11institutetext: 1University of Oulu, Finland
2Preon Health Oy, Finland
11email<EMAIL_ADDRESS>
# SiNGR: Brain Tumor Segmentation via Signed Normalized Geodesic Transform
Regression

Trung Dang,1 Equal contributions Huy Hoang Nguyen⋆,1 Aleksei Tiulpin1,2

###### Abstract

One of the primary challenges in brain tumor segmentation arises from the
uncertainty of voxels close to tumor boundaries. However, the conventional
process of generating ground truth segmentation masks fails to treat such
uncertainties properly. Those “hard labels” with 0s and 1s conceptually
influenced the majority of prior studies on brain image segmentation. As a
result, tumor segmentation is often solved through voxel classification. In
this work, we instead view this problem as a voxel-level regression, where the
ground truth represents a certainty mapping from any pixel based on the
distance to tumor border. We propose a novel ground truth label
transformation, which is based on a signed geodesic transform, to capture the
uncertainty in brain tumors’ vicinity, while maintaining a margin between
positive and negative samples. We combine this idea with a Focal-like
regression L1-loss that enables effective regression learning in high-
dimensional output space by appropriately weighting voxels according to their
difficulty. We thoroughly conduct an experimental evaluation to validate the
components of our proposed method, compare it to a diverse array of state-of-
the-art segmentation models, and show that it is architecture-agnostic. The
code of our method is made publicly available at https://github.com/Oulu-
IMEDS/SiNGR/.

###### Keywords:

Semantic Segmentation Soft Labels Brain Tumor Signed Geodesic Transform

##  1 Introduction

Glioma is the most prevalent type of brain tumor in adults, which is also the
leading cause of cancer deaths for men under 40 years old and women under 20
years old [17]. Timely detection and characterization of gliomas is crucial
for patient survival. Fortunately, the widespread availability of magnetic
resonance imaging (MRI) enables non-invasive quantitative brain assessments,
enabling healthcare professionals to detect and closely monitor the
progression of brain tumors [21]. In response to this, various studies have
been conducted to develop deep learning (DL)-based methods for segmentation of
brain tumors’ volumes from MR images [20, 8, 7, 25]. In this paper, so as it
is done conventionally, by segmentation we imply semantic segmentation (SSEG),
the goal of which is to categorize each element (voxel or pixel) within an
image into background and foreground classes.

In the existing literature on brain tumor segmentation, the first line of
research aims to improve the effectiveness of DL architectures by increasing
the model capacity and/or embedding domain knowledge into architecture design
[7, 16, 26, 23, 8]. The second line of research studies novel segmentation
losses that are preferably correlated with metrics of interest such as Dice
score or Intersection-of-Union (IoU) [3, 2, 15, 25, 24]. Finally, the third
group of studies investigates alternative ways to define the ground truth (GT)
masks through _soft labels_ [20, 12]. Our study is at the intersection of the
last two directions, as we aim to define the GT masks through soft labels, as
well as to develop a new loss function for the problem.

SSEG is typically modeled as voxel-wise _classification_ [7, 16, 26, 23, 8].
This might be inherited from the annotation process of GT masks, where each
voxel is assigned to one or more classes by human annotators. However, such
strictly defined categories (e.g. $0$ vs $1$ in the binary case) in the
segmentation masks111Hereinafter, we use the terms segmentation masks, 0-1 GT
masks, and hard labels interchangeably. imply an equal role for all voxels
when it comes to determining the edges of an object of interest in an image.
Such an approach fails to capture the intra-class uncertainty in the
annotation masks. Delineating complex non-rigid structures such as tumors in
low-resolution MR images is highly challenging, and the uncertainty one can
easily identify could come from technical image quality, visibility, detail
complexity, and the knowledge of the annotator.

Figure 1: The overview of Signed Normalized Geodesic transform Regression
(SiNGR).

Some studies [20, 12, 25, 24] attempted to develop soft labels for SSEG, which
allows the model to learn that it should not always be a certainty. One such
example is label smoothing. This technique relies on the number of classes
[18], which is not suitable for the aforementioned intra-class uncertainty.
Liu et al. [11] proposed to produce soft labels using the signed Euclidean
distance map (SDM). Nevertheless, the SDM regression was merely considered as
a regularizer for the segmentation task. Vasudeva et al. [20] employed the
unsigned geodesic distance (GeoDT) transform [19] to generate a novel type of
soft labels that can characterize both spatial distance and image gradient.
Yet unsigned GeoDT-based soft labels were then integrated into the cross-
entropy (CE) loss, which is a classification loss.

We observe that the complexity of the segmentation annotation process is in
the labeling of voxels around the boundaries of objects of interest (OOI). The
uncertainty of a voxel’s label is proportional to the inverse of its distance
to the nearest boundary, as well as the visual blurriness in this region.
Although unsigned GeoDT naturally allows us to capture these properties,
additional signals are needed to differentiate foreground from background
voxels. Moreover, voxels significantly distant from the OOI exhibit very low
uncertainty; thus, these voxels should be marked in a manner that directs the
model to pay less attention to them.

In this study, we formulate the SSEG problem as _voxel regression_ (see Figure
1). We propose an extension of the unsigned GeoDT [19], termed _Signed
Normalized Geodesic_ (SiNG) transform that aims to approximate modeling of
segmentation annotations done by human from input images and the corresponding
GT masks. The SiNG transform is designed to primarily focus on the vicinity of
the OOI by assigning values in $(0,1]$ to foreground (FG) regions, values in
$(-1,0]$ to nearby background (BG) regions, and $-1$’s to distant voxels. As
we perform SSEG via _SiNG transform Regression_ , our method is named SiNGR.
To handle the imbalance of FG and BG voxels, we introduce a novel regression
loss, termed Focal-L1 loss. We conduct standardized and thorough experiments
to demonstrate the effectiveness of our method on the BraTS and LGG FLAIR
datasets. The empirical evidence shows that our method is beneficial for
various DL architectures.

##  2 Method

###  2.1 Overview

We approach the SSEG problem via signed soft-label regression. Specifically,
we utilize the SiNG transform presented in Section 2.2 to convert 0-1 GT masks
to signed soft labels, where regions of interest and BG regions are
represented by positive and negative values respectively. The proposed
transform is designed in such a way that FG and BG voxels are marginally
separate across $0$, which allows a simple $0$-threshold post-processing step
to produce final predicted masks. To effectively train the regression task, we
introduce a novel loss, named _Focal-L1_ , in Section 2.3.

###  2.2 Signed Normalized Geodesic Transform

####  2.2.1 Unsigned transform.

Given an input image $I$ with spatial dimensions of $H\times W\times L$, and
an arbitrary-shaped region $R\subset\Omega=[H]\times[W]\times[L]$ with
$[N]=\\{1,\dots,N\\}$, an L1-based _unsigned geodesic distance transform_
(GeoDT) from a point $i\in\Omega$ to $R$ is defined as [19, 22]

$\displaystyle G^{\lambda}(i;R,I)$ $\displaystyle=\min_{j\in
R}D^{\lambda}(i,j,I),$ (1) $\displaystyle D^{\lambda}(i,j;I)$
$\displaystyle=\min_{p\in
P_{i,j}}\int_{0}^{1}(1-\lambda)\|p^{\prime}(s)\|_{1}+\lambda\|\nabla
I(p(s))\cdot u(s)\|_{1}$ (2)

where $\lambda\in[0,1]$ is a weighting hyperparameter, $P(i,j)$ is the set of
all paths between locations $i$ and $j$, $p$ is a feasible parameterized path,
$u(s)=\frac{p^{\prime}(s)}{\|p^{\prime}(s)\|_{1}}$, and $\nabla I(p(s))$
represents image gradient at $p(s)$. Here, we have that $G^{\lambda}(i,R,I)=0\
\forall i\in R,\ \forall\lambda\in[0,1]$. Intuitively, the unsigned GeoDT
calculates the cost of the shortest path from point $i$ to the region $R$
based on both distance and image gradient information. The integral makes
GeoDT computationally expensive when we apply the transform for the whole
image. Its cost is proportional to $R$’s cardinality for Eq. 1 and $I$’s size
due to Eq. 2.

####  2.2.2 Signed version.

For _signed GeoDT_ , one typically runs the GeoDT transform twice for the
foreground and background regions, which is intensively costly [6]. To speed
up the signed GeoDT, we merely consider a set $R_{B}$ of boundary voxels of
OOI extracted from a 0-1 segmentation mask $M$ using the Canny edge detector
[5]. Hereinafter, as $R_{B}$ is our primary region of interest, we omit
$R_{B}$ and $I$ from Eq. 1 for simplicity, that is
$G^{\lambda}_{i}=G^{\lambda}(i;R_{B},I)$. Given $R_{B}$, we apply unsigned
GeoDT for the whole image to produce an unsigned map. Afterwards, we rely on
$M$ to specify the signs of the obtained map, that is
$s_{i}G^{\lambda}_{i},\forall i\in\Omega$, where
$s_{i}=\mathrm{sign}(2M_{i}-1)$ is the sign of voxel $i$. As the uncertainty
of human annotations is primarily around the boundary $R_{B}$, we thus ignore
regions substantially far from the boundary. As such, we let $\lambda=0$, and
define the neighboring region of the boundary

$\displaystyle\mathcal{B}$ $\displaystyle=\left\\{i\in\Omega\mid
G^{0}_{i}\leq\beta\right\\}$ (3) $\displaystyle\mathrm{s.t.}\ \beta$
$\displaystyle=\max_{i\in\Omega:M_{i}=1}G^{0}_{i}$ (4)

where $\beta$ represents the maximum spatial distance (i.e. the latter term in
Eq. 2 is omitted when $\lambda=0$). Finally, we propose the SiNG transform as
follows

$\displaystyle\mathcal{S}_{i}=\left\\{\begin{matrix}\frac{1}{\tau}(1-\delta)s_{i}G^{\lambda}_{i}+\delta
s_{i}&\quad\ \mathrm{if}\ i\in\mathcal{B}\\\ -1&\quad\
\mathrm{otherwise}\end{matrix}\right.$ (5)

where $\tau=\max_{j\in\mathcal{B}}G^{\lambda}_{j}$ is a normalization constant
for each pair $(I,M)$, and $\delta\in[0,1)$ is a margin hyperparameter. The
margin $\delta$ is needed to ensure the discrepancy in the signed normalized
map between foreground and background voxels.

(a) L1 loss (b) Focal-L1 loss ($\gamma=1$) (c) Focal-L1 behavior

Figure 2: Comparison between L1 and Focal-L1 losses: (a-b) 2D loss surfaces,
and (c) a behavior of the proposed Focal-L1 loss. Colors in (a-b) represent
loss magnitudes. In (c), we assume that $|\mathcal{S}_{i}|=1$ or
$|\mathcal{Z}_{i}|=1,\forall i\in\Omega$.

###  2.3 Focal-L1 Loss

We formulate the tumor segmentation task as image-level “regression” rather
than voxel “classification”, where one commonly utilizes the cross-entropy
(CE) or focal loss for the optimization. In alignment with the preceding
subsection, we also consider the significance of diversity across various
regions. As such, the regression loss is supposed to prioritize hard regions
over easy ones. Inspired by the CE-based focal loss [10], we propose the
following L1-based focal loss, namely Focal-L1, for the regression task.

Given an arbitrary pair of inputs $(I,M)$, we utilize the SiNG transformation
to produce a corresponding map $\mathcal{S}$ with a certain $\lambda$. In
addition, assuming that we have a parametric function $f_{\theta}$ with
parameters $\theta$, and $f_{\theta}(I)$ denotes a predicted mask from the
input image $I$. We then utilize the tanh function to convert values of
$f_{\theta}(I)$ into the range $[-1,1]$, that is
$\mathcal{Z}=\tanh(f_{\theta}(I))$. To this end, we propose the Focal-L1 loss
as follows

$\mathcal{L}_{\mathrm{FocalL1}}(\mathcal{S},\mathcal{Z};\theta)=\frac{1}{|\Omega|}\sum_{i\in\Omega}|\mathcal{S}_{i}-\mathcal{Z}_{i}|\underbrace{\frac{|\mathcal{S}_{i}-\mathcal{Z}_{i}|^{\gamma\mathbbm{I}(\mathcal{S}_{i}\mathcal{Z}_{i}\geq
0)}}{\max(|\mathcal{S}_{i}|,|\mathcal{Z}_{i}|)+\varepsilon}}_{\mathrm{Sample\
weighting}},$ (6)

where $\varepsilon$ is a positive constant to avoid numerical issues, $\gamma$
is a positive hyperparameter, and $\mathbb{I}(\cdot)$ is the indicator
function. A graphical comparison between our loss and the L1 loss is presented
in Figure 2. Note that the backpropagation is not applied to the weighting
term. Whereas the weighting term’s numerator is fixed at $1$ for hard cases
($\mathcal{S}_{i}\mathcal{Z}_{i}<0$), it is scaled down to less than $1$ for
easy ones ($\mathcal{S}_{i}\mathcal{Z}_{i}\geq 0$). Meanwhile, the denominator
reduces the importance of highly certain voxels (with high absolute value), as
they are straightforward to predict. Overall, the sample weighting helps the
loss to prioritize enforcing difficult voxels over simple ones.

##  3 Experiments

Table 1: Performance comparisons between our methods and the SOTA baselines on
the BraTS test set (means and SEs over $5$ random seeds). The best results
with substantial differences are highlighted in bold.

Method | GT | Dice score (%) $\uparrow$ | HD95 ($mm$) $\downarrow$
---|---|---|---
ET | TC | WT | Avg | ET | TC | WT | Avg
TransBTS [23] |  Hard label | 81.0±0.3 | 83.0±0.4 | 90.4±0.1 | 84.8±0.2 | 4.2±0.3 | 6.4±0.2 | 6.4±0.1 | 5.7±0.2
SegResnet [14] | 81.1±0.3 | 85.5±0.3 | 90.7±0.1 | 85.8±0.2 | 3.3±0.1 | 5.5±0.3 | 5.7±0.4 | 4.9±0.2
UNETR [8] | 82.3±0.2 | 82.0±0.4 | 90.0±0.1 | 84.8±0.2 | 4.0±0.3 | 7.4±0.2 | 6.2±0.4 | 5.9±0.2
EoFormer [16] | 82.5±0.2 | 84.8±0.4 | 91.3±0.0 | 86.2±0.2 | 3.7±0.2 | 6.4±0.4 | 5.9±0.3 | 5.3±0.2
UNet++3D [28] | 83.1±0.1 | 86.0±0.2 | 90.9±0.2 | 86.7±0.1 | 4.4±0.4 | 6.3±0.3 | 5.7±0.3 | 5.5±0.3
UNet3D [9] | 83.1±0.2 | 86.1±0.3 | 90.4±0.2 | 86.5±0.1 | 3.8±0.4 | 5.9±0.3 | 6.5±0.6 | 5.4±0.4
NestedFormer [26] | 83.5±0.1 | 85.4±0.1 | 91.2±0.1 | 86.7±0.1 | 4.4±0.4 | 7.4±0.4 | 6.6±0.4 | 6.1±0.4
Swin-UNETR [7] | 84.1±0.2 | 85.7±0.4 | 90.9±0.1 | 86.9±0.2 | 3.7±0.3 | 6.4±0.2 | 6.0±0.2 | 5.4±0.2
Ours (Swin-UNETR) | SiNG | 85.1±0.3 | 88.0±0.1 | 91.3±0.1 | 88.1±0.1 | 2.3±0.1 | 4.2±0.2 | 4.8±0.0 | 3.8±0.1

###  3.1 Setup

Datasets. We thoroughly conducted experiments on two public brain tumor
datasets: BraTS 2020 and LGG FLAIR. BraTS 2020 [13] consists of multi-modal MR
images from $369$ subjects. Each image was aligned across four modalities and
was standardized to a volume size of $240\times 240\times 155$. The
segmentation targets are enhancing tumor (ET), tumor core (TC), and whole
tumor (WT). We divided the dataset into three portions with $236$, $59$, and
$74$ samples for training, validation, and test sets, respectively. LGG FLAIR
[4] has $110$ 3-channel FLAIR MR images and corresponding abnormality
segmentation masks. The number of axial slices of each MR scan varies from
$20$ to $80$, and they have a common spatial dimension of $256\times 256$. We
split the data into training, validation, and test sets with $70$, $18$, and
$22$ samples, respectively.

Implementation details. All models were trained on Nvidia V100 GPUs. We
implemented our pipeline using Pytorch and followed a standard data processing
configuration for all methods. We used the FastGeodist library to compute
unsigned GeoDT maps [1]. During the training process on BraTS, the 3D MR
images were randomly cropped to $128\times 128\times 128$ cubes, and augmented
by flipping, intensity scaling, and intensity shifting. We performed the
window-slicing technique with a window size of $128\times 128\times 128$ in
the validation and test stages. We utilized the Adam optimizer with an initial
learning rate of $10^{-4}$, a weight decay of $10^{-5}$, and a batch size of
$2$. Regarding SiNGR-specific hyperparameters, we empirically found
$\lambda=0.5$ and $\gamma=1$ working the best. The hyperparameter
$\varepsilon$ in Eq. 6 was set to $0.0$. We performed $0$-thresholding to
binarize our predicted maps. For performance evaluation, we utilized image-
wise IoU, Dice score, and $95\%$ Hausdorff distance (HD95).

We compared our method to a diverse array of references specializing in 3D
data as listed in Tables 1 and 2. In addition, we validated the SiNG transform
and the Focal-L1 loss against other soft-label baselines – namely label
smoothing (LS) [18] and Geodesic label smoothing (GeoLS) [20] – together with
Jaccard metric loss (JML) [24], which specializes in optimizing with soft
labels.

Table 2: Performance comparisons between our methods and the SOTA baselines on the LGG FLAIR test set (means and SEs over $5$ random seeds). The best results are highlighted in bold. Method | GT | IoU (%) $\uparrow$ | Dice (%) $\uparrow$ | HD95 ($mm$) $\downarrow$
---|---|---|---|---
EoFormer [16] |  Hard label | 38.8±11.3 | 68.4±3.0 | 27.1±5.1
NestedFormer [26] | 46.6±3.1 | 60.4±1.1 | 58.4±1.7
UNETR [8] | 49.5±3.2 | 62.8±1.0 | 49.5±1.8
SegResnet [14] | 50.5±1.5 | 64.9±0.5 | 46.3±2.1
UNet++3D [28] | 59.0±3.3 | 71.1±1.1 | 34.6±1.3
Swin-UNETR [7] | 54.4±2.3 | 67.8±0.8 | 32.2±1.8
TransBTS [23] | 58.5±1.1 | 71.9±0.5 | 24.8±2.4
UNet3D [9] | 57.9±4.0 | 69.1±1.3 | 33.6±2.0
Ours (UNet3D) | SiNG | 71.4±0.7 | 82.3±0.5 | 11.4±1.6

(a) Dice on BraTS (b) HD95 on BraTS (c) Dice on LGG (d) HD95 on LGG

Figure 3: Pair-wise comparisons between baselines using hard labels (in blue)
and our method (in orange) across different architectures and metrics. NF, SU,
and TBTS mean NestedFormer, Swin-UNETR, and TransBTS respectively.

###  3.2 Results

BraTS 2020 dataset. The quantitative results on the BraTS test set are
presented in Table 1. Accordingly, the top-3 methods with the highest IoU used
the UNet3D, NestedFormer, and Swin-UNETR architectures. When we applied our
SiNGR method to those three architectures, we observed consistent and
substantial improvements in all the metrics (see Figures 3a and 3b). For
instance, compared to the best baseline, Swin-UNETR, our corresponding method
achieved $1.2\%$ and $1.6mm$ better in average Dice score and average HD95,
respectively. We present a qualitative comparison of the methods in Suppl.
Figure S1.

LGG FLAIR dataset. We present the detailed results on the LGG FLAIR test set
in Table 2. On this small dataset, the impact of SiNGR was particularly
significant on UNet3D, and this setting also acquired the highest IoU and Dice
score. Compared to the UNet3D reference, our method resulted in significant
gains of $13.5\%$ IoU, $13.2\%$ Dice score, and $22.2mm$ HD95. Using our
method also consistently led to substantial improvements for Swin-UNETR and
TransBTS (see Figures 3c and 3d). The qualitative results are presented in
Suppl. Figure S2.

Impact of SiNG and Focal-L1. We investigated the effects of the components of
our method and demonstrated the results on the BraTS test set in Table 3. Two
unsigned label smoothing techniques [18, 20] did not perform well on the task.
Compared to LS, our method outperformed with differences of $8.4\%$ and
$7.8mm$ in average Dice score and average HD95, respectively. Additionally,
among L1, L2, and product [27] losses, L1 was the best combination with the
SiNG transform. However, L1 setting achieved an average Dice of $1.1\%$ lower
than our method. Moreover, the empirical evidence showed the importance of the
margin in the SiNG transform as well as the sample weighting coefficient in
Focal-L1 loss. Notably, excluding the margin $\delta$ in SiNG led to a
substantial drop of $3.4\%$ average Dice score and an increase of $1.5mm$
average HD95.

Table 3: Performance comparisons between different combinations of soft labels
and loss functions on BraTS. UNet3D was the common architecture. $\delta$ is
the margin in SiNG. Focal-L1† indicates that the sample weighting in Eq. 6 is
simplified to $|\mathcal{S}_{i}-\mathcal{Z}_{i}|^{\gamma}$. Our optimal
setting is highlighted in cyan.

Label | $\delta$ | Loss | Dice score (%) $\uparrow$ | HD95 ($mm$) $\downarrow$
---|---|---|---|---
ET | TC | WT | Avg | ET | TC | WT | Avg
LS [18] | - | JML [24] | 75.5±1.2 | 75.5±0.7 | 87.5±0.3 | 79.5±0.6 | 5.7±0.4 | 17.8±4.5 | 11.2±1.5 | 11.6±2.1
GeoLS [20] | - | JML [24] | 67.0±0.9 | 63.3±2.3 | 80.8±0.7 | 70.4±1.1 | 20.5±1.7 | 12.2±0.9 | 24.3±2.7 | 19.0±1.5
SiNG | 0.5 | L2 | 80.9±0.1 | 84.3±0.2 | 90.4±0.1 | 85.2±0.1 | 3.9±0.4 | 5.9±0.3 | 6.3±0.4 | 5.4±0.3
Product [27] | 82.5±0.4 | 86.9±0.1 | 90.9±0.1 | 86.8±0.2 | 3.9±0.5 | 5.6±0.4 | 5.5±0.3 | 5.0±0.4
L1 | 82.7±0.1 | 87.1±0.3 | 90.7±0.1 | 86.8±0.1 | 3.3±0.3 | 4.9±0.2 | 5.6±0.2 | 4.6±0.2
Focal-L1† | 83.4±0.3 | 87.1±0.2 | 91.0±0.1 | 87.1±0.2 | 3.1±0.4 | 4.8±0.4 | 5.2±0.2 | 4.3±0.3
0 |  | 80.0±0.5 | 83.8±0.5 | 89.9±0.1 | 84.5±0.3 | 4.3±0.3 | 5.8±0.4 | 5.7±0.1 | 5.3±0.3
0.5 | Focal-L1 | 84.4±0.1 | 88.1±0.1 | 91.1±0.1 | 87.9±0.0 | 2.5±0.1 | 4.2±0.1 | 4.8±0.0 | 3.8±0.0

##  4 Conclusions

We have introduced a simple approach to segmentation of brain tumors through
voxel-wise regression.We proposed the novel SiNG transform that allows us to
convert 0-1 annotated masks to soft labels that take into account the
uncertainty of the labeling process. In addition, we have introduced the
Focal-L1 loss to effectively weight voxels according to their difficulty. Our
empirical findings indicate that our method consistently enhances performance
across different DL architectures. We will make our implementation publicly
accessible to the community.

## References

  * [1] Asad, M., Dorent, R., Vercauteren, T.: Fastgeodis: Fast generalised geodesic distance transform. arXiv preprint arXiv:2208.00001 (2022)
  * [2] Berman, M., Triki, A.R., Blaschko, M.B.: The lovász-softmax loss: A tractable surrogate for the optimization of the intersection-over-union measure in neural networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 4413–4421 (2018)
  * [3] Bertels, J., Eelbode, T., Berman, M., Vandermeulen, D., Maes, F., Bisschops, R., Blaschko, M.B.: Optimizing the dice score and jaccard index for medical image segmentation: Theory and practice. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22. pp. 92–100. Springer (2019)
  * [4] Buda, M., Saha, A., Mazurowski, M.A.: Association of genomic subtypes of lower-grade gliomas with shape features automatically extracted by a deep learning algorithm. Computers in biology and medicine 109, 218–225 (2019)
  * [5] Canny, J.: A computational approach to edge detection. IEEE Transactions on pattern analysis and machine intelligence (6), 679–698 (1986)
  * [6] Fu, K., Gu, I.Y., Ödblom, A., Liu, F.: Geodesic distance transform-based salient region segmentation for automatic traffic sign recognition. In: 2016 IEEE Intelligent Vehicles Symposium (IV). pp. 948–953. IEEE (2016)
  * [7] Hatamizadeh, A., Nath, V., Tang, Y., Yang, D., Roth, H.R., Xu, D.: Swin unetr: Swin transformers for semantic segmentation of brain tumors in mri images. In: International MICCAI Brainlesion Workshop. pp. 272–284. Springer (2021)
  * [8] Hatamizadeh, A., Tang, Y., Nath, V., Yang, D., Myronenko, A., Landman, B., Roth, H.R., Xu, D.: Unetr: Transformers for 3d medical image segmentation. In: Proceedings of the IEEE/CVF winter conference on applications of computer vision. pp. 574–584 (2022)
  * [9] Kerfoot, E., Clough, J., Oksuz, I., Lee, J., King, A.P., Schnabel, J.A.: Left-ventricle quantification using residual u-net. In: Statistical Atlases and Computational Models of the Heart. Atrial Segmentation and LV Quantification Challenges: 9th International Workshop, STACOM 2018, Held in Conjunction with MICCAI 2018, Granada, Spain, September 16, 2018, Revised Selected Papers 9. pp. 371–380. Springer (2019)
  * [10] Lin, T.Y., Goyal, P., Girshick, R., He, K., Dollár, P.: Focal loss for dense object detection. In: Proceedings of the IEEE international conference on computer vision. pp. 2980–2988 (2017)
  * [11] Liu, Z., He, X., Lu, Y.: Combining unet 3+ and transformer for left ventricle segmentation via signed distance and focal loss. Applied Sciences 12(18), 9208 (2022)
  * [12] Ma, J., Wang, C., Liu, Y., Lin, L., Li, G.: Enhanced soft label for semi-supervised semantic segmentation. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 1185–1195 (2023)
  * [13] Menze, B.H., Jakab, A., Bauer, S., Kalpathy-Cramer, J., Farahani, K., Kirby, J., Burren, Y., Porz, N., Slotboom, J., Wiest, R., et al.: The multimodal brain tumor image segmentation benchmark (brats). IEEE transactions on medical imaging 34(10), 1993–2024 (2014)
  * [14] Myronenko, A.: 3d mri brain tumor segmentation using autoencoder regularization. In: Brainlesion: Glioma, Multiple Sclerosis, Stroke and Traumatic Brain Injuries: 4th International Workshop, BrainLes 2018, Held in Conjunction with MICCAI 2018, Granada, Spain, September 16, 2018, Revised Selected Papers, Part II 4. pp. 311–320. Springer (2019)
  * [15] Salehi, S.S.M., Erdogmus, D., Gholipour, A.: Tversky loss function for image segmentation using 3d fully convolutional deep networks. In: International workshop on machine learning in medical imaging. pp. 379–387. Springer (2017)
  * [16] She, D., Zhang, Y., Zhang, Z., Li, H., Yan, Z., Sun, X.: Eoformer: Edge-oriented transformer for brain tumor segmentation. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 333–343. Springer (2023)
  * [17] Siegel, R.L., Miller, K.D., Fuchs, H.E., Jemal, A.: Cancer statistics, 2022. CA: a cancer journal for clinicians 72(1), 7–33 (2022)
  * [18] Szegedy, C., Vanhoucke, V., Ioffe, S., Shlens, J., Wojna, Z.: Rethinking the inception architecture for computer vision. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 2818–2826 (2016)
  * [19] Toivanen, P.J.: New geodosic distance transforms for gray-scale images. Pattern Recognition Letters 17(5), 437–450 (1996)
  * [20] Vasudeva, S.A., Dolz, J., Lombaert, H.: Geols: Geodesic label smoothing for image segmentation. In: Medical Imaging with Deep Learning. pp. 468–478. PMLR (2024)
  * [21] Verduin, M., Compter, I., Steijvers, D., Postma, A.A., Eekers, D.B., Anten, M.M., Ackermans, L., Ter Laan, M., Leijenaar, R.T., van de Weijer, T., et al.: Noninvasive glioblastoma testing: multimodal approach to monitoring and predicting treatment response. Disease markers 2018 (2018)
  * [22] Wang, G., Zuluaga, M.A., Li, W., Pratt, R., Patel, P.A., Aertsen, M., Doel, T., David, A.L., Deprest, J., Ourselin, S., et al.: Deepigeos: a deep interactive geodesic framework for medical image segmentation. IEEE transactions on pattern analysis and machine intelligence 41(7), 1559–1572 (2018)
  * [23] Wang, W., Chen, C., Ding, M., Yu, H., Zha, S., Li, J.: Transbts: Multimodal brain tumor segmentation using transformer. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part I 24. pp. 109–119. Springer (2021)
  * [24] Wang, Z., Blaschko, M.B.: Jaccard metric losses: Optimizing the jaccard index with soft labels. arXiv preprint arXiv:2302.05666 (2023)
  * [25] Wang, Z., Popordanoska, T., Bertels, J., Lemmens, R., Blaschko, M.B.: Dice semimetric losses: Optimizing the dice score with soft labels. arXiv preprint arXiv:2303.16296 (2023)
  * [26] Xing, Z., Yu, L., Wan, L., Han, T., Zhu, L.: Nestedformer: Nested modality-aware transformer for brain tumor segmentation. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 140–150. Springer (2022)
  * [27] Xue, Y., Tang, H., Qiao, Z., Gong, G., Yin, Y., Qian, Z., Huang, C., Fan, W., Huang, X.: Shape-aware organ segmentation by predicting signed distance maps. In: Proceedings of the AAAI Conference on Artificial Intelligence. vol. 34, pp. 12565–12572 (2020)
  * [28] Zhou, Z., Siddiquee, M.M.R., Tajbakhsh, N., Liang, J.: Unet++: Redesigning skip connections to exploit multiscale features in image segmentation. IEEE transactions on medical imaging 39(6), 1856–1867 (2019)

Figure S1: Visualization of predictions of our method and the baselines on the
samples from the BraTS test set. The overlaid numbers are averaged Dice
scores.

Figure S2: Visualization of predictions of our method and the baselines on the
samples from the LGG FLAIR test set. The overlaid numbers are Dice scores.


# Strongly convergent unitary representations
of limit groups

Larsen Louder and Michael Magee
With appendix by Will Hide and Michael Magee

###### Abstract

We prove that all finitely generated fully residually free groups (limit
groups) have a sequence of finite dimensional unitary representations that
‘strongly converge’ to the regular representation of the group. The
corresponding statement for finitely generated free groups was proved by
Haagerup and Thorbjørnsen in 2005. In fact, we can take the unitary
representations to arise from representations of the group by permutation
matrices, as was proved for free groups by Bordenave and Collins.

As for Haagerup and Thorbjørnsen, the existence of such representations
implies that for any non-abelian limit group, the Ext-invariant of the reduced
$C^{*}$-algebra is not a group (has non-invertible elements).

An important special case of our main theorem is in application to the
fundamental groups of closed orientable surfaces of genus at least two. In
this case, our results can be used as an input to the methods previously
developed by the authors of the appendix. The output is a variation of our
previous proof of Buser’s 1984 conjecture that there exist a sequence of
closed hyperbolic surfaces with genera tending to infinity and first
eigenvalue of the Laplacian tending to $\frac{1}{4}$. In this variation of the
proof, the systoles of the surfaces are bounded away from zero and the
surfaces can be taken to be arithmetic.

###### Contents

  1. 1 Introduction
    1. 1.1 Further consequences I: Spectral gaps
    2. 1.2 Further consequences II: $\mathrm{Ext}(\Gamma)$ is not a group
  2. 2 Background
  3. 3 Proof of Proposition 1.7
  4. 4 Proof of Theorem 1.6
  5. 5 Proof of Theorem 1.1
    1. 5.1 Proof when $\Gamma=\mathbf{Z}^{r}$
    2. 5.2 Proof for non-abelian limit groups
  6. 6 Proof of Proposition 1.3
  7. 7 Proof of Corollary 1.9
  8. A Spectral gaps of hyperbolic surfaces
    1. A.1 Set up
    2. A.2 Function spaces
    3. A.3 Operators on $\mathbb{H}$
    4. A.4 Proof of Theorem A.1

##  1 Introduction

A discrete group $\Gamma$ is _fully residually free_(FRF)__ if for any finite
set $S\subset\Gamma$, there exists a homomorphism
$\Gamma\to\mathrm{\mathbf{F}}$ that is injective on $S$ where
$\mathrm{\mathbf{F}}$ is a free group. Finitely generated FRF groups are known
to coincide with Sela’s _limit groups_[Sel01]_,_ so we use these two notions
interchangeably in the sequel.

For $N\in\mathbf{N}$ let $\mathbf{U}(N)$ denote the group of $N\times N$
complex unitary matrices. For a discrete group $\Gamma$,
$\lambda_{\Gamma}:\Gamma\to\mathrm{End}(\ell^{2}(\Gamma))$ is the left regular
representation. It was an open problem for some years, popularized by
Voiculescu in [Voi93, Qu. 5.12], whether for a finitely generated free group
$\mathrm{\mathbf{F}}$, there exists a sequence of unitary representations
$\\{\rho_{i}:\mathrm{\mathbf{F}}\to\mathbf{U}(N_{i})\\}_{i=1}^{\infty}$ such
that for any element $z\in\mathbf{C}[\mathrm{\mathbf{F}}]$,

$\limsup_{i\to\infty}\|\rho_{i}(z)\|\leq\|\lambda_{\mathrm{\mathbf{F}}}(z)\|.$

The norm on the left is the operator norm on $\mathbf{C}^{N_{i}}$ with respect
to the standard Hermitian metric, and the norm on the right is the operator
norm on $\ell^{2}(\Gamma)$. This problem was solved in the affirmative in a
huge breakthrough by Haagerup and Thorbjørnsen [HT05].

In fact, following [Voi93], given that the reduced $C^{*}$-algebra of
$\mathrm{\mathbf{F}}$ is simple by a result of Powers [Pow75], the inequality
above can be improved automatically111See proof of Theorem 1.1 below for
details. to

$\lim_{i\to\infty}\|\rho_{i}(z)\|=\|\lambda_{\mathrm{\mathbf{F}}}(z)\|\quad\forall
z\in\mathbf{C}[\mathrm{\mathbf{F}}].$ (1.1)

This notion of convergence of a sequence of finite dimensional unitary
representations given by (1.1) applies equally as well to any discrete group
$\Gamma$ and we refer to this as _strong convergence._

######  Theorem 1.1.

Any limit group $\Gamma$ has a sequence of finite dimensional unitary
representations that strongly converge to the regular representation of
$\Gamma$. In fact, these unitary representations can be taken to factor
through

$\Gamma\to S_{N}\xrightarrow{\mathrm{std}}\mathbf{U}(N-1)$ (1.2)

for some varying $N$, where $S_{N}$ is the group of permutations of $N$
letters, and $\mathrm{std}$ is the $N-1$ dimensional irreducible component of
the representation of $S_{N}$ by 0-1 matrices.

It was proved by G. Baumslag in [Bau62] that the fundamental groups
$\Lambda_{g}$ of closed orientable surfaces are FRF, and it is also known
[Bau67, pp. 414-415] that the fundamental groups of non-orientable surfaces
$S$ with $\chi(S)\leq-2$ are FRF. This gives the following corollary of
Theorem 1.1.

######  Corollary 1.2.

Let $\Gamma$ denote the fundamental group of a connected closed surface $S$
that is either orientable with no constraint on $\chi(S)$, or non-orientable
with $\chi(S)\leq-2$. Then $\Gamma$ has a sequence of finite dimensional
unitary representations that strongly converge to the regular representation.
Moreover, they can be taken to be of the form (1.2) for some varying $N$.

Corollary 1.2 leaves open the cases of connected non-orientable surfaces with
$\chi=1$ ($\mathbf{R}P^{2}$), $\chi=0$ (the Klein bottle), and $\chi=-1$. In
all these cases the corresponding fundamental groups are not FRF222The first
two cases are easy to check, and the case of $\chi=-1$ is due to Lyndon
[Lyn59].. The fundamental group of $\mathbf{R}P^{2}$ is
$\mathbf{Z}/2\mathbf{Z}$, and its regular representation is finite
dimensional. We prove the following.

######  Proposition 1.3.

The Klein bottle group $\langle\,a,b\,|\,b^{-1}=aba^{-1}\,\rangle$ has a
sequence of finite dimensional unitary representations that strongly converge
to its regular representation.

This leaves open the seemingly significant problem of extending Theorem 1.1 to
the fundamental group of the connected sum of three copies of the real
projective plane.

The proof of Theorem 1.1 revolves around the following potential property of
discrete groups that we introduce here.

######  Definition 1.4.

A discrete group $\Gamma$ is _$C^{*}$ -residually free_ if for any finite set
$S$ and $\epsilon>0$, there is a homomorphism
$\phi:\Gamma\to\mathrm{\mathbf{F}}$ with $\mathrm{\mathbf{F}}$ free such that

$\|\lambda_{\mathrm{\mathbf{F}}}(\phi(z))\|\leq\|\lambda_{\Gamma}(z)\|+\epsilon.$

for all $z\in\mathbf{C}[\Gamma]$ supported on $S$ with unit $\ell^{1}$ norm.

######  Example 1.5.

Any extension $N\to G\xrightarrow{\phi}\mathrm{\mathbf{F}}$ of a free group by
an amenable group $N$ is $C^{*}$-residually free. Indeed, since $N$ is
amenable $1$ is weakly contained in the regular representation of $N$. Then by
Fell’s continuity of induction ([Fel62], [BdlHV08, Thm. F.3.5]) we have that
the quasi-regular representation of $G$ on $\ell^{2}(G/N)$ is weakly contained
in the regular representation $G$, hence by [BdlHV08, Thm. F.4.4] for any
$z\in\mathbf{C}[\mathrm{\mathbf{F}}]$

$\|\lambda_{G/N}(zN)\|=\|\lambda_{\mathrm{\mathbf{F}}}(\phi(z))\|\leq\|\lambda_{G}(z)\|.$

Here we prove the following.

######  Theorem 1.6.

Limit groups are $C^{*}$-residually free.

The converse to Theorem 1.6 does not hold: Example 1.5 shows that
$\mathbf{Z}\times\mathrm{\mathbf{F}}$ is $C^{*}$-residually free, but it is
easy to see that it is not FRF. It is, however, also easy to see that it is
residually free333It is an interesting question, not pursued here, to give an
alternative characterization of a group being $C^{*}$-residually free. .

Given a free group $\mathrm{\mathbf{F}}$, and a basis $X$ of
$\mathrm{\mathbf{F}}$, we write $|f|_{X}$ for the word length of $f$ in the
basis $X$. In any discrete group $\Gamma$ with generating set we write
$B_{Y}(r)$ for the elements of $\Gamma$ that can be written a product of as
most $r$ elements of $Y\cup Y^{-1}$. The proof of Theorem 1.6 relies on the
following key proposition.

######  Proposition 1.7.

Let $\Gamma$ be a limit group with a fixed finite generating set $Y$. There is
$D=D(\Gamma,Y)>0$ and $C=C(\Gamma,Y)>0$ such that for any $r>0$ there is an
epimorphism $f:\Gamma\to\mathrm{\mathbf{F}}$ with $\mathrm{\mathbf{F}}$ free,
which is injective on $B_{Y}(r)$, and a basis $X$ of $\mathrm{\mathbf{F}}$
such that

$\max_{g\in B_{Y}(r)}|f(g)|_{X}\leq Cr^{D}.$

###  1.1 Further consequences I: Spectral gaps

A hyperbolic surface is a complete Riemannian surface (without boundary) of
constant curvature -1 . Given a hyperbolic surface $X$, we write $\Delta_{X}$
for the Laplace-Beltrami operator on $L^{2}(X)$. If $X$ is closed this
operator’s spectrum $\mathrm{spec}(\Delta_{X})$ consists of eigenvalues
$0=\lambda_{0}(X)\leq\lambda_{1}(X)\leq\cdots\leq\lambda_{k}(X)\leq\cdots$
with $\lambda_{k}(X)\to\infty$ as $k\to\infty$. It was a conjecture of Buser
[Bus84] whether there exist a sequence of closed hyperbolic surfaces $X_{i}$
with genera tending to infinity and with

$\lambda_{1}(X_{i})\to\frac{1}{4}$

where $\lambda_{1}$ denotes the first non-zero eigenvalue of the Laplacian.
The value $\frac{1}{4}$ is the asymptotically optimal one by a result of Huber
[Hub74]. See [HM, Introduction] for an overview of the rich history of this
problem. Buser’s conjecture was settled in [HM]. One interesting feature of
the proof therein is that the surfaces constructed may have very short closed
curves, so there is no control on the systole: the shortest closed curve on
the surface. The results of this work in conjunction with the ideas in [HM]
allow us, along with Hide, to prove:

######  Theorem 1.8.

There exists a sequence of closed hyperbolic surfaces
$\left\\{X_{i}\right\\}_{i\in\mathbb{N}}$ with $g(X_{i})\to\infty$, systoles
uniformly bounded away from zero, and with

$\lambda_{1}(X_{i})\to\frac{1}{4}.$

In fact the $X_{i}$ can be taken to be covering spaces of a fixed hyperbolic
surface $X$. This base surface $X$, and hence all the $X_{i}$, can be taken to
be arithmetic.

Theorem 1.8 is proved in the Appendix444In fact, the Appendix proves a more
general statement about coverings of _any_ hyperbolic surface; see Theorem
A.1. by the second named author (MM) and Hide, as a consequence of the
following corollary of Theorem 1.1.

######  Corollary 1.9 (Matrix coefficients version of Theorem 1.1).

Let $\Gamma$ be a non-abelian limit group. There exist a sequence of finite
dimensional representations $\rho_{i}$ of the form (1.2) such that for any
$r\in\mathbf{N}$ and finitely supported map $a:\Gamma\to\mathrm{Mat}_{r\times
r}(\mathbf{C})$, we have

$\limsup_{i\to\infty}\|\sum_{\gamma\in\Gamma}a(\gamma)\otimes\rho_{i}(\gamma)\|\leq\|\sum_{\gamma\in\Gamma}a(\gamma)\otimes\lambda(\gamma)\|.$

The norm on the left hand side is the operator norm for the tensor product of
($r$ and $N_{i}$-dimensional) $\ell^{2}$ norms. The norm on the right is the
operator norm for the tensor product of $\ell^{2}$ and the inner product on
$\ell^{2}(\Gamma)$.

The proof of Corollary 1.9 from Theorem 1.1 is explained in $\S$7.

###  1.2 Further consequences II: $\mathrm{Ext}(\Gamma)$ is not a group

In [BDF73, BDF77], Brown, Douglas, and Fillmore introduced and studied a
homological/K-theoretic invariant $\mathrm{Ext}(\mathcal{A})$ of a unital
separable $C^{*}$-algebra $\mathcal{A}$. By definition,
$\mathrm{Ext}(\mathcal{A})$ is the collection of $*$-homomorphisms

$\pi:\mathcal{A}\to B(\ell^{2}(\mathbf{N}))/\mathcal{K}$

modulo conjugation of unitary operators on $\ell^{2}(\mathbf{N})$, where
$B(\ell^{2}(\mathbf{N}))$ is the bounded operators on $\ell^{2}(\mathbf{N})$
and $\mathcal{K}$ is the ideal of compact operators therein. This is naturally
a semigroup with multiplication arising from
$(\pi_{1},\pi_{2})\mapsto\pi_{1}\oplus\pi_{2}$ composed with an isomorphism
$\ell^{2}(\mathbf{N})\oplus\ell^{2}(\mathbf{N})\cong\ell^{2}(\mathbf{N})$.

One of the motivations of the work of Haagerup and Thorbjørnsen [HT05] was to
prove that there are non-invertible elements of
$\mathrm{Ext}(C_{r}^{*}(\mathrm{\mathbf{F}}))$ when $\mathrm{\mathbf{F}}$ is a
finitely generated non-abelian free group, i.e.,
$\mathrm{Ext}(C_{r}^{*}(\mathrm{\mathbf{F}}))$ is not a group.

The passage from the existence of strongly convergent unitary representations
of $\mathrm{\mathbf{F}}$ to this statement uses the following result proved by
Voiculescu in [Voi93, §§5.14] (see [HT05, Rmk. 8.6] for another exposition).

######  Proposition 1.10.

If $\Gamma$ is a discrete, countable, non-amenable group with a sequence of
finite dimensional unitary representations that strongly converge to the
regular representation of $\Gamma$, then $\mathrm{Ext}(C_{r}^{*}(\Gamma))$ is
not a group.

Since non-abelian limit groups $\Gamma$ are $C^{*}$-simple (Lemma 5.2), they
are non-amenable. Indeed, an amenable group $\Gamma$ has a $C^{*}$-algebra
morphism $C_{r}^{*}(\Gamma)\to\mathbf{C}$ by [BdlHV08, Thm. F.4.4] whose
kernel contradicts simplicity. Hence combining Theorem 1.1 with Proposition
1.10 we obtain the following extension of
‘$\mathrm{Ext}(C_{r}^{*}(\mathrm{\mathbf{F}}))$ is not a group’:

######  Corollary 1.11.

If $\Gamma$ is a non-abelian limit group, then
$\mathrm{Ext}(C_{r}^{*}(\Gamma))$ is not a group.

### Acknowledgments

We thank Benoît Collins, Simon Marshall, and Dan Voiculescu for comments
around this work.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 949143).

##  2 Background

#### Groups

We write $e$ for the identity in any group. For any group $\Gamma$,
$\mathbf{C}[\Gamma]$ denotes the group algebra of $\Gamma$ with complex
coefficients. For a free group $\mathrm{\mathbf{F}}$ with a fixed set of
generators $X,$ for each $h\in\mathrm{\mathbf{F}}$, we write $|h|_{X}$ for the
reduced word length of $h$ with respect to $X$. If $Y$ is a symmetric
generating set of any group $\Gamma$, we write $B_{Y}(r)\subset\Gamma$ for the
elements of $\Gamma$ that can be written as the product of at most $r$
elements of $Y$.

#### Analysis

Given a discrete group $\Gamma$,
$\lambda_{\Gamma}:\Gamma\to\mathrm{End}(\ell^{2}(\Gamma))$ is the _left
regular representation_

$\lambda_{\Gamma}(g)[f](h)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f(g^{-1}h).$

This representation extends by linearity to one of the convolution algebra
$\ell^{1}(\Gamma$). For $\psi\in\ell^{1}(\Gamma)$, since $\lambda$ is unitary
we have the basic inequality

$\|\lambda(\psi)\|\leq\|\psi\|_{\ell^{1}}$ (2.1)

where the norm on the left is operator norm. The _reduced $C^{*}$-algebra_ of
$\Gamma$, denoted $C_{r}^{*}(\Gamma)$, is the closure of
$\lambda(\ell^{1}(\Gamma))$ with respect to the operator norm topology. A
_tracial state_ on a unital $C^{*}$ algebra $\mathcal{A}$ is a linear
functional $\tau$ such that $\tau(1)=1$, $\tau(a^{*}a)\geq 0$ (in particular,
is real) for all $a\in\mathcal{A}$ , and $\tau(ab)=\tau(ba)$ for all
$a,b\in\mathcal{A}$.

An important inequality due to Haagerup [Haa79] links the operator norm in
$\mathrm{End}(\ell^{2}(\mathrm{\mathbf{F}}))$ and the $\ell^{2}$ norm in
$\mathbf{C}[\mathrm{\mathbf{F}}]$.

######  Lemma 2.1 (Haagerup).

Let $X$ denote a finite generating set for a free group $\mathrm{\mathbf{F}}$.
Suppose that $a\in\mathbf{C}[\mathrm{\mathbf{F}}]$ is supported on $B_{X}(r)$.
Then

$\|\lambda_{\mathrm{\mathbf{F}}}(a)\|\leq(r+1)^{\frac{3}{2}}\|a\|_{\ell^{2}}.$

###### Proof.

Haagerup in [Haa79, Lemma 1.4] proved that

$\|\lambda_{\mathrm{\mathbf{F}}}(a)\|\leq\sum_{i=0}^{\infty}(i+1)\|a_{i}\|_{\ell^{2}}$

where $a_{i}$ is the function $a$ multiplied pointwise by the indicator
function of $B_{X}(i)\backslash B_{X}(i-1)$, i.e. the sphere of radius $i$. If
$a$ is supported on $B_{X}(r)$ then using Cauchy-Schwarz above gives the
result, since $\sum_{i=0}^{r}\|a_{i}\|_{\ell^{2}}^{2}=\|a\|_{\ell^{2}}^{2}$. ∎

There is also a more basic inequality in the reverse direction that holds for
arbitrary discrete groups. Suppose that $\Gamma$ is a discrete group. Then let
$\delta_{e}\in\ell^{2}(\Gamma)$ denote the indicator function of the identity.
We have for $a\in\mathbf{C}[\Gamma]$

$\|a\|_{\ell^{2}}^{2}=\langle\lambda(a)\delta_{e},\lambda(a)\delta_{e}\rangle\leq\|\lambda_{\Gamma}(a)\|^{2}.$
(2.2)

##  3 Proof of Proposition 1.7

The proof of Proposition 1.7 relies on the deep fact that any limit group
embeds in an iterated extension of centralizers of a free group, and
quantified versions of theorems of Gilbert and Benjamin Baumslag.

######  Definition 3.1.

Let $\Gamma$ be a limit group, $A<\Gamma$ a maximal abelian subgroup. A group
$\Gamma^{\prime}=\Gamma*_{A}B$, $B=A\times\langle t\rangle$ is an _extension
of centralizers_ of $\Gamma$. A group $\Gamma$ is an _iterated extension of
centralizers_ if there is a chain of subgroups

$\mathbf{F}=\Gamma_{0}<\Gamma_{1}<\dotsb<\Gamma_{n}=\Gamma$

such that $\Gamma_{i+1}$ is an extension of centralizers of $\Gamma_{i}$. The
_height_ of the extension is $n$.

Any iterated extension of centralizers is fully residually free, and so are
their finitely generated subgroups, hence such subgroups are limit groups.
Amazingly, the converse holds: any limit group actually embeds in a (finitely)
iterated extension of centralizers. This was first claimed by Kharlampovich
and Myasnikov in their papers on the Tarski problem [KM98, Theorem 4]. For a
proof following Sela see [CG05, Theorem 4.2]. The forward implication seems to
be contained in Lyndon’s original paper on his free exponential group [Lyn60,
last two paragraphs, page 533], which is the direct limit over the family of
all iterated extensions of centralizers of $\mathbf{F}$, ordered by inclusion.
See also [BMR02, Theorem C1].

Let $\Gamma$ be a limit group with some fixed generating set $Y$. The
_distortion function of $\Gamma$ with respect to $Y$_ is the function

$d_{Y}(r)=\min_{\begin{subarray}{c}f\colon\Gamma\to\mathbf{F}\\\
X\subset\mathrm{\mathbf{F}}\end{subarray}}\max_{g\in B_{Y}(r)}|f(g)|_{X}\,,$

where the minimum is over all $f\colon\Gamma\to\mathbf{F}$ which are injective
on $B_{Y}(r)$ and $X$ which are bases of $\mathrm{\mathbf{F}}$. The proof of
Proposition 1.7 is a recapitulation of the proof that an iterated extension of
centralizers is fully residually free in a way that lets us bound the
distortion function by a polynomial whose degree depends on the height. We
start with an improvement of Baumslag’s power lemma.

######  Lemma 3.2 (cf. [Bau62, Proposition 1]).

Let $u,b_{1},\dotsc,b_{n}$, reduced words in $\mathbf{F}$, with $u$ also
cyclically reduced, nontrivial, and not a proper power of another element. If

$w=\prod_{i=0}^{n}u^{k_{i}}b_{i}=e$ (3.1)

for

$\min_{i>0}\\{|k_{i}|\\}>(8n+2)\cdot\max_{i\geq 0}\\{1,|b_{i}|/|u|\\}\,,$

then $\left[u,b_{i}\right]=e$ for some $i$.

G. Baumslag proved the same thing if $w=e$ for infinitely many integral values
of each of the $k_{i}$. See also the proof of [Wil09, Lemma 4.13], which has,
implicitly, an effective version of Lemma 3.2 in it.

###### Proof.

The proof is by induction on $n$. Clearly for $n=0$, if $u^{k_{0}}b_{0}=e$
then $b_{0}$ is a power of $u$ and hence commutes with $u$.

We begin by manipulating our hypothesis to a more convenient form for the
induction. If

$\min_{i>0}\\{|k_{i}|\\}>(8n+2)\cdot\max_{i\geq 0}\\{1,|b_{i}|/|u|\\}$

then

$|w|>|u|\cdot\sum_{i>0}|k_{i}|>(8n+2)\cdot\max_{i\geq 0}\\{|u|,|b_{i}|\\}.\,$
(3.2)

(Here $|w|$ is the non-reduced length of $w$.)

Let $T$ be the tree of cancellations for $w$. A vertex of $T$ is _special_ if
it either corresponds to an endpoint of one of the subwords $u^{k_{i}}$, one
of the $b_{i}$, or has valence at least three. An embedded segment in $T$ with
special endpoints and no special vertices in its interior is a _long edge_.

Every valence one vertex in $T$ is special, so there are at most $2n+2$ of
them. We now work out the maximal number of long edges in a tree with at most
$2n+2$ valence one vertices, which will happen when the number $q_{\geq 3}$ of
vertices of valence at least 3 is maximized. Let $q_{m}$ be the number of
valence $m$ vertices in $T$. Then

$1=\chi(T)=\sum q_{m}(1-m/2)\leq\frac{2n+2}{2}-\frac{1}{2}q_{\geq 3}$

implies $q_{\geq 3}\leq 2n$, there are at most $4n+2$ special vertices, and
there are at most $4n+1$ long edges.

The sum of the lengths of the long edges is $|w|/2$, so there is a long edge
of length at least $|w|/(8n+2)$, which from (3.2) is at least

$\max_{i\geq 0}\\{|u|,|b_{i}|+1\\}\,.$

If this is the case, since the endpoints of the $b_{i}$ are special, the long
edge is covered only by subsegments of powers of $u$. Because $u$ is not a
proper power, the segment (with a fixed direction) corresponds to a unique
reduced expression of the form $u_{0}u^{a}u_{1}$ where $u_{0}$ and $u_{1}$ are
proper subwords of $u$ and $a>0$. (Otherwise, one is led to the conclusion
that $u$ can be written as a reduced product of reduced words $u=pq=qp$, and
by [Raz14, Lemma 2.2], this contradicts $u$ being a proper power.) Let us now
fix the direction of the long edge so $a>0$.

The upshot of this unique expression is that the term $u_{0}$ corresponds to a
terminal subsegment of a $u$ as written in (3.1) (part of a $u^{k_{i}}$ with
$k_{i}>0$), for each time the long edge is traversed in its given direction.
If the long edge is traversed in the other way by the path of $w$, then the
$u_{0}$ segment corresponds to an initial subsegment of a $u^{-1}$ in a
$u^{k_{i}}$ with $k_{i}<0$.

Fix an endpoint $v$ of the $u_{0}$ segment in the long edge. Consider the
subpaths of the path of $w$ punctuated by returns to $v$. After cutting the
tree at $v$, there must be at least one $b_{i}$ subpath on either half of the
resulting forest. So there must be some closed subpath of $w$ beginning and
ending at $v$ and corresponding, possibly after cyclic rotation of $w$, to a
subsequence

$u^{k_{0}}b_{0}u^{k_{1}}b_{1}\dotsb
u^{k_{j}-a}\underbrace{u^{a}b_{j}u^{k_{j+1}}\dotsb
u^{k_{l}}b_{l}u^{c}}_{\downarrow}u^{k_{l+1}-c}b_{l+1}\dotsb u^{k_{n}}b_{n}$

with $l-j<n$, and

$u^{a}b_{j}u^{k_{j+1}}\dotsb u^{k_{l}}b_{l}u^{c}=e\,,$

which implies

$u^{a+c}b_{j}u^{k_{j+1}}\dotsb u^{k_{l}}b_{l}=e.$

Reducing $a+c$, we can use the inductive hypothesis to conclude that for some
$j$, $[u,b_{j}]=e$. (Note that this is where the minimum of $k_{i}$ only over
$i>0$ is useful in the induction; $a+c$ could in principle be very small.) ∎

A similar result holds when $u$ is not necessarily cyclically reduced and is a
power, e.g., $u=ps^{l}p^{-1}$, with $s$ cyclically reduced and
$|u|=2|p|+l|s|$. Rewrite the expression for $w$ as

$e=w=\prod ps^{lk_{i}}p^{-1}b_{i}$

conjugate by $p^{-1}$, and absorb the $p$’s into the $b$’s to get

$e=w^{\prime}=\prod s^{lk_{i}}b_{i}^{\prime}\,.$

Then the same conclusion clearly holds when

$l\min_{i\geq 0}\\{|k_{i}|\\}>(8n+2)\cdot\max_{i\geq
0}\\{1,(|b_{i}|+2|p|)/|s|\\}\,.$

For the applications, since $|u|=l|s|+2|p|>2|p|$ so we can use instead the
easier to use yet still sufficient inequality

$\min_{i\geq 0}\\{|k_{i}|\\}\geq(8n+2)\cdot\max_{i\geq 0}\\{|b_{i}|+|u|\\}\,,$
(3.3)

which gives the same conclusion. Note that this minimum of $k_{i}$ is now over
all $i$, and not just $i>0$. (The latter was just more convenient for the
previous induction.)

In what follows, $\Gamma$ is a limit group with a fixed finite generating set
$Y$, $A$ is a maximal abelian subgroup in $\Gamma$, and $\Gamma^{\prime}$ is
the extension of centralizers $\Gamma^{\prime}=\Gamma*_{A}B$, where
$B=A\times\langle t\rangle$. Let $Y^{\prime}=Y\cup t$. The standard fact about
amalgamated products lets us write any element of $\Gamma^{\prime}$ in normal
form:

$\prod_{i=0}^{n}\beta_{i}\gamma_{i}$

with $\gamma_{i}\in\Gamma$, $\beta_{i}\in B$, so that the only elements which
are allowed to be trivial are $\beta_{0}$ and $\gamma_{n}$, and if any of them
are in $A$ then the expression has length one – if, say, $\beta_{i}\in A$,
then $\gamma_{i}\beta_{i}\gamma_{i+1}\in\Gamma$ and we group these together.
Normal forms are unique (up to insertion of $\alpha\alpha^{-1}$ pairs), but we
will not use this fact. Given an element as above, write each $\beta_{i}$
(uniquely) as $t^{n_{i}}\alpha_{i}$ with $\alpha_{i}\in A$

$\prod_{i=0}^{n}t^{n_{i}}\alpha_{i}\gamma_{i}$

and absorb the $\alpha_{i}$ into an adjacent $\gamma_{i}$ or $\gamma_{i+1}$ to
get

$\prod_{i=0}^{n}t^{n_{i}}v_{i}\>$

with each $v_{i}\in\Gamma$. If this isn’t possible, leave it alone. In this
case the word is of the form

$t^{n}\alpha\,,$

for some $\alpha\in A$. For the purposes of this argument, a word is in normal
form if it is of either of these two types.

######  Lemma 3.3 (QI lemma).

There is a constant $K$ such that if $w$ is a word in $Y^{\prime}$, then there
is a word $w^{\prime}=_{\Gamma}w$ in normal form such that $|w^{\prime}|\leq
K\cdot|w|$.

###### Proof.

The word $w$ can be put in normal form by replacing subwords which are
elements of $A$ in one step. Since $A$ is maximal abelian in $\Gamma$ it is
quasiconvex by [Ali05, Theorem 3.4], and the rewritten word can only increase
in length by a factor of $K$, where $1/K$ is the shrinking factor of the
embedding $A\hookrightarrow\Gamma^{\prime}$. ∎

######  Lemma 3.4 (cf. [Bau67, Lemma 7, Theorem 8]).

Let $K$ be the constant from Lemma 3.3, and fix $a\in A\backslash\\{e\\}$.
Then

$d_{Y^{\prime}}(r/K)\leq(8r^{2}+4r)d_{Y}(2(r+|a|))^{2}\,.$

If $d_{Y}(r)$ is a polynomial of degree $D$ then $d_{Y^{\prime}}(r)$ is
bounded above by a polynomial of degree $2D+2$.

This is essentially a version of B. Baumslag’s generalization of G. Baumslag’s
version of Lemma 3.2 from free groups to limit groups, where we keep track of
the constants and avoid the phrases “sufficiently large” and “as large as we
like.”

###### Proof.

Let $\pi\colon\Gamma^{\prime}\to\Gamma$ be the retraction to $\Gamma$ defined
by $\pi(t)=e$, let $\tau$ be the automorphism of $\Gamma^{\prime}$ fixing
$\Gamma$ with

$\displaystyle\tau(t)$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}ta.$

We will find an $h\colon\Gamma^{\prime}\to\mathbf{F}$ which is injective on
normal forms in $Y^{\prime}$ of length at most $r$ and doesn’t stretch too
much.

Suppose first that $g$ has the normal form

$\prod t^{n_{i}}v_{i}\,,$

which is a product of at most $\lfloor r/2\rfloor$ terms $t^{n_{i}}v_{i}$.
Then

$f\circ\pi\circ\tau^{m}(g)=\prod f(a)^{mn_{i}}f(v_{i})\,.$

In order to use Lemma 3.2 we need to choose $f$ so that
$f(\left[a,v_{i}\right])\neq e$. In the worst case the commutator
$\left[a,v_{i}\right]$ has length at most

$L\stackrel{{\scriptstyle\mathrm{def}}}{{=}}2(r+|a|).$

Choose $f\colon\Gamma\to\mathbf{F}$ and a basis $X$ of $\mathrm{\mathbf{F}}$
such that

$d_{Y}(L)=\max_{g\in B_{Y}(L)}|f(g)|_{X}$

and $f$ embeds $B_{Y}(L)\subset\Gamma$. By (3.3), with $k_{i}=mn_{i}$,
$n=\lfloor r/2\rfloor$, $u=f(a)$, and $b_{i}=f(v_{i})$, as long as

$\min\\{|mn_{i}|\\}\geq(4r+2)\cdot\max\\{|f(v_{i})|_{X}+|f(a)|_{X}\\}$

$f\circ\pi\circ\tau^{m}(g)$ is nontrivial. In the worst case $n_{i}=1$ for all
$i$ and $f(a)$ and $f(v_{i})$ have length $d_{Y}(L)$, so choose
$m=m(r,|a|)=(4r+2)\cdot 2d_{Y}(L)$ and let

$\displaystyle h$ $\displaystyle=f\circ\pi\circ\tau^{m}.$

We continue to use the same basis $X$ for $\mathrm{\mathbf{F}}$. Now
overestimate the length of $h(g)$: the normal form which can be expanded the
most is $t^{r}$, so we have $r\cdot m$ terms whose images have length at most
$d_{Y}(L)$, and therefore

$|h(g)|_{X}\leq r\cdot\underbrace{(8r+4)\cdot
d_{Y}(L)}_{m}\cdot\,d_{Y}(L)=(8r^{2}+4r)d_{Y}(L)^{2}\,.$

If $g$ is of the form $t^{n}\alpha$ then the worst that can happen is $n=1$
and $\alpha$ has length at most $r-1$, $h(g)=f(a)^{m}f(\alpha)$, but in this
case

$m\cdot d_{Y}(L)\geq m\cdot|f(a)|_{X}\geq|h(t)|_{X}\geq
m>d_{Y}(r-1)\geq|f(\alpha)|_{X}\,,$

so $h(g)$ is nontrivial – $h(t)$ and $f(\alpha)$ cannot fully cancel since
$m>d_{Y}(r-1)$, and is not longer than $m\cdot d_{Y}(L)+d_{Y}(r-1)$, which is
less than $r\cdot m\cdot d_{Y}(L)$.

Now by Lemma 3.3 if $g\in B_{Y^{\prime}}(r/K)$, it has a normal form of length
at most $r$ and $h(g)\neq e$, so $h$ embeds $B_{Y^{\prime}}(r/K)$,
$|h(g)|_{X}\leq(8r^{2}+4r)d_{Y}(L)^{2}$, and the first part of the lemma
follows.

The statement about degrees is obvious since $L$ is linear in $r$. ∎

######  Corollary 3.5.

Let $\Gamma$ be a limit group, and suppose $\Gamma$ embeds in an extension of
centralizers of height $n$. Then $d_{Y}(r)$ is bounded above by a polynomial
in $r$ of degree

$D(n)=2^{n+2}-2^{n}-2\,.$

###### Proof.

For height $0$, the distortion function is just $r$. Clearly by Lemma 3.4 and
induction a polynomial of degree $D(n)$ suffices. Now embed $\Gamma$ in an
iterated extension of centralizers of height $n$:

$\Gamma\hookrightarrow\Gamma_{n}>\Gamma_{n-1}>\dotsb>\Gamma_{1}>\mathbf{F}\,.$

Since the embedding $\Gamma\hookrightarrow\Gamma_{n}$ expands lengths at most
linearly, $\Gamma$ has distortion function bounded above by a polynomial of
degree $D(n)$ as well. ∎

Proposition 1.7 follows immediately.

##  4 Proof of Theorem 1.6

###### Proof of Theorem 1.6.

Fix a set $Y$ of generators of $\Gamma$. It suffices to prove the theorem for
the finite set $B_{Y}(R)$ for arbitrary $R>0$. We are given $\epsilon>0$. Let
$S_{Y}(R)\subset\mathbf{C}[\Gamma]$ denote the $\ell^{1}$-unit sphere of the
elements supported on $B_{Y}(R)$. Our task is to prove that there is a
homomorphism $\phi:\Gamma\to\mathrm{\mathbf{F}}$ with $\mathrm{\mathbf{F}}$
free such that

$\|\lambda_{\mathrm{\mathbf{F}}}(\phi(a))\|\leq\|\lambda_{\Gamma}(a)\|+\epsilon.$
(4.1)

for all $a\in S_{Y}(R)$. The set $S_{Y}(R)$ is compact with respect to the
$\ell^{1}$ norm. Take a finite $\frac{\epsilon}{3}$-net
$\\{a_{i}\\}_{i\in\mathcal{I}}$ for $S_{Y}(R)$ w.r.t. the $\ell^{1}$ norm.

Due to the inequality (2.1) and triangle inequality, the functions
$a\mapsto\|\lambda_{\mathrm{\mathbf{F}}}(a)\|$ and
$a\mapsto\|\lambda_{\Gamma}(a)\|$ are 1-Lipschitz on $S_{Y}(R)$ with respect
to the $\ell^{1}$ norm and hence if we can prove the existence of
$\phi:\Gamma\to\mathrm{\mathbf{F}}$ with $\mathrm{\mathbf{F}}$ free such that

$\|\lambda_{\mathrm{\mathbf{F}}}(\phi(a_{i}))\|\leq\|\lambda_{\Gamma}(a_{i})\|+\frac{\epsilon}{3}$
(4.2)

for all $i\in\mathcal{I}$ then (4.1) will follow for all $a\in S_{Y}(R)$ as
required. So we now set out to prove (4.2).

Let $C$ and $D$ be the constants from Proposition 1.7. Choose
$m=m(\epsilon)\in\mathbf{N}$ large enough so that

$[C(2mR)^{D}]^{\frac{3}{4m}}\leq 1+\frac{\epsilon}{3}.$ (4.3)

We apply Proposition 1.7 with $r=2mR$ to get an epimorphism
$\phi:\Gamma\to\mathrm{\mathbf{F}}$ injective on $B_{Y}(2mR)$, and a
generating set $X$ of $\mathrm{\mathbf{F}}$ such that

$\phi(B_{Y}(2mR))\subset B_{X}\left(C(2mR)^{D}\right).$ (4.4)

Let $b_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\phi(a_{i})$ for each
$i\in\mathcal{I}$.

Note that

$\|\lambda_{\Gamma}(a_{i})\|^{2m}=\|\lambda_{\Gamma}\left(a_{i}^{*}a_{i}\right)\|^{m}=\|\lambda_{\Gamma}\left(a_{i}^{*}a_{i}\right)^{m}\|$

and similarly,
$\|\lambda_{\mathrm{\mathbf{F}}}(b_{i})\|^{2m}=\|\lambda_{\mathrm{\mathbf{F}}}\left(b_{i}^{*}b_{i}\right)^{m}\|$.
Each $\left(b_{i}^{*}b_{i}\right)^{m}$ is supported on $B_{X}(C(2mR)^{D})$ by
(4.4), hence by Haagerup’s inequality (Lemma 2.1) we have

$\displaystyle\|\lambda_{\mathrm{\mathbf{F}}}(b_{i})\|^{2m}$
$\displaystyle=\|\lambda_{\mathrm{\mathbf{F}}}\left(b_{i}^{*}b_{i}\right)^{m}\|$
$\displaystyle\leq[C(2mR)^{D}]^{\frac{3}{2}}\|\left(b_{i}^{*}b_{i}\right)^{m}\|_{\ell^{2}}$
$\displaystyle=[C(2mR)^{D}]^{\frac{3}{2}}\|\left(a_{i}^{*}a_{i}\right)^{m}\|_{\ell^{2}}$
$\displaystyle\leq[C(2mR)^{D}]^{\frac{3}{2}}\|\lambda_{\Gamma}(a_{i})\|^{2m}.$

The equality on the third line used that $\phi$ is injective on $B_{Y}(2mR)$,
and the final inequality used (2.2). Hence

$\displaystyle\|\lambda_{\mathrm{\mathbf{F}}}(b_{i})\|$
$\displaystyle\leq[C(2mR)^{D}]^{\frac{3}{4m}}\|\lambda_{\Gamma}(a_{i})\|$
$\displaystyle\leq\left(1+\frac{\epsilon}{3}\right)\|\lambda_{\Gamma}(a_{i})\|\leq\|\lambda_{\Gamma}(a_{i})\|+\frac{\epsilon}{3}$

by our choice of $m$ in (4.3); the last inequality used that
$\|a_{i}\|_{\ell^{1}}=1$ and (2.1). ∎

##  5 Proof of Theorem 1.1

Here we split into cases when $\Gamma$ is abelian or not. Limit groups cannot
have torsion, so abelian limit groups are of the form $\mathbf{Z}^{r}$ for
some $r\in\mathbf{N}$.

###  5.1 Proof when $\Gamma=\mathbf{Z}^{r}$

The case when $\Gamma=\mathbf{Z}^{r}$ must be dealt with by hand here.

######  Lemma 5.1.

Theorem 1.1 holds when $\Gamma=\mathbf{Z}^{r}$.

###### Proof.

Let $T^{r}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(S^{1})^{r}$ be the
standard $r$-dimensional flat torus. The Fourier transform gives an
isomorphism of $C^{*}$-algebras

$\mathcal{F}\colon C(T^{r})\to C_{r}^{*}(\mathbf{Z}^{r}).$

For $q\in\mathbf{N}$ let $T_{q}^{r}$ denote the subtorus
$(\mathbf{Z}/q\mathbf{Z})^{r}\subset T^{r}$. We obtain, via restriction and
Fourier transform, a finite dimensional representation

$C_{r}^{*}(\mathbf{Z}^{r})\xrightarrow{\rho_{q}}C(T_{q}^{r})$

that restricts to finite dimensional unitary representation of
$\mathbf{Z}^{r}$. For any $z\in\mathbf{C}[\mathbf{Z}^{r}]$ we have

$\|\rho_{q}(z)\|=\max_{x\in T_{q}^{r}}|\mathcal{F}^{-1}[z](x)|\to\max_{x\in
T^{r}}|\mathcal{F}^{-1}[z](x)|=\|\lambda_{\mathbf{Z}^{r}}(z)\|$

as $q\to\infty$. We have only used here the fact that $T_{q}^{r}$ Hausdorff
converges to $T^{r}$ as $q\to\infty$. ∎

###  5.2 Proof for non-abelian limit groups

In the following, $\mathrm{\mathbf{F}}$ will always denote some (not always
the same) free group, and $\Gamma$ will be a fixed limit group.

######  Lemma 5.2.

If $\Gamma$ is a non-abelian limit group, then the reduced $C^{*}$ algebra of
$\Gamma$ is simple (has no non-trivial closed ideals) and has a unique tracial
state.

###### Proof.

We claim that any non-abelian FRF group $\Gamma$ has the $P_{\mathrm{nai}}$
property of Bekka, Cowling, and de la Harpe [BCdlH94, Def. 4]. This states
that for any finite set $S\subset\Gamma\backslash\\{e\\}$, there is
$y\in\Gamma$ of infinite order such that for every $x\in S$, $x$ and $y$ are
free generators of a free rank 2 subgroup of $\Gamma$.

_Proof of Claim._ It is easy to check that since $\Gamma$ is FRF, two elements
$x$ and $y$ are free generators of a free rank $2$ subgroup of $\Gamma$ if and
only if they do not commute. So to check property $P_{\mathrm{nai}}$ above, it
remains to check that given any finite subset
$S\subset\Gamma\backslash\\{e\\}$, there is an infinite order $y$ not
commuting with any element of $S$.

Because $\Gamma$ is non-abelian, there are two elements $a,b\in\Gamma$ with
$[a,b]\neq e$. By the FRF condition, there is a epimorphism
$\phi:\Gamma\to\mathrm{\mathbf{F}}$ that is an injection on
$S\cup\\{e\\}\cup\\{[a,b]\\}$. In particular, the rank of
$\mathrm{\mathbf{F}}$ must be at least 2. Since $\phi(S)$ is a finite subset
of $\mathrm{\mathbf{F}}$ not containing the identity, there is an (necessarily
infinite order) element $f$ not commuting with any element of $\phi(S)$. Then
any preimage of $f,$ say $y$, is infinite order and does not commute with any
element of $S$. _This ends the proof of the claim._

The proof of Lemma 5.2 now concludes by using [BCdlH94, Lemmas 2.1 and 2.2]. ∎

###### Proof of Theorem 1.1.

The upshot of Lemma 5.2 is that proving the existence of a sequence of unitary
representations $\\{\rho_{i}:\Gamma\to\mathbf{U}(N_{i})\\}_{i=1}^{\infty}$
strongly converging to the regular representation reduces to proving the
existence of a sequence with

$\limsup_{i\to\infty}\|\rho_{i}(z)\|\leq\|\lambda_{\Gamma}(z)\|$ (5.1)

for all $z\in\mathbf{C}[\Gamma]$ of unit $\ell^{1}$ norm. We give a proof of
this passage that was also mentioned in the Introduction.

Suppose (5.1) holds. Then for any non-principal ultrafilter $\mathcal{F}$, we
form the ultraproduct555For background on ultrafilters and ultraproducts, see
[BO08, Appendix A]. $C^{*}$-algebra
$\mathcal{U}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\prod_{\mathcal{F}}\rho_{i}(\mathbf{C}[\Gamma])$.
There is a natural $*$-algebra map $\iota:\mathbf{C}[\Gamma]\to\mathcal{U}$.
The inequality (5.1) implies

$\|\iota(z)\|_{\mathcal{U}}\leq\|\lambda_{\Gamma}(z)\|$ (5.2)

for all $z\in\mathbf{C}[\Gamma]$. If $\mathcal{U}_{1}$ denotes the closure of
$\iota(\mathbf{C}[\Gamma])$ in $\mathcal{U}$, then inequality (5.2) implies
that the map $\iota$ extends continuously to $C^{*}$-algebra map from
$C_{r}^{*}(\Gamma)$ to $\mathcal{U}_{1}$. But since we know
$C_{r}^{*}(\Gamma)$ is simple by Lemma 5.2, this map must be injective. But
injective $C^{*}$-algebra maps are isometries (to their images), so we have
for all $z\in\mathbf{C}[\Gamma]$

$\|\lambda_{\Gamma}(z)\|=\|\iota(z)\|_{\mathcal{U}}=\lim_{i\to\mathcal{U}}\|\rho_{i}(z)\|.$

Since this holds for arbitrary non-principal ultrafilters, it holds also that
$\|\lambda_{\Gamma}(z)\|=\lim_{i\to\infty}\|\rho_{i}(z)\|.$

This reduces our task to proving 5.1, which we begin now. Given $\epsilon>0$
we will prove that there is a unitary representation
$\rho=\rho(U,\epsilon):\Gamma\to\mathbf{U}(N)$ with $N=N(U,\epsilon)$ such
that

$\|\rho(z)\|\leq\|\lambda_{\Gamma}(z)\|+\epsilon$

for $z\in\mathbf{C}[\Gamma]$ with support in
$B\left(\frac{1}{\epsilon}\right)$ and $\|z\|_{\ell^{1}}=1$. By taking
$\epsilon\to 0$, this will imply the existence of a sequence $\rho_{i}$
satisfying (5.1) for any $z$.

As in the proof of Theorem 1.6 $(\text{\S}$4), by taking an
$\frac{\epsilon}{3}$-net of the unit $\ell^{1}$ sphere of the elements in
$\mathbf{C}[\Gamma]$ supported on $B\left(\frac{1}{\epsilon}\right)$, it
suffices to prove

$\|\rho(a_{i})\|\leq\|\lambda_{\Gamma}(a_{i})\|+\frac{\epsilon}{3}$

for a finite collection $\\{a_{i}\\}_{i\in\mathcal{I}}$ of elements of
$\mathbf{C}[\Gamma]$ with $\|a_{i}\|_{\ell^{1}}=1$.

We apply Theorem 1.6 with $S=B\left(\frac{1}{\epsilon}\right)$ to obtain a
homomorphism $\phi:\Gamma\to\mathrm{\mathbf{F}}$ with $\mathrm{\mathbf{F}}$
free such that

$\|\lambda_{\mathrm{\mathbf{F}}}(\phi(a_{i}))\|\leq\|\lambda_{\Gamma}(a_{i})\|+\frac{\epsilon}{6}.$
(5.3)

for all $i\in\mathcal{I}$. Let
$b_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\phi(a_{i})\in\mathbf{C}[\mathrm{\mathbf{F}}]$.

The remainder of the proof splits into three cases.

A. If $\mathrm{\mathbf{F}}$ is rank 1, i.e. $\mathrm{\mathbf{F}}=\mathbf{Z}$
then Lemma 5.1 tells that there is a finite dimensional unitary representation
$\pi$ of $\mathrm{\mathbf{F}}$ such that

$\|\pi(b_{i})\|\leq\|\lambda_{\mathrm{\mathbf{F}}}(b_{i})\|+\frac{\epsilon}{6}$
(5.4)

for all $i\in\mathcal{I}$.

B. Otherwise, if one only wants unitary representations in Theorem 1.1, then
by Haagerup and Thorbjørnsen [HT05, Thm. B] there is a finite dimensional
unitary representation $\pi$ of $\mathrm{\mathbf{F}}$ such that (5.4) holds
for all $i\in\mathcal{I}$.

C. If one wants the full strength of Theorem 1.1 and $\mathrm{\mathbf{F}}$ has
rank at least $2$, then unitary representations satisfying 5.4 for all
$i\in\mathcal{I}$ exist by the work of Bordenave and Collins [BC19].

Then let $\rho\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\pi\circ\phi$, a
finite dimensional unitary representation of $\Gamma$. Since
$\rho(a_{i})=\pi(b_{i})$, using (5.3) we obtain

$\|\rho(a_{i})\|=\|\pi(b_{i})\|\leq\|\lambda_{\mathrm{\mathbf{F}}}(b_{i})\|+\frac{\epsilon}{6}\leq\|\lambda_{\Gamma}(a_{i})\|+\frac{\epsilon}{3}$

for all $i\in\mathcal{I}$ as required. ∎

##  6 Proof of Proposition 1.3

We now turn to the Klein bottle group, $K=\langle t,a\mid
tat^{-1}=a^{-1}\rangle$. Let $A<K$ be the rank two free abelian index two
subgroup of $K$ generated by $a$ and $t^{2}$, and $T^{2}=\hat{A}$ (a two-
torus) the Pontryagin dual of $A$. By [Tay89, Proposition 1], $C_{r}^{*}(K)$
is isomorphic (via an inverse fourier transform $\hat{\mathcal{F}}$) to a
$C^{*}$-subalgebra of $\mathrm{Mat}_{2\times 2}(C(T^{2}))\cong
C(T^{2};\mathrm{Mat}_{2\times 2}(\mathbf{C}))$ with norm defined by

$\|f\|_{\infty}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{x\in
T^{2}}\|f(x)\|_{\mathrm{op}}.$

The norm on the right hand side is the operator norm on $\mathrm{Mat}_{2\times
2}(\mathbf{C})$ coming from the standard Hermitian form on $\mathbf{C}^{2}$.
As in the proof of Lemma 5.1, for $q\in\mathbf{N}$ let
$T_{q}^{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathbf{Z}/q\mathbf{Z}$.
Restriction gives a $C^{*}$-algebra map

$C_{r}^{*}(K)\xrightarrow{\rho_{q}}C(T_{q}^{2};\mathrm{Mat}_{2\times
2}(\mathbf{C}))$

with finite dimensional right hand side. The restriction to $K$ is unitary and
for any $z\in\mathbf{C}[K]$ we have as $q\to\infty$

$\|\rho_{q}(z)\|=\max_{x\in
T_{q}^{2}}\|\mathcal{F}[z](x)\|_{\mathrm{op}}\to\max_{x\in
T^{2}}\|\mathcal{F}[z](x)\|_{\mathrm{op}}=\|\lambda_{K}(z)\|.$

##  7 Proof of Corollary 1.9

We wish to appeal to results in the literature to deduce Corollary 1.9 from
Theorem 1.1. To do so, we first establish the following.

######  Lemma 7.1 (Strong convergence implies weak convergence).

Let $\Gamma$ be a finitely generated discrete group such that
$C_{r}^{*}(\Gamma)$ has a unique tracial state. If
$\\{\rho_{i}:\Gamma\to\mathbf{U}(N_{i})\\}_{i=1}^{\infty}$ are a sequence of
finite dimensional unitary representations that strongly converge to the
regular representation of $\Gamma$, then for any $z\in\mathbf{C}[\Gamma]$

$\lim_{i\to\infty}\frac{\mathrm{tr}(\rho_{i}(z))}{N_{i}}=\tau(z),$

where $\tau$ is the unique tracial state on $C_{r}^{*}(\Gamma)$. $\mathrm{tr}$
denotes the usual matrix trace on $\mathbf{U}(N_{i})$ extended linearly to
$\mathbf{C}[\mathbf{U}(N_{i})]$.

We heard this lemma stated by Benoît Collins in a talk in Northwestern
University in June 2022. The proof is to our knowledge not in the literature
so we give it here.

###### Proof of Lemma 7.1.

Consider any non-principal ultrafilter $\mathcal{F}$ on $\mathbf{N}$, and form
the ultraproduct $C^{*}$-algebra
$\mathcal{U}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\prod_{\mathcal{F}}\rho_{i}(\mathbf{C}[\Gamma])$.
Let $\mathcal{U}_{1}$ denote the $C^{*}$-subalgebra in $\mathcal{U}$ generated
by the images $\hat{\gamma_{i}}$ in $\mathcal{U}$ of the generators
$\gamma_{i}$ of $\Gamma$. Strong convergence implies that the natural map from
$\mathbf{C}[\Gamma]$ to $\mathcal{U}_{1}$ is an isometric embedding with
respect to the norm on $\mathbf{C}[\Gamma]$ coming from $C_{r}^{*}(\Gamma)$,
and hence extends to an isomorphism between $C_{r}^{*}(\Gamma)$ and
$\mathcal{U}_{1}$. On the other hand,

$\lim_{i\to\mathcal{F}}\frac{\mathrm{tr}\circ\rho_{i}}{N_{i}}$

defines a tracial state on $\mathcal{U}_{1}$, and when transferred to
$C_{r}^{*}(\Gamma)$ must coincide with the unique tracial state there. Since
the convergence holds for all non-principal ultrafilters, the convergence must
hold in general. ∎

###### Proof of Corollary 1.9.

Since non-abelian limit groups $\Gamma$ have unique tracial states on
$C_{r}^{*}(\Gamma)$ by Lemma 5.2, Lemma 7.1 implies that

$\lim_{i\to\infty}\frac{\mathrm{tr}(\rho_{i}(z))}{N_{i}}=\tau(z)$ (7.1)

where $\tau(z)$ is the unique tracial state on $C_{r}^{*}(\Gamma)$ (induced by
$\tau(e)=1$ and $\tau(g)=0$ for $e\neq g\in\Gamma$).

Note that for unitary matrices, $u_{1},\ldots,u_{r}$, any non-commutative
polynomial (possibly with matrix coefficients) in the $u_{i}$ is another
polynomial (possibly with matrix coefficients) in the Hermitian matrices
$u_{i}+u_{i}^{*}$ and $i(u_{i}-u_{i}^{*})$, and vice versa.

Given this observation, (7.1) together with strong convergence of the sequence
$\rho_{i}$ provided by Theorem 1.1 used as inputs to [Mal12, Prop. 7.3] yield
Corollary 1.9. ∎

##  Appendix A Spectral gaps of hyperbolic surfaces

The purpose of this appendix is to explain how the following theorem can be
deduced from Corollary 1.9.

######  Theorem A.1.

Let $X$ be a compact hyperbolic surface. There exists a sequence of Riemannian
covers $\left\\{X_{i}\right\\}_{i\in\mathbb{N}}$ of $X$ with genera
$g(i)\to\infty$ as $i\to\infty$ such that for any $\epsilon>0$, for $i$ large
enough depending on $\epsilon$,

$\mathrm{spec}\left(\Delta_{X_{i}}\right)\cap\left[0,\frac{1}{4}-\epsilon\right)=\mathrm{spec}\left(\Delta_{X}\right)\cap\left[0,\frac{1}{4}-\epsilon\right),$

where the multiplicities are the same on either side.

To see that we can take all surfaces to be arithmetic we use the following
argument. Let

$\Gamma_{0}(15)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathrm{SL}_{2}(\mathbf{Z})\,:\,c\equiv 0\bmod
15\right\\}.$

The cusped hyperbolic surface
$Y_{0}(15)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{0}(15)\backslash\mathbb{H}$
has no spectrum in $(0,\frac{1}{4})$ by a result of Huxley [Hux85, Thm., pg.
250]. Let $D_{3,5}$ denote the quaternion algebra over $\mathbf{Q}$ generated
by $i,j,k$ such that

$i^{2}=3,\,j^{2}=5,\,ij=-ji=k.$

Then $D_{3,5}$ is a division algebra with discriminant 15 [Ber16, Ex. 8.27].
Let $\mathcal{O}$ denote a maximal order666See [Ber16, Ex. 8.27] for an
explicit maximal order. in $D_{3,5}$ and $\mathcal{O}^{1}$ the elements of
norm 1 in $\mathcal{O}$. Then $\mathcal{O}^{1}$ embeds as a cocompact subgroup
of $\mathrm{PSL}_{2}(\mathbf{R})$; let
$X=\mathcal{O}^{1}\backslash\mathbb{H}$. By the work of Jacquet and Langlands
[JL70] (see [Ber16, Thm. 8.18] for a convenient concise reference) every
eigenvalue of $X$ is an eigenvalue of $Y_{0}(15)$ and hence $X$ has no
eigenvalues in $(0,\frac{1}{4})$.

Taking this $X$ in Theorem A.1, one obtains a different proof of [HM,
Corollary 1.3] with a slightly stronger conclusion, i.e. there exists a
sequence of compact _arithmetic_ hyperbolic surfaces
$\left\\{X_{i}\right\\}_{i\in\mathbb{N}}$ with genera
$g\left(X_{i}\right)\to\infty$ and
$\lambda_{1}\left(X_{i}\right)\to\frac{1}{4}$. Such a sequence of covering
surfaces also have systoles uniformly bounded away from $0$, also in contrast
to the proof of [HM, Corollary 1.3] (this conclusion on the systole is
independent of arithmeticity).

###  A.1 Set up

For any $n\in\mathbb{N}$, let
$[n]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{1,\ldots,n\\}$ and $S_{n}$
denote the group of permutations of $[n]$. Let $X$ be a fixed compact
hyperbolic surface with genus $g\geqslant 2$. We view $X$ as

$X=\Gamma\backslash\mathbb{H},$

where $\Gamma$ is a discrete, torsion free subgroup of
$\text{PSL}_{2}\left(\mathbb{R}\right)$, isomorphic to the surface group
$\Lambda_{g}$. Given any $\phi\in\text{Hom}\left(\Gamma,S_{n}\right)$ we
define an action of $\Gamma$ on $\mathbb{H}\times[n]$ by

$\gamma\left(z,x\right)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(\gamma
z,\phi(\gamma)[x]\right).$

Then we obtain a degree $n$ covering space $X_{\phi}$ of $X$ by

$X_{\phi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma\backslash_{\phi}\left(\mathbb{H}\times[n]\right).$
(A.1)

Let $V_{n}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\ell^{2}\left([n]\right)$
and $V_{n}^{0}\subset V_{n}$ the subspace of functions with zero mean. Then
$S_{n}$ acts on $V_{n}$ via std, the standard representation by $0$-$1$
matrices, and $V_{n}^{0}$ is the $n-1$ dimensional irreducible component.
Throughout this appendix, we let $\left\\{\rho_{i}\right\\}_{i\in\mathbb{N}}$
be a sequence of $N_{i}$-dimensional unitary representations of $\Gamma$ that
factor through $S_{N_{i}}$ by

$\Gamma\xrightarrow{\phi_{i}}S_{N_{i}}\xrightarrow{\text{std}}\text{End}\left(V_{N_{i}}^{0}\right),$
(A.2)

such that for any $r\in\mathbf{N}$ and finitely supported map
$a:\Gamma\to\mathrm{Mat}_{r\times r}(\mathbf{C})$, we have

$\limsup_{i\to\infty}\|\sum_{\gamma\in\Gamma}a(\gamma)\otimes\rho_{i}(\gamma)\|\leq\|\sum_{\gamma\in\Gamma}a(\gamma)\otimes\lambda(\gamma)\|,$
(A.3)

as provided by Corollary 1.9. Note that by approximation by finite-rank
operators on either side (as in [HM, Proof of Prop. 6.3]) the property in
(A.3) extends easily to the case of

$a:\Gamma\to\mathcal{K}$

where $\mathcal{K}$ are the compact operators on a separable Hilbert space. We
use this extension in the sequel.

Then through $\left\\{\rho_{i}\right\\}_{i\in\mathbb{N}}$, we obtain a
sequence of degree-$N_{i}$ covering surfaces
$\left\\{X_{i}\right\\}_{i\in\mathbb{N}}$ from (A.1).

###  A.2 Function spaces

For the convenience of the reader we recall the following function spaces from
[HM, Section 2.2]. We define $L_{\mathrm{new}}^{2}\left(X_{i}\right)$ to be
the space of $L^{2}$ functions on $X_{i}$ orthogonal to all lifts of $L^{2}$
functions from $X$. Then

$L^{2}\left(X_{i}\right)\cong
L_{\text{$\mathrm{new}$}}^{2}\left(X\right)\oplus L^{2}\left(X\right).$

We fix $F$ to be a Dirichlet fundamental domain for $X$. Let
$C^{\infty}\left(\mathbb{H};V_{N_{i}}^{0}\right)$ denote the smooth
$V_{N_{i}}^{0}$-valued functions on $\mathbb{H}$. There is an isometric linear
isomorphism between

$C^{\infty}\left(X_{i}\right)\cap
L_{\text{$\mathrm{new}$}}^{2}\left(X_{i}\right),$

and the space of smooth $V_{N_{i}}^{0}$-valued functions on $\mathbb{H}$
satisfying

$f\left(\gamma z\right)=\rho_{i}\left(\gamma\right)f\left(z\right),$

for all $\gamma\in\Gamma$, with finite norm

$\|f\|_{L^{2}(F)}^{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\int_{F}\|f(z)\|_{V_{N_{i}}^{0}}^{2}d\mu_{\mathbb{H}}\left(z\right)<\infty.$

We denote the space of such functions by
$C_{\phi_{i}}^{\infty}\left(\mathbb{H};V_{N_{i}}^{0}\right).$ The completion
of $C_{\phi_{i}}^{\infty}\left(\mathbb{H};V_{N_{i}}^{0}\right)$ with respect
to $\|\bullet\|_{L^{2}(F)}$ is denoted by
$L_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)$; the isomorphism above
extends to one between $L_{\text{$\mathrm{new}$}}^{2}\left(X_{i}\right)$ and
$L_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)$.

We introduce the following Sobolev spaces. Let $H^{2}\left(\mathbb{H}\right)$
denote the completion of $C_{c}^{\infty}\left(\mathbb{H}\right)$ with respect
to the norm

$\|f\|_{H^{2}\left(\mathbb{H}\right)}^{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\|f\|_{L^{2}\left(\mathbb{H}\right)}^{2}+\|\Delta
f\|_{L^{2}\left(\mathbb{H}\right)}^{2}.$

Let $C_{c,\phi_{i}}^{\infty}\left(\mathbb{H};V_{N_{i}}^{0}\right)$ denote the
subset of $C_{\phi_{i}}^{\infty}\left(\mathbb{H};V_{N_{i}}^{0}\right)$
consisting of functions which are compactly supported modulo $\Gamma$. We let
$H_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)$ denote the completion
of $C_{c,\phi_{i}}^{\infty}\left(\mathbb{H};V_{N_{i}}^{0}\right)$ with respect
to the norm

$\|f\|_{H_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)}^{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\|f\|_{L^{2}(F)}^{2}+\|\Delta
f\|_{L^{2}(F)}^{2}.$

We let $H^{2}\left(X_{i}\right)$ denote the completion of
$C_{c}^{\infty}\left(X_{i}\right)$ with respect to the norm

$\|f\|_{H^{2}(X_{i})}^{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\|f\|_{L^{2}(X_{i})}^{2}+\|\Delta
f\|_{L^{2}(X_{i})}^{2}.$

Viewing $H^{2}\left(X_{i}\right)$ as a subspace of
$L^{2}\left(X_{\phi_{i}}\right)$, we let

$H_{\mathrm{new}}^{2}\left(X_{i}\right)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}H^{2}\left(X_{i}\right)\cap
L_{\mathrm{new}}^{2}\left(X_{\phi_{i}}\right).$

There is an isometric isomorphism between
$H_{\mathrm{new}}^{2}\left(X_{i}\right)$ and
$H_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)$ that intertwines the
two relevant Laplacian operators.

###  A.3 Operators on $\mathbb{H}$

For $s\in\mathbb{C}$ with $\text{Re}(s)>\frac{1}{2}$, let

$\displaystyle R_{\mathbb{H}}(s)$
$\displaystyle:L^{2}\left(\mathbb{H}\right)\to L^{2}\mathbb{\left(H\right)},$
$\displaystyle R_{\mathbb{H}}(s)$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(\Delta_{\mathbb{H}}-s(1-s)\right)^{-1},$

be the resolvent on the upper half plane. Then $R_{\mathbb{H}}(s)$ is an
integral operator with radial kernel $R_{\mathbb{H}}(s;r)$. Let
$\chi_{0}:\mathbb{R}\to\left[0,1\right]$ be a smooth function such that

$\chi_{0}\left(t\right)=\begin{cases}0&\text{if $t\leqslant 0$, }\\\
1&\text{if $t\geqslant 1$. }\end{cases}.$

For $T>0$, we define a smooth cutoff function $\chi_{T}$ by
$\chi_{T}(t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\chi_{0}(t-T).$We then
define the operator $R_{\mathbb{H}}^{(T)}(s):L^{2}\left(\mathbb{H}\right)\to
L^{2}\left(\mathbb{H}\right)$ to be the integral operator with radial kernel

$R_{\mathbb{H}}^{(T)}(s;r)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\chi_{T}(r)R_{\mathbb{H}}(s;r).$

Following [HM, Section 5.2] we define
$L_{\mathbb{H}}^{(T)}(s):L^{2}\left(\mathbb{H}\right)\to
L^{2}\left(\mathbb{H}\right)$ to be the integral operator with radial kernel

$\mathbb{L}^{(T)}(s;r)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(-\frac{\partial^{2}}{\partial
r^{2}}\left[\chi_{T}\right]-\frac{1}{\tanh r}\frac{\partial}{\partial
r}\left[\chi_{T}\right]\right)R_{\mathbb{H}}(s;r)-2\frac{\partial}{\partial
r}\left[\chi_{T}\right]\frac{\partial R_{\mathbb{H}}}{\partial r}(s;r).$

It is proved in [HM, Lemma 5.3] that for any $f\in
C_{c}^{\infty}\left(\mathbb{H}\right)$and $s\in\left[\frac{1}{2},1\right]$, we
have

  1. 1.

$R_{\mathbb{H}}^{(T)}(s)f\in H^{2}\left(\mathbb{H}\right).$

  2. 2.

$\left(\Delta-s(1-s)\right)R_{\mathbb{H}}^{(T)}(s)f=f+\mathbb{L}_{\mathbb{H}}^{(T)}(s)f$
as equivalence classes of $L^{2}$ functions.

It is also proved, as a consequence of [HM, Lemma 5.2], that for any
$s_{0}>\frac{1}{2}$ we can choose a $T=T(s_{0})$ such that for all
$s\in[s_{0},1]$ we have

$\|\mathbb{L}_{\mathbb{H}}^{(T)}(s)\|_{L^{2}}\leq\frac{1}{4}.$ (A.4)

###  A.4 Proof of Theorem A.1

Recall that $\left\\{\rho_{i}\right\\}_{i\in\mathbb{N}}$ is a sequence of
strongly convergent representations of the form (A.2) that satisfy (A.3) as
guaranteed by Corollary 1.9. As in [HM, Section 5.3], we define

$\displaystyle R_{\mathbb{H},i}^{(T)}(s;x,y)$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}R_{\mathbb{H}}^{(T)}(s;x,y)\mathrm{Id}_{V_{N_{i}}^{0}},$
$\displaystyle\mathbb{L}_{\mathbb{H},i}^{(T)}(s;x,y)$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathbb{L}_{\mathbb{H}}^{(T)}(s;x,y)\mathrm{Id}_{V_{N_{i}}^{0}}.$

We define $R_{\mathbb{H},i}^{(T)}\left(s\right)$,
$\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)$ to be the corresponding
integral operators. We have the following analogue of [HM, Lemma 5.5].

######  Lemma A.2.

For all $s\in\left[\frac{1}{2},1\right]$,

  1. 1.

The integral operator $R_{\mathbb{H},i}^{(T)}\left(s\right)$ is well-defined
on $C_{c,\phi_{i}}^{\infty}\left(\mathbb{H};V_{N_{i}}^{0}\right)$ and extends
to a bounded operator

$R_{\mathbb{H},i}^{(T)}\left(s\right):L_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)\to
H_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right).$

  2. 2.

The integral operator $\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)$ is well-
defined on $C_{c,\phi_{i}}^{\infty}\left(\mathbb{H};V_{N_{i}}^{0}\right)$ and
and extends to a bounded operator on
$L_{\phi}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)$.

  3. 3.

We have

$\left[\Delta-s(1-s)\right]R_{\mathbb{H},i}^{(T)}\left(s\right)=1+\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)$

as an identity of operators on
$L_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)$.

The proof of Lemma A.2 easily follows from the proof of [HM, Lemma 5.5],
simplified in places by the compactness of the fundamental domain $F$ in the
current setting.

We have an isomorphism of Hilbert spaces

$L_{\phi_{i}}^{2}\left(\mathbb{H};V_{N_{i}}^{0}\right)\cong
L^{2}\left(F\right)\otimes V_{N_{i}}^{0},$

given by

$f\mapsto\sum_{e_{i}}\langle f\lvert_{F},e_{i}\rangle\otimes e_{i},$

where $\\{e_{j}\\}_{j=1}^{N_{i}-1}$ is some choice of basis for
$V_{N_{i}}^{0}$. After conjugation by this isomorphism, the operator
$\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)$ becomes

$\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)\cong\sum_{\gamma\in
S}a_{\gamma}^{(T)}\left(s\right)\otimes\rho_{i}\left(\gamma^{-1}\right),$
(A.5)

where

$\displaystyle a_{\gamma}^{(T)}\left(s\right)$
$\displaystyle:L^{2}\left(F\right)\to L^{2}\left(F\right),$ $\displaystyle
a_{\gamma}^{(T)}\left(s\right)\left[f\right]\left(x\right)$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\int_{y\in
F}\mathbb{L}_{\mathbb{H}}^{(T)}\left(s;\gamma
x,y\right)f\left(y\right)d\mu_{\mathbb{H}}\left(y\right).$

Since $\mathbb{L}_{\mathbb{H}}^{(T)}\left(s,\gamma x,y\right)$ is only non-
zero when $d\left(\gamma x,y\right)\leqslant T+1$, in (A.5) one can take
$S=S\left(T\right)\subset\Gamma$ to be finite. Since
$\mathbb{L}_{\mathbb{H}}^{(T)}\left(s;\gamma x,y\right)$ is smooth and bounded
it follows that the operators $a_{\gamma}^{(T)}\left(s\right)$ are Hilbert-
Schmidt and therefore compact. We define

$\mathcal{L}_{s,\infty}^{(T)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{\gamma\in
S}a_{\gamma}^{(T)}(s)\otimes\lambda\left(\gamma^{-1}\right).$

Under the isomorphism

$\displaystyle L^{2}\left(F\right)\otimes\ell^{2}\left(\Gamma\right)$
$\displaystyle\cong L^{2}\left(\mathbb{H}\right),$ $\displaystyle
f\otimes\delta_{\gamma}$ $\displaystyle\mapsto f\circ\gamma^{-1},$

(with $f\circ\gamma^{-1}$ extended by zero from a function on $\gamma F$) the
operator $\mathcal{L}_{s,\infty}^{(T)}$ is conjugated to

$\mathbb{L}_{\mathbb{H}}^{(T)}(s):L^{2}\left(\mathbb{H}\right)\to
L^{2}\left(\mathbb{H}\right).$

To prove Theorem A.1, we need to replace the probabilistic bound [HM, Lemma
6.3] by a deterministic one.

######  Proposition A.3.

For any $s_{0}>\frac{1}{2}$ there is a $T=T(s_{0})>0$ such that for any fixed
$s\in[s_{0},1]$ there is an $I\left(s_{0},s\right)$ with

$\|\mathcal{L}_{s,\phi_{i}}^{(T)}\|_{L^{2}(F)\otimes
V_{N_{i}}^{0}}\leq\frac{1}{4},$

for all $i\geqslant I$.

###### Proof.

Let $s_{0}>\frac{1}{2}$ and a fixed $s\in[s_{0},1]$ be given. By (A.4) we can
find a $T\left(s_{0}\right)$ such that

$\|\mathcal{L}_{s,\infty}^{(T)}\|_{L^{2}(F)\otimes\ell^{2}(\Gamma)}\leq\frac{1}{8}.$
(A.6)

Recall that the coefficients $a_{\gamma}(s)$ are supported on a finite set
$S=S(T)\subset\Gamma$. Because the $a_{\gamma}(s)$ are compact, we apply (A.3)
(and the following remark) to the operators
$\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)$ to find that there is
$I\in\mathbf{N}$ such that for all $i\geq I$

$\|\mathcal{L}_{s,\phi_{i}}^{(T)}\|_{L^{2}(F)\otimes
V_{N_{i}}^{0}}\leq\|\mathcal{L}_{s,\infty}^{(T)}\|_{L^{2}(F)\otimes\ell^{2}(\Gamma)}+\frac{1}{8}\leq\frac{1}{4}.$

∎

We can now prove Theorem A.1.

###### Proof of Theorem A.1.

Given $\epsilon>0$ let $s_{0}=\frac{1}{2}+\sqrt{\epsilon}$ so that
$s_{0}\left(1-s_{0}\right)=\frac{1}{4}-\epsilon$. Let $T=T\left(s_{0}\right)$
be the value provided by Proposition A.3 for this $s_{0}$.

We use a finite net to control all values of $s\in\left[s_{0},1\right]$. Using
[HM, Lemma 6.1] as in [HM, Proof of Thm. 1.1] tells us that there is a finite
set $Y=Y(s_{0})$ of points in $[s_{0},1]$ such that for any $s\in[s_{0},1]$
there is $s^{\prime}\in Y$ with

$\|\mathcal{L}_{s,\phi_{i}}^{(T)}-\mathcal{L}_{s^{\prime},\phi_{i}}^{(T)}\|\leq\frac{1}{4}$
(A.7)

for all $i$.

Combining (A.7) with Proposition A.3 applied to
$\mathcal{L}_{s,\phi_{i}}^{(T)}$ for every $s\in Y$ we find that there is an
$I\left(s_{0}\right)$ such that for all $s\in[s_{0},1]$ and $i\geq I(s_{0})$

$\|\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)\|_{L_{\mathrm{new}}^{2}(X_{i})}=\|\mathcal{L}_{s,\phi_{i}}^{(T)}\|_{L^{2}(F)\otimes
V_{N_{i}}^{0}}\leq\frac{1}{2}.$ (A.8)

By Lemma A.2, for $s>\frac{1}{2}$ $R_{\mathbb{H},i}^{(T)}\left(s\right)$ is a
bounded operator from $L_{\text{$\mathrm{new}$}}^{2}\left(X_{i}\right)$ to
$H_{\mathrm{new}}^{2}\left(X_{i}\right)$. By Lemma A.2 we have that

$\left(\Delta_{X_{\phi}}-s(1-s)\right)R_{\mathbb{H},i}^{(T)}\left(s\right)=1+\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right),$

on $L_{\text{new}}^{2}\left(X_{i}\right)$. From (A.8) for all $i\geqslant
I\left(s_{0}\right)$ and $s\in\left[s_{0},1\right]$
$\left(1+\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)\right)^{-1}$ exists as
a bounded operator on $L_{\mathrm{new}}^{2}\left(X_{i}\right)$. We now get
that for all $i\geqslant I\left(s_{0}\right)$ and all
$s\in\left[s_{0},1\right],$

$\left(\Delta_{X_{i}}-s\left(1-s\right)\right)R_{\mathbb{H},i}^{(T)}\left(s\right)\left(1+\mathbb{L}_{\mathbb{H},i}^{(T)}\left(s\right)\right)^{-1}=1,$

and we conclude that $\left(\Delta_{X_{i}}-s\left(1-s\right)\right)$ has a
bounded right inverse from $L_{\text{$\mathrm{new}$}}^{2}\left(X_{i}\right)$
to $H_{\mathrm{new}}^{2}\left(X_{i}\right)$, implying that for $i\geqslant
I\left(s_{0}\right)$, $\Delta_{X_{i}}$ has no new eigenvalues $\lambda$ with
$\lambda\leq s_{0}\left(1-s_{0}\right)=\frac{1}{4}-\epsilon$. ∎

## References

  * [Ali05] Emina Alibegović, _A combination theorem for relatively hyperbolic groups_ , Bull. London Math. Soc. 37 (2005), no. 3, 459–466. MR 2131400
  * [Bau62] Gilbert Baumslag, _On generalised free products_ , Math. Z. 78 (1962), no. 3, 423–438. MR 140562
  * [Bau67] Benjamin Baumslag, _Residually free groups_ , Proc. London Math. Soc. (3) 17 (1967), 402–418. MR 215903
  * [BC19] C. Bordenave and B. Collins, _Eigenvalues of random lifts and polynomials of random permutation matrices_ , Ann. of Math. (2) 190 (2019), no. 3, 811–875. MR 4024563
  * [BCdlH94] M. Bekka, M. Cowling, and P. de la Harpe, _Some groups whose reduced $C^{*}$-algebra is simple_, Inst. Hautes Études Sci. Publ. Math. (1994), no. 80, 117–134 (1995). MR 1320606
  * [BDF73] L. G. Brown, R. G. Douglas, and P. A. Fillmore, _Unitary equivalence modulo the compact operators and extensions of C ∗\- algebras_, Proc. Conf. Operator Theory, Dalhousie Univ., Halifax 1973, Lect. Notes Math. 345, 58-128 (1973)., 1973.
  * [BDF77] , _Extensions of $C^{*}$-algebras and $K$-homology_, Ann. Math. (2) 105 (1977), 265–324 (English).
  * [BdlHV08] Bachir Bekka, Pierre de la Harpe, and Alain Valette, _Kazhdan’s property (t)_ , New Math. Monogr., vol. 11, Cambridge: Cambridge University Press, 2008 (English).
  * [Ber16] N. Bergeron, _The spectrum of hyperbolic surfaces_ , Universitext, Springer, Cham; EDP Sciences, Les Ulis, 2016, Appendix C by Valentin Blomer and Farrell Brumley, Translated from the 2011 French original by Brumley [2857626]. MR 3468508
  * [BMR02] Gilbert Baumslag, Alexei Myasnikov, and Vladimir Remeslennikov, _Discriminating completions of hyperbolic groups_ , Geom. Dedicata 92 (2002), 115–143, Dedicated to John Stallings on the occasion of his 65th birthday. MR 1934015
  * [BO08] Nathanial P. Brown and Narutaka Ozawa, _$C^{*}$ -algebras and finite-dimensional approximations_, Grad. Stud. Math., vol. 88, Providence, RI: American Mathematical Society (AMS), 2008 (English).
  * [Bus84] P. Buser, _On the bipartition of graphs_ , Discrete Appl. Math. 9 (1984), no. 1, 105–109. MR 754431
  * [CG05] Christophe Champetier and Vincent Guirardel, _Limit groups as limits of free groups_ , Israel J. Math. 146 (2005), 1–75. MR 2151593
  * [Fel62] J. M. G. Fell, _Weak containment and induced representations of groups_ , Can. J. Math. 14 (1962), 237–268 (English).
  * [Haa79] Uffe Haagerup, _An example of a nonnuclear $C^{\ast}$-algebra, which has the metric approximation property_, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930
  * [HM] W. Hide and M. Magee, _Near optimal spectral gaps for hyperbolic surfaces_ , preprint, arXiv:2107.05292.
  * [HT05] U. Haagerup and S. Thorbjørnsen, _A new application of random matrices: ${\rm Ext}(C^{*}_{\rm red}(F_{2}))$ is not a group_, Ann. of Math. (2) 162 (2005), no. 2, 711–775. MR 2183281
  * [Hub74] H. Huber, _Über den ersten Eigenwert des Laplace-Operators auf kompakten Riemannschen Flächen_ , Comment. Math. Helv. 49 (1974), 251–259. MR 365408
  * [Hux85] M. N. Huxley, _Introduction to Kloostermania_ , Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 217-306 (1985)., 1985.
  * [JL70] H. Jacquet and R. P. Langlands, _Automorphic forms on ${\rm GL}(2)$_, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970\. MR 0401654
  * [KM98] O. Kharlampovich and A. Myasnikov, _Irreducible affine varieties over a free group. II. Systems in triangular quasi-quadratic form and description of residually free groups_ , J. Algebra 200 (1998), no. 2, 517–570. MR 1610664
  * [Lyn59] R. C. Lyndon, _The equation $a^{2}b^{2}=c^{2}$ in free groups_, Mich. Math. J. 6 (1959), 89–95 (English).
  * [Lyn60] Roger C. Lyndon, _Groups with parametric exponents_ , Trans. Amer. Math. Soc. 96 (1960), 518–533. MR 151502
  * [Mal12] Camille Male, _The norm of polynomials in large random and deterministic matrices_ , Probab. Theory Relat. Fields 154 (2012), no. 3-4, 477–532 (English).
  * [Pow75] Robert T. Powers, _Simplicity of the $C^{\ast}$-algebra associated with the free group on two generators_, Duke Math. J. 42 (1975), no. 1, 151–156. MR 374334
  * [Raz14] Alexander A. Razborov, _A product theorem in free groups_ , Ann. of Math. (2) 179 (2014), no. 2, 405–429. MR 3152939
  * [Sel01] Zlil Sela, _Diophantine geometry over groups. I. Makanin-Razborov diagrams_ , Publ. Math. Inst. Hautes Études Sci. (2001), no. 93, 31–105. MR 1863735
  * [Tay89] Keith F. Taylor, _C*-algebras of crystal groups_ , Oper. Theory, Adv. Apply. 41, 511-518 (1989)., 1989.
  * [Voi93] Dan Voiculescu, _Around quasidiagonal operators_ , Integral Equations Operator Theory 17 (1993), no. 1, 137–149. MR 1220578
  * [Wil09] Henry Wilton, _Solutions to Bestvina & Feighn’s exercises on limit groups_, Geometric and cohomological methods in group theory, London Math. Soc. Lecture Note Ser., vol. 358, Cambridge Univ. Press, Cambridge, 2009, pp. 30–62. MR 2605175

Will Hide,
Department of Mathematical Sciences,
Durham University,
Lower Mountjoy, DH1 3LE Durham,
United Kingdom

<EMAIL_ADDRESS>

Larsen Louder,
Department of Mathematics,
University College London,
Gower Street, London WC1E 6BT,
United Kingdom
<EMAIL_ADDRESS>

Michael Magee,
Department of Mathematical Sciences,
Durham University,
Lower Mountjoy, DH1 3LE Durham,
United Kingdom

<EMAIL_ADDRESS>