# Introducing Reducing-Width-QNNs, an AI-inspired Ansatz design pattern

Jonas Stein† †These authors contributed equally. 0000-0001-5727-9151 LMU
Munich
<EMAIL_ADDRESS>Tobias Rohe† LMU Munich
<EMAIL_ADDRESS>Francesco Nappi  LMU Munich
<EMAIL_ADDRESS>Julian Hager  LMU Munich
<EMAIL_ADDRESS>David Bucher 0009-0002-0764-9606 Aqarios GmbH
<EMAIL_ADDRESS>Maximilian Zorn 0009-0006-2750-7495 LMU Munich
<EMAIL_ADDRESS>Michael Kölle 0000-0002-8472-9944 LMU Munich
<EMAIL_ADDRESS>Claudia Linnhoff-Popien 0000-0001-6284-9286 LMU
Munich
<EMAIL_ADDRESS>
###### Abstract

Variational Quantum Algorithms are one of the most promising candidates to
yield the first industrially relevant quantum advantage. Being capable of
arbitrary function approximation, they are often referred to as Quantum Neural
Networks (QNNs) when being used in analog settings as classical Artificial
Neural Networks (ANNs). Similar to the early stages of classical machine
learning, known schemes for efficient architectures of these networks are
scarce. Exploring beyond existing design patterns, we propose a _reducing-
width_ circuit Ansatz design, which aims at mitigating the common problem of
vanishing gradients caused by barren plateaus in the parameter training of
QNNs. Our design of gradually width-reduced ansatz-layers is inspired by the
similar reduction of layer-width in classical ANNs such as the encoder
component in autoencoders. We evaluate our approach in a VQE ansatz to the
maximum cut problem and identify its potential for increasingly deep circuits
in terms of training time and result quality.

###### Index Terms:

Quantum Computing, Variational Quantum Circuits, Circuit Optimization,
Variational Quantum Eigensolver

##  I Introduction

A central goal of current quantum computing research is achieving an
industrially relevant quantum advantage. As immense computational requirements
of known quantum algorithms enabling speedups exceed prevailing hardware
capabilities in the present NISQ era [1], less demanding approaches like
Quantum Neural Networks (QNNs) are of great interest, especially as they have
already shown very promising results towards a possible quantum advantage [2].
QNNs closely resemble classical Artificial Neural Networks (ANNs), as they are
composed of concatenated, parameterized functions capable of approximating
arbitrary functions [3]. The core difference between QNNs and ANNs is the type
of utilized functions: While ANNs employ typically non-linear _activation_
functions mapping vectors to scalar values [4], QNNs consist of predominantly
linear but exponentially high dimensional functions in terms of the number of
qubits used [3].

A key advantage of QNNs compared to established quantum algorithms like those
of Shor [5] or Grover [6] is their flexibility towards respecting restricted
hardware capabilities. A specific example to this can be observed in
optimization problems, as QNNs enable us to utilize specifically optimized
quantum circuits to solve given problem instances with a smaller circuit depth
than the comparatively deep standard quantum optimization algorithm QAOA [7].
Analogous to the historical progress on ANNs, where many useful and performant
network architectures and design patterns emerged, recent literature on QNNs
explores productive quantum circuit design patterns [8, 9, 10, 11, 12, 13, 14,
15]. Similar to ANNs, these techniques are used to find a sufficiently good
trade-off between _expressibility_ and _trainability_ , i.e., either the model
is simple to train but cannot represent difficult functions or it is
sufficiently expressive to model a complex function, but is hard to train [9].
Aiming to find a suitable design pattern for QNNs that supports sufficiently
expressive yet trainable quantum circuits, we set out to investigate the
transferability of existing techniques for ANNs to QNNs.

We realize this task exemplary by studying the transferability of the well-
established design pattern of reducing the number of applied functions for
subsequent layers in ANNs to QNNs. This design pattern is ubiquitous in many
ANN classes such as the encoder component of autoencoders [16]. Apart from its
success in ANNs, our motivation in choosing this design pattern also targets
improving the trainability of the QNN, which is largely influenced by the
number of parameters, which in turn depends on the number of used functions.
To test the effectiveness of our approach, we compare QNNs with a reducing
number of functions per layer with QNNs with the exact same network
architecture, but a constant number of functions per layer. Our main
contributions in this paper can be summarized as the following:

  * •

We propose a novel, AI-inspired QNN design pattern, that can notably also be
applied as a pruning technique for a given QNN.

  * •

We conduct a case study of this design pattern for the maximum cut
optimization problem, including noise effects on the circuits created based on
our design pattern.

This work is structured as follows. Sec. II outlines necessary background
knowledge on solving optimization problems with QNNs and Sec. III gives an
overview of known design patterns from literature. Subsequently, we present
our novel design pattern in Sec. IV. Following this, Sec. V displays an
extensive evaluation including the noise analysis. Finally, Sec. VI concludes
our findings and states possible future work.

##  II Background

In the following, we describe the core algorithms and techniques employed in
our methodology. We show how QNNs can be utilized to solve specific
optimization problems like the maximum cut problem as part of the Varational
Quantum Eigensolver (VQE) [7].

###  II-A Quantum Neural Networks

In current literature, the term Quantum Neural Network is predominantly used
to describe a Parameterized Quantum Circuit (PQC) acting similar to a
classical Artificial Neural Network [17, 18]. Although the terminology
suggests a close similarity between QNNs and ANNs, there exist significant
differences. The main distinction is that QNNs do not consist of individual
neurons, instead they are comprised of qubits, which can interact with each
other via (parameterized) quantum gates. Despite the substantial differences
between ANNs and QNNs, there also exist strong commonalities, foremost the
shared capability of universal function approximation, i.e., ANNs and QNNs can
both be described as parameterized functions, that can approximate arbitrary
functions under certain conditions [10]. Moreover, representing QNNs and
feedforward ANNs as mathematical functions, we can observe clear similarities.
A feedforward neural network is mathematically represented by concatenations
of typically non-linear activation functions for all neurons in each layer
[19]. In contrast, QNNs model concatenations of typically linear layers with a
single non-linear layer at the end, which is responsible for the measurement,
i.e., generating a classical output [3]. For examples of feedforward ANNs and
QNNs, see Figs. 1 and 2.

Figure 1: Example network architecture of the encoder in an autoencoder.

Mathematically, the final state of a QNN composed of $n\in\mathbb{N}$ qubits
can be formalized as denoted in Eq. 1111This formalization explicitly
disregards intermediate measurements, in-line with predominant literature.. In
this formalization, we started in the standard initial state
$\left|0\right\rangle^{\otimes n}$ and subsequently applied a parameterized
unitary operation $U\left(\theta\right)$, with parameters
$\theta\in\mathbb{R}^{m}$, where $m\in\mathbb{N}$ denotes the number of
parameters. Notably, any desired data input $x\in\mathbb{C}^{k}$ has to be
conducted with the application of $U\left(\theta\right)$, which is thus often
denoted as $U\left(\theta,x\right)$. In practice, $U$ is predominantly
decomposed into parameterized single-qubit and two-qubit gates to match native
gate sets of existing quantum computers.

$\displaystyle\left|\psi\left(\theta\right)\right\rangle\coloneqq
U\left(\theta\right)\left|0\right\rangle^{\otimes n}$ (1)

To extract classical information from the final state of the QNN, we tyically
measure each qubit $\left|\psi_{i}\right\rangle$ in the computational basis
$\left\\{\left|0\right\rangle,\left|1\right\rangle\right\\}$ using the
hermitian measurement operator $M=I^{i-1}\otimes\sigma_{z}\otimes I^{n-i}$,
where $\sigma_{z}$ denotes the Pauli-$Z$ operator. These measurements yield a
probability distribution over bitstrings
$\left|x\right\rangle\in\mathbb{C}^{2^{n}}$ representing the output of the
quantum circuit, where $\left|x\right\rangle$ occurs at probability
$\left|\left\langle x\middle|\psi\left(\theta\right)\right\rangle\right|^{2}$.

Having presented an overview of the circuit components of a QNN, we now
address parameter training approaches. Analogous to ANNs, two main types of
parameter optimization approaches exist: gradient-based and derivative-free
methods [3]. While QNNs can be trained with essentially the same techniques as
ANNs, i.e., using approaches like gradient descent, genetic algorithms, and
others, the gradient computation for QNNs is fundamentally more expensive.
Using backpropagation, the gradient in feedforward ANNs can be calculated as
fast as a forward pass of the ANN, whereas all known quantum gradient
computations need $\mathcal{O}\left(m\right)$ executions of the QNN, where
$m\in\mathbb{N}$ denotes the number of parameters [3, 20]. The standard method
to calculate the gradient of a parameterized quantum circuit is the parameter
shift rule, which is used to calculate each partial derivative
$\nicefrac{{\partial}}{{\partial\theta_{i}}}U\left(\theta\right)\left|0\right\rangle^{\otimes
n}$ based on Eq. 2, given that all parameters are enclosed in single qubit
gates and
$\theta^{\pm}\coloneqq\left(\theta_{1},...,\theta_{i-1},\theta_{i}\pm\nicefrac{{\pi}}{{2}},\theta_{i+1},...,\theta_{m}\right)$
[3, 20].

$\displaystyle\dfrac{\partial}{\partial\theta_{i}}U\left(\theta\right)\left|0\right\rangle^{\otimes
n}=\dfrac{U\left(\theta^{+}\right)\left|0\right\rangle^{\otimes
n}-U\left(\theta^{-}\right)\left|0\right\rangle^{\otimes n}}{2}$ (2)

Given this linear runtime increase in the gradient computation for QNNs,
achieving better solution quality or a quick convergence to sufficiently good
parameters is essential for QNNs.

###  II-B The Maximum Cut Problem in Quantum Optimization

The _maximum cut_ problem, is an NP-hard combinatorial optimization problem
with the objective of partitioning the nodes $v_{i}\in V$ of a given graph
$G\left(V,E\right)$ into two complementary sets $A\dot{\cup}B=V$ such that
number of edges between $A$ and $B$ is maximal. Encoding a partitioning via a
binary vector $s\in\left\\{-1,1\right\\}^{n}$, where
$n\coloneqq\left|V\right|$ and $s_{i}=1$ $\Leftrightarrow$ $v_{i}\in A$ and
$s_{i}=-1$ $\Leftrightarrow$ $v_{i}\in B$, the search for the optimal
partition can be formulated as shown in Eq. II-B [21]. This formulation
naturally evolves after observing that
$\nicefrac{{\left(1-s_{i}s_{j}\right)}}{{2}}\in\left\\{0,1\right\\}$ equals
one iff the edge $\left(v_{i},v_{j}\right)\in E$ connects $A$ and $B$.

$\displaystyle\operatorname*{arg\,max}_{s\in\left\\{-1,1\right\\}^{n}}\sum_{(v_{i},v_{j})\in
E}\dfrac{1-s_{i}s_{j}}{2}$ $\displaystyle=$
$\displaystyle\operatorname*{arg\,min}_{s\in\left\\{-1,1\right\\}^{n}}\sum_{(v_{i},v_{j})\in
E}s_{i}s_{j}$ (3)

We have chosen this $\pm 1$ encoding of the solution, as it can directly be
transformed into a quantum Hamiltonian $\hat{H}$ that represents the solution
space of this optimization problem, which is the standard approach in quantum
optimization. For this, we merely need to substitute $s_{i}$ using the map
$s_{i}\mapsto\sigma^{z}_{i}\coloneqq I^{\otimes i-1}\otimes\sigma_{z}\otimes
I^{\otimes n-i}$, where $I$ denotes the two dimensional identity matrix.
Therefore, we can define $\hat{H}\coloneqq\sum_{\left(v_{i},v_{j}\right)\in
E}\sigma^{z}_{i}\sigma^{z}_{j}$.

Switching to a $0/1$ encoding, i.e., $x\in\left\\{0,1\right\\}^{n}$ with
$x_{i}=0\Leftrightarrow v_{i}\in A$ and $x_{i}=1\Leftrightarrow v_{i}\in B$,
$\hat{H}$ can be understood as a mapping between solutions $x$ and the value
of the objective function. More specifically, its eigenvalues for a given
eigenvector $\left|x\right\rangle$ are the values of the objective function,
where $x$ represents the $0/1$ encoding of the solution.

###  II-C The Variational Quantum Eigensolver

The Variational Quantum Eigensolver (VQE) is an algorithm targeted towards
finding the ground state of a given Hamiltonian $\hat{H}$ using the
_variational method_ [7]. The term _variational method_ describes the process
of making small changes to a function $f$ to find minima or maxima of a given
function $g$ [22]. In the case of the VQE, $f:\theta\mapsto
U\left(\theta\right)\left|0\right\rangle^{\otimes n}$ describes a
parameterized quantum circuit yielding the state
$\left|\psi\left(\theta\right)\right\rangle\coloneqq
U\left(\theta\right)\left|0\right\rangle^{\otimes n}$ which represents a guess
of the argmin of the function
$g:\left|\varphi\right\rangle\mapsto\left\langle\varphi\right|\hat{H}\left|\varphi\right\rangle$.
More specifically, the VQE uses parameter training techniques on the given
_Ansatz_ $U\left(\theta\right)$ to generate an output state
$\left|\psi\left(\theta\right)\right\rangle$ that approximates the ground
state of $\hat{H}$.

For our purposes, i.e., solving an optimization problem, we need to (1) define
a Hamiltonian encoding our objective function and (2) select a suitable Ansatz
in order to use the VQE algorithm. While we can use the procedure exemplified
in section II-B to define a Hamiltonian $\hat{H}$, the selection of the Ansatz
and thus a circuit architecture is not thoroughly understood. Although
literature on automatized quantum architecture search tools is emerging [23],
most Ansätze are designed based on expert knowledge [24]. For more details on
Ansatz design, see Sec. III.

##  III Related Work

A suitable circuit design in QNNs is essential for achieving good results in
variational quantum computing and is actively researched in literature [24].
While some design patterns like a layerwise circuit structure with alternating
single qubit rotations and entanglement gates, have become established
concepts [24], the list of known circuit elements still is considerably
smaller than that in classical neural networks. A central problem when trying
to design circuits is the trade-off between expressibility and trainablity.
More specifically, reasonably complex models demand for expressive circuits,
which in turn can quickly lead to the problem of _barren plateaus_ , a
trainability issue where the probability that the optimizer-gradient becomes
exponentially small is directly correlated to the number of qubits [18].
Moreover, barren plateaus are found to be dependent on the choice of the cost
function itself [25] as well as the number of parameters to optimize
simultaneously [26, 27].

Classical deep learning faced a similar problem in _vanishing gradients_ for
recurrent neural networks [28], which eventually lead to the adoption of more
specialized network architectures like the (long short-term memory units
(LSTM) [29] for sequence processing. Similarly, the current trend of improving
variational architectures focuses on optimizing the designs of quantum circuit
Ansätze, e.g., via optimization procedures starting from randomized circuits
[30]. In our work, we also study the optimization of circuit expressibility
and trainability via gradual reduction of the width of the Ansatz. Our method
for this reduction of newly-constructed or given circuits can also be likened
to the process of circuit pruning. This is similar to the adaptive technique
of pruning certain sections of the circuits employed in [31], with the
difference that our pruning has the very specific structure of gradually
reducing the circuit width.

##  IV Methodology

In the subsequent sections, we describe our reducing-width Ansatz design
pattern and two different baselines: A full-width variant of the Ansatz and
one with randomly removed gates. Concluding our methodology, we describe our
approach to parameter training and problem instance generation.

###  IV-A Full-width Design

In the following, we introduce the design of our first baseline Ansatz, which
we coin _full-width design_. We choose a very simple Ansatz architecture
consisting of BasicEntanglerLayers222For details and the implementation, see
https://docs.pennylane.ai/en/stable/code/api/pennylane.BasicEntanglerLayers.html.,
as it resembles the most basic circuit architecture used for building
variational quantum circuits in quantum machine learning.

More specifically, this baseline circuit is constructed of two components. The
first is a sequence of parameterized gates, i.e., Pauli-$X$, Pauli-$Z$ and
another Pauli-$X$ gate, applied to each qubit, allowing us to describe
arbitrary single qubit rotations. The second is a circular CNOT-entanglement,
i.e., CNOTs applied to all neighboring qubit pairs, using the maximum width of
the circuit. We combine these single- and two-qubit gate components into a
single layer, where every circuit is composed of four such layers to enable a
sufficient scaling analysis. The entire circuit with all its layers is
illustrated in Fig. 2. In the following sections, we will refer to this
circuit design as the full-width circuit or full-width design, as it uses the
same maximum width throughout all layers. All the circuits utilized in this
study are derived from this full-width circuit, where variations are
introduced through the removal of gates.

Figure 2: The full-width design in which gates are applied over the full width
of the circuit throughout all layers. The circuit is displayed with four
layers, in-line with its configuration in our evaluation. Figure 3: The
reducing-width design in which the width is reduced from layer to layer by a
magnitude of two. With this design approach, we prune the circuit to a version
which uses much fewer gates. The circuit is displayed with four layers, in-
line with its configuration in our evaluation. Figure 4: The random-width
design in which a specific number of gates are deleted from the full-width
design at random positions. The number of gates removed gates equals the
number of removed gates in the reducing-width circuit. The circuit is
displayed with four layers, in-line with its configuration in our evaluation.

###  IV-B Reducing-width Design

Motivated from the typically reducing width of layers in many ANNs such as the
encoder component of autoencoders, we propose a circuit design that emulates
this architecture, i.e., the _reducing-width_ Ansatz desgin. This circuit
design progressively reduces the width of each appended layer, and hence
reduces the number of trained parameters in each training step. As the
occurrence of barren plateaus is directly proportional to the number of
parameters to be optimised, the reducing-width Ansatz desgin hence decreases
the risk of vanishing gradients intrinsically [26]. Especially when employing
layerwise learning [26], which is a useful learning technique restricting the
number of parameters concurrently trained to specific layers, this also allows
to increase the number of layers to be trained simultaneously. Additionally,
we argue that when entanglement between the qubits is sufficiently high (i.e.,
usually after some Ansatz layers), every gate executed on a subset of these
qubits intrinsically also acts on all qubits, as they must be regarded as one
system. A well known example in which exactly this happens, is the ancilla
quantum encoding as part of the quantum algorithm for solving linear systems
of equations [32], in which controlled rotations on a single ancilla qubit
substantially change the state of an entire qubit register.

In our implementation, we reduce the circuit width in every layer by a
magnitude of two in a symmetric manner. This implies that the outermost qubits
manipulated in the previous layer are no longer altered directly in the
appended layer. This reduction applies to both single-qubit gates and two-
qubit gates. Consequently, the circular entanglement is applied solely to the
remaining inner qubits, so that gates are not only deleted, but also
reconnected. A visualization of our reducing-width Ansatz can be found in Fig.
3.

The number of reduced single- and two-qubits gates is dependent on the number
of qubits $n$, the number of layers $l$, and the magnitude by which the
circuit width is reduced per layer $m$. Further, we have to distinguish
between the reduction in one-qubit gates $\sum_{n=1}^{l}(i-1)3m$, and the
reduction in two-qubit gates $\sum_{n=1}^{l}(i-1)m$. The proposed reduction is
naturally restricted by the number of qubits in the circuit, i.e., $(l-1)m<n$
must be satisfied, so that the reduction can be conducted.

In our case (i.e., using eight qubits, four Ansatz layers, and a reduction of
the circuit width per layer of two in magnitude), we reduce the number of one-
qubit gates by 36, and the number of two-qubit gates by 12. Respectively, this
corresponds to a reduction of one-qubit gates by 37.5%, and a 50% reduction
for two-qubit gates.

Beside the hypothesized positive effects a reduction of the number of gates
has on the parameter optimization process, the proposed circuit design is
naturally also more noise tolerant. Particularly the reduced number of gates
and consequently the lower circuit depth reduce the total noise accumulated
from imperfect gate fidelities and decoherence.

As the expressivity of the circuit is unclear a priori, we discuss its
corresponding performance in the evaluation.

###  IV-C Random-width Design

In addition to the previously shown design proposals, we introduce a random-
width design, which works as a control-design. This design is necessary to
control for the pure effect of removing gates and thus to isolate the effect
the removal-structure has. We therefore remove the same number and the same
kind of gates from the full-width design, as the reducing-width design does.
However, instead of removing these gates in a systematic way, as done in the
reducing-width design, we randomly determine the gates for removal. Therefore,
the number and kind of gates compared to the reducing-width design stays the
same, but the sequence and position of the gates differ. An exemplary circuit
resulting from the random-width design is shown in Fig. 4.

###  IV-D Training Method

Taking inspiration from the work of Skolik et al. [26], we train our quantum
circuits layer by layer. We therefore start with the first layer of the
respective circuit individually, i.e., without any other layers present in the
circuit. Subsequently, the parameterized gates get initialized with zeroes to
approximate an identity layer similar to the approach proposed in [33] and the
parameter training is started. For the training process we use the COBYLA
optimizer [34], where the maximum number of iterations is set to 300 to allow
for the learning curve to converge sufficiently, according to empirical tests
conducted for this study. After the training of the first layer is completed,
the trained parameters get saved, and a new quantum circuit is constructed.
The new circuit now implements two layers and re-uses the previously trained
and saved parameters for the initialization of the first layer. The parameters
of the gates added in the next layer of the circuit are initialized with zero,
while the subsequent training-process only trains the newly added layer of the
respective circuit. This training method now repeats itself until all four
layers are implemented and trained. With this training approach we aim to
improve on every previously achieved solution iteratively, similar to
techniques like reverse annealing [35].

###  IV-E Problem Instance Generation

To evaluate the performance of the three different design approaches, we solve
the maximum cut problem for 40 randomly generated Erdős-Rényi graphs [36]
using a VQE setup with the respective circuit design as the Ansatz. These
graphs are generated in a random manner, where the number of nodes, as well as
the probability that an edges exists between two nodes, are given as input. We
use eight nodes to match the qubit count and choose an edge probability of 50%
to yield hard problem instances compared to fully random graphs [37].

##  V Evaluation

For our experiments we have used the Atos Quantum Learning Machine (QLM),
stationed at the Leibniz Supercomputing Centre in Munich. Our experiments are
conducted with the three different circuits shown in Figs. 2, 3 and 4 aiming
to solve 40 maximum cut problem instances using a VQE approach and layerwise
learning. To rudimentarily account for the targeted NISQ-hardware, we included
depolarizing noise effects in our circuit simulations: The one-qubit and two-
qubit gate error rate were both set to 1%, which roughly coincides with the
gate fidelities of recent IBM QPUs.

In the following two sections, we investigate the runtime and result quality
of the proposed reducing-width Ansatz design pattern.

###  V-A Execution Times

Figure 5: Average Execution Time necessary to train each layer for the
different circuit design schemes.

We have measured the time the quantum simulator needs to simulate the circuit
and find the optimal parameters. For all circuit-designs and their different
number of layers, we have calculated this Average Execution Time (AET) in
seconds for the 40 simulated problem instances. The results are displayed in
Fig. 5.

For the full-width design and the random-width design, we observe increasing
AETs with increasing number of layers. This effect is stronger for the full-
width design than for the random-width design. We explain those results by the
two major tasks the simulator is performing in the measured time: The circuit
simulation and the parameter optimization. With increasing number of layers,
the number of gates to simulate increases, which in turn leads to increasing
circuit simulation times. Since the random-width design implements fewer gates
overall and as the reduction of gates is distributed randomly across all
layers, the AET curve has a flatter slope than the curve of the full-width
design, assuming that the parameter optimization time stays roughly constant.

When analyzing the AET development of the reducing-width approach, we observe
another pattern. In layer one, the AET coincides with that of the full-width
design, as both designs implement the same circuit in the first layer. On the
other hand, the reducing-width approach has a lower AET here, as its pruning
already starts in layer one. From layer one to layer two, the AETs slightly
increase for the reducing-width design, while for the remaining layers we
observe significant decreases of its AETs. We can also explain this behavior
by looking at the supposed development of the two main components of the AET.
We assume, that the circuit simulation time increases throughout the layers at
a decreasing pace, however, the absolute parameter optimization time decreases
faster, as fewer gates are appended to the circuit and therefore need to be
trained. While the effect of increasing circuit simulation time is still
stronger in the step from the first layer to the second layer than the effect
of decreasing optimisation time, this ratio changes constantly, starting at
the transition from layer two to three. The AET is almost halved from layer
one to layer four. These results indicate good scalability properties of the
reducing-width design in regards to training time.

###  V-B Average Approximation Ratios

Figure 6: Average Approximation Ratio of the different circuit design schemes.
Higher is better. Figure 7: Layerwise learning evolution. The numbers denote
the layer and the arrows show the progress from layer to layer.

In order to assess the performance of different design schemes qualitatively,
we investigate their Average Approximation Ratios (AARs) of the best possible
graph-segmentation. The optimal solution of every problem instance is
calculated by brute force and the AAR describes the deviation of given
solution from it. An AAR of $1$ indicates perfect results, while an AAR of $0$
is the worst possible solution.

Fig. 6 illustrates the evolution of the different AARs depending on the number
of layers they implement. The plotted results show that the AARs generally
decrease with an increasing number of layers, but no clear systematic
difference between the curves is apparent. Notably, the reducing-width design
shows a slightly better ARR performance in layer four compared to the two
baselines. Based on the general downtrend of AARs, we come to the conclusion,
that the problem instances are too small and easy to solve, as a single layer
of gates already achieves the highest AARs.

Fig. 7 finally lets us combines our observations in regards to execution time
and results quality. The general deterioration of ARRs is also clearly
observable here.

##  VI Conclusion

In this work, we proposed the novel _reducing-width_ Ansatz design pattern,
which similar to some classical ANN architectures, reduces the number of
applied functions (i.e., quantum gates) in each layer. Our experiments have
shown that this Ansatz design has potential to help escaping the problem of
barren plateaus in QNNs, as it performed better than both baseline approaches
for relatively deep circuits in terms of solution quality and runtime. Taking
its considerably better general properties in noisy quantum circuits into
account, further investigation on its scaling behavior is of significant
interest, especially for solving harder problem instances in which more layers
generally increase the solution quality.

For generalizing our results in future work, an investigation of the
performance with other learning approaches apart from the conducted single-
layer-wise strategy is necessary. Moreover, exploring the performance for
different problems beyond optimization is needed to verify its general
applicability as an Ansatz design pattern.

## Acknowledgment

This work was partially funded by the German BMWK project QCHALLenge
(01MQ22008A).

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# Real hyperbolic representations of $PU(1,n)$

Gonzalo Ruiz Stolowicz EGG, Institute of Mathematics, EPFL,
<EMAIL_ADDRESS>
###### Abstract

It is shown that $PU(1,n)$, for $n\geq 2$, does not admit non-elementary
representations into the group of isometries of an infinite-dimensional real
hyperbolic space.

00footnotetext: 2020 _Mathematics Subject Classification._ 51F99;
20C99.00footnotetext: _Key words and phrases._ Hyperbolic spaces,
representation theory.

## Introduction

The principal contribution of this article is the following.

######  Theorem.

If $n\geq 2$, the group of holomorphic isometries of
${\mathbf{H}}^{n}_{\mathbf{C}}$, the complex hyperbolic space of dimension n,
does not admit non-elementary representations into the group of isometries of
${\mathbf{H}}^{\infty}_{\mathbf{R}}$, a separable infinite-dimensional real
hyperbolic space.

Contrary to the finite-dimensional case, this statement is not an instance of
a general principle such as the Mostow-Karpelevich theorem. Indeed, there do
exist exotic non-elementary representations of the group of holomorphic
isometries of ${\mathbf{H}}^{n}_{\mathbf{C}}$, for every $n\geq 1$, on the
infinite-dimensional complex hyperbolic space (see for example Theorem 1.15 in
[21]) and likewise from real hyperbolic to real hyperbolic (see for example
Theorem B in [22]).

The main idea used to prove this appears in [8] and [11]: the existence of
smooth harmonic maps
${\mathbf{H}}^{n}_{\mathbf{C}}\rightarrow{\mathbf{H}}^{\infty}_{\mathbf{R}}$
associated to a uniform lattice and a non-elementary representation of the
group of isometries of the domain, together with the strong restrictions on
the rank of such maps (see [25]).

This paper is a follow up of [21] and [22]. In the first paper the author
studies general representations of groups into groups of isometries of
hyperbolic spaces. In the second one, among other results, the authors
classify the non-elementary representations of $PO(n,1)$ into the group of
isometries of an infinite-dimensional real hyperbolic space. In the process of
trying to unify the two different perspectives of the aforementioned articles,
the main question addressed here arises naturally.

##  1 Preliminaries and notations.

Suppose $\mathbf{F}=\mathbf{R},\mathbf{C}$ and let $B$ be a non-degenerate
bilinear form, $\mathbf{F}$-linear in the first argument and
$\mathbf{F}$-antilinear in the second, defined on $H$, a Hilbert space over
$\mathbf{F}$. Following [4], the form $B$ is called strongly non-degenerate of
signature $(1,n)$, with $n\in\mathbf{N}\cup\infty$, if $H$ admits a
$B$-orthogonal decomposition $\mathbf{F}\oplus H^{\prime}$, with
$\dim_{\mathbf{F}}(H^{\prime})=n$ and where $B$ restricted to $\mathbf{F}$ is
the usual inner product and $\left(H^{\prime},-B|_{H^{\prime}}\right)$ is a
Hilbert space.

Let $(H,B)$ be a Hilbert space over $\mathbf{F}$ and let $B$ be a strongly
non-degenerate bilinear form of signature $(1,n)$. For $v\in H$ , define
$[v]=\mathbf{F}v$. The $n$-$dimensional$ hyperbolic space over $\mathbf{F}$,
associated to $(H,B)$, is defined as

${\mathbf{H}}^{n}_{\mathbf{F}}=\\{[v]\mid B(v,v)>0\\},$

provided with the metric,

$\cosh(d([v],[w]))=\frac{|B(v,w)|}{B(v,v)^{1/2}B(w,w)^{1/2}}.$

For further reading about hyperbolic spaces of infinite dimensions see [10]
and [4].

Let $X$ be a metric space. Given three points $x,y,z\in X$ define the Gromov
product of $y$ and $z$ with respect to $x$ as,

$(y,z)_{x}=\frac{1}{2}\left(d(y,x)+d(z,x)-d(y,z)\right).$

A sequence $(x_{i})$ in $X$ is called a Gromov sequence if for $z_{0}$ a (any)
base point,

$\lim_{n,m\to\infty}(x_{n},x_{m})_{z_{0}}=\infty.$

Two Gromov sequences, $(x_{i})$ and $(y_{i})$, are called equivalent if for
$z_{0}$ a (any) base point,

$\lim_{n,m\to\infty}(x_{n},y_{m})_{z_{0}}=\infty.$

The relation defined above in the set of Gromov sequences is an equivalence
relation. Denote $\partial_{g}X$ the set of equivalence classes of Gromov
sequences in $X$. The set $\partial_{g}X$ will be called the boundary at
infinity of X.

In this work CAT(-1) spaces will always be considered complete. For
definitions and an extensive study of these spaces and the definition of
Gromov hyperbolicity see [3].

Every CAT(-1) space is hyperbolic in the sense of Gromov (Proposition 3.3.4 in
[10]). Therefore there are two natural ways to define and topologize a
boundary at infinity for a complete CAT(-1) space. The first one is
considering $X$ as a Gromov hyperbolic space and taking $\partial_{g}X$. The
second is considering $X$ as a CAT(0) space and defining the boundary at
infinity as the set of equivalence classes of asymptotic geodesic rays. It is
a classical result that for a CAT(-1) space these two notions are equivalent.
A sketch of proof will be given later due to the author’s lack of knowledge of
a reference in the literature.

This is Lemma 3.4.10 in [10].

######  Lemma 1.1.

Let $X$ is a $CAT(-1)$ space and suppose $\xi,\eta\in\partial_{g}X$ and
$z,w\in X$. If $(x_{i})\in\xi$ and $(y_{i})\in\eta$, the limits

$(\xi,\eta)_{z}=\lim_{n,m\to\infty}(x_{n},y_{m})_{z}$

and

$(\xi,w)_{z}=\lim_{n\to\infty}(x_{n},w)_{z}$

exist and do not depend on the choice of representatives.

Define $\mathcal{T}_{g}$ as the unique topology on $X\cup\partial_{g}X$ such
that for $S\subset X\cup\partial_{g}X$, $S$ is open if, and only if, $S\cap X$
is open for the metric topology and for every $\xi\in S\cap\partial_{g}X$,
there exists $t\geq 0$ such that $N_{t}(\xi)\subset S$, where

$N_{t}(\xi)=\\{y\in X\cup\partial_{g}X\mid(y,\xi)_{x_{0}}>t\\}.$

The following is Lemma 3.4.22 in [10].

######  Proposition 1.2.

Let $X$ be a CAT(-1) space. Suppose $(z_{n})$ is a sequence in $X$ and suppose
$(x_{n})$ and $(y_{n})$ are sequences in $X\cup\partial_{g}X$ converging with
the topology $\mathcal{T}_{g}$ to $z\in X$ and $x,y\in X\cup\partial_{g}X$,
respectively. Therefore

$\lim_{n\to\infty}(x_{n},y_{n})_{z_{n}}=(x,y)_{z}.$

Let $X$ be a complete CAT(0) space and $x_{0}\in X$ a base point. Given to
geodesic rays $\sigma$ and $\tau$ issuing from $x_{0}$, the map $t\mapsto
d(\sigma(t),\tau(t))$ is a convex non-negative function that vanishes at $0$,
therefore if it is bounded, then it has to be constant. This observation gives
sense to the following definitions.

For $s>r$ there is a projection

$\overline{B(x_{0},s)}\xrightarrow{p_{r,s}}\overline{B(x_{0},r)}.$

This defines and inverse system of topological spaces indexed by the positive
numbers. Let

$\overline{X}=\\{[0,\infty)\xrightarrow{\sigma}X\mid\sigma(0)=x_{0}\,\,\text{and}\,\sigma\,\text{is
a generalized geodesic ray}\\}$

be the inverse limit associated to this inverse system. Here a generalized
geodesic is either a geodesic ray issuing from $x_{0}$ or a geodesic segment
issuing from $x_{0}$ defined in an interval $[0,r]$, which is considered
constant in $[r,\infty)$.

The topology of inverse limit in $\overline{X}$ (the subspace topology of the
product $X^{\mathbf{R}_{\geq 0}}$) is the same as the topology of uniform
convergence in compact sets. This topology on $\overline{X}$, often called the
cone topology, and here denoted as $\mathcal{T}_{c}$, restricts to the metric
topology on $X$ and it does not depend on the choice of the base point $x_{0}$
(see II.8.8 in [3]). Denote as $\partial_{c}X$ the set of geodesic (infinite)
rays with base point in $x_{0}$ with the topology of subspace of the cone
topology.

For every $r>0$ let

$\overline{X}\xrightarrow{p_{r}}\overline{B(x_{0},r)}$

be the function that is the identity in $\overline{B(x_{0},r)}$ and
$p_{r}(\sigma)=\sigma(r)$, for any $\sigma$ generalized geodesic ray that is
not constant on $[r,\infty)$.

Given a geodesic ray $\xi$, let $U(\xi,R,\epsilon)$ be the set of generalized
rays $\tau$ such that $\tau|_{[R,\infty)}$ is not constant and
$d(p_{R}(\tau),p_{R}(\xi))<\epsilon.$ Observe that given a geodesic ray $\xi$,
the sets $U(\xi,R,\epsilon)$ are a neighborhood basis for the cone topology.

The following result is often called the finite approximation Lemma, see for
example Theorem 1 in Chapter 8 of [9].

######  Lemma 1.3.

Suppose $(X,x_{0})$ is a $\delta$-hyperbolic geodesic space and consider

$\\{x_{1},\dots,x_{n}\\}\subset X\cup\partial X.$

Here a point at infinity is understood as the limit of a geodesic ray. Define
$Y$ as the union of the geodesic segments or geodesic rays $[x_{0},x_{i}]$. If
$2n\leq 2^{k}+1$, there exists a simplicial tree $Tr(Y)$ and a map
$Y\xrightarrow{f}Tr(Y)$ with the following properties:

  1. 1.

For every $i$, the restriction of $f$ to $[x_{0},x_{i}]$ is an isometry.

  2. 2.

For every $x,y\in Y$,

$d(x,y)-2k\delta\leq d(f(x),f(y))\leq d(x,y).$

When $n=2$ the tree of the finite approximation Lemma is a tripod where the
extremes are $f(x_{i})$, with $i=0,1,2$ (see Proposition 3.1 of Chapter 1 in
[9]).

As it was mentioned before, the following theorem is a classic result for
which the author could not find a reference in the literature for non-proper
spaces.

######  Theorem 1.4.

Let $X$ be a CAT(-1) space. There is a natural homeomorphism

$((X,\partial_{c}X),\mathcal{T}_{c})\xrightarrow{\Psi}((X,\partial_{g}X),\mathcal{T}_{g}).$

###### Proof.

Fix a base point $z_{0}\in X$. Observe that for every geodesic ray $\tau$ with
$\tau(0)=z_{0}$, the sequence $(\tau(t_{n}))$ is a Gromov sequence for any
sequence $(t_{n})\to\infty$ and the class of equivalence of this Gromov
sequence does not depend on the choice of the sequence $(t_{n})$. Therefore
for every geodesic ray $\tau$ with starting point at $z_{0}$ there is a well
defined Gromov sequence $[\tau]$. Let $\Psi$ be the map such that
$\partial_{c}X\xrightarrow{}\partial_{g}X$ is defined by
$\Psi(\sigma)=[\sigma]$ and the identity in $X$. In Proposition 4 of Chapter 7
in [15] the authors showed, for proper CAT(-1) spaces, that
$\Psi|_{\partial_{c}X}$ is a bijection. The same proof can be applied in this
context if convergence arguments of Arzelà-Ascoli type are exchanged by
properties of convergence of Gromov sequences and applications of the finite
approximation Lemma.

The claim now is that $\Psi$ is a homeomorphism. Fix $N_{t}([\sigma])$ for
$t>0$ and a geodesic ray $\sigma$ issuing from $z_{0}$. Call $C$ the general
constant error coming from the tree approximation for $3$ points. Fix
$R,\epsilon>0$ such that $R-\epsilon-C>t+1$. Let $\tau\neq\sigma$ be a
geodesic ray from $x_{0}$ such that $d(\tau(R),\sigma(R))<\epsilon$ and
consider any $s>R$. If $(\sigma(s),\tau(s))_{z_{0}}>t+1$, then $\sigma(s)\in
N_{t}([\sigma])$. If this is not the case, then
$R>(\sigma(s),\tau(s))_{z_{0}}$ and from the tripod approximation for the
points $\\{z_{0},\sigma(s),\tau(s)\\},$

$|(\sigma(s),\tau(s))_{z_{0}}-(\sigma(R),\tau(R))_{z_{0}}|<C.$

But $(\sigma(R),\tau(R))_{z_{0}}>R-\frac{\epsilon}{2},$ therefore
$(\sigma(s),\tau(s))_{x_{0}}>t+1,$ which is a contradiction. This shows that
$(\sigma(s),\tau(s))_{x_{0}}>t+1$ and that

$([\tau],[\sigma])_{x_{0}}=\lim_{s\to\infty}(\sigma(s),\tau(s))_{x_{0}}\geq
t+1,$

or in other words, that $[\tau]\in N_{t}([\sigma])$. To finish just observe
that for every $r>0$,

$\begin{array}[]{rcl}(\sigma(s+r),\tau(s))_{x_{0}}&=&\frac{1}{2}\big{(}2s+r-d(\sigma(s+r),\tau(s))\big{)}\\\
&\geq&\frac{1}{2}\big{(}2s-d(\sigma(s),\tau(s))\big{)}\\\
&=&(\sigma(s),\tau(s))_{z_{0}}.\end{array}$

This implies that

$\lim_{r\to\infty}(\sigma(s+r),\tau(s))_{x_{0}}=([\sigma],\tau(s))_{x_{0}}\geq
t+1,$

that shows $\Psi(U(\sigma,R,\epsilon))\subset N_{t}([\sigma])$.

Fix $R,\epsilon>0$ and consider $U(\sigma,R,\epsilon).$ Suppose that for every
$t>0$

$N_{t}([\sigma])\not\subset U(\sigma,R,\epsilon).$

Thus, for every $n\in\mathbf{N}$ there exists $x_{n}\in
N_{n}([\sigma])\setminus U(\sigma,R,\epsilon).$ This means that for every $n$,
$(x_{n},[\sigma])_{z_{0}}\geq n$. Choose $s_{n}$ such that for every $r\geq
s_{n}$,

$(x_{n},\sigma(r))_{z_{0}}\geq n.$

Without lost of generality, suppose that $(s_{n})_{n}$ and
$(d(x_{n},x_{0}))_{n}$ are increasing sequences. Using the finite
approximation lemma for

$\\{z_{0},x_{n},x_{n+r},\sigma(s_{n+r})\\},$

it is possible to show that $(x_{n})$ is a Gromov sequence. If $\sigma_{n}$ is
the geodesic segment that connects $z_{0}$ to $x_{n}$, then

$\gamma(t)=\lim_{n\to\infty}\sigma_{n}(t)$

is a geodesic, in fact $\gamma$ is such that $\Psi(\gamma)=[(x_{n})]$. Here an
abuse of notation is made because only for $n$ bigger than $t$ it is possible
to assume that $\sigma_{n}(t)$ is defined. By construction
$d(\gamma(R),\sigma(R))\geq\epsilon$, therefore $\gamma\neq\sigma$, but this
is a contradiction because $(x_{n})$ belongs to $[\gamma]$ and $[\sigma]$.
Therefore there exists $t>0$ such that

$N_{t}([\sigma])\subset U(\sigma,R,\epsilon).$

∎

######  Lemma 1.5.

If $X$ is a CAT(-1) space there exists a constant $C>0$ such that for every
$x,y,z\in X$,

$|d(x,[y,z])-(x,y)_{z}|<C.$

###### Proof.

This is just an application of Lemma 1.3 for $w,x,y,z\in X$ where $w\in[y,z]$
is the point that minimizes the distance between $x$ and the geodesic segment
connecting $y$ and $z$. ∎

In Theorem 1.1 of [6] the authors proved the main statement of the following
lemma in a more general setting. Also in Proposition 2.1 of [1] there is a
similar result for locally compact CAT(0) spaces, using the idea of that
proof, here an elementary argument is given.

######  Lemma 1.6.

Let $X$ be a CAT(-1) space and let $\mathcal{C}=\\{C_{i}\\}_{i\in\mathbf{N}}$
be a family of non-empty, closed and convex subsets of $X$ such that for every
$n$, $C_{n+1}\subset C_{n}$. Suppose that for some (any) $z_{0}\in X$,
$\lim_{n\to\infty}d(z_{0},C_{n})=\infty$, then there exists $\xi\in\partial X$
such that,

$\\{\xi\\}=\bigcap_{n}\partial C_{n}.$

In particular if there is a group $G$ acting by isometries on $X$ and
permuting the elements of $\mathcal{C}$, then $\xi$ is a G-fixed point.

###### Proof.

For every $n$ there is $x_{n}\in C_{n}$ such that
$d(z_{0},x_{n})=d(z_{0},C_{n}).$ There is a constant $C>0$ coming from the
finite approximation lemma such that for every $n,m\in\mathbf{N}$,

$|d(z_{0},[x_{n},x_{m}])-(x_{n},x_{m})_{z_{0}}|<C.$

If $m$ is bigger than $n$,

$d(z_{0},[x_{n},x_{m}])\geq d(z_{0},x_{n}),$

therefore $(x_{n})$ is a Gromov sequence. If $\xi$ is its equivalence class,
then $\xi\in\bigcap_{n}\partial C_{n}$.

Suppose there is $\eta\neq\xi$ such that $\eta\in\bigcap_{n}\partial C_{n}$.
If $\tau$ is the unique geodesic connecting $\eta$ and $\xi$ (see Proposition
4.4.4 of [10]), then the image of $\tau$ is contained in every $C_{n}$, this
is a contradiction because $\bigcap_{n}C_{n}=\emptyset$.

The last claim of the Lemma follows from the fact that $G$ also permutes the
elements of $\\{\partial C_{n}\\}_{n}.$ ∎

Let $G$ be a group acting on a space $X$. A function
$X\xrightarrow{f}\mathbf{R}$ is called quasi-invariant if for every $g$ there
exists a constant $c(g)$ such that for every $x\in X$,

$f(gx)-f(x)=c(g).$

Observe that the map $c$ in the previous definition has to be a homomorphism.
The statement of the next lemma, but in the context of proper CAT(0) spaces,
appears in Section 2 of [1]. The arguments there work also for CAT(-1) spaces
given the statement of Lemma 1.6 and the following observation. Let
$\\{C_{i}\\}_{i\in\mathbf{N}}$ be a family of non-empty, convex and closed
sets in a complete CAT(0) space $X$ such that for every $n$, $C_{n+1}\subset
C_{n}$. Therefore, $\bigcap_{n}C_{n}=\emptyset$ if, and only if, for every
$x_{0}\in X$, $\lim\limits_{n\to\infty}d(x_{0},C_{n})=\infty$ (see Proposition
1.2 of [19]).

######  Lemma 1.7.

Let a group $G$ act by isometries on a CAT(-1) space $X$. If the action does
not have fixed points in $X\cup\partial X$, then every quasi-invariant convex
function defined on $X$ is $G$-invariant, has a lower bound and the non-empty
sublevel sets of it are $G$-invariant and unbounded.

Let $G$ be a topological group and let $X$ be a topological space. An action
of $G$ on $X$ is called orbitally continuous if for every $x\in X$, the map
$g\mapsto g\cdot x$ is continuous. If $X$ is a $CAT(-1)$ space an orbitally
continuous representation $G\xrightarrow{\rho}Isom(X)$ is called non-
elementary if it does not have finite orbits in $X\cup\partial X$. From now on
all the representations will be considered orbitally continuous.

If $X$ is $\text{CAT}(-1)$ space, $x_{0}$ is a base of point of $X$ and
$\xi\in\partial X$, the  Busemann function based on $\xi$ and normalized in
$x_{0}$ is defined as follows. If $\sigma$ is the geodesic ray that starts at
$x_{0}$ and points towards $\xi$,

$b_{\xi,x_{0}}(y)=\lim_{t\to\infty}d(y,\sigma(t))-t.$

Observe that

$\begin{array}[]{rcl}b_{\xi,x_{0}}(y)+2(y,\xi)_{x_{0}}&=&\lim\limits_{t\to\infty}(d(y,\sigma(t))-t)+\lim\limits_{t\to\infty}\big{(}d(y,x_{0})+t-d(y,\sigma(t))\big{)}\\\
&=&d(y,x_{0}).\end{array}$

The following lemma is well known but a reference in the literature is unknown
to the author.

######  Lemma 1.8.

Let $X$ be a CAT(-1) space. A representation $G\xrightarrow{\rho}Isom(X)$ is
non-elementary if, and only if, it does not fix a point in $X\cup\partial X$
and it does not preserve a geodesic.

###### Proof.

Suppose that $\rho$ does not have fixed points in $X\cup\partial X$ and that
it does not preserve a geodesic. If $\rho$ has a finite orbit in $X$, then it
has a fixed point in $X$ (see Corollary II.2.7 of [3]). Suppose that there is
$\\{\xi_{1},\dots,\xi_{l}\\}$ a $G$-invariant set in $\partial X$ with $n\geq
3$. Fix a base point $x_{0}\in X$ and consider the function
$f=\sum_{i=1}^{n}b_{\xi_{i},x_{0}}$.

Observe that $b_{\xi,x_{0}}(gy)=b_{g^{-1}\xi,g^{-1}x_{0}}(y)$. As the set
$\\{\xi_{1},\dots,\xi_{l}\\}$ is invariant, there is a permutation of
$\\{1,\dots,l\\}$ defined by $g^{-1}\xi_{i}=\xi_{\varphi(i)}$. Therefore

$\begin{array}[]{rcl}b_{\xi,x_{0}}(gy)&=&b_{g^{-1}\xi_{i},g^{-1}x_{0}}(y)\\\
&=&b_{\xi_{\varphi(i)},g^{-1}x_{0}}(y)\\\
&=&b_{\xi_{\varphi(i)},x_{0}}(y)-b_{\xi_{\varphi(i)},x_{0}}(g^{-1}x_{0}).\end{array}$

As a consequence, the convex function $f$ is quasi-invariant because

$f(gy)=\sum_{i=1}^{n}b_{\xi_{i},x_{0}}(gy)=\sum_{i=1}^{n}b_{\xi_{i},x_{0}}(y)-\sum_{i=1}^{n}b_{\xi_{i},x_{0}}(g^{-1}x_{0}).$

By Lemma 1.7, any non-empty sublevel set of $f$ is unbounded. Fix one non-
empty sublevel set $C_{r}$ and let $(y_{n})\in C_{r}$ be an unbounded
sequence. Up to taking a subsequence, we can suppose that $(y_{n})$ converges
to at most one point at infinity $\eta.$ Observe that for every $\xi_{i}$,

$b_{\xi_{i},x_{0}}(y_{n})=d(y_{n},x_{0})-2(y_{n},\xi_{i})_{x_{0}},$

therefore if $\eta\neq\xi_{1},\dots,\xi_{l}$, there exists $C>0$ such that for
every $n$,

$|f(y_{n})-ld(y_{n},x_{0})|<C.$

This is a contradiction because $\min(f)\leq f(y_{n})\leq r$ and
$\lim_{n}d(y_{n},x_{0})=\infty$.

Now suppose that $(y_{n})$ converges to $\eta=\xi_{1}$. Observe that
$b_{\xi_{1},x_{0}}(y)\geq-d(y,x_{0})$ and that there exists $C^{\prime}>0$
such that for every $y_{n}$,

$\begin{array}[]{rcl}f(y_{n})&=&b_{\xi_{1},x_{0}}(y_{n})+b_{\xi_{2},x_{0}}(y_{n})+\cdots+b_{\xi_{l},x_{0}}(y_{n})\\\
&\geq&-d(y_{n},x_{0})+(l-1)d(y_{n},x_{0})-C^{\prime}\\\
&\geq&(l-2)d(y_{n},x_{0})-C^{\prime}.\end{array}$

Therefore $\\{d(y_{n},x_{0})\\}_{n}$ is bounded, which is a contradiction. ∎

##  2 The main result.

Let $G$ be a (Hausdorff) locally compact group. A discrete subgroup $\Gamma$
is called a lattice if the space $G/H$ admits a non-zero finite $G$-invariant
Radon measure.

The next proposition appears in Proposition 2.1 of [7] in the context of
proper CAT(0) spaces. The ideas in that article can be used with slight
modifications for the case of CAT(-1) spaces.

######  Proposition 2.1.

Suppose that $G$ is a locally compact and $\sigma$-compact group, $\Gamma\leq
G$ is a lattice and $X$ is a CAT(-1) space. If $G\xrightarrow{\rho}Isom(X)$ is
a non-elementary representation and $\rho|_{\Gamma}$ does not have fixed
points in $X$, then $\rho|_{\Gamma}$ is a non-elementary representation.

###### Proof.

The proof will be by contradiction. Suppose that there exists $\eta\in\partial
X$ fixed by the action of $\Gamma$. Using the continuous map
$G/\Gamma\rightarrow\partial X$, induced by the orbit map $g\mapsto g\eta$, it
is possible to define a $G$-invariant probability measure $\mu$ in $\partial
X.$ Fix a point $x_{0}\in X$ and consider the function

$F(y)=\int_{\partial
X}b_{\xi,x_{0}}(y)d\mu(\xi)=\int_{G/\Gamma}b_{g\eta,x_{0}}(y)d\nu(g\Gamma),$

where $\nu$ is the $G$-invariant probability measure in $G/\Gamma$. The
function $\xi\mapsto b_{\xi,x_{0}}(y)$ is continuous (see Lemma 3.4.22 in
[10]) and for every $\xi\in\partial X$, $|b_{\xi,x_{0}}(y)|\leq d(y,x_{0})$.
This shows that the integral makes sense.

Every function $b_{\xi,x_{0}}$ is convex, therefore $F$ is convex too.
Moreover, for every $g\in G$,

$\begin{array}[]{rcl}F(g^{-1}y)&=&\int_{\partial
X}b_{\xi,x_{0}}(g^{-1}y)d\mu(\xi)\\\ &=&\int_{\partial
X}b_{g\xi,gx_{0}}(y)d\mu(\xi)\\\ &=&\int_{\partial
X}\Big{(}b_{g\xi,x_{0}}(y)-b_{g\xi,gx_{0}}(x_{0})\Big{)}d\mu(\xi)\\\
&=&\int_{\partial
X}\Big{(}b_{g\xi,x_{0}}(y)-b_{\xi,x_{0}}(g^{-1}x_{0})\Big{)}d\mu(\xi)\\\
&=&F(y)-F(g^{-1}x_{0}),\end{array}$

where the last equality holds because the measure $\mu$ is $G$-invariant.
Therefore $F$ is quasi-invariant, and by Lemma 1.7, it is a $G$-invariant
function.

Notice that $x_{0}\in C_{0}$, the sublevel set of $F$ associated to $0$.
Observe that for every $n\in\mathbf{N}$ there exists $x_{n}\in C_{0}$ such
that $d(x_{0},x_{n})>n$. Up to taking a subsequence, it is possible to suppose
that $(x_{n})$ converges at most to $\xi_{0}\in\partial X$. The claim is that
$F(x_{n})\to\infty$, which would be a contradiction. The proof for this
statement will follow the ideas of Lemma 2.4 in [5].

By Lemma 1.8, the orbit of every $\eta\in\partial X$ is infinite, hence $\mu$
is a non-atomic measure, therefore

$F(y)=\int\limits_{\partial X\setminus\xi_{0}}b_{\xi,x_{0}}(y)d\mu(\xi).$

For every $y,z\in X$,

$(y,z)_{x_{0}}\leq\min\\{d(y,x_{0}),d(z,x_{0})\\},$

thus, for every $\eta\in\partial X$, $(y,\eta)_{x_{0}}\leq d(y,x_{0}).$
Therefore, for every $y\in X$ and $\eta\in\partial X$,

$b_{\eta,x_{0}}(y)=d(y,x_{0})-2(y,\eta)_{x_{0}}\geq-d(y,x_{0}).$

Define for every $n\in\mathbf{N}$ the measurable set

$V(n)=\\{\eta\in\partial
X\mid\sup_{m\in\mathbf{N}}\\{2(x_{m},\eta)_{x_{0}}\\}\leq n\\}.$

The sequence $(x_{n})$ belongs to at most $\xi_{0}$, therefore

$\partial X\setminus\xi_{0}\subset\bigcup_{n}V(n).$

For every $n$, $V(n)\subset V(n+1)$, thus there exists some $n_{0}$ such that
$\mu(V(n_{0}))>\frac{1}{2}.$ Therefore for every $x_{m}$,

$\begin{array}[]{rcl}F(x_{m})&=&\int\limits_{V(n_{0})\setminus\xi_{0}}b_{\xi,x_{0}}(x_{m})d\mu(\xi)+\int\limits_{(\partial
X\setminus\xi_{0})\setminus V(n_{0})}b_{\xi,x_{0}}(x_{m})d\mu(\xi)\\\
&\geq&\big{(}d(x_{m},x_{0})-n_{0}\big{)}\mu(V(n_{0}))-\big{(}1-\mu(V(n_{0}))\big{)}d(x_{m},x_{0})\\\
&=&\big{(}2\mu(V(n_{0}))-1\big{)}d(x_{m},x_{0})-n_{0}\mu(V(n_{0})).\end{array}$

Thus $F(x_{m})\to\infty,$ which is a contradiction.

If $\rho|_{\Gamma}$ permutes two points at infinity there is an index two
subgroup of $\Gamma$ that preserves a point at infinity. A finite index
subgroup of a lattice is a lattice (see for example Lemma 1.6 in [24]), thus
this assumption leads to a contradiction.

If $\Gamma$ has a fixed point $x\in X$, the orbit map $g\mapsto g\cdot x$
induces in $X$ a $G$-invariant probability measure $\mu$. Consider a nested
family of compact sets $\\{K_{i}\\}_{i\in\mathbf{N}}$ such that
$\bigcup_{i}K_{i}=G$. There exists $i$ such that $\mu(K_{i}\cdot x)>1/2$,
therefore for every $g\in G$,

$gK_{i}\cdot x\cap K_{i}\cdot x\neq\emptyset,$

or in other words, there are $k_{1},k_{2}\in K_{i}$ such that $gk_{1}\cdot
x=k_{2}\cdot x$. Observe that

$\begin{array}[]{rcl}d(g\cdot x,x)&\leq&d(g\cdot x,gk_{1}\cdot x)+d(k_{2}\cdot
x,x).\end{array}$

This shows that $x$ has a bounded orbit, but this is a contradiction because
$G$ does not fix any point in $X$. ∎

Observe that the arguments in the previous proof show that if $G$ has a non-
elementary representation on a CAT(-1) space $X$, then neither $\partial X$
nor $X$ admit a $G$-invariant probability measure. This property characterizes
the non-elementary representations.

######  Proposition 2.2.

Let $G$ be a locally compact and $\sigma$-compact group and let $X$ be a
CAT(-1) space. If $G\xrightarrow{\rho}Isom(X)$ is a representation, then
$\rho$ is non-elementary if, and only if, neither $X$ nor $\partial X$ admit a
$G$-invariant probability measure.

###### Proof.

The implication that has not been discussed can be proved by considering Dirac
masses. ∎

######  Lemma 2.3.

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two uniform lattices of a locally compact
group $G$ and let $X\xrightarrow{f_{i}}Y$, $i=1,2$ be two continuous functions
between $X$ a topological space and $Y$ a metric space. Suppose $G$ acts
transitively on $X$ with compact stabilizers, by isometries on $Y$ and
orbitally continuously on both. If $f_{i}$ is $\Gamma_{i}$-equivariant, then
there exists $C>0$ such that for every $x\in X$, $d(f_{1}(x),f_{2}(x))<C.$

###### Proof.

There exist compact sets $K_{i}\subset G$ such that $\Gamma_{i}K_{i}=G$ (see
for example Lemma 2.46 in [13]). Fix $x_{0}\in X$ and take $y\in X$. There
exist $\gamma_{i}\in\Gamma_{i}$ and $k_{i}\in K_{i}$, such that
$\gamma_{i}k_{i}x_{0}=y.$ Therefore,

$\begin{array}[]{rcl}d(f_{1}(y),f_{2}(y))&=&d\big{(}\gamma_{1}f_{1}(k_{1}x_{0}),\gamma_{2}f_{2}(k_{2}x_{0})\big{)}\\\
&=&d\big{(}\gamma_{2}^{-1}\gamma_{1}f_{1}(k_{1}x_{0}),f_{2}(k_{2}x_{0})\big{)}\\\
&\leq&\sup\\{d\big{(}zf_{1}(l_{1}x_{0}),f_{2}(l_{2}x_{0})\big{)}\mid z\in
K_{2}\text{Stab}(x_{0})K_{1}^{-1},l_{i}\in K_{i}\\}.\end{array}$

∎

######  Lemma 2.4.

Let $\\{H_{n}\\}_{n\in\mathbf{N}_{\geq 1}}$ be a sequence of finite-
dimensional hyperbolic spaces embedded in
${\mathbf{H}}^{\infty}_{\mathbf{R}}$, where for $n\geq 2$, $H_{n}$ is
isometric to ${\mathbf{H}}^{n}_{\mathbf{R}}$ and $H_{1}$ is a geodesic.
Suppose that for every $n\geq 1$, $H_{n}\subset H_{n+1}$ and

$\overline{\bigcup_{n\geq 1}H_{n}}={\mathbf{H}}^{\infty}_{\mathbf{R}}.$

Therefore, for every $n\geq 2$ and
$y_{1},y_{2}\in{\mathbf{H}}^{\infty}_{\mathbf{R}}$, there exists
$\varphi\in\text{Isom}({\mathbf{H}}^{\infty}_{\mathbf{R}})$ such that,
$\varphi|_{H_{n}}=Id$ and $\varphi(\\{y_{1},y_{2}\\})\subset H_{n+2}$.

###### Proof.

Given $H_{n}\subset H_{n+2}$ and
$y_{1},y_{2}\in{\mathbf{H}}^{\infty}_{\mathbf{R}}$, there exists $m\geq n+2$
and $\mathbf{H}$ isometric to $\mathbf{H}^{m}_{\mathbf{R}}$ such that
$y_{i}\in\mathbf{H}$ and $H_{n+2}\subset\mathbf{H}.$ Observe that every
isometry of $\mathbf{H}$ can be extended to an isometry of
${\mathbf{H}}^{\infty}_{\mathbf{R}}.$ Therefore the problem can be reduced to
a statement about $\mathbf{H}^{m}_{\mathbf{R}}$, where the claim is clear. ∎

Let $M$ be a Riemannian manifold and let $U\subset M$ be an open set contained
in a chart $(V,\phi)$. Suppose that $\overline{U}\subset V$ and
$\phi^{-1}(U)=B(p,r)\subset\mathbf{R}^{m}$. For every
$\varphi\in\mathcal{C}^{0}(\partial U)$, there exists a unique
$h_{\varphi}\in\mathcal{C}^{0}(\overline{U})\cap\mathcal{C}^{2}(U)$ which
solves the Dirichlet problem, in other words, $h_{\varphi}$ is harmonic in $U$
($\Delta h_{\varphi}|_{U}=0$) and $h_{\varphi}|_{\partial
U}=\varphi|_{\partial U}$ (see Lemma 6.10 in [16]). For references about
harmonic maps in the Riemannian setting see [23] and for harmonic maps with a
CAT(0) codomain see [18] and [19].

For every $x\in U$, the claim is that the map

$\begin{array}[]{rcl}\mathcal{C}^{0}(\partial U)&\rightarrow&\mathbf{R}\\\
\varphi&\mapsto&h_{\varphi}(x)\end{array}$

is a positive linear functional, in other words, it defines a probability
measure $p^{U}_{x}$ in $\partial U$. Indeed, in every $U$ as above, a harmonic
map defined on $\overline{U}$ achieves its maximum (minimum) in $\partial U$
and if there exists $u\in U$ such that the maximum (minimum) of $h$ is
achieved in $u$, then $h$ is constant in $\overline{U}$ (see Theorem 3.1 in
[16]). Thus

$h_{\varphi}\leq\max\limits_{y\in\partial
U}h_{\varphi}(y)=\max\limits_{y\in\partial U}\varphi(y),$

therefore the linear map $\varphi\mapsto h_{\varphi}(x)$ is positive and
continuous for every $x\in U$.

A continuous function $M\xrightarrow{\varphi}\mathbf{R}$ is called subharmonic
if for every $U$ as above and every $x\in U$,

$\varphi(x)\leq\int\limits_{\partial U}\varphi dp^{U}_{x}.$

If $\varphi\in\mathcal{C}^{2}(M)$, then $\varphi$ is subharmonic if, and only
if, $\Delta f\geq 0$ (see page 103 of [16]).

Observe that every non-constant subharmonic function $\varphi$ defined on
$\overline{U}$ satisfies a maximum principle: the maximum of $\varphi$ is
achieved only in the boundary.

######  Lemma 2.5.

Let $M$ be a Riemannian manifold and let $(\varphi_{n})$ be a sequence of
subharmonic functions defined in $M$. If $(\varphi_{n})\to\varphi$ uniformly
on compact sets, then $\varphi$ is subharmonic.

The proof of the next lemma follows some of the ideas in Theorem 2.3 in [20].

######  Lemma 2.6.

Let $X$ be a homogeneous and complete Riemannian manifold and let

$u,v:X\rightarrow{\mathbf{H}}^{\infty}_{\mathbf{R}}$

be two harmonic and Lipschitz continuous functions of class $\mathcal{C}^{2}$.
If there exists $C>0$ such that for every $x\in X$, $d(u(x),v(x))<C$, then
either $f=g$ or the images of $f$ and $g$ are contained in one geodesic.

###### Proof.

Suppose that $K>0$ is a Lipschitz constant for $u$ and $v$. Let
$\\{y_{i}\\}_{i\in\mathbf{N}}\subset{\mathbf{H}}^{\infty}_{\mathbf{R}}$ be
such that if for every $n\geq 1$, $H_{n}$ is the smallest hyperbolic space
that contains $\\{y_{0},\dots,y_{n}\\}$, then the family $\\{H_{n}\\}_{n\geq
1}$ satisfies the hypothesis of Lemma 2.4.

Let $(x_{n})_{n\geq 1}$ be a sequence in $X$ such that

$d(u,v)=\sup_{x\in
X}\\{d(u(x),v(x))\\}=\lim_{n\to\infty}d(u(x_{n}),v(x_{n})).$

Fix $x_{0}\in X$ and for every $i$ choose
$\varphi_{i}\in\text{Isom}({\mathbf{H}}^{\infty}_{\mathbf{R}})$ such that
$\varphi_{i}(x_{0})=x_{i}$. Define $u_{i}=u\circ\varphi_{i}$ and
$v_{i}=v\circ\varphi_{i}$. For every $i$ there exist an isometry $T_{i}^{1}$
such that $T_{i}^{1}\circ u_{i}(x_{0})=y_{0}$ and $T_{i}^{1}\circ
v_{i}(x_{0})\in H_{1}$. Observe that for every $i$,

$d(T_{i}^{1}\circ u_{i}(x_{0}),T_{i}^{1}\circ v_{i}(x_{0}))\leq d(u,v).$

$H_{1}$ is locally compact, therefore there exists a subsequence
$\big{(}T_{1,i}^{1}\circ v_{1,i}(x_{0})\big{)}_{i\in\mathbf{N}}$ of
$\big{(}T_{i}^{1}\circ v_{i}(x_{0})\big{)}_{i\in\mathbf{N}}$ which is
convergent.

Let $\\{z_{i}\\}_{i\in\mathbf{N}_{\geq 1}}$ be a dense subset of $X$. Observe
that for every $i$, there exists an isometry $T_{i}^{2}$ such that
$T_{i}^{2}|_{H_{1}}=Id$ and

$\\{T_{i}^{2}\circ T_{1,i}^{1}\circ u_{1,i}(z_{1}),T_{i}^{2}\circ
T_{1,i}^{1}\circ v_{1,i}(z_{1})\\}\subset H_{3}.$

Notice that for every $i$,

$d\Big{(}T_{i}^{2}\circ T_{1,i}^{1}\circ u_{1,i}(z_{1}),T_{i}^{2}\circ
T_{1,i}^{1}\circ u_{1,i}(x_{0})\Big{)}\leq Kd(z_{1},x_{0}),$

but

$T_{i}^{2}\circ T_{1,i}^{1}\circ u_{1,i}(x_{0})=T_{1,i}^{1}\circ
u_{1,i}(x_{0})=y_{0}.$

Therefore

$\Big{(}T_{i}^{2}\circ T_{1,i}^{1}\circ
u_{1,i}(z_{1})\Big{)}_{i\in\mathbf{N}_{\geq 1}}$

is a bounded sequence in $H_{3}$. Also, for every $i$,

$d\Big{(}T_{i}^{2}\circ T_{1,i}^{1}\circ u_{1,i}(z_{1}),T_{i}^{2}\circ
T_{1,i}^{1}\circ v_{1,i}(z_{1})\Big{)}\leq d(u,v).$

Thus,

$\Big{(}T_{i}^{2}\circ T_{1,i}^{1}\circ v_{1,i}(z_{1})\Big{)}_{i\geq 1}$

is again a bounded sequence in $H_{3}$. So it is possible to chose respective
subsequences,

$\Big{(}T_{2,i}^{2}\circ T_{2,i}^{1}\circ u_{2,i}(z_{1})\Big{)}_{i{\geq 1}}$

and

$\Big{(}T_{2,i}^{2}\circ T_{2,i}^{1}\circ v_{2,i}(z_{1})\Big{)}_{i{\geq 1}}$

that are convergent.

By induction on $n$, suppose that for every for every $2\leq m\leq n$ and for
every $i\geq 1$ there are isometries $T^{m}_{m,i}$, and $T^{1}_{n,i}$ such
that

  1. 1.

$T^{1}_{n,i}\circ u_{i}(x_{0})=y_{0}$ and $\big{(}T^{1}_{n,i}\circ
v_{i}(x_{0})\big{)}_{i\geq 1}$ is a convergent sequence in $H_{1}$.

  2. 2.

$T^{m}_{n,i}|_{H_{1+2(m-2)}}=Id.$

  3. 3.

$\Big{(}T^{m}_{n,i}\circ\cdots\circ T^{1}_{n,i}\circ
u_{n,i}(z_{m-1})\Big{)}_{i\geq 1}$ and $\Big{(}T^{m}_{n,i}\circ\cdots\circ
T^{1}_{n,i}\circ v_{n,i}(z_{m-1})\Big{)}_{i\geq 1}$ are converging sequences
in $H_{1+2(m-1)}$.

For every $i\geq 1$, let $T^{n+1}_{i}$ be an isometry with the following
properties,

  1. 1.

$T^{n+1}_{i}|_{H_{1+2(n+1-2)}}=Id$.

  2. 2.

$T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ u_{n,i}(z_{n})$ and
$T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ v_{n,i}(z_{n})$ are elements of
$H_{1+2(n+1-1)}$.

Observe that

$\Big{(}T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ
u_{n,i}(z_{n})\Big{)}_{i\geq 1}$

is a bounded sequence in $H_{1+2(n+1-1)}$, indeed

$d\Big{(}T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ
u_{n,i}(z_{n})\,,\,T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ
u_{n,i}(x_{0})\Big{)}\leq Kd(z_{n},x_{0}),$

but

$T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ u_{n,i}(x_{0})=y_{0}.$

Moreover, for every $i$,

$d\Big{(}T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ
u_{n,i}(z_{n})\,,\,T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ
v_{n,i}(z_{n})\Big{)}\leq d(u,v).$

Therefore

$\Big{(}T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ
u_{n,i}(z_{n})\Big{)}_{i\geq 1}$

and

$\Big{(}T^{n+1}_{i}\circ\cdots\circ T_{n,i}^{1}\circ
v_{n,i}(z_{n})\Big{)}_{i\geq 1}$

are bounded sequences in $H_{1+2(n+1-1)}$. Hence it is possible to choose
convergent subsequences

$\Big{(}T^{n+1}_{n+1,i}\circ\cdots\circ T_{n+1,i}^{1}\circ
u_{n+1,i}(z_{n})\Big{)}_{i\geq 1}$

and

$\Big{(}T^{n+1}_{n+1,i}\circ\cdots\circ T_{n+1,i}^{1}\circ
v_{n+1,i}(z_{n})\Big{)}_{i\geq 1}.$

Define now,

$U(z_{n})=\lim\limits_{i\to\infty}T^{i}_{i,i}\circ\cdots\circ T_{i,i}^{1}\circ
u_{i,i}(z_{n})$

and

$V(z_{n})=\lim\limits_{i\to\infty}T^{i}_{i,i}\circ\cdots\circ T_{i,i}^{1}\circ
v_{i,i}(z_{n}).$

Observe that there exists $M>0$ such that,

$\begin{array}[]{rcl}U(z_{n})&=&\lim\limits_{i\to\infty}T^{i}_{i,i}\circ\cdots\circ
T_{i,i}^{1}\circ u_{i,i}(z_{n})\\\
&=&\lim\limits_{i\to\infty}T^{M}_{i,i}\circ\cdots\circ T_{i,i}^{1}\circ
u_{i,i}(z_{n})\\\ &=&\lim\limits_{i\to\infty}T^{M}_{M,i}\circ\cdots\circ
T_{M,i}^{1}\circ u_{M,i}(z_{n})\\\ \end{array}$

and

$V(z_{n})=\lim\limits_{i\to\infty}T^{M}_{M,i}\circ\cdots\circ T_{M,i}^{1}\circ
v_{M,i}(z_{n}).$

Given $z_{n}$ and $z_{m}$, there exists $M^{\prime}>0$ such that

$\begin{array}[]{rcl}d(U(z_{n}),U(z_{m}))&=&\lim\limits_{i\to\infty}d(u\circ\varphi_{M^{\prime},i}(z_{n}),u\circ\varphi_{M^{\prime},i}(z_{m}))\\\
&\leq&Kd(z_{n},z_{m}),\end{array}$

and with the same reasoning,

$d(V(z_{n}),V(z_{m}))\leq K(d(z_{n},z_{m})).$

Therefore $U$ and $V$ can be extended to $X$.

For every $m\geq 1$, define

$R_{m}=T^{m}_{m,m}\circ\cdots\circ T^{1}_{m,m}\circ u_{m,m}$

and

$S_{m}=T^{m}_{m,m}\circ\cdots\circ T^{1}_{m,m}\circ v_{m,m}.$

Observe that for every $m$, $R_{m}$ and $S_{m}$ are Lipschitz continuous
functions with Lipschitz constant smaller or equal than $K$. Therefore
$\\{R_{n}\\}_{n}$ and $\\{S_{n}\\}_{n}$ are equicontinuous families. If the
function $L_{n}$ is defined as $L_{n}(z)=d(R_{n}(z),S_{n}(z)),$ then the
family $\\{L_{n}\\}_{n}$ is equicontinuous and pointwise convergent to
$z\mapsto d(U(z),V(z))$, thus by Arzelà-Ascoli Theorem, the convergence is
uniform on compact sets.

The functions $u$ and $v$ are $\mathcal{C}^{2}$, and for every $i$,
$\varphi_{i}$ is an isometry, therefore $u_{i}$ and $v_{i}$ are harmonic
functions (see for example Proposition 2.2 in [17]). Moreover, for every
$i,j$, the map $T_{i,i}^{j}$ is an isometry, therefore for every $m$, the
functions $R_{m}$ and $S_{m}$ defined above are harmonic. For one reference
for the last statement see the corollary at the end of page 131 of [12].

The distance function
${\mathbf{H}}^{\infty}_{\mathbf{R}}\times{\mathbf{H}}^{\infty}_{\mathbf{R}}\xrightarrow{d}\mathbf{R}$
is a (geodesically) convex function and for every $m$, the map $x\mapsto
d(R_{m}(x),S_{m}(x))$ is harmonic (see the second example in page 133 of
[12]). Therefore, for every $m$ the function $L_{m}$ is subharmonic (see
Theorem 3.4 in [17]) and by Lemma 2.5, the map $z\mapsto d(U(z),V(z))$ is
subharmonic.

Notice that for every $z\in X$ $d(u,v)\geq d(U(z),V(z))$, also

$\begin{array}[]{rcl}d(U(x_{0}),V(x_{0}))&=&\lim\limits_{m}d(T_{m}(x_{0}),S_{m}(x_{0}))\\\
&=&\lim\limits_{m}d(u_{m,m}(x_{0}),v_{m,m}(x_{0}))\\\
&=&\lim\limits_{m}d(u(x_{m,m}),v(x_{m,m}))=d(u,v).\end{array}$

Therefore $d(U(z),V(z))$ is constant as a consequence of the maximum principle
for subharmonic maps. By construction, for every $z$,

$d(U(z),V(z))=d(u(z),v(z)),$

hence, by Lemma 2.2 in [20], either $u=v$ or the images of $u$ and $v$ are
contained in a geodesic. ∎

######  Lemma 2.7.

If $\Gamma$ is a torsion free uniform lattice of $SU(1,n)$, then the following
hold:

  1. 1.

All the non-trivial elements act as hyperbolic isometries of
${\mathbf{H}}^{n}_{\mathbf{C}}$.

  2. 2.

If $\textit{l}(g)$ is the translation length of $g$ acting as an isometry of
${\mathbf{H}}^{n}_{\mathbf{C}}$, then
$\inf\\{\textit{l}(\gamma)\mid\gamma\in\Gamma\setminus e\\}>0$.

  3. 3.

There exists $g\in SU(1,n)$ such that $g\Gamma g^{-1}$ and $\Gamma$ are non-
commensurable.

###### Proof.

For 1) and 2) see Proposition II.6.10 in [3] and observe that if
$g\in\Gamma\setminus e$ acts as an elliptic isometry, then it is contained in
a compact (finite) subgroup of $\Gamma$ and this cannot be the case.

For 3) observe that every $\gamma\in\Gamma\setminus e$ preserves a unique axis
in ${\mathbf{H}}^{n}_{\mathbf{C}}$ and that $\Gamma$ is finitely generated
(see Theorem 6.15 and Remark 6.18 in [24]). Define

$X=\\{\xi\in\partial{\mathbf{H}}^{n}_{\mathbf{C}}\mid\gamma\cdot\xi=\xi\,\text{for
some}\,\,\gamma\in\Gamma\\}.$

Let $x\in X$ and $g\in SU(1,n)$ be such that $g\cdot x\not\in X$. This is
possible because $X$ is countable. The claim is that $g\Gamma g^{-1}$ and
$\Gamma$ are not commensurable. Indeed, $g\cdot x$ is fixed by some $\theta\in
g\Gamma g^{-1}$, but for every $n$, $\theta$ and $\theta^{n}$ share the axis,
therefore the two lattices cannot be commensurable. ∎

The existence of uniform lattices in connected, non compact and semisimple
groups is due to Borel, for one reference see Chapter XIV in [24]. Any of
these lattices is finitely generated and as a consequence of Selberg’s Lemma
(see [2]) they are also virtually torsion free. This two facts together with
the previous observations show that there exist $\Gamma_{1}$ and $\Gamma_{2}$,
non-commensurable uniform lattices in $SU(1,n)$.

Following [14], a pair $(G,H)$ is called a Borel pair if $G$ does not admit
non-trivial homomorphisms to a compact group, $H$ is a closed subgroup and
$G/H$ admits a finite $G$-invariant measure. In this article the author showed
that if $(G,H)$ is a Borel pair, where $G$ is a connected real algebraic
group, then $H$ is Zariski dense in $G$ (see Corollary 4 in [14]).

######  Lemma 2.8.

Given two non-commensurable lattices $\Gamma_{1}$ and $\Gamma_{2}$ of
$SU(1,n)$ (or any connected real semisimple linear algebraic group without
compact factors), the group $H$ generated by $\Gamma_{1}\cup\Gamma_{2}$ is
dense in $SU(1,n)$.

###### Proof.

Observe that $\overline{H}$, the closure of $H$ for the usual topology, is
Zariski dense in $SU(1,n)$. Consider $\mathfrak{h}$ the Lie subalgebra of
$\overline{H}$. This space is invariant under the action of ${H}$, therefore
it is $SU(1,n)$-invariant because the action is Zariski continuous. This means
that $\overline{H}_{0}$ is a normal subgroup of $SU(1,n)$, but $SU(1,n)$ is
simple. Suppose $\overline{H}_{o}$ is the trivial group. Observe that
$\overline{H}/\Gamma_{i}$ carries a finite invariant measure (see Lemma 1.6 in
[24]), therefore $\Gamma_{1}$ and $\Gamma_{2}$ have finite index in
$\overline{H}$. This implies that $\Gamma_{1}$ and $\Gamma_{2}$ are
commensurable, which is a contradiction. ∎

Let $SU(1,n)\xrightarrow{\phi}Isom({\mathbf{H}}^{n}_{\mathbf{C}})$ be the
projectivization map. This is a surjective homomorphism onto
$Isom_{\mathbf{C}}({\mathbf{H}}^{n}_{\mathbf{C}})$, the group of holomorphic
isometries of ${\mathbf{H}}^{n}_{c}$. The map $\phi$ has finite kernel,
therefore if $\Gamma_{1}$ and $\Gamma_{2}$ are as above, $\phi(\Gamma_{1})$
and $\phi(\Gamma_{2})$ are two uniform non-commensurable lattices of
$Isom_{\mathbf{C}}({\mathbf{H}}^{n}_{\mathbf{C}})$. Indeed, observe that
$\Gamma_{i}\cdot\ker(\phi)$ is closed and countable (discrete), therefore
there is $U$ an open subset of $SU(1,n)$ such that $U\cap(\Gamma_{i}\cdot
ker(\phi))=\\{e\\}.$ This shows that $\phi(\Gamma_{i})$ is a discrete subgroup
of $Isom_{\mathbf{C}}({\mathbf{H}}^{n}_{\mathbf{C}})$. For the existence of a
finite $\phi(SU(1,n))$-invariant measure observe that there is a natural
continuous $G$-equivariant bijection

$SU(1,n)/\Gamma_{i}\rightarrow\phi(SU(1,n))/\phi(\Gamma_{i})$

where the domain is compact. The lattices $\phi({\Gamma_{1}})$ and
$\phi(\Gamma_{2})$ are not commensurable because $ker(\phi)$ is finite. The
group generated by $\phi(\Gamma_{1})$ and $\phi(\Gamma_{2})$ is dense because
$\Gamma_{1}$ and $\Gamma_{2}$ generate a dense subgroup of $SU(1,n)$.

######  Theorem 2.9.

For $n\geq 2,$ the group of holomorphic isometries of the complex hyperbolic
space of dimension $n$, $Isom_{\mathbf{C}}({\mathbf{H}}^{n}_{\mathbf{C}})$,
does not admit non-elementary representations into
$Isom({\mathbf{H}}^{\infty}_{\mathbf{R}})$, the group of isometries of the
infinite-dimensional real hyperbolic space.

###### Proof.

Let $\rho$ is a non-elementary representation, given a uniform lattice
$\Gamma$ of $Isom_{\mathbf{C}}({\mathbf{H}}^{n}_{\mathbf{C}})$, the
restriction of $\rho$ to $\Gamma$ is non-elementary. Therefore there exists a
$\Gamma$-equivariant, harmonic and Lipschitz continuous map
${\mathbf{H}}^{n}_{\mathbf{C}}\xrightarrow{u}{\mathbf{H}}^{\infty}_{\mathbf{R}}$
(see Theorem 2.3.1 of [19]). In Section 3.2 of [11], the authors showed that
this map is $\mathcal{C}^{\infty}$.

Given $\Gamma_{1}$ and $\Gamma_{2}$ two non-commensurable and uniform lattices
of $Isom_{\mathbf{C}}({\mathbf{H}}^{n}_{\mathbf{C}})$, there are
$\mathcal{C}^{2}$, harmonic, Lipschitz and $\Gamma_{i}$-equivariant functions,
${\mathbf{H}}^{n}_{\mathbf{C}}\xrightarrow{u_{i}}{\mathbf{H}}^{\infty}_{\mathbf{R}}$.
Therefore it follows from Lemmas 2.3 and 2.6 that $u_{1}=u_{2}$. This implies
that the function $u=u_{i}$ is
$Isom_{\mathbf{C}}({\mathbf{H}}^{n}_{\mathbf{C}})$-equivariant. In Proposition
8 of [11], the authors showed that the real rank of $u$ is at most 2. The
arguments used there go back to the work of Sampson (see [25]). If
$x\in{\mathbf{H}}^{n}_{\mathbf{C}}$, the kernel of $df_{x}$ is nontrivial. The
group $Stab(x)$ acts transitively in spheres of the tangent space of $x$ and
$u$ is $Isom_{\mathbf{C}}({\mathbf{H}}^{n}_{\mathbf{C}})$-equivariant,
therefore $u$ is constant, but this is a contradiction.

### Acknowledgments

I would like to thank Nicolas Monod for suggesting me the main question that
this paper answers, as well as for the countless fruitful and enlightening
discussions. Without his wisdom it would have been impossible for me to
complete this text. I would also like to thank Pierre Py for suggesting the
way to solve this problem as well as for clarifying so many questions that
arose in the process of creating this article. But above all for all his
generosity during my stay in Strasbourg in the fall of 2021, without which
this article would not have been possible. ∎

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