5},\dots\\}$ so
$X-T=\\{x_{2},x_{4},x_{6},\ldots\\}$.

In [13], Anick gave the name mild group to a discrete group with defining
relators whose initial forms are strongly free and gave many examples using
his criterion of combinatorial freeness to prove mildness. The most important
of these is the fundamental group $G$ of the complement of a pure braid link
$\mathcal{L}$ in $S^{3}$ which is obtained from a pure braid by identifying
the top and bottom of each strand. If $m$ is the number of strands of the
braid then $G$ has the presentation $G=F/(r_{1},\ldots,r_{m-1})$, where
$r_{i}=[x_{i},y_{i}]$ with

$y_{i}=x_{i}^{a_{i}}\prod_{j=1}^{m}x_{j}^{a_{ij}}\ {\rm mod}\ [F,F].$

so that

$r_{i}=\prod_{j=1}^{n}[x_{i},x_{j}]^{a_{ij}}\ {\rm mod}\ [F,[F,F]].$

The matrix $(a_{ij})$ is a symmetric matrix with zero diagonal because
$a_{ij}$ is the linking number between the $i$-th and $j$-th unknot of the
link. We assume that the matrix $a_{ij}$ has no zero rows so that the image
$\rho_{i}$ in ${\rm gr}_{2}(F)$ is non-zero. We then have

$\rho_{i}=\sum_{j=1}^{m}a_{ij}[\xi_{i},\xi_{j}],$

where $\xi_{i}$ is the image of $x_{i}$ in ${\rm gr}_{2}(F)$. Anick uses a
weighted graph associated to the matrix $(a_{ij})$ to give a criterion for the
strong freeness of the sequence $\rho_{1},\ldots,\rho_{m-1}$; note that
$\sum\rho_{i}=0$. This graph, which Anick calls a linking diagram, has as
vertices the set $\\{\xi_{1},\ldots,\xi_{m}\\}$ with $\xi_{i},\xi_{j}$ being
joined if $a_{ij}\neq 0$; in this case, the weight being $a_{ij}$. The graph
is connected mod $p$ if and only if there is a spanning subtree whose vertices
are not congruent to zero modulo $p$ where $p$ can be any prime.

Anick then shows that if the linking diagram of the matrix $(a_{ij})$, or of
the link $\mathcal{L}$, is connected mod $p$ then the sequence
$\rho_{1},\ldots,\rho_{i}$ is strongly free mod $p$ and strongly free if it is
connected mod $p$ for any prime $p$.

##  4\. Back to pro-$p$-groups

In [10] Koch uses Lazard filtrations of the completed group algebra
${\mathbb{F}}_{p}[[F]]$ to show that if
$G=F(x_{1},\ldots,x_{m})/(r_{1},\ldots,r_{d})$ we have ${\rm cd}(G)=2$ if the
initial forms of the relators form a strongly free sequence of Lie elements in
the free Lie subalgebra $\mathfrak{L}$ of ${\rm gr}(F)$ on
$\xi_{,}\ldots,\xi_{m}$, the initial forms of $x_{1},\ldots,x_{m}$. The
${\mathbb{F}}_{p}$-algebra $\mathfrak{A}={\rm gr}(F)$ is the free associative
algebra over ${\mathbb{F}}_{p}$ on the elements $\xi_{1},\ldots,\xi_{m}$. For
Koch, strong freeness means that, if $\mathfrak{R}$ is the ideal of
$\mathfrak{A}$ generated by $\rho_{1},\ldots,\rho_{d}$ and $\mathfrak{I}$ is
the augmentation ideal of $\mathfrak{A}$, then
$\mathfrak{A}/\mathfrak{I}\mathfrak{R}$ is a free $\mathfrak{A}/\mathfrak{R}$
module on the images of the $\rho_{i}$. But
$\mathfrak{B}=\mathfrak{A}/\mathfrak{R}$ is the enveloping algebra of
$\mathfrak{g}=\mathfrak{L}/\mathfrak{r}$, where $\mathfrak{r}$ is the ideal of
$\mathfrak{L}$ generated by the $\rho_{i}$ and
$\mathfrak{R}/\mathfrak{I}\mathfrak{R}$ is isomorphic to
$\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$, the isomorphism being induced by
the inclusion $\mathfrak{r}\subset\mathfrak{R}$. So his definition was the
same as ours but this connection was not made in [10].

Koch also gives a criterion for strong freeness. To describe it, let

$\rho_{i}=\sum_{j=1}^{d}a_{ij}\xi_{j}$

where $a_{ij}=\partial_{j}\rho_{i}\in\mathfrak{A}$ (Fox derivative). The free
commutative, associative ${\mathbb{F}}_{p}$-algebra $\tilde{\mathfrak{A}}$ on
$\xi_{1},\ldots,\xi_{m}$ is naturally a quotient of $\mathfrak{A}$; we
identify $\xi$ with its image in $\tilde{\mathfrak{A}}$. Koch’s Criterion is
that the rank of the ${d\times m}$ matrix $M_{K}=(\tilde{a}_{ij})$ be equal to
$d$. If Koch’s Criterion holds, one has $d<m$ because of the relations

$\sum_{j=1}^{j=d}\partial_{j}\rho_{i}\xi_{j}=0,\ 1\leq i\leq m$

which shows that the columns of $M_{K}$ are linearly dependent.

For the relators
$\rho_{1}=[\xi_{1},\xi_{2}],\rho_{2}=[\xi_{2},\xi_{3}],\rho_{3}=[\xi_{3},\xi_{4}]$
where $m=4$ we have

$M_{K}=\begin{bmatrix}-\xi_{2}&\xi_{1}&0&0\\\ 0&-\xi_{3}&\xi_{2}&0\\\
0&0&-\xi_{4}&\xi_{3}\end{bmatrix}$

which is of rank $3$ and so the relators are strongly free.

In [9], using local and global classfield theory, Koch gives a presentation
for the Galois group $G$ of the maximal $p$-extension of ${\mathbb{Q}}$ which
is unramified outside a finite set $S$ of primes which is analogous to that of
the fundamental group of the complement of a tame link in $S_{3}$. If
$S=\\{q_{1},\ldots,q_{m}\\}$ with $q_{i}\equiv 1$ mod $p$, he shows that
$G=F(x_{1},\ldots,x_{m})/(r_{1},\dots,r_{m}$ where

$r_{i}=x_{i}^{q_{i}-1}\prod_{j\neq i}[x_{i},x_{j}]^{\ell_{ij}}\ {\rm mod}\
F_{3}$

where for $j\leq m$, $\ell_{ij}$ is the image in ${\mathbb{Z}}/pZ$ is any
integer $r$ with

$q_{i}\equiv g_{j}^{-r}\ {\rm mod\ q_{j}}$

where $g_{i}$ is a primitive root mod $q_{i}$.

If $S=\\{q_{1},\ldots,q_{m},p\\}$ the presentation is the same except for one
additional variable $x_{m+1}$ but with $\ell_{ij}$ defined as before for
$i,j\leq m$ while for $i\leq m,j=m+1$, $\ell{ij}$ is defined by

$q_{i}\equiv(1+p)^{-\ell_{ij}}\ {\rm mod}\ p.$

In both cases there is a linking diagram associated to the matrix
$(\ell_{ij})$. There is a striking similarity between the latter presentation
and that of the complement of a pure braid link in $S^{3}$. In the Galois case
we have ${\rm cd}(G)=2$, cf. [3], while we can’t always prove this for the
link group.

The first presentation is strikingly different since $\ell_{ij}\neq\ell_{ji}$
in general, the group has the same number of generators as relators and $G$ is
fab. If $\rho_{i}$ be the image of $r_{i}$ in ${\rm gr}_{2}(F)$ we have

$\rho_{i}=c_{i}\pi\xi_{i}+\sum_{j\neq i}\ell_{ij}[\xi,\xi_{j}]$

where $c_{i}=(q_{i}-1)/p$. Reducing mod $\pi$ we get the elements

$\overline{\rho}_{i}=\sum_{j\neq i}\ell_{ij}[\xi,\xi_{j}]$

in the free lie algebra over ${\mathbb{F}}_{p}$ on the $\xi_{i}$, where
$\xi_{i}$ is the image of $x_{i}$ in ${\rm gr}_{1}(F)$. The relators
$(\rho_{i})$ are strongly free iff the relators $(\overline{\rho}_{i})$ are
strongly free.

At this point in time there was not even one example of a pro-$p$-group $G$
with $G/[G,G]$ finite and ${\rm cd}(G)=2$. Even worse the Galois group in
question has all of its derived factors finite because by class field theory,
the maximal abelian $p$-extension of a number field which is unramified
outside a finite set of primes not divisible by $p$ is of finite degree since
the ramification is tame for such primes. Such a group is called is a fab
group. In this case the sentiment was that such a group must have torsion, so
could not have finite cohomological dimension.

##  5\. The lightening bolt hits

While on leave at Western University in London, Ontario in the Fall of 2004,
discussing with Jan Minác̆ criteria for strong freeness of the relators in the
Koch presentation for $G_{S}$ with $p\in S$, it suddenly dawned on me while
reviewing the criteria for strong freeness in my paper [12] that, if $m$ was
even and $\geq 4$, I could prove that the relators

$\rho_{1}=[\xi_{1},\xi_{2}],\rho_{2}=[\xi_{2},\xi_{3}]\ldots\rho_{m-1}=[\xi_{m-1},\xi_{m}],\rho_{m}=[\xi_{m},\xi_{1}]$

are strongly free by the Elimination Theorem with $K={\mathbb{F}}_{p}$ and
$S=\\{\xi_{i}|i\ {\rm odd}\\}$. Note in this case, the linking diagram is a
circuit. This meant that I was able to prove that the relators

$x_{1}^{p}[x_{1},x_{2}],x_{2}^{p}[x_{2},x_{3}],\ldots,x_{m-1}^{p}[x_{m-1},x_{m}],x_{m}^{p}[x_{m},x_{1}]$

are also strongly free giving the first example of a pro-$p$-group $G$ of
cohomological dimension $2$ with $G/[G,G]$ finite.

Motivated by this I started to search for circuits in the linking diagram of
the Galois group of the maximal $p$-extension unramified outside
$S=\\{q_{1},\ldots,q_{m}\\}$ with $m=4$, $p=3$. For $q_{1}=7$, $q_{2}=19$,
$q_{3}=61$, $q_{4}=163$ I found the circuit

$[\xi_{1},\xi_{2}],[\xi_{2},\xi_{3}],[\xi_{3},\xi_{4}],[\xi_{4},\xi_{1}]$

in the linking diagram of its presentation (mod $\pi$)

$\displaystyle\overline{\rho}_{1}$
$\displaystyle=[\xi_{1},\xi_{2}]+[\xi_{1},\xi_{4}]$
$\displaystyle\overline{\rho}_{2}$
$\displaystyle=[\xi_{2},\xi_{1}]+[\xi_{2},\xi_{3}]+[\xi_{2},\xi_{4}]$
$\displaystyle\overline{\rho}_{3}$ $\displaystyle=[\xi_{3},\xi_{4}]$
$\displaystyle\overline{\rho}_{4}$
$\displaystyle=-[\xi_{4},\xi_{1}]-[\xi_{4},\xi_{3}].$

To prove that theses relators are strongly free it is enough, by our criterion
for strong freeness, to prove that they are part of a basis of
$\mathfrak{s}=(\xi_{2},\xi_{4})$. But because our Lie agebras are graded over
${\mathbb{F}}_{p}$, it is enough to prove the linear independence of their
images $\overline{\rho}_{i}$ in $\mathfrak{s}/[\mathfrak{s},\mathfrak{s}]$ are
linearly independent. Since the $\overline{\rho}_{i}$ lie in the subspace
spanned by the images of

$[\xi_{1},\xi_{2}],[\xi_{2},\xi_{3}],[\xi_{3},\xi_{4}],[\xi_{4},\xi_{1}].$

we just have to prove that the rank of the matrix of the $\overline{\rho}_{i}$
with respect to this basis is $4$. But this matrix is

$\begin{bmatrix}1&0&0&-1\\\ -1&1&0&0\\\ 0&0&1&0\\\ 0&0&-1&1\\\ \end{bmatrix}$

whose determinant is $1$ and we have won! The proof in the general case of
Theorem 1 is quite similar.

######  Corollary 7.

Let $A,B$ be a partition of $\\{1,\ldots,d\\}$. Let $\mathfrak{T}$ consist of
the elements $[\xi_{i},\xi_{j}]={\rm ad}(\xi_{i})(\xi_{j})$ with $i\in A$,
$j\in B$. Let $\rho_{1},\ldots,\rho_{m}$ be the elements of $\mathfrak{s}$
with

$\rho_{k}=\sum_{1\leq i<j\leq m}\overline{a}_{ijk}[\xi_{i},\xi_{j}].$

If they satisfy conditions

  1. (a)

$\overline{a}_{ijk}=0$ when $i,j\in A$,

  2. (b)

the rank of $M_{L}$ is $m$,

the sequence $\rho_{1},\ldots,\rho_{m}$ is strongly free in $L$.

Condition (a) implies that the $\rho_{k}$ lie in $\mathfrak{T}$ modulo
$[\mathfrak{s},\mathfrak{s}]$ and condition (b) implies the linear
independence of the sequence $\rho_{1},\ldots,\rho_{m}$ modulo
$[\mathfrak{s},\mathfrak{s}]$.

Going back to Theorem 1, the Koch presentation of $G_{S}$ has $d=m$ and the
relators are

$r_{i}=x_{i}^{q_{i}-1}\prod_{j\neq i}[x_{i},x_{j}]^{\ell_{ij}}w_{i}$

with $w_{i}$ in the third term of the descending $p$-central series of $F$.
This implies that the holomony relators are

$\rho_{i}=\sum_{j\neq i}\ell_{ij}[\xi_{i},\xi_{j}].$

If the conditions of Theorem 1 are satisfied, that $\rho_{1},\ldots,\rho_{m}$
is a strongly free sequence follows from Corollary 7 if we take $S$ to be the
$\xi_{i}$ with $i$ even. Indeed, if we index the columns of the the matrix
$M_{L}$ by $(1,2),(2,3),\ldots,(m-1,m),(1,m)$ we have

$M_{L}=\begin{bmatrix}\ell_{12}&0&0&\cdots&0&\ell_{1m}\\\
-\ell_{21}&\ell_{23}&0&\cdots&0&0\\\ 0&-\ell_{32}&\ell_{34}&\cdots&0&0\\\
0&0&-\ell_{43}&\cdots&0&0\\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&\cdots&\ell_{m,m-1}&0\\\
0&0&0&\cdots&-\ell_{m,m-1}&-\ell_{m1}\end{bmatrix},$

whose determinant is
$\ell_{12}\ell_{23}\cdots\ell_{m-1,m}\ell_{m1}-\ell_{21}\ell_{32}\cdots\ell_{m,m-1}\ell_{1m}$.

So the conditions of Theorems 1 are conditions for the strong freeness of the
holonomy relators of a given presentation of a given pro-$p$-group.

##  6\. Uravelling the Statement of Theorem 3

Let $G$ be a finitely generated pro-p-group. The cohomology group
$H^{i}(G,{\mathbb{Z}}/p{\mathbb{Z}})$ will be denoted by $H^{i}(G)$; it is a
vector space over the finite field ${\mathbb{F}}_{p}$. The pro-p-group $G$ is
said to be of cohomological dimension $n$ if $H^{n}(G)\neq 0$ and $H^{i}(G)=0$
for $i>n$. The cohomological dimension of $G$ is said to be infinite if no
such $n$ exists. The cohomological dimension of $G$ is denoted by ${\rm
cd}(G)$. The cohomological dimension of a free pro-$p$-group is 1. In [4] we
show that If $\dim H^{2}(G)=1$ and the cup-product

$H^{1}(G)\otimes H^{1}(G)\rightarrow H^{2}(G)$

is a non-degenerate bilinear form then ${\rm cd}(G)=2$. This is the case when
$G$ is the Galois group of the maximal $p$-extension of
${\mathbb{Q}}_{p}(\zeta_{p})$, where ${\mathbb{Q}}_{p}$ is the $p$-adic number
field and $\zeta_{p}$ is a primitive $p$-th root of unity.

We now unravel the statement of Theorem 3 in the context of a presentation of
a pro-$p$-group $G$. If $F=F(x_{1},\ldots,\\_d)$ is the free pro-$p$-group on
$x_{1},....,x_{d}$ and

$1\rightarrow R\rightarrow F\rightarrow G\rightarrow 1$

is a presentation of $G$ we have an exact sequence

$0\rightarrow H^{1}(G)\rightarrow H^{1}(F)\rightarrow H^{1}(R)^{F}\rightarrow
H^{2}(G)\rightarrow H^{2}(F),$

where $H^{1}(G)$ is the dual of $G/G^{p}[G,G]$. Here $[G,G]$ is the closed
subgroup of $G$ generated by the commutators $[x,y]=x^{-1}y^{-1}xy$ with $x,y$
in $G$. This implies that $\dim H^{1}(G)$ is the minimal number of generators
of $G$. If $\dim H^{1}(F)=\dim H^{1}(G)$ the presentation is called minimal,
in which case $H^{2}(G)=H^{1}(R)^{F}=H^{1}(R/R^{p}[R,F])$. We let $\xi_{i}$ be
the image of $x_{i}$ in $F/F^{p}[F,F]=G/G^{p}[G,G]$ and
$\chi_{1},\ldots,\chi_{d}\in H^{1}(G)=H^{1}(F)$ the basis of
$H^{1}(F)=H^{1}(G)$ dual to $\xi_{1},\ldots,\xi_{d}$. Thus, for a minimal
presentation, $\dim H^{2}(G)$ is the minimal number of elements of $R$ that
generate $R$ as a closed normal subgroup of $R$. The presentation is minimal
if and only if $R\leq F^{p}[F,F]$.

Let $r_{1},\ldots,r_{m}$ be elements of $F^{p}[F,F]$, let
$R=(r_{1}\ldots,r_{m})$ be the closed normal subgroup of $F$ generated by
$r_{1},\ldots,r_{m}$ and let $G=F/R$. We have

$r_{k}=\prod_{j=1}^{d}x_{j}^{pc_{kj}}\prod_{1\leq i<j\leq
d}[x_{i},x_{j}]^{a_{ijk}}s_{k}$

with $s_{k}\in[F,[F,F]]$ and $c_{kj},a_{ijk}\in{\mathbb{Z}}_{p}$. Let
$\overline{r}_{k}$ be the image of $r_{k}$ in $R/R^{p}[R,F]=H^{2}(G)^{*}$, the
dual of $H^{2}(G)$. Then $\overline{r}_{1},\ldots,\overline{r}_{m}$ generate
$H^{2}(G)^{*}$ so that $m\geq\dim H^{2}(G)$ which is $\geq 1$ if $G$ is not a
free pro-$p$-group. An important basic fact is that
$\overline{r}_{k}(\chi_{i}\cup\chi_{j})=\overline{a}_{ijk}$, the image of
$a_{ijk}$ in ${\mathbb{F}}_{p}={\mathbb{Z}}_{p}/p{\mathbb{Z}}_{p}$. Dualizing
the cup product mapping

$\phi:H^{1}(G)\otimes H^{1}(G)\rightarrow H^{2}(G),$

we get, when $p\neq 2$, a mapping

$\phi^{*}:H^{2}(G)^{*}\rightarrow\bigwedge^{2}W=\bigoplus_{1\leq i<j\leq
d}{\mathbb{F}}_{p}\xi_{i}\wedge\xi_{j}$

where $W=F/F^{p}[F,F]$. We have
$\langle\chi_{i}\cup\chi_{j},\overline{r}_{k}\rangle=\rho_{k}(\chi_{i}\cup\chi_{j})=\overline{a}_{ijk}$,
But $\chi_{i}\cup\chi_{j}=\phi(\chi_{i}\otimes\chi_{j})$ so that
$\overline{a}_{ijk}=\langle\chi_{i}\otimes\chi_{j},\phi^{*}(\overline{r}_{k})\rangle$
which implies that

$\phi^{*}(\overline{r}_{k})=\sum_{1\leq 1<j\leq
d}\overline{a}_{ijk}\xi_{i}\wedge\xi_{j}.$

If $H^{1}(G)$ is the direct sum of non-trivial subspaces $U,V$ we have, after
a possible change of basis,

$U=\bigoplus_{i\in A}{\mathbb{F}}_{p}\chi_{i}\ \ V=\bigoplus_{i\in
B}{\mathbb{F}}_{p}\chi_{i},$

where $A,B$ is a partition of $\\{1,\ldots,d\\}$. If $\psi$ is the restriction
of $\phi$ to $U\otimes V$, then $\psi$ maps $U\otimes V$ surjectiively onto
$H^{2}(G)$ if and only if

$\psi^{*}:H^{2}(G)^{*}\rightarrow Z=\bigoplus_{i\in A,\ j\in
B}{\mathbb{F}}_{p}\xi_{i}\wedge\xi_{j}.$

is injective; note that

$\psi^{*}(\overline{r}_{k})=\sum_{i\in A,\ j\in
B}a_{ijk}\xi_{i}\wedge\xi_{j}.$

If $\sigma$ is the projection of $\bigwedge^{2}W$ onto $Z$ we have
$\psi^{*}=\sigma\phi^{*}$ so that $\psi^{*}$ is injective if and only if
$\phi^{*}$ is injective, the latter being equivalent to $m$ being equal to the
rank of the matrix $M_{L}$ whose columns are indexed by the pairs $(i,j)$ with
$i\in A,j\in B$ and whose entry in the $k$-th row and $(i,j)$-th column is
$\overline{a}_{ijk}$. Thus, to prove Theorem 3, we have to prove that ${\rm
cd}(G)=2$ if the following two conditions

  1. (a)

$\overline{a}_{ijk}=0$ when $i,j\in A$,

  2. (b)

the rank of $M_{L}$ is $m$,

hold for the elements

$\rho_{k}=\sum_{1\leq 1<j\leq d}\overline{a}_{ijk}\xi_{i}\wedge\xi_{j}.$

Note that that ${\rm rank}\ M_{L}=m$ implies $\dim H^{2}(G)=m$. If $L=L(X)$ is
the free ${\mathbb{F}}_{p}$-Lie algebra on $X=\\{\xi_{1},\ldots,\xi_{d}\\}$ we
can identify $\bigwedge^{2}W$ the space of degree $2$ elements of $L$ with
$\xi_{i}\wedge\xi_{j}$ correspnding to $[\xi_{i},\xi_{j}].$ So that

$\rho_{k}=\sum_{1\leq i<j\leq d}\overline{a}_{ijk}[\xi_{i},\xi_{j}].$

But, by our criterion for strong freeness, the sequence
$\rho_{1},\ldots,\rho_{m}$ is strongly free and hence ${\rm cd}(G)=2$.

##  7\. Epilog

In [15] Alexander Schmidt extended our results to the case of global fields.
In [17] our results for $p$ odd were extended to $p=2$. In [16] Forré gave an
independent treatment of our results which covered the case $p=2$. In [18]
Gartner, using higher Masssey products, extended our cohomological criterion
for mildness to obtain mild pro-$p$-groups defined by relators of arbitrary
degree.

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