# Computational toolkit for predicting thickness of 2D materials using machine
learning and autogenerated dataset by large language model

C.E. Ekuma<EMAIL_ADDRESS>Department of Physics, Lehigh University,
Bethlehem, PA 18015, USA

###### Abstract

The thickness of 2D materials not only plays a crucial role in determining the
performance of nanoelectronic and optoelectronic devices but also introduces
complexities in predicting volume-dependent properties such as energy storage
capacity, due to the intrinsic vacuum within these materials. Although a
plethora of experimental techniques, including but not limited to optical
contrast, Raman spectroscopy, nonlinear optical spectroscopy, near-field
optical imaging, and hyperspectral imaging, facilitate the measurement of 2D
material thickness, comprehensive data for many materials remains elusive.
Over the last decade, the exponential proliferation of 2D materials and their
heterostructures has outstripped the capabilities of conventional experimental
and computational approaches. In this evolving landscape, machine learning
(ML) has emerged as an indispensable tool, offering novel avenues to augment
these traditional methodologies. Addressing the critical gap, we introduce
THICK2D — Thickness Hierarchy Inference and Calculation Kit for 2D Materials.
This Python-based computational framework harnesses an autogenerated thickness
database, developed using large language models (LLMs), and advanced ML
algorithms to facilitate the rapid and scalable estimation of material
thickness, relying solely on crystallographic data. To demonstrate the utility
and robustness of THICK2D, we successfully employed the toolkit to predict the
thickness of more than 8000 2D-based materials, sourced from two extensive 2D
material databases. THICK2D is disseminated as an open-source utility,
accessible on GitHub atgithub.com/gmp007/THICK2D, and archived on Zenodo at
10.5281/zenodo.11216648.

###### keywords:

2D materials; thickness; machine learning; large language model; deep neural
network; data augmentation

††journal: Computer Physics Communications

PROGRAM SUMMARY
Program Title: THICK2D
Catalog identifier: THICK2D
Licensing provisions: GNU General Public License, version 3
Programming language: Python 3
Computer: Any computer that can run Python (versions 3.8 and later).
Operating system: Any operating system that can run Python.
Nature of problem: A computational toolkit that can efficiently compute the
thickness of 2D materials using advanced machine learning algorithms.
Solution method: THICK2D leverages machine learning, deep neural networks and
data augmentation to accurately predict the thickness of 2D materials using
only the crystal information.
Additional Information: THICK2D utilizes crystal structure data in CIF format,
supporting structural optimization via integration with Vienna Ab initio
Simulation Package (VASP) and Quantum Espresso.
Running time: THICK2D runs in less than 30 seconds for basic runs. Structure
optimization and temperature considerations may extend this duration.
Furthermore, training the machine learning model — using either standard ML
techniques or advanced deep neural networks — adds an extra 2-5 minutes, based
on the augmented data size.

##  1 Introduction

The advent of two-dimensional (2D) materials [1, 2, 3, 4] has ushered in a new
era in materials science, offering unprecedented opportunities for
advancements in electronics, photonics, energy storage, and nanotechnology.
Among their unique properties, the thickness of 2D materials plays a pivotal
role in determining their electronic structure, optical properties, mechanical
behavior, and chemical reactivity. Accurate knowledge of the thickness of 2D
materials is crucial for harnessing their full potential in various
applications. Materials such as graphene, transition metal dichalcogenides
(TMDCs), and hexagonal boron nitride (hBN) exhibit properties highly sensitive
to their atomic thickness. For instance, the bandgap of TMDCs, such as MoS2,
transitions from indirect to direct when reduced to a single layer,
significantly enhancing its photoluminescence efficiency. This property is
vital for the development of optoelectronic devices, including photodetectors,
light-emitting diodes (LEDs), and solar cells. Similarly, the mechanical
strength, flexibility, and transparency of graphene, essential for flexible
electronics and wearable devices, are influenced by its thickness. Moreover,
the surface-to-volume ratio, a critical factor in catalysis and sensor
applications, is directly affected by the thickness of 2D materials. Thinner
materials offer a higher surface area, making them more effective for
catalytic processes and increasing their sensitivity as biosensors or chemical
sensors. In energy storage applications, such as batteries and
supercapacitors, the thickness of 2D materials influences ion transport and
storage capacity, impacting the efficiency and energy density of these
devices.

Given the importance of thickness in determining the properties and
functionalities of 2D materials, accurate and reliable methods for predicting
material thickness are indispensable. Traditional experimental techniques for
measuring thickness, such as atomic force microscopy (AFM), hyperspectral
imaging, and transmission electron microscopy (TEM) [5, 6, 7], while precise,
are time-consuming, labor-intensive, and not always feasible for large-scale
or high-throughput analysis. Furthermore, the rapid increase in the number of
2D-based structures renders any experimental approach impractical. On the
computational front, there is a notable absence of any robust method for the
effective determination of the thickness of 2D materials. This gap underscores
the urgent need for innovative computational strategies that can complement
traditional techniques and facilitate efficient, scalable analysis of material
thickness in the burgeoning field of 2D materials.

In this paper, we introduce THICKD2D, a Python-based computational framework
that leverages advanced machine learning algorithms and the strategic
application of conversational large language models (LLMs) as implemented in
PropertyExtractor [8] to autonomously generate a thickness database for 2D
materials. Given the scarcity of reported thickness for 2D-based materials,
our rigorous curation using PropertyExtractor and validation process resulted
in high-quality data for approximately 60 unique 2D materials. To enhance our
dataset, we implemented data augmentation and noise introduction, expanding
the dataset size to $N+N\times M$, where $N=60$ represents the original
dataset size, and $M=50$ signifies the augmentation magnitude. This enriched
dataset is employed to train a deep neural network (DNN) and classical ML
models to predict the thickness of 2D materials based solely on features
derived from the chemical composition, such as element fraction and valence
electron count. Our framework seamlessly integrates with two widely used
electronic structure codes, VASP [9] and Quantum Espresso [10], facilitating
both zero- and finite-temperature predictions of the thickness of 2D-based
materials. Utilizing this computational framework and the trained ML models,
we efficiently screened over 8000 materials, predicting their thickness using
two expansive online 2D materials databases: the Computational 2D Materials
Database (C2DB) [11] and the 2D materials encyclopedia (2DMatPedia) [12]. This
approach accelerates the characterization process and facilitates the
screening and selection of materials for specific applications, expediting the
development of next-generation technologies.

##  2 Theoretical Background

2D materials are characterized by ultra-thin crystals, typically consisting of
a single layer or a few atomic layers of atoms. The thickness of these
materials, understood to be a fundamental property, is a crucial parameter
that significantly influences their physical behaviors, such as electronic,
optical, and mechanical properties. Understanding and accurately measuring the
thickness of 2D materials is essential for their application in integration
into devices. The intrinsic vacuum in 2D materials, due to the lack of atomic
layers along the z-axis, poses a unique challenge in predicting volume-
dependent properties like energy storage capacity. Several experimental
techniques are employed to measure the thickness of 2D materials, including:

  * •

Optical contrast microscopy: Utilizes the contrast difference between a 2D
material and its substrate to estimate thickness.

  * •

Raman spectroscopy: Analyzes the vibrational modes of the material, which
shift in frequency based on thickness.

  * •

Nonlinear optical spectroscopy: Measures higher-order optical responses that
are sensitive to the thickness of the material.

  * •

Near-field optical imaging: Provides high-resolution imaging of surface
features, including thickness variations.

  * •

Hyperspectral imaging: Captures spectral data across a wide range of
wavelengths to determine material properties, including thickness.

Despite various experimental techniques available, comprehensive thickness
data for many 2D materials remains incomplete due to the rapid expansion of
the field and the limitations of conventional methods. Unlike other properties
of 2D materials, a dedicated thickness database does not currently exist.
Additionally, there are no scalable computational approaches for predicting
the thickness of 2D materials. As a result, machine learning (ML) and
artificial intelligence (AI) have become indispensable tools for augmenting
traditional approaches to predict the thickness of 2D materials accurately.
These advanced computational methods offer a promising solution to fill the
gaps in existing data, enabling rapid and reliable predictions that support
the ongoing development and application of 2D materials in various
technological fields.

##  3 Software Description

THICK2D framework integrates PropertyExtractor conversational LLM-based
toolkit for the initial dataset generation, which is then expanded through
domain-specific augmentation and noise injection. This enriched dataset is
used to train our ML models in THICK2D to learn the mechanistic relationships
between features, characterized only by the material composition and the
target, thickness. Once trained, the model can predict the thickness of new
materials, requiring only the crystal structure information. The framework is
enhanced by integrating with electronic structure codes like VASP and QE for
structural optimization and finite-temperature simulations of the thickness of
2D-based materials.

###  3.1 Input Parameter Description of THICK2D

The essential parameters required to successfully use the THICK2D package are
described in Table 1 and are detailed below.

  * •

code_type: Electronic structure calculator (VASP or QE) in THICK2D.

  * •

use_dnn: Toggle for deep neural network thickness prediction (T/F).

  * •

use_ml_model: Toggle for using a pre-trained ML model for thickness prediction
(T/F).

  * •

throughput: Enables batch processing of materials (T/F); use the auxiliary
Python code throughput_thickness_calc.py if T.

  * •

structure_file: Crystal structure input (CIF format or VASP POSCAR for QE).

  * •

optimize: Optimize crystal structure before calculation (T/F).

  * •

nlayers: Number of material layers (integer; ‘1’ for monolayer).

  * •

vdwgap: van der Waals gap for interlayer distances (floating-point).

  * •

potential_dir: Pseudopotential directory for VASP/QE.

  * •

num_augmented_samples: Augmented sample count for model training.

  * •

add_thickness_data: Add new thickness data (T/F); requires mat_thickness.txt.

  * •

parallel_submit_command: Ab initio code submission command (serial or
parallel).

Table 1: The control parameters and their possible values for THICK2D Parameters | Values
---|---
code_type | vasp/qe
use_dnn | T/F
use_ml_model | T/F
throughput | T/F
structure_file | cif extension
optimize | T/F
nlayers | integer
vdwgap | float
potential_dir | potential directory
num_augmented_samples | integer
add_thickness_data | T/F
parallel_submit_command | run command

###  3.2 Installation and Requirements

THICK2D is optimized to use the latest Python release and integrates several
essential libraries, including NumPy [13], Spglib [14], the Atomic Simulation
Environment (ASE) [15], and Pandas [16], ensuring seamless dependency
management and installation. The preferred installation method for THICK2D is
via pip, the Python package manager, which streamlines the process and
automatically installs any required libraries. To install, execute pip install
-U thick2d, ensuring the installation of the most recent version of the code.
For those opting to download the source from our GitHub repository,
installation via pip remains straightforward: navigate to the source directory
and run pip install . — the period indicating the current directory. While
python setup.py install is an alternative, pip installation is recommended for
its efficiency and ease.

###  3.3 Running THICK2D

THICK2D requires only the crystal structure in CIF format as input. For users
who wish to optimize the crystal structure before thickness prediction,
THICK2D offers integration with the electronic structure codes such as VASP
[9] and Quantum Espresso [10] for structural optimization. Configuration of
calculations within the THICK2D framework is facilitated through the
thick2dtool.in file, which can be automatically generated in the calculation
directory by executing thick2d -0. Users are then advised to modify this
primary input file to delineate the desired type of calculation. To prepare
for thickness calculations, especially after structural optimization, the
command thick2d -0 should be rerun to generate the requisite input files. It
is imperative that the crystal structure be provided in CIF format, denoted
with the ‘.cif’ file extension. THICK2D is adept at automatically detecting
and extracting crystal structure information from these files. Upon specifying
the calculation type and adjusting other necessary parameters within
thick2dtool.in, users must define the paths to their VASP or Quantum Espresso
pseudopotential directories. THICK2D will then automatically select the most
suitable pseudopotentials for the material, based on its crystal system. To
initiate the thickness calculation, the command thick2d should be executed
within the job directory, thereby starting the calculation process with the
previously configured settings. It is important to note that specifying the
pseudopotential directory and other related settings is essential only if
structural optimization is required before thickness calculation. For enhanced
throughput in materials modeling, THICK2D provides an additional Python script
that facilitates the high-throughput computation of material thickness across
extensive datasets, thereby optimizing the workflow for large-scale studies.
To use the high-throughput capabilities, run the python auxiliary code
throughput_thickness_calc.py.

##  4 Results and Discussions

###  4.1 Database generation

The database for the thickness of 2D materials currently does not exist. On
the other hand, generative AI is enabling the extraction of structured data
from the vast and expanding unstructured data in diverse fields, including
material science. We programmatically obtained unstructured scientific
literature with thickness-related keywords using the Elsevier ScienceDirect
API, CrossRef REST API, and PubMed API. These unstructured scientific texts
were pre-processed and passed to the LLM-based computational toolkit,
PropertyExtractor, to obtain the initial thickness database. After cleaning to
delineate only thickness values relevant to 2D materials, the data is
seamlessly embedded into the THICK2D architecture. Details of the structural
data extraction are provided in the article describing the PropertyExtractor
software [8].

###  4.2 Data Augmentation and Incorporation of Noise

With the initial thickness database obtained using the PropertyExtractor
software and refined through exploratory data analysis, we proceed to develop
the classical ML and deep neural network models. However, we are faced with
the common challenge of data scarcity, a limitation that significantly impacts
the ability of the model to learn and generalize. To address the limited
dataset size, we employ data augmentation and noise incorporation strategies,
crucial for enhancing the robustness and generalizability of machine learning
models in data-scarce scenarios [17]. These strategies not only artificially
enlarged the training dataset to $N+N\times M$ (where $N\approx 60$ and $M=50$
represent the original dataset size and augmentation magnitude, respectively)
but also improved model performance and out-of-sample statistics.

Data augmentation is a technique used to increase the diversity of training
data without collecting new data. This is particularly useful in materials
science, where data may be scarce or expensive to obtain. It involves
generating a synthetic dataset via domain-specific techniques that introduce
slight modifications to the input data without changing its meaning or
associated label. Historically, in the realm of computer vision, techniques
such as rotating, cropping, and flipping images have successfully enhanced
model performance by providing varied data representations. These methods
mimic the various scenarios under which a model might be deployed in real-
world applications [18]. In addition, the systematic incorporation of noise
through Bayesian optimization into the training data plays a crucial role in
mitigating overfitting and enhancing the performance of machine learning
models on unseen data. This involves introducing variability into the dataset,
which forces models to learn generalized features rather than memorizing
specific instances. This strategy is essential in various domains, including
materials science, leading to the development of models that are both
resilient and capable of adapting to the unpredictability of real world data
[19, 20, 17].

###  4.3 Machine Learning Architecture

One of the essential components in designing a robust ML model is its
transferability, scalability, and generalizability. Often, the features
required to characterize ML models are difficult to obtain without additional
computations. To address this, we designed our ML models using only the
crystal structure information. Specifically, we utilized element fraction
characterized by the stoichiometry of the material and valence electron count.
Hence, users only need to pass the crystal structure information to the
software to predict the thickness of a 2D material.

Table 2: Performance metrics of various machine learning algorithms for predicting the thickness of 2D materials. RF: RandomForest, DT: DecisionTree, ETs: ExtraTrees, AB: AdaBoost, GB: GradientBoosting, ET: ExtraTree, CB: CatBoost regressors. Metrics: Algo (Algorithm), Model-Sc (Training $R^{2}$), Adj-Sc (Adjusted $R^{2}$), CV-Sc (Cross Validation Score), MSE (Mean Squared Error), MAE (Mean Absolute Error), and STD (Standard Deviation). Algo | Model-Sc | Adj-Sc | CV-Sc | MSE | MAE | STD
---|---|---|---|---|---|---
RF | 99.94 | 99.94 | 99.27 | 1.11 | 0.70 | 0.00
DT | 100.00 | 100.00 | 99.09 | 1.38 | 0.81 | 0.01
ETs | 100.00 | 100.00 | 99.99 | 0.67 | 0.57 | 0.00
AB | 90.88 | 90.70 | 91.66 | 0.74 | 0.66 | 0.01
GB | 99.63 | 99.62 | 99.15 | 1.16 | 0.77 | 0.00
ET | 100.00 | 100.00 | 99.70 | 0.70 | 0.70 | 0.00
CB | 99.96 | 99.95 | 99.36 | 3.10 | 1.00 | 0.00
DNN | 90.78 | - | 95.96 | 0.43 | - | -
Figure 1: Distribution of top 20 materials by space group in the 2D materials
encyclopedia database.

In developing the ML models for predicting the thickness of 2D materials, we
employed a variety of classical machine learning models, including boosting
and bagging techniques. Boosting models, such as Gradient Boosting and
CatBoost, iteratively build an ensemble by focusing on the errors of previous
models, thereby enhancing prediction accuracy. Bagging models, like Random
Forest, create multiple subsets of data, train individual models on these
subsets, and then aggregate their predictions to improve robustness and reduce
variance. The performance of each model was rigorously evaluated using cross-
validation techniques to ensure reliability and generalizability. Metrics such
as the mean squared error (MSE) were used to quantify prediction errors, and
the model with the optimal combination of cross-validation scores and error
metrics was automatically selected for making the final predictions. This
approach ensured that the most accurate and reliable model was utilized for
thickness estimation. Additionally, for the deep neural network network, we
designed an architecture with four hidden layers, each comprising 500 neurons.
This architecture was chosen to balance complexity and computational
efficiency while capturing the intricate patterns in the data. Each hidden
layer employs rectified linear unit (ReLU) activation functions to introduce
non-linearity, enabling the model to learn from complex feature interactions.
The network was trained on the thickness dataset, with a particular emphasis
on minimizing the MSE between the predicted and actual thickness values.
During training, techniques such as dropout were employed to prevent
overfitting, ensuring the model’s robustness across diverse datasets.
Additionally, we used batch normalization to stabilize and accelerate the
training process.

###  4.4 Performance Evaluation

The primary statistical metrics employed to evaluate the performance of
THICK2D are listed in Table 2. These metrics include the coefficient of
determination ($R^{2}$), cross-validation score, mean squared error (MSE),
mean absolute error (MAE), and standard deviation (STD). Our methodology
encompassed the development of both conventional machine learning (ML) models
and deep neural network models. Among the conventional models, the ExtraTrees
regression model demonstrated the best combination of accuracy and error
metrics. Although the accuracy score of the DNN is comparatively lower
($\sim$95.96%) than those of conventional ML models, it exhibits a much
smaller MSE of 0.43, indicating a different aspect of model performance.
Notably, smaller error metrics like MSE and MAE can be more informative than
the $R^{2}$ value because they directly measure the prediction errors,
providing insights into how close the predictions are to the actual values.
Lower MSE values indicate more precise predictions, which is crucial for
applications where accuracy is paramount. In contrast, a high $R^{2}$ value,
while indicative of a good fit, does not always guarantee low prediction
errors, especially in cases where the model may overfit the training data but
perform poorly on unseen data. This is particularly important in practical
scenarios where generalization to new, unseen data is essential. Without data
augmentation, the performance of the ML models was significantly poor, with
some even showing negative cross-validation scores, and the DNN accuracy was
only $\sim$17.60%. We observed that the performance and scalability of THICK2D
significantly improved as the size of the augmented dataset increased,
achieving optimal performance around 40-50 times the size of the original
data.

Figure 2: Scatter plot (a) and (b) density distribution plot of the predicted
thickness of 2D materials with the colors indicating thickness variation using
the 2D materials encyclopedia database.

##  5 Benchmarking SMATool

We have employed the THICK2D toolkit to predict the thickness properties of
over 8000 2D materials using the Materials Cloud 2D Database (MC2D) [21] and
the 2D materials encyclopedia (2DMatPedia) [12]. The efficacy of the toolkit
in predicting the thickness was assessed without performing structural
optimization on the structures from the databases. A representative plot of
the distribution of the specific space groups using the 2D materials
encyclopedia database is presented in Figure 1. This distribution provides
insights into the variety of structural symmetries present in the dataset. The
predicted thickness profiles of the various 2D materials are shown in Figure
2. The full list of the predicted values is available on the code’s GitHub
repository. The predicted thickness values span from approximately 2 Å to over
13.5 Å (Figure 2a), with a majority clustering around 4.6 - 7.5 Å. Notably,
the predicted thicknesses for well-known 2D materials such as MoS2, GeSe,
WSe2, graphene, hexagonal BN, TiO2, and GaN are 6.77, 5.58, 6.67, 3.32, 3.35,
4.19, and 3.42 Å, respectively. These predictions are within 5% of the
experimentally reported thickness values for these materials [22, 23, 24, 25].
The THICK2D toolkit thus demonstrates robust performance and reliability in
accurately predicting the thickness of a wide range of 2D materials, providing
a valuable resource for both integration in high-throughput materials design
and the prediction of thickness of 2D materials.

##  6 Conclusions

THICK2D addresses a critical gap in the field of 2D materials by providing a
scalable and accurate toolkit for predicting material thickness. The toolkit
has successfully predicted the thickness of over 8000 2D materials from the
Materials Cloud 2D Database (MC2D) and the 2D materials encyclopedia
(2DMatPedia). This achievement underscores the importance of THICK2D in
bridging the gap to support traditional experimental and computational
methods, which are often time-consuming, labor-intensive, and not scalable for
high-throughput analysis. By leveraging advanced machine learning algorithms,
THICK2D provides accurate thickness predictions, with values spanning from
approximately 2 Åto over 13.5 Å. The majority of thicknesses clustered around
4.6 - 7.5 Å, and predictions for well-known 2D materials such as MoS2, GeSe,
WSe2, graphene, hexagonal BN, TiO2, and GaN were within 5% of experimentally
reported values. This level of precision is crucial for the application of 2D
materials in various technological fields, including nanoelectronics,
optoelectronics, and energy storage. The comprehensive thickness data
generated by THICK2D enhances our understanding of 2D materials and
facilitates their integration into advanced applications.

## Acknowledgments

This work was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences under Award DE-SC0024099. Computational support is
provided by Computational materials group at Lehigh University.

## Data Availability

The autogenerated thickness database of 2D materials is integrated into the
THICK2D computational toolkit. The predicted thickness values for over 8000 2D
materials from two 2D materials databases are shared along with the THICK2D
software on GitHub at github.com/gmp007/THICK2D and Zenodo at
10.5281/zenodo.11216648. Pre-trained ML models for both the best conventional
models and the DNN architecture are also shared alongside the code.
Additionally, the repository provides detailed instructions on setting up and
using THICK2D. The availability of these resources ensures that researchers
can easily access, implement, and benefit from the THICK2D toolkit to
accurately predict the thickness of 2D materials, thereby bridging the gap in
providing scalable and precise thickness determinations for various
applications in materials science.

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$(\delta_{\boldsymbol{\mu}})_{*}b$ is
$\operatorname{Aut}(\tilde{\mathfrak{g}}/\delta_{\boldsymbol{\mu}}K)$
invariant up to scaling. Indeed, $\delta_{\boldsymbol{\mu}}$ is a graded
automorphism of $\tilde{g}$ that sends the ideal $K$ to
$\delta_{\boldsymbol{\mu}}K$. Thus $\delta_{\boldsymbol{\mu}}$ induces a
graded isomorphism from $\tilde{g}/K$ to $\tilde{g}/{\delta_{\bf\mu}K}$.
Moreover
$\delta_{\boldsymbol{\mu}}:(\tilde{V}_{2},b)\rightarrow(\tilde{V}_{2},(\delta_{\boldsymbol{\mu}})_{*}b)$
is a linear isometry and
$\operatorname{Stab}(\delta_{\boldsymbol{\mu}}K,\tilde{\mathfrak{g}})=\delta_{\boldsymbol{\mu}}\circ\operatorname{Stab}(K,\mathfrak{g})\circ\delta_{\boldsymbol{\mu}}^{-1}$.
Hence $\operatorname{Stab}(\delta_{\boldsymbol{\mu}}K,\tilde{\mathfrak{g}})$
preserves $(\delta_{\boldsymbol{\mu}})_{*}b$ up to scaling.

We now choose a special dilation by taking $\mu_{i}=(b(Y_{i},Y_{i}))^{1/2}$.
Then

(7.2)
$(\delta_{\boldsymbol{\mu}})_{*}b(\tilde{Y}_{i},\tilde{Y}_{i})=b(\delta_{\boldsymbol{\mu}}^{-1}\tilde{Y}_{i},\delta_{\boldsymbol{\mu}}^{-1}\tilde{Y}_{i})=\mu_{i}^{-2}b(\tilde{Y}_{i},\tilde{Y}_{i})=1.$

The action of
$\Phi\in\operatorname{Stab}(\delta_{\boldsymbol{\mu}}K,\operatorname{Aut}(\tilde{\mathfrak{g}}))$
on the second layer can be written as a product of a permutation and an
invertible diagonal operator. In view of (7.2) permutations preserve
$(\delta_{\boldsymbol{\mu}})_{*}b$. Since $\Phi$ preserves
$(\delta_{\boldsymbol{\mu}})_{*}b$ up to a scalar factor the diagonal operator
must also preserve that scalar product up to a scalar factor. It follows that
the norm of the eigenvalues of the diagonal operator is constant. Then the
diagonal operator also preserves the standard scalar product up to a scalar
factor. Since permutations preserve the standard scalar product the assertion
follows. ∎

To simplify the notation we will assume from now on without loss of generality

(7.3) $\operatorname{Stab}(K,\operatorname{Aut}(\tilde{\mathfrak{g}}))$ acts
conformally on the second layer with respect to the standard scalar product.

######  Proposition 7.4.

Let $\Phi\in\operatorname{Stab}(K,\operatorname{Aut}(\tilde{\mathfrak{g}}))$
and let $\sigma=\sigma_{\Phi}$ be the induced permutation of the subspaces
$V_{1,i}$ (and $V_{2,i}$). Then there exist $\lambda_{\Phi}\in(0,\infty)$, a
diagonal matrix $S_{\Phi}$ with entries $s_{i}\in\\{-1,1\\}$, and a
permutation matrix $P_{\Phi}$ with the following properties. If
$m\in\mathbb{R}^{n}$ and

$Y=\sum_{j=1}^{n}m_{j}Y_{j},\quad Z=\sum_{j=1}^{n}m_{j}Z_{j}$

then

$\Phi(Y)=\sum_{i=1}^{n}(\lambda_{\Phi}S_{\Phi}P_{\Phi}m)_{i}Y_{i},\quad\Phi_{*}(Z)=\sum_{i=1}^{n}(\lambda_{\Phi}S_{\Phi}P_{\Phi}m)_{i}Z_{i}.$

Moreover $(P_{\Phi})_{ij}=\delta_{i\sigma_{\Phi}(j)}$ and $S_{\Phi}$ and
$\lambda_{\Phi}$ are uniquely determined and

(7.5) $S_{\Phi}\in O(n),\quad P_{\Phi}\in O(n).$

(Note that this definition of $P$ differs from the one in Sections 8 and 6 by
a transpose.)

###### Proof.

There exist $\mu_{i}\neq 0$ such that
$\Phi(Y_{j})=\mu_{\sigma(j)}Y_{\sigma(j)}$. It follows from (7.3) that there
exist a $\lambda\in(0,\infty)$ such that $|\mu_{\sigma(j)}|=\lambda$ for all
$j=1,\ldots,n$. Set $s_{i}=\mu_{i}/\lambda$. Then $s_{i}\in\\{-1,1\\}$ and
$\Phi(Y_{j})=\sum_{i=1}^{n}(\lambda SP)_{ij}Y_{i}$. Thus the formula for
$\Phi(Y)$ follows by linearity.

The formula for $\Phi(Z)$ is proved similarly. Alternatively one can use the
following facts. There is a unique linear map $L_{[\,]}:\Lambda_{2}V_{1}\to
V_{2}$ such that $L_{[\,]}(X\wedge Y)=[X,Y]$. This map commutes with every
graded homomorphism $\Phi$, i.e., $\Phi\circ L_{[\,]}=L_{[\,]}\circ\Phi_{*}$.
For the first Heisenberg group the restriction of $L_{[\,]}$ to
$\oplus_{i=1}^{n}\Lambda_{2}V_{1,i}$ is a linear isomorphism onto $V_{2}$ and
$L_{[\,]}Z_{i}=-Y_{i}$.

Uniqueness of $\lambda=|\Phi(Y_{j})|$ and the $s_{i}$ is verified easily. The
property (7.5) follows directly from the definition of $P$ and $S$. ∎

###  7.3. Restrictions on the Pansu differential arising from the pullback of
$2$-forms

For the remainder of Section 7 we fix for some $p>\nu$ a
$W^{1,p}_{\operatorname{loc}}$-mapping $f:G\supset U\rightarrow G$, where $U$
is open and the sign of the determinant of the Pansu differential
$D_{P}f(x):\mathfrak{g}\rightarrow\mathfrak{g}$ is constant almost everywhere.
At each point $x\in U$ of Pansu differentiability the Pansu differential,
viewed as an automorphism of the algebra
$\mathfrak{g}=\tilde{\mathfrak{g}}/K$, can be identified with an element of
$\operatorname{Stab}(K,\operatorname{Aut}(\tilde{\mathfrak{g}}))$, which we
denote by

(7.6)
$\tilde{D}_{P}f(x)\in\operatorname{Stab}(K,\operatorname{Aut}(\tilde{\mathfrak{g}})).$

Recalling the notation in Proposition 7.4 we set

(7.7) $\lambda(x)=\lambda_{\tilde{D}_{P}f(x)},\quad
S(x)=S_{\tilde{D}_{P}f(x)},\quad P(x)=P_{\tilde{D}_{P}f(x)}.$

We also define

(7.8) $\hat{K}:=\\{m\in\mathbb{R}^{n}:\sum_{i=1}^{n}m_{i}Y_{i}\in K\\}.$

Note that $\tilde{D}_{P}f(x)$ preserves $K$ and thus by Proposition 7.4

(7.9) $S(x)P(x)\hat{K}=\hat{K}.$

The key calculation is contained in the following result. Recall that
$\gamma_{i}=\theta_{2i-1}\wedge\theta_{2i}$ and $Z_{i}=X_{3i-2}\wedge
X_{3i-1}$.

######  Lemma 7.10.

Let $k\in\\{1,\ldots,n\\}$, $X\in V_{1,k}$ and $Z=\sum_{i=1}^{n}m_{i}Z_{i}$.
Then

(7.11) $\displaystyle d(i_{Z}\omega)$ $\displaystyle=$ $\displaystyle
0\quad\forall m\in\hat{K},$ (7.12) $\displaystyle d(i_{X}i_{Z}\omega)$
$\displaystyle=$ $\displaystyle 0\quad\forall m\in\hat{K}\cap e_{k}^{\perp}.$

For $\varphi\in C_{c}^{\infty}(U)$ we have

(7.13) $f_{P}^{*}(\gamma_{i})\wedge d(\varphi i_{X}i_{Z}\omega)=(\lambda
SPm)_{i}X\varphi\,\omega\quad\forall m\in\hat{K}\cap e_{k}^{\perp}.$

###### Proof.

Since $Z_{i}=X_{3i-2}\wedge X_{3i-1}$ and $[X_{3i-2},X_{3i-1}]=-\tilde{Y}_{i}$
we get from (2.6) and (2.14)

$d(i_{Z_{i}}\omega)=d(i_{X_{3i-1}}i_{X_{3i-2}}\omega)=i_{Y_{i}}\omega$

where $i_{Y_{i}}\omega$ is a shorthand for $i_{Y_{i}+K}\omega$ (see the end of
Section 2.2). If $m\in\hat{K}$ then $Y:=\sum_{i=1}^{n}m_{i}Y_{i}\in K$ and
thus

$d(i_{Z}\omega)=i_{Y}\omega=0$

since $\omega$ annihilates $K$. Similarly to prove (7.12) we use (2.15) and
get

$d(i_{X}i_{Z_{i}}\omega)=-i_{X}i_{Y_{i}}\omega\quad\forall i\neq k.$

Multiplying by $m_{i}$, summing over $i$ and using that $m_{k}=0$ we get
$d(i_{X}i_{Z}\omega)=-i_{X}i_{Y}\omega=0$ since $Y\in K$.

To show (7.13) we compute

$\displaystyle\,f_{P}^{*}(\gamma_{i})\wedge
d(\varphi\,i_{X}i_{Z}\omega)=f_{P}^{*}(\gamma_{i})\wedge d\varphi\wedge
i_{X}i_{Z}\omega$ $\displaystyle=$
$\displaystyle\,(f_{P}^{*}(\gamma_{i})\wedge d\varphi)(Z\wedge
X)\,\omega=f_{P}^{*}(\gamma_{i})(Z)\,d\varphi(X)\,\omega.$

To get the second equality we used the identity $i_{X}\circ i_{Z}=i_{Z\wedge
X}$ (see (2.6)) and (2.8). For the last equality we used $m_{k}=0$ and
$f_{P}^{*}(\gamma_{i})(X,Y)=0$ if $X\in V_{1,k}$ and $Y\in V_{1,j}$ with
$j\neq k$ since $f_{P}^{*}(\gamma_{i})$ is a linear combination of the forms
$\gamma_{j}$. Now it follows from Proposition 7.4 that

$f_{P}^{*}(\gamma_{i})(Z)=\gamma_{i}((D_{P}f)_{*}Z)=\sum_{j=1}^{m}(\lambda
SP)_{ij}m_{j}=(\lambda SPm)_{i}.$

This concludes the proof of (7.13). ∎

######  Lemma 7.14.

Let $k\in\\{1,\ldots,n\\}$. Then

(7.15) $X(\lambda SPm)=0\quad\forall X\in V_{1,k},\,\,m\in\hat{K}\cap
e_{k}^{\perp},$

in the sense of distributions on $U$ and

(7.16) $\lambda$ is locally constant almost everywhere in $U$.

###### Proof.

By Lemma 7.10, the form $i_{X}i_{Z}\omega$ is a closed codegree $3$ form with
weight $\leq-\nu+3$. Thus $d(\varphi i_{X}i_{Z}\omega)=d\varphi\wedge
i_{X}i_{Z}\omega$ is a codegree $2$ with weight $\leq-\nu+2$. Moreover
$\gamma_{i}$ is a closed $2$-form of weight $-2$. Hence the identity (7.15)
follows from the Theorem 2.1 and equation (7.13).

To prove (7.16) it suffices to show $X\lambda=0$ for all $X\in V_{1,k}$ and
all $k=1,\ldots,n$. Fix $k$. Since $\dim K\geq 2$ the set $\hat{K}\cap
e_{k}^{\perp}$ contains an element $m$ with $|m|=1$. Since $P$ and $S$ are
isometries and $\lambda>0$ we have $\lambda=|\lambda SPm|$. Thus the desired
assertion $X\lambda=0$ follows from Lemma 2.25 by taking $h(v)=|v|$. ∎

###  7.4. Restrictions from codegree $2$ forms and constant action on $K$

We continue the analysis of the Sobolev map $f:U\rightarrow G$. The map $SP$
is an isometry of $\mathbb{R}^{n}$ and maps $\hat{K}$ to itself. Hence
$(SP)^{T}=(SP)^{-1}$ also maps $\hat{K}$ to itself. Thus (7.15) is equivalent
to

(7.17) $X((\lambda SP)^{T}a,m)=0\quad\forall a\in\hat{K},\,m\in\hat{K}\cap
e_{k}^{\perp},\,X\in V_{1,k}.$

We already know that $\lambda$ is locally constant. To show that
$SP|_{\hat{K}}$ is locally constant we need to show that in addition

(7.18) $X((SP)^{T}a,e_{k})=0\quad\forall a\in\hat{K},\,X\in V_{1,k}.$

This will be achieved by pulling back suitable closed codegree $2$ forms. The
main identities are contained in the following lemma.

######  Lemma 7.19.

Let $\Phi\in\operatorname{Stab}(K,\operatorname{Aut}(\tilde{\mathfrak{g}}))$,
viewed as an element of $\operatorname{Aut}(\mathfrak{g})$, let
$\lambda_{\Phi}$, $S_{\Phi}$ and $P_{\Phi}$ be the quantities introduced in
Proposition 7.4. Let $m\in\hat{K}$ and set $Z=\sum_{i=1}^{n}m_{i}Z_{i}$. Then

(7.20) $\displaystyle|\det\Phi|$ $\displaystyle=$
$\displaystyle\lambda_{\Phi}^{2n-\dim K},$ (7.21) $\displaystyle
d(i_{Z}\omega)$ $\displaystyle=$ $\displaystyle 0,$

and for every two-form $\alpha$

(7.22)
$\Phi^{*}(i_{Z}\omega)\wedge\alpha=\det\Phi\,\alpha(\Phi^{-1}Z)\,\omega.$

###### Proof.

The determinant of $\Phi$ is characterised by the identity
$\Phi^{*}(\omega)=\det\Phi\,\omega$. It follows from Proposition 7.4 that

(7.23)
$\Phi^{*}(\wedge_{i=1}^{n}\gamma_{i})=\lambda^{n}\det(SP)\wedge_{i=1}^{n}\gamma_{i}=\pm\lambda^{n}\wedge_{i=1}^{n}\gamma_{i}.$

Moreover $V_{2}=\tilde{V}_{2}/K$ can be identified with $K^{\perp}$ and
$\lambda^{-1}\Phi$, viewed as a map on $\tilde{V}_{2}$, is an isometry and
preserves $K$ and hence $K^{\perp}$. Thus $\lambda^{-1}\Phi$ restricted to
$K^{\perp}$ has determinant $\pm 1$ and hence

$\Phi^{*}(\tau)=\pm\lambda^{n-\dim K}\tau$

for every volume form $\tau$ on $V_{2}$. Together with (7.23) this implies
(7.20).

The second assertion is just (7.11).

To show (7.22) we compute

$\displaystyle\,\Phi^{*}(i_{Z}\omega)\wedge\alpha=\Phi^{*}(i_{Z}\omega\wedge(\Phi^{-1})^{*}\alpha)$
$\displaystyle=$
$\displaystyle\,\det\Phi\,i_{Z}\omega\wedge(\Phi^{-1})^{*}\alpha$
$\displaystyle=$
$\displaystyle\,\det\Phi\,\big{(}(\Phi^{-1})^{*}\alpha\big{)}(Z)\,\omega=\det\Phi\,\alpha(\Phi^{-1}Z)\,\omega.$

∎

######  Theorem 7.24.

The restriction of $\lambda SP$ to $\hat{K}$ is locally constant almost
everywhere in $U$.

###### Proof.

Let $Z$ be as in Lemma 7.19, and pick $k\in\\{1,\ldots,n\\}$. Let
$\eta=\varphi\theta_{3k-1}$ and $\varphi\in C^{\infty}_{c}(U)$. Thus
$d\eta=d\varphi\wedge\theta_{3k-1}$. Now $i_{Z}\omega$ has codegree $2$ and
weight $-\nu+2$ while the form $d\eta$ has weight $\leq-2$. Thus the pullback
theorem applies to $f_{P}^{*}(i_{Z}\omega)\wedge d\eta$. Since the sign of
$\det D_{P}f$ is constant almost everywhere by hypothesis, after post-
composing with a graded automorphism if necessary, we may assume without loss
of generality that $\det D_{P}f>0$ a.e. and thus Lemma 7.19 implies that

$f_{P}^{*}(i_{Z}\omega)\wedge d\eta=\lambda^{2n-\dim
K}(d\varphi\wedge\theta_{3k-1})((D_{P}f)^{-1}(Z))\,\omega.$

Since $(D_{P}f)^{-1}(Z)\in\oplus_{i=1}^{n}\Lambda_{2}V_{1,i}$ only the term
with $i=k$ contributes and we get

$\displaystyle(d\varphi\wedge$
$\displaystyle\theta_{3k-1})((D_{P}f)^{-1}(Z))=(X_{3k-2}\varphi)\,\gamma_{k}((D_{P}f)^{-1}Z)$
$\displaystyle=$
$\displaystyle(X_{3k-2}\varphi)\,\lambda^{-1}((SP)^{-1}m,e_{k}).$

Since $\lambda$ is locally constant it follows that

$X_{3k-2}((SP)^{-1}m,e_{k})=0\quad\forall m\in\hat{K}.$

Using the form $\eta=\varphi\theta_{3k-2}$ we get the same assertion with
$X_{3k-2}$ replaced by $X_{3k-1}$. Using that $(SP)^{-1}=(SP)^{T}$ we thus
conclude that

(7.25) $X((SP)^{T}a,e_{k})=0\quad\forall a\in\hat{K},\,X\in V_{1,k}.$

Combining this with (7.17) we deduce that $X(SP)^{T}a=0$ for all
$a\in\hat{K}$, all $X\in V_{1,k}$ and all $k$. Therefore $\lambda SP$ is
locally constant, as desired. ∎

###  7.5. Constancy of the permutation

It follows from Theorem 7.24 that after composing $f$ with a graded
automorphism of $G$ and shrinking $U$ we may assume without loss of generality
that $\tilde{D}_{P}f$ acts as the identity on $K$. In this subsection we
analyse the graded automorphisms which act as the identity on $K$ and then
show that the permutation induced by $\tilde{D}_{P}f$ is locally constant.

To simplify notation, we let $H:=\operatorname{Aut}(\mathfrak{g})$ be the
graded automorphism group of $\mathfrak{g}$. By Lemma 4.16 we know that $H$
canonically embeds in $\operatorname{Aut}(\tilde{\mathfrak{g}})$ as the
stabilizer of $K\subset\tilde{V}_{2}\subset\tilde{\mathfrak{g}}$. Therefore we
have actions $H\operatorname{\curvearrowright}\mathfrak{g}$,
$H\operatorname{\curvearrowright}\tilde{\mathfrak{g}}$ by graded
automorphisms, as well as the induced action
$H\operatorname{\curvearrowright}I=\\{1,\ldots,n\\}$ on the (set of indices of
the) summands $\tilde{\mathfrak{g}}_{i}$, and the restriction action
$H\operatorname{\curvearrowright}K$. We let

$H^{\prime}:=\\{h\in H\mid h\cdot W=W\,,\;\forall W\in K\\}\subset H$

be the ineffective kernel of the action $H\operatorname{\curvearrowright}K$;
since $H^{\prime}$ is the kernel of a homomorphism, it is a normal subgroup of
$H$.

We let $I=I_{1}\sqcup\ldots\sqcup I_{k}$ be the decomposition of $I$ into the
distinct orbits of the action $H^{\prime}\operatorname{\curvearrowright}I$; as
$H^{\prime}$ is normal in $H$, the $H^{\prime}$-orbit decomposition is
respected by $H$.

We let $\tilde{V}_{2,I_{j}}=\oplus_{i\in I_{j}}\tilde{V}_{2,i}$, so we have
direct sum decomposition $\tilde{V}_{2}=\oplus_{j}\tilde{V}_{2,I_{j}}$. For
every $j\in\\{1,\ldots,k\\}$, let $K_{j}:=\pi_{\tilde{V}_{2,I_{j}}}(K)$.

######  Lemma 7.26.

$\dim K_{j}=1$ for all $j\in\\{1,\ldots,k\\}$.

###### Proof.

We first claim that $H$ preserves the collection $\\{K_{j}\\}_{1\leq j\leq
k}$, and acts transitively on it. To see this, note that $H$ preserves $K$ and
the direct sum decomposition $\tilde{V}_{2}=\oplus_{j}\tilde{V}_{2,I_{j}}$.
Therefore if $h\in H$ and $h(I_{j})=I_{j^{\prime}}$, then
$h\circ\pi_{\tilde{V}_{2,I_{j}}}=\pi_{\tilde{V}_{2,I_{j^{\prime}}}}\circ h$,
so $h(K_{j})=K_{j^{\prime}}$. Since
$H\operatorname{\curvearrowright}\\{I_{1},\ldots,I_{k}\\}$ is transitive, this
proves the claim.

It follows that all the $K_{j}s$ have the same dimension, and since
$K\subset\oplus_{j}K_{j}$ we must have $\dim K_{j}\geq 1$ for all $j$.

Now suppose $\dim K_{j}\geq 2$ for some $j$. Choose $i_{0}\in I_{j}$. Since
$|I_{j}|\geq\dim K_{j}\geq 2$, the set $I_{j}\setminus\\{i_{0}\\}$ is
nonempty, and it follows that $K_{j}$ intersects the hyperplane $\oplus_{i\in
I_{j}\setminus\\{i_{0}\\}}\tilde{V}_{2,i}$ in some vector $W\neq 0$. Let
$J:=\\{i\in I_{j}\mid\pi_{\tilde{V}_{2,i}}(W)\neq 0\\}$. Note that $J$ is
invariant under $H^{\prime}$, since $H^{\prime}$ fixes $W$ and preserves the
direct sum decomposition $\tilde{V}_{2}=\oplus_{i}\tilde{V}_{2,i}$. Since
$I_{j}$ is an $H^{\prime}$-orbit, we must have $J=I_{j}$. This contradicts the
fact that $J\subset I_{j}\setminus\\{i_{0}\\}$. ∎

Using an argument similar to Lemma 6.5, up to a graded isomorphism, we may
assume without loss of generality that each $K_{j}\subset V_{2,I_{j}}$ is
“diagonal”, i.e. $K_{j}=\operatorname{span}(\sum_{i\in I_{j}}Y_{i})$ for all
$1\leq j\leq k$.

######  Theorem 7.27.

The permutation $\sigma_{x}$ induced by the Pansu differential
$\tilde{D}_{P}f(x)$ is locally constant almost everywhere in $U$.

###### Proof.

Recall that we have reduced to the case when $\tilde{D}_{P}f(x)$ acts as the
identity on $K$ and $K_{j}=\operatorname{span}(\sum_{i\in I_{j}}Y_{i})$ for
all $j\in\\{1,\ldots k\\}$. Thus $\tilde{D}_{P}f(x)\in H^{\prime}$ for a.e.
$x\in U$. If $H^{\prime}=\\{\operatorname{id}\\}$ we are done. So we assume
that $H^{\prime}\neq\\{\operatorname{id}\\}$ and will show that
$\tilde{D}_{P}f(x)$ acts as a constant permutation on each $H^{\prime}$-orbit
$I_{j}$, for $j\in\\{1,\ldots,k\\}$. Then we can argue as in the case $\dim
K=1$ to deduce constancy of the permutation.

We provide some details for the convenience of the reader. Let $I$ be an orbit
and $\tilde{V}_{2,I}=\oplus_{i\in I}\tilde{V}_{2,i}$. The map
$\tilde{D}_{P}f(x)$ preserves $\tilde{V}_{2,I}$ and acts as the identity on
the subspace $K_{I}=\operatorname{span}\\{\sum_{i\in I}\tilde{Y}_{2,i}\\}$.
Thus the argument in the proof of Proposition 6.7 shows that the identities in
that proposition hold with $\lambda=1$. Recall that
$P_{mi}(x)=\delta_{m\sigma_{x}^{-1}(i)}$. Applying the pullback theorem to the
forms $f_{P}^{*}\omega_{ij}\wedge d(\varphi i_{X}\beta)$ where $\omega_{ij}$
and $\beta$ are as in (6.10) and (6.13) with $i,j,m\in I$ and where $X\in
V_{1,l}$ with $l\neq m$ we get as in the proof of Lemma (6.18) the
distributional identity $XP_{mi}=0$ for all $m,i\in I$ and all $X\in V_{1,l}$
with $l\neq m$. Since $\sigma_{x}(I)=I$ we have $\sum_{m\in I}P_{mi}=1$ for
all $i\in I$. Now for $X\in V_{1,m}$ we get $XP_{mi}=X(1-\sum_{m^{\prime}\in
I\setminus\\{m\\}}P_{m^{\prime}i})=0$. Thus $P_{mi}$ is constant for $m,i\in
I$ and hence $\sigma_{x}|_{I}$ is constant. ∎

##  8\. Product quotients of complex Heisenberg groups and higher real
Heisenberg groups

In this section we prove Theorem 5.1 for product quotients
$G=\tilde{G}/\exp(K)$ where $\tilde{G}$ is a product of copies of a higher
real Heisenberg group or complex Heisenberg group. We retain the notation for
product quotients from Section 5. The approach used in this section is
different, and conceptually simpler, than the ones used in Sections 6 and 7.
The main difference between these two situations is that for the higher real
and the complex Heisenberg group there exist linearly independent vectors
$X,Y$ in $V_{1,i}$ with $[X,Y]=0$ while this is not the case for the first
Heisenberg group. More systematically, consider the map

(8.1) $L_{[\,]}:\Lambda_{2}V_{1}\to V_{2},\qquad L_{[\,]}(X\wedge Y)=[X,Y].$

For the complex Heisenberg groups and higher real Heisenberg groups

(8.2) $\ker_{i}:=\ker L_{[\,]}\cap V_{1,i}$

is non-trivial (as follows from a dimension count) while for the first real
Heisenberg group $L_{[\,]}:\Lambda_{2}V_{1,i}\to V_{2}$ is injective. Any
graded automorphism – in particular the Pansu differential – permutes the
subspaces $\ker_{i}$. Now for complex or higher real Heisenberg product
quotients we can pull back a collection of forms $\alpha\in\Lambda^{2}V_{1,i}$
which restrict to a basis of the dual space $\ker_{i}^{\prime}$ and take
$\beta=i_{X}i_{Z}\omega$ where $X\in\ker_{k}$ and $Z\in\oplus_{k^{\prime}\neq
k}V_{1,k^{\prime}}$. The choice of this collection of forms $\alpha$ may look
somewhat adhoc at this point, but it is in fact related to our general
strategy to look for closed forms modulo exact forms. The point is that the
map $L_{[]}$ is the dual of the exterior differential
$d:\Lambda^{1}V_{2}\to\Lambda^{2}V_{1}\simeq(\Lambda_{2}V_{1})^{\prime}$, see
Remark 8.14 after the end of the proof. With the above choice of $\alpha$ and
$\beta$ a short argument similar to the one used for products shows that the
permutation of the subspaces induced by $D_{P}f$ must be locally constant. The
subspace $K$ plays no role in this argument. In fact the argument can also be
used to obtain an alternative proof for products of the complex or the higher
real Heisenberg groups.

We now turn to the formal proof. Note first that (2.15), (2.16), the relation
$[V_{1,i},V_{1,j}]=0$ for $i\neq j$ and (2.6) imply that

(8.3)
$d(i_{Y}i_{Y^{\prime}}i_{Z}\omega)=-i_{Z}i_{[Y,Y^{\prime}]}\omega\quad\hbox{$\forall\,\,Y,Y^{\prime}\in
V_{1,i},\,\,Z\in V_{1,j}$ with $i\neq j$.}$

######  Lemma 8.4.

Let $k\neq k^{\prime}$, $Y,Y^{\prime}\in V_{1,k}$, $Z\in V_{1,k^{\prime}}$ and
$\gamma\in\oplus_{j=1}^{n}\Lambda^{2}(V_{1,j})$. For $U\subset G$ open let
$\varphi\in C^{1}(U)$. Then the following identities hold in $U$:

(8.5) $\displaystyle d(\varphi\,i_{Y}i_{Y^{\prime}}i_{Z}\omega)=$
$\displaystyle\,(Y\varphi)\,i_{Y^{\prime}}i_{Z}\omega-\,(Y^{\prime}\varphi)i_{Y}i_{Z}\omega+(Z\varphi)\,i_{Y}i_{Y^{\prime}}\omega$
$\displaystyle+$ $\displaystyle\,\varphi\,d(i_{Y}i_{Y^{\prime}}i_{Z}\omega),$
(8.6) $\gamma\wedge
d(\varphi\,i_{Y}i_{Y^{\prime}}i_{Z}\omega)=-\gamma(Y,Y^{\prime})\,(Z\varphi)\,\omega+\varphi\,\gamma\wedge
d(i_{Y}i_{Y^{\prime}}i_{Z}\omega).$

If $X\in\Lambda_{2}V_{1,k}$ then

(8.7) $\gamma\wedge
d(\varphi\,i_{X}i_{Z}\omega)=\gamma(X)\,(Z\varphi)\,\omega+\varphi\,\gamma\wedge
d(i_{X}i_{Z}\omega).$

###### Proof.

We have

$d(\varphi\,i_{Y}i_{Y^{\prime}}i_{Z}\omega)-\varphi\,d(i_{Y}i_{Y^{\prime}}i_{Z}\omega)=d\varphi\wedge
i_{Y}i_{Y^{\prime}}i_{Z}\omega.$

Now the graded Leibniz rule (2.7) for the interior derivative gives

$\displaystyle d\varphi\wedge i_{Y}i_{Y^{\prime}}i_{Z}\omega$ $\displaystyle=$
$\displaystyle(Y\varphi)i_{Y^{\prime}}i_{Z}\omega-i_{Y}(d\varphi\wedge
i_{Y^{\prime}}i_{Z}\omega)$ $\displaystyle=$
$\displaystyle(Y\varphi)i_{Y^{\prime}}i_{Z}\omega-
i_{Y}((Y^{\prime}\varphi)i_{Z}\omega)+i_{Y}i_{Y^{\prime}}(d\varphi\wedge
i_{Z}\omega)$ $\displaystyle=$
$\displaystyle(Y\varphi)i_{Y^{\prime}}i_{Z}\omega-(Y^{\prime}\varphi)i_{Y}i_{Z}\omega+(Z\varphi)i_{Y}i_{Y^{\prime}}\omega.$

In the last step we used (2.8). This proves (8.5).

The assertion (8.6) now follows from (2.8) and the fact that
$\gamma(Y^{\prime},Z)=\gamma(Y,Z)=0$ since $Z$ lies in a different first layer
subspace than $Y$ and $Y^{\prime}$. Note that (2.6) gives the identity
$i_{Y}\circ i_{Y^{\prime}}=-i_{Y\wedge Y^{\prime}}$ and thus
$i_{Y}i_{Y^{\prime}}\gamma=-\gamma(Y,Y^{\prime})$.

Finally for $X=-Y\wedge Y^{\prime}$ (8.7) is the same as (8.6) and for general
$X$ the identity (8.7) follows by linearity. ∎

We now assume that $f:G\supset U\rightarrow G$ is a
$W^{1,p}_{\operatorname{loc}}$-mapping for some $p>\nu$ such that $D_{P}f(x)$
is an isomorphism for a.e. $x\in U$. Recall that $\ker_{i}$ is defined in
(8.2). The main point in the proof of Theorem 5.1 for product quotients of the
higher groups is to show the following identity

(8.8)
$\displaystyle\int_{U}\alpha((D_{P}f)_{*}X)\,(Z\varphi)=0\qquad\forall\varphi\in
C_{c}^{\infty}(U),$
$\displaystyle\forall\alpha\in\oplus_{j=1}^{n}\Lambda^{2}V_{1,j},\,\,X\in\ker_{k},\,\,Z\in
V_{1,k^{\prime}}\quad\hbox{with $k^{\prime}\neq k$.}$

Then we can deduce easily that the permutation induced by $D_{P}f$ is locally
constant.

To prove (8.8) fix $\alpha$, $X$, $Z$ and $\varphi$ and consider the codegree
$3$ forms

$\beta=i_{X\wedge Z}\omega\quad\hbox{and}\quad\eta=\varphi\beta.$

It follows from (8.3) and linearity that $\beta$ is closed. Moreover
$\operatorname{wt}(\beta)=-\nu+3$. Thus Theorem 2.1 implies that

(8.9) $\int_{U}f_{P}^{*}\alpha\wedge d\eta=0.$

Since $D_{P}f$ permutes the first layer subspaces $V_{1,j}$ it follows that
$f_{P}^{*}\alpha\in\oplus_{j=1}^{n}\Lambda^{2}V_{1,j}$. Thus we get from (8.7)
and (8.3)

$f_{P}^{*}\alpha\wedge d\eta=f_{P}^{*}\alpha(X)\,(Z\varphi)\,\omega.$

This yields (8.8).

To deduce properties of the permutation $\sigma_{x}$ induced by $D_{P}f(x)$
recall that $D_{P}f(x)$ maps $\ker_{k}\subset\Lambda_{2}V_{1,k}$ to
$\ker_{\sigma_{x}(k)}$. Let

$P_{kl}(x)=\delta_{\sigma_{x}(k)l}.$

If we choose basis vectors $X_{k,m}$ of $\ker_{k}$ then there exist measurable
functions $G_{mm^{\prime}}:U\to\mathbb{R}$ such that

$(D_{P}f)_{*}X_{k,m}=\sum_{l=1}^{n}P_{kl}\sum_{m^{\prime}}G_{m^{\prime}m}X_{l,m^{\prime}}$

and the matrix $G(x)$ is invertible for a.e. $x$. Let
$\alpha_{k,m^{\prime}}\in\Lambda^{2}V_{1,k}$ be chosen such that the
restrictions to $\ker_{k}$ yield a basis of the dual space $\ker_{k}^{\prime}$
which is dual to the basis $X_{k,m}$, i.e.
$\alpha_{k,m^{\prime}}(X_{k,m})=\delta_{mm^{\prime}}$ for each $k=1,\ldots,n$.
Applying (8.8) with $X=X_{k,m}$ and $\alpha=\alpha_{l,m^{\prime}}$ we get for
each $k$ and $l$

(8.10) $\int_{U}P_{kl}G\,(Z\varphi)\,\omega=0\quad\forall\varphi\in
C_{c}^{\infty}(U),\,\,Z\in V_{1,k^{\prime}}\quad\hbox{with $k^{\prime}\neq
k$}.$

Define a function $h$ on the space of matrices by $h(G)=1$ if $\det G\neq 0$
and $h(G)=0$ else. Then $h$ is a Borel function. Since the matrix $G$ in
(8.10) is invertible a.e. in $U$, we have $h(P_{kl}G)=P_{kl}$ a.e. and it
follows from Lemma 2.25 that

(8.11) $\int_{U}P_{kl}\,(Z\varphi)\,\omega=0\quad\forall\varphi\in
C_{c}^{\infty}(U),\,\,Z\in V_{1,k^{\prime}}\quad\hbox{with $k^{\prime}\neq
k$}.$

In other words,

(8.12) $ZP_{kl}=0\quad\forall l,\quad\forall Z\in
V_{1,k^{\prime}},\,\,k^{\prime}\neq k\,\,$

in the sense of distributions. By exchanging $k$ and $k^{\prime}$ we get

(8.13) $ZP_{k^{\prime}l}=0\quad\forall l,\quad\forall Z\in
V_{1,k},\,\,k^{\prime}\neq k.\,\,$

Since $P_{kl}=1-\sum_{k^{\prime}\neq k}P_{k^{\prime}l}$ we deduce that
$ZP_{kl}=0$ for all $Z\in V_{1}$ and all $k,l$. Thus $P_{kl}$ is locally
constant and hence the permutation of the first layer subspaces induced by
$D_{P}f(x)$ is locally constant. This completes the proof.

######  Remark 8.14.

The choice of the forms $\alpha_{k,l}$ can be motivated by our guiding
principle to look for closed left invariant forms modulo exact left invariant
forms (see [KMX20, Lemma 4.8]) ). First note by (2.22) we have
$d\alpha(X,Y)=-\alpha([X,Y])$ for $\alpha\in\Lambda^{1}V_{2}$. If we identify
two-forms on $V_{1}$ with the dual space of two-vectors by setting
$\alpha(X\wedge Y)=\alpha(X,Y)$, then we see that exterior differentiation
$d:\Lambda^{1}V_{2}\to(\Lambda_{2}V_{1})^{\prime}$ is dual map of $-L_{[]}$.
Now consider first the case that $\mathfrak{g}=\oplus_{i}\mathfrak{h}_{i}$ is
a direct sum of $n$ copies of the same (higher) Heisenberg group. A natural
space of closed left invariant forms to detect the permutation $\sigma_{x}$
induced by the Pansu differential is the space $\oplus_{i}\Lambda^{2}V_{1,i}$.
In this case the relevant space of exact forms is given by
$d\Lambda^{1}V_{2}=\oplus_{i}d\Lambda^{1}V_{2,i}$. By the duality between $d$
and $-L_{[]}$ all elements of $d\Lambda^{1}V_{2}$ vanish on all the space
$\ker_{i}$ and the sets $\ker_{i}$ are characterized by this condition. Thus
to find a basis for the quotient space
$\oplus_{i}\Lambda^{2}V_{1,i}/d\Lambda^{1}V_{2}$ it is natural to look for
elements of $\Lambda^{2}V_{1,i}$ whose restriction to $\ker_{i}$ does not
vanish. This leads to the basis $\alpha_{k,l}$ used in the proof. The fact
that there exist non-zero $X\in\ker_{i}$ is crucial for finding closed
codegree forms $\beta=i_{Z}i_{X}\omega$ which interact well with the forms
$\alpha_{k,l}$.

If $\mathfrak{g}$ is a product quotient $\oplus_{i}\tilde{\mathfrak{g}}_{i}/K$
then it is still true that each element of $d\Lambda^{1}V_{2}$ vanishes on
each space $\ker_{i}$. Hence the collection of cosets
$\alpha_{k,l}+d\Lambda^{1}V_{2}$ is still linearly independent in the quotient
space and the proof shows that it is sufficiently rich to detect the
permutation $\sigma_{x}$.

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