# Interrelation of equivariant Gaussian processes and convolutional neural
networks

Andrey Demichev and Alexander Kryukov  Skobeltsyn Institute of Nuclear
Physics, M.V. Lomonosov Moscow State University,
1(2) Leninskie gory, Moscow 119991, Russian Federation
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Currently there exists rather promising new trend in machine leaning (ML)
based on the relationship between neural networks (NN) and Gaussian processes
(GP), including many related subtopics, e.g., signal propagation in NNs,
theoretical derivation of learning curve for NNs, QFT methods in ML, etc. An
important feature of convolutional neural networks (CNN) is their equivariance
(consistency) with respect to the symmetry transformations of the input data.
In this work we establish a relationship between the many-channel limit for
CNNs equivariant with respect to two-dimensional Euclidean group with vector-
valued neuron activations and the corresponding independently introduced
equivariant Gaussian processes (GP).

##  1 Introduction

In recent years, a correspondence has been established between the appropriate
asymptotics of deep neural networks (DNNs), including convolutional ones
(CNNs), and the machine learning methods based on Gaussian processes (GP), see
[1] and refs. therein. The ultimate goal of establishing such interrelations
is to achieve a better theoretical understanding of various methods of machine
learning (ML) and their improvement. Since Gaussian processes are
mathematically similar to Euclidean quantum field theory (QFT), one of the
intriguing consequences of this correspondence is the potential for using a
vast arsenal of QFT methods to analyze deep neural networks [2], in particular
for predictions of learning curves and DNN outputs.

An important feature of convolutional networks is their equivariance
(consistency) with respect to the symmetry transformations of the input data
[3, 4]. Equivariance guarantees that exactly the same filters are applied to
each part of the input image regardless of position and that the network can
detect any given object equally well regardless of its location respecting
data symmetry properties.

It is important that the above mentioned works on establishing the
interrelations between CNNs and GPs deal only with translational equivariance
of images. On the other hand there exists investigations of more general
(e.g., including rotations) equivariant neural Gaussian processes [5] but
without established relations with CNNs in the appropriate limit. In our
preceding work [6], initiated in the framework of Russian–German Astroparticle
Data Life Cycle Initiative [7], we derived the many-channel limit for a toy 1D
CNNs with $SO(2)$ symmetry and scalar activations keeping explicit
equivariance at each step of the derivation and calculated the corresponding
equivariant GP kernel. In the present work we provide the derivation of the
many-channel limit for the practically important case of CNNs equivariant with
respect to Euclidean group of motions of a two-dimensional input space and
vector-valued neuron activations. Such DNNs are described in the framework of
steerable convolutional neural networks (SCNN) [3, 8] which use induced
representations of the respective symmetry groups. The result obtained can be
applied for theoretical analysis of CNNs and improvement of their performance
with the account of a prior information about symmetries of input data (see,
e.g., [2] for such an analysis without account of any symmetry).

##  2 Many-channel CNNs with vector-valued activations and $SE(2)$ symmetry

We consider a series of $L+1$ convolutional layers, $l=0,\dots,L$ in a
steerable convolutional neural network (SCNN) [3] with $SE(2)\equiv
T_{2}\rtimes SO(2)$ symmetry on a $2D$-plane [8]. Here $T_{2}$ and $SO(2)$ are
groups of two-dimensional translations and rotations, respectively. Such a
SCNN defines feature spaces as spaces of fields
$y:\mathbb{R}^{2}\to\mathbb{R}^{c}$ which associate a $c$-dimensional feature
vector $y({\vec{r}})\in\mathbb{R}^{c}$ to each point $\vec{r}$ of a base
space, in the considered case the plane $\mathbb{R}^{2}$. The transformation
law of a general feature field $f:\mathbb{R}^{2}\to\mathbb{R}^{c}$ is fully
characterized by the $SE(2)$ group representation $\rho:\ SO(2)\mapsto
GL(\mathbb{R}^{c})$ specifying how the $c$ channels of each feature vector
$y(\vec{r})$ mix under transformations. A simplest example are scalar feature
fields ${s:\mathbb{R}^{2}\to\mathbb{R}}$ with the trivial representation
$\rho(g)=\mathbbm{1}$. Such feature fields describe, e.g., gray-scale images,
temperature, pressure, etc. Infinite-channel limit for CNNs with scalar fields
was derived in the works [1], and the issue of equivariance (for 1D analog)
was discussed in [6]. Vector fields
${\mathbf{y}:\mathbb{R}^{2}\to\mathbb{R}^{2}}$ transform as
${\mathbf{y}(\vec{r})\mapsto
g\cdot\mathbf{y}\left(g^{-1}(\vec{r}-\vec{t})\right)}$, where $g$ is the
matrix of two-dimensional rotations. Examples include fields of wind direction
and speed, electric fields, etc. More details and examples can be found, e.g.,
in [8] (in the context of SCNNs) and in [5] (in the context of GPs). In this
paper we consider the case of vector feature fields.

More generally, fields transforms under the induced representation (see [8]
and refs. therein)

$\displaystyle y^{l}(\vec{r})\ \ \mapsto\ \
\left(\left[\operatorname{Ind}_{SO(2)}^{T_{2}\rtimes
SO(2)}\rho\right]\\!\\!(tg)\cdot y^{l}\right)\\!(\vec{r})\ \ :=\ \
\rho(g)\cdot y^{l}\left(g^{-1}(\vec{r}-\vec{t})\right).$ (2.1)

It is important that the $\rho$-representation can be reducible. Moreover, as
it is well known, all complex-valued irreducible representattion (irreps) of
the Abelian group $SO(2)$ are one-dimensional.

The most general _equivariant linear map_ between steerable feature spaces
(SCNN’s layers), transforming under $\rho_{\text{in}}$ and
$\rho_{\text{out}}$, is given by convolutions with $SO(2)$-steerable kernels
(filters) $\omega:\mathbb{R}^{2}\to\mathbb{R}^{c_{\text{out}}\times
c_{\text{in}}}$, satisfying a kernel constraint (see [9], [8] and refs
therein)

$\displaystyle\omega(g\vec{r})\ =\
\rho_{\text{out}}(g)\omega(\vec{r})\rho_{\text{in}}(g^{-1})\quad\forall g\in
SO(2),\ \vec{r}\in{\mathbb{R}}^{2}\,.$ (2.2)

From the point of view of limiting transition to Gaussian processes, SCNNs
poses the following main problems to be solved: (1) the application-important
vectors (like, e.g., wind speed and direction) are also treated as channels,
so the question is how one can go to the infinite-channel limit given that
such vectors must have a finite dimensionality defined by an application task;
(2) since the filter $\omega(\vec{r})$ satisfies the condition (2.2), its
values are related at different points of the base space $\mathbb{R}^{2}$ and
cannot be initialized independently, the latter being an essential ingredient
for the derivation of the infinite-channel limit.

The first problem is solved in the present work by using in (2.1) a reducible
representation so that real-valued activations have two indices
$y^{l}_{i\alpha}(\vec{r})$, the first one (latin) numerating $n^{l}$ different
irreps within the $\rho$-representation, while the second (Greek) index
indicating components within the $i$-th two-dimensional irrep. In the group
theory the number of irreps $n^{l}$ is often called multiplicity. Thus the
infinite-channel limit can be harmlessly carried out over the multiplicity
$n^{l}$ keeping the application-meaningful vector dimensionality fixed. The
second problem is solved by separation of longitudinal and angle modes of the
filter $\omega(\vec{r})$ and intializing independently only the longitudinal
modes which are not constraint by the condition (2.2) (see below).

An essential technical simplification of the infinite-channel derivation can
be achieved by using instead of two-dimensional vectors $y_{i\alpha}(\vec{r})\
(\alpha=1,2;\,i=1,\dots,n^{l})$ complex-valued feature field $y_{i}(\vec{r}):\
\mathbb{R}^{2}\mapsto\mathbb{C}$. In particular, this can be clearly seen from
a comparison of the solution derivation for the constraint (2.2) in the
complex [9] and real-valued [8] forms. Certainly, in general two-dimensional
vectors and matrices are not fully equivalent to complex numbers. In
particular, a general $2\times 2$ matrix $\omega_{ij,\alpha\beta}$ (for fixed
$i,j$) cannot be represented as a complex number. But the filter $\omega$
satisfying (2.2) proves to be a unique angular frequency (mode) [8]

$\displaystyle\omega_{ij}(\vec{r})\
=R_{ij}(r)\left[\begin{array}[]{cc}\cos((m-n)\phi)&-\sin((m-n)\phi)\\\
\sin((m-n)\phi)&\cos((m-n)\phi)\end{array}\right]\ ,$ (2.5)

where $\\{r,\phi\\}$ are the polar coordinates for the vector $\vec{r}$. This
matrix and its action on a two-dimensional vectors can be represented as the
multiplication of two complex numbers: $\sum_{j}\omega_{ij}(\vec{r})y_{j}$;
$\omega_{ij},y_{j}\in\mathbb{C}$. Further discussion on the relation between
real and complex valued representations in the context of SCNNs can be found
in [8], Appendix F.5.

The network has activations $y^{l}(\vec{r})$ and pre-activations
$z^{l}(\vec{r})$ for each input $x_{i}(\vec{r})$. They are defined by the
following mappings from layer to layer:

$\displaystyle y^{l}_{i}(\vec{r})$
$\displaystyle=\left\\{\begin{array}[]{cc}x_{i}(\vec{r})&l=0\\\
\varphi^{(CS)}\left(z^{l-1}_{i}(\vec{r})\right)&l>0\end{array}\right.,$
$\displaystyle z^{l}_{i}(\vec{r})$
$\displaystyle=\sum_{j=1}^{n^{l}}[\omega_{ij}\star y_{j}](\vec{r})\ ,$ (2.8)

the bar over a character denotes the complex conjugate and $\varphi^{(CS)}$ is
a nonlinearity in the coordinate space. As it is seen, the pre-activations
$z^{l}_{i}(\vec{r})$ are defined by the cross-correlation $\star$ on
$\mathbb{R}^{2}$: $[\omega_{ij}\star y_{j}](\vec{r})\equiv\int
d^{2}r^{\prime}\,\bar{\omega}_{ij}(\vec{r}^{\,\prime})y_{j}(\vec{r}+\vec{r}^{\,\prime})$.
One can show that the cross-correlation in (2.8) with a filter satisfying
(2.2) preserve the equivariance of the SCNN.

However since we have to detach angular modes, working with the mapping (2.8)
in coordinate space proves to be very cumbersome, in particular because we
have to sum vectors in polar coordinates in the cross-correlation, cf. [10].
To overcome this difficulty, we use Fourier transform of the feature fields
$F(\vec{p})=\int
d^{2}r\,\exp\left\\{-{\rm{i}}\,\vec{r}\vec{p}\right\\}f(\vec{r})$ (here
$\\{f,F\\}$ stands for $\\{y,Y\\}$, $\\{z,Z\\}$ or $\\{\omega,\Omega\\}$,
respectively). In the terms of the Fourier transform the induced
representation (2.1) reads

$\displaystyle Y^{l}_{i}(\vec{p})$ $\displaystyle\mapsto
Y^{\prime\,l}_{i}(\vec{p})={\rm{e}}^{{\rm{i}}\,(k\theta-\vec{t}\vec{p})}\,Y^{l}_{i}(g^{-1}_{\theta}\vec{p})\
,$ (2.9)

where $\vec{t}$ is the parameter of the translation subgroup $T_{2}\in SE(2)$,
$\theta$ is the parameter of the $SO(2)$ subgroup and $k$ is the index of the
induced representation111It is worth noting that while the coordinate space
representations (2.1) are defined in the space of functions on
$\mathbb{R}^{2}=SE(2)/SO(2)$, the Fourier space representations (2.9) are
defined in the space of functions on a circle $S^{1}=SE(2)/T_{2}$. Details on
the latter representation can be found in [11]., cf. (2.1).

Since we have to detach the angular modes of the filter $\omega,\Omega$, we
expand it and the feature fields over the angles:
$F(\vec{p})=\sum_{m}F_{m}(p)\,\exp\left\\{-{\rm{i}}\,m\psi\right\\}$, where
$\\{p,\psi\\}$ are the polar coordinates for $\vec{p}$. In these Fourier-space
and angular-mode fields the mappings (2.8) reads as follows

$\displaystyle Y^{l}_{i,n}(p)$
$\displaystyle=\left\\{\begin{array}[]{ccc}X_{i,n}(p)&&l=0\\\
\varphi^{(FS)}\left(Z^{l-1}_{i,\cdot}(p)\right)&&l>0\end{array}\right.,$
$\displaystyle Z^{l}_{i,n}(p)$
$\displaystyle=\sum_{j=1}^{n^{l}}{\bar{\Omega}}_{ij,q_{l}}(p)Y^{l}_{j,n+q_{l}}(p)\
.$ (2.12)

The $\varphi^{(FS)}$ denotes a nonlinearity in the Fourier space, $q_{l}$ is
an arbitrarily chosen mode of the filter $\Omega$ so that the latter satisfies
the constraint (2.2), precisely because of the selection of a unique mode. It
is worth noting that the angular modes in the coordinate and Fourier spaces
for the same mode numbers are related to each other by the Hankel transform
[10]. Transformation properties of the angular modes can be easily deduced
from (2.9) but they are rather cumbersome and we do not display them here. As
it is readily seen, the cross-product in (2.8) is substituted in (2.12) by the
the pointwise product; this is a consequence of the well-known convolution
theorem.

For the illustration of the general approach we choose the polynomial
nonlinearity in the Fourier space preserving the explicit equivariance of the
SCNN (cf. [4] and [8]):
$Y^{l}_{i}(\vec{p})=\varphi^{(FS)}\left(Z^{l-1}_{i}(p)\right)\equiv\bar{Z}^{l-1}_{i}(\vec{p})Z^{l-1}_{i}(\vec{p})Z^{l-1}_{i}(\vec{p})\
,$ or, equivalently, in terms of the angular modes

$\displaystyle Y^{l}_{i,m}(p)$
$\displaystyle=\varphi_{m}^{(FS)}\left(Z^{l-1}_{i,\cdot}(p)\right)\equiv\sum_{n,k}\,\bar{Z}^{l-1}_{i,k}(p)Z^{l-1}_{i,n}(p)Z^{l-1}_{i,m+k-n}(p)\
.$ (2.13)

Actually, the latter can be presented as a 1D discrete cross-product
$\star_{1}$ (cf. [10]) with respect to angular mode indices:
$Y^{l}_{i,m}(p)=[Z^{l-1}_{i,\cdot}(p)\star_{1}Z^{l-1}_{i,\cdot}(p)\star_{1}Z^{l-1}_{i,\cdot}(p)]$.

According to the general approach to NN-GP correspondence [1], it is assumed
the Gaussian prior on the independent filter weights in the Fourier space,
$\Omega^{l}_{ij,q_{l}}(p)\sim{\cal
N}\left(0,{\sigma^{2}_{\omega}}/{n^{l}}\right)\ ,$ where $\sigma^{2}_{\omega}$
is the weight variance. Since $\Omega^{l}_{ij,q_{l}}(p)$ are i.i.d. variables,
they satisfy
$\mathbb{E}[\Omega^{l}_{ij,q_{l}}(p)\Omega^{l}_{i^{\prime}j^{\prime},q_{l}}(p^{\prime})]=\mathbb{E}[\bar{\Omega}^{l}_{ij,q_{l}}(p)\bar{\Omega}^{l}_{i^{\prime}j^{\prime},q_{l}}(p^{\prime})]=0$;
$\mathbb{E}[\bar{\Omega}^{l}_{ij,q_{l}}(p)\Omega^{l}_{i^{\prime}j^{\prime},q_{l}}(p^{\prime})]=\mathbb{E}[\Omega^{l}_{ij,q_{l}}(p)\bar{\Omega}^{l}_{i^{\prime}j^{\prime},q_{l}}(p^{\prime})]=({\sigma^{2}_{w}}/{2n^{l}})\delta_{ii^{\prime}}\delta_{jj^{\prime}}\delta_{pp^{\prime}}.$The
pre-activations $Z^{l}$ are linear combinations of the Gaussian variables
$\Omega^{l}$, specified by the previous layer’s activations $Y^{l}$. Thus for
the conditioned on $Y^{l}$ pre-activations one has
$\mathbb{E}[Z_{i,n^{\prime}}^{l}(p^{\prime})\bar{Z}_{i,n}^{l}(p)]=\frac{\sigma^{2}_{w}}{2}\delta_{pp^{\prime}}K^{l}_{n+q_{l},n^{\prime}+q_{l}}(p,p^{\prime})$,
where the uncentered covariance matrix $K^{l}$ of the activations $Y^{l}$ is
defined as (cf. [1]):

$\displaystyle K^{l}_{n,n^{\prime}}\left(p,p^{\prime}\right)$
$\displaystyle\,\stackrel{{\scriptstyle\rm
def}}{{\equiv}}\,\frac{1}{n^{l}}\sum_{i=1}^{n^{l}}Y^{l}_{i,n}(p)\bar{Y}^{l}_{i,n^{\prime}}(p^{\prime})\
.$ (2.14)

It is important to emphasize that in general $K^{l}$ is constructed as an
outer product of two two-dimensional vectors
$K^{l}\sim\mathbf{Y}\otimes\mathbf{Y}$ and cannot be represented as a complex
valued function. But we are looking for an equivariant Gauss kernel which is
presumably can be obtained as an appropriate limit for $K^{l}$. A key
observation, greatly simplifying the following calculations, is that according
to Theorem 1 of the paper [5] an equivariant Gauss kernel satisfies the same
constraint condition (2.2) as the SCNN’s filter. Therefore we suppose that the
covariance matrix $K^{l}$ is of the form (2.5) and it can indeed be
represented as a complex valued function.

Since a linear combination of Gaussian variables is itself a Gaussian, we can
conclude that $\left(Z^{l}|Y^{l}\right)\sim{\cal N}_{\cal
C}(0,\widetilde{\Gamma})$, where ${\cal N}_{\cal C}(0,\widetilde{\Gamma})$ is
the circularly-symmetric central complex normal distribution, and the
covariance $\widetilde{\Gamma}$ reads
$\widetilde{\Gamma}^{l}_{n,n^{\prime}}(p,p^{\prime})=\frac{\sigma^{2}_{w}}{2}\delta_{pp^{\prime}}K^{l}_{n+q_{l},n^{\prime}+q_{l}}(p,p^{\prime})\otimes
I_{n^{l+1}}$. Thus the normal distribution of $\left(Z^{l}|Y^{l}\right)$
depends only on $K^{l}$ and similarly to [1] we can conclude that the random
variable $\left(Z^{l}|K^{l}\right)$ has the same distribution:
$(Z^{l}|K^{l})\sim{\cal N}_{\cal C}(0,\widetilde{\Gamma})$.

Now by means of the relation (2.12) we can express the covariance (2.14) via
the pre-activations, take $n^{l}\rightarrow\infty$ limit and use the weak law
of large numbers (cf. [1]):

$\displaystyle
K^{l}_{n,n^{\prime}}(p,p^{\prime})=\frac{1}{n^{l}}\sum_{i=1}^{n^{l}}{\varphi}_{n}^{(FS)}\left(\bar{Z}^{l-1}_{i,\cdot}(P),Z^{l-1}_{i,\cdot}(p)\right)\bar{\varphi}_{n^{\prime}}^{(FS)}\left(\bar{Z}^{l-1}_{i,\cdot}(p^{\prime}),Z^{l-1}_{i,\cdot}(p^{\prime})\right)$
$\displaystyle\ \raise
1.29167pt\hbox{$---\longrightarrow$\kern-27.0pt\raise-4.73611pt\hbox{$\scriptstyle{n^{l}\rightarrow\infty}$}}\
\ \mathbb{E}_{Z\sim{\cal N}_{{\cal
C}}(0,\Gamma^{l-1})}\left[{\varphi}_{n}^{(FS)}\left(\bar{Z}^{l-1}_{i,\cdot}(P),Z^{l-1}_{i,\cdot}(p)\right)\bar{\varphi}_{n^{\prime}}^{(FS)}\left(\bar{Z}^{l-1}_{i,\cdot}(p^{\prime}),Z^{l-1}_{i,\cdot}(p^{\prime})\right)\right]\
,$ (2.15)

where
$\Gamma^{l-1}_{n,n^{\prime}}(p,p^{\prime})=({\sigma^{2}_{w}}/{2})\delta_{pp^{\prime}}K^{l}_{n+q_{l},n^{\prime}+q_{l}}(p,p^{\prime})$,
and we have used the nonlinearity (2.13). The mean value in (2) is equal to
the Gaussian-like integral which can be calculated by adding the terms with
external sources and using Feynman technique. The result allows us to show
that:

  * •

if
$K^{l-1}_{n,n^{\prime}}(p,p^{\prime})=\delta_{nn^{\prime}}\delta_{pp^{\prime}}K^{l-1}_{n}(p)$
(diagonal kernel), then the same is true for $K^{l}$; this property guarantees
the translational invariance for the kernel, cf. [5], Theorem 1;

  * •

if $K^{l-1}(\vec{p})$ contains only one angular mode, say with some number
$s_{l-1}$, then the same is true for $K^{l}$; this property guarantees the
rotational equivariance ([5], Theorem 1), similarly to the solution for the
filter with the constraint (2.2).

As a result of the calculations, for the $K^{l}_{n}(p)$ the simple recursive
relation has been found

$\displaystyle
K_{s}^{L}(p)=\left(6\left({\sigma_{w}^{2}}/{2}\right)^{3}\right)^{(3^{L}-1)/2}\Big{(}K^{0}_{s+q_{0}+\dots+q_{L}}(p)\Big{)}^{3^{L}}\
.$ (2.16)

Thus we have derived the self-consistent solution for the equivariant kernel
of the Gaussian process obtained as $n_{l}\rightarrow\infty$ limit of the
SCNN. The development of a general method for such a derivation was the main
goal of this work.

##  3 Conclusion

In this work we have derived the many-channel limit for $2D$ steerable CNNs
with $SE(2)\equiv T_{2}\rtimes SO(2)$ symmetry and vector-valued neuron
activations with explicit equivariance at each step of the derivation and
calculated the corresponding equivariant GP kernel. All the subtleties and
mathematically rigorous proofs for the expressions obtained are quite similar
to the case of classical CNNs and can be found in [1]. Thereby this work
filled the gap between many-channel equivariant CNNs and independently
introduced equivariant GPs [5].

In order to derive the limit for application-important vector-valued feature
fields we separated channels indices in two categories: the indices that
numerate the vector components within an irrep and used to describe their
transformations under matrix representations of a symmetry group and the
indices that numerate different irreducible representations (of the same or
different types). The second type of the indices are not restricted and can be
used for the limiting transition to the corresponding GP. Also it was
technically important to use the Fourier space for convenient detachment of
independent components of SCNN’s filters.

In the subsequent works the result obtained will be applied for theoretical
analysis of equivariant CNNs with the aim of improvement of their performance.

This work was funded by the Russian Science Foundation (grant No.
22-21-00442).

## References

## References

  * [1] Novak R, Xiao L, Lee J, Bahri Y, Yang G, Hron J, Abolafia D A, Pennington J and Sohl-Dickstein J 2018 arXiv preprint arXiv:1810.05148
  * [2] Cohen O, Malka O and Ringel Z 2019 arXiv preprint arXiv:1906.05301
  * [3] Cohen T S and Welling M 2016 arXiv preprint arXiv:1612.08498
  * [4] Kondor R, Lin Z and Trivedi S 2018 Advances in Neural Information Processing Systems vol 31 ed Bengio S, Wallach H, Larochelle H, Grauman K, Cesa-Bianchi N and Garnett R (Curran Associates, Inc.)
  * [5] Holderrieth P, Hutchinson M and Teh Y W 2020 arXiv preprint arXiv:2011.12916
  * [6] Demichev A 2021 Proceedings of Science 410 002 URL https://pos.sissa.it/410/002/pdf
  * [7] Bychkov I, Demichev A, Kryukov A et al. 2018 Data 3 56
  * [8] Weiler M and Cesa G 2019 arXiv preprint arXiv:1911.08251
  * [9] Worrall D E, Garbin S J, Turmukhambetov D and Brostow G J 2017 Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition pp 5028–5037
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# Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces

Thomas Hill

###### Abstract.

The work of Mann and Rafi gives a classification surfaces $\Sigma$ when
$\operatorname{Map}(\Sigma)$ is globally CB, locally CB, and CB generated
under the technical assumption of _tameness_. In this article, we restrict our
study to the pure mapping class group and give a complete classification
without additional assumptions. In stark contrast with the rich class of
examples of Mann–Rafi, we prove that $\operatorname{PMap}(\Sigma)$ is globally
CB if and only if $\Sigma$ is the Loch Ness monster surface, and locally CB or
CB generated if and only if $\Sigma$ has finitely many ends and is not a Loch
Ness monster surface with (nonzero) punctures.

##  1\. Introduction

The mapping class group of a surface $\Sigma$ is the group of orientation-
preserving homeomorphisms up to isotopy and is denoted by
$\operatorname{Map}(\Sigma)$. The subgroup that fixes the ends of the surface
pointwise is called the pure mapping class group and is denoted by
$\operatorname{PMap}(\Sigma)$. These groups form a short exact sequence with
the set of homeomorphisms of the end space:

$1\to\operatorname{PMap}(\Sigma)\to\operatorname{Map}(\Sigma)\to\operatorname{Homeo}(E(\Sigma),E_{G}(\Sigma))\to
1$

From this sequence, we see that understanding $\operatorname{Map}(\Sigma)$
involves understanding two components:
$\operatorname{Homeo}(E(\Sigma),E_{G}(\Sigma))$ thought of as the group of
“external symmetries” of the surface, and $\operatorname{PMap}(\Sigma)$ the
group of “internal symmetries”. The set of ends is homeomorphic to a subset of
a Cantor set, rendering $\operatorname{Homeo}(E(\Sigma),E_{G}(\Sigma))$ a
highly complex object. On the other hand, there are many well-established
results concerning pure mapping class groups of infinite-type surfaces (see
[APV20],[Dom22],[PV18]).

This article aims to describe the large-scale geometry of pure mapping class
groups of infinite-type surfaces. Mapping class groups of _finite_ -type
surfaces have straightforward large-scale geometry. The Dehn-Lickorish theorem
[FM12, see Chapter 4] establishes that the (pure) mapping class group of a
finite-type surface is finitely generated. This finite generating set induces
a word metric on $\operatorname{Map}(\Sigma)$. Similarly, a different set of
generators induces a (possibly) different word metric on
$\operatorname{Map}(\Sigma)$. The finiteness of both generating sets allows
one to establish a quasi-isometry between these two metric spaces, thereby
showing that $\operatorname{Map}(\Sigma)$ possesses a well-defined _quasi-
isometry type_.

Characterizing the large-scale geometry of big mapping class groups,
specifically identifying whether they possess a well-defined quasi-isometry
type and determining that type, presents a significantly greater challenge
compared to mapping class groups of surfaces with finite type. This disparity
arises because the conventional techniques of geometric group theory,
applicable to finitely generated groups, extend to locally compact, compactly
generated topological groups, but big mapping class groups do not fall into
this category. Fortunately, Rosendal’s work [Ros22] provides a framework to
extend the tools of geometric group theory to big mapping class groups and,
more generally, to Polish groups. This framework replaces the compactness
conditions with the notion of _locally coarsely bounded_ (abbreviated locally
CB) and _CB-generated_ (i.e., generated by a coarsely bounded set).

Under the additional assumption of tameness, Mann and Rafi described the
large-scale geometry of the full mapping class group by classifying which
surfaces have a CB, locally CB, or CB-generated mapping class group [MR22].
Within their classification, there are infinitely many surfaces that have a CB
mapping class group, and an even larger class of surfaces have a locally CB
and CB-generated mapping class group. In contrast, we show that a relatively
small class of surfaces have a CB, locally CB, or CB-generated _pure_ mapping
class group. Furthermore, by restricting to the pure mapping class group, we
give a CB classification of $\operatorname{PMap}(\Sigma)$ without the
additional tameness condition:

######  Theorem 1.1.

Let $\Sigma$ be an infinite-type surface. Then $\operatorname{PMap}(\Sigma)$
is

  * _(a)_

globally CB if and only if $\Sigma$ is the Loch Ness monster surface;

  * _(b)_

locally CB if and only if $|E(\Sigma)|<\infty$ and $\Sigma$ is not a Loch Ness
monster surface with (nonzero) punctures;

  * _(c)_

CB-generated if and only if $\operatorname{PMap}(\Sigma)$ is locally CB.

Many of the results of Mann–Rafi restrict nicely to the pure mapping class
group and serve as tools to prove 1.1, including using nondisplaceable
subsurfaces to disprove the CB-ness of the mapping class group. However, a
fundamental difference in the setting of $\operatorname{PMap}(\Sigma)$ is that
a surface with more than three (and possibly infinitely many) ends always
contains a nondisplaceable subsurface:

######  Proposition 1.2.

Let $\Sigma$ be an infinite-type surface, possibly with an infinite end space.
If $|E(\Sigma)|\geq 3$ then $\Sigma$ contains a nondisplaceable subsurface
with respect to $\operatorname{PMap}(\Sigma).$

This proposition drastically reduces the possible candidates of surfaces with
a CB pure mapping class group, and the remaining possibilities are considered
case-by-case. 1.2 is also an important new tool used to classify locally CB
and CB-generated pure mapping class groups.

### Acknowledgements

The author extends appreciation to Priyam Patel for her insightful project
suggestion, invaluable feedback, and mentorship. The author also acknowledges
Sanghoon Kwak for his extensive and insightful conversations, along with his
meticulous comments and feedback on multiple drafts of the paper. Appreciation
also goes to Mladen Bestvina and Rebecca Rechkin for their helpful discussions
that significantly contributed to the development of this project. The author
acknowledges support from RTG DMS–1840190.

##  2\. Preliminaries

###  2.1. Surfaces and Mapping Class Groups

Surfaces are 2-dimensional manifolds, and they come in two flavors: finite-
type surfaces and infinite-type surfaces. Finite-type surfaces are those with
a finitely generated fundamental group. These surfaces are classified by their
genus $g$, number of boundary components $b$, and number of punctures $n$. On
the other hand, those that do not have a finitely generated fundamental group
are called infinite-type surfaces. For infinite-type surfaces, the triple
$(g,b,n)$ does not contain enough information to distinguish between distinct
surfaces. For example, the Loch Ness monster surface and the ladder surface
both have $(g,b,n)=(\infty,0,0)$, see Figure 1. The classification of
infinite-type surfaces also depends on the _end space_ of the surface. Roughly
speaking, the end space represents the distinct ways to move toward infinity
on the surface.

Figure 1. The ladder surface (left) has two distinct ends accumulated by
genus, while the Loch Ness monster surface (center) has only one. The Cantor
tree surface (right) is a sphere with a Cantor set removed and has uncountably
many planar ends.

Put precisely, an end is an equivalence class of nested connected subsurfaces
of $S$, denoted by $[U_{1}\supset U_{2}\supset\cdots]$, such that each $U_{i}$
has a compact boundary, and any compact subsurface $K\subset S$ is disjoint
from $U_{n}$ for large enough $n$. Two sequences $U_{1}\supset
U_{2}\supset\cdots$ and $V_{1}\supset V_{2}\supset\cdots$ are equivalent if
they are both eventually contained in one another; that is, if for any $n$
there is an $m$ such that $U_{m}\subset V_{n}$ and $V_{m}\subset U_{n}$. An
end $[U_{1}\supset U_{2}\supset\cdots]$ is called planar if the $U_{i}$ are
all eventually homeomorphic to an open subset of $\mathbb{R}^{2}$. If instead
all $U_{i}$ in the sequence have infinite genus, the end is said to be
accumulated by genus or non-planar. The collection of all ends is a
topological space called the end space of $\Sigma$ and is denoted by
$E(\Sigma)$. The subset of ends accumulated by genus is denoted by
$E_{G}(\Sigma)$. As a topological space, the points of $E(\Sigma)$ are ends
$[U_{1}\supset U_{2}\supset\cdots]$, and open sets of $E(\Sigma)$ can be
thought of as subsurfaces of $\Sigma$ with compact boundary. To be precise,
the collection of all ends eventually contained in a subsurface
$U\subset\Sigma$ with compact boundary, forms a basis for the topology of
$E(\Sigma)$. An end $[U_{1}\supset U_{2}\supset\cdots]$ is said to be
contained in the open set $U$ if $U_{i}\subset U$ for large enough $i$. With
the additional data of the end space, infinite-type surfaces can be completely
classified by the homeomorphism type of the tuple
$(g,b,E(\Sigma),E_{G}(\Sigma))$, see [Ric63]. We will only consider surfaces
with empty boundary.

The group of orientation-preserving homeomorphisms up to isotopy is called the
mapping class group and is denoted by $\operatorname{Map}(\Sigma)$. The
subgroup of $\operatorname{Map}(\Sigma)$ that restricts to the identity on
$E(\Sigma)$ is called the pure mapping class group and is denoted by
$\operatorname{PMap}(\Sigma)$. Mapping class groups of infinite-type surfaces
are often called _big_ mapping class groups.

###  2.2. Large-Scale Geometry and Coarse Boundedness

Describing the large-scale geometry, or well-defined quasi-isometry type, of
big mapping class groups is more challenging than for finite-type surfaces, as
standard geometric group theory tools for finitely generated groups do not
directly apply to big mapping class groups. Rosendal’s work [Ros22] introduces
the notion of _locally coarsely bounded_ and _CB-generated_ groups, providing
a framework to extend the tools of geometric group theory to a larger class of
groups. This subsection aims to summarize Rosendal’s key contributions.

######  Definition 2.1.

A subset $A$ of a topological group $G$ is globally coarsely bounded in $G$,
or simply CB, if for every continuous isometric action of $G$ on a given
metric space $X$, the orbit $A\cdot x$ is bounded for all $x\in X$.

A topological group is Polish if it is separable and completely metrizable.
Both $\operatorname{Map}(\Sigma)$ and $\operatorname{PMap}(\Sigma)$ are
examples of Polish groups [AV20]. For Polish groups, Rosendal provides a
useful alternative criterion for coarse boundedness:

######  Fact 2.2 ([Ros22, Prop. 2.7 (5)]).

A subset $A$ of a Polish group $G$ is CB if and only if for every neighborhood
of the identity $\mathcal{V}$ there is a finite subset $\mathcal{F}\subset G$
and $k\geq 1$, so that $A\subset(\mathcal{F}\mathcal{V})^{k}$.

In the remark below, we use 2.2 to show that length functions on CB Polish
groups are always bounded. This is key in the proof of 2.15, where the
existence of an unbounded length function is used to disprove CB-ness. Recall
that a length function on a group $G$ is a continuous map $\ell\colon
G\to[0,\infty)$ that satisfies $\ell(\operatorname{Id})=0$,
$\ell(g)=\ell(g^{-1})$, and $\ell(gh)\leq\ell(g)+\ell(h)$ for all $g,h\in G$.

######  Remark 2.3.

If $G$ is a CB Polish group, then any length function $\ell\colon
G\to[0,\infty)$ is bounded. Indeed, take
$\mathcal{V}=B_{r}(\operatorname{Id})$, then by CB-ness of $G$ there is a
finite set $\mathcal{F}$ and $k\geq 0$ so that $G\subset(\mathcal{F}\cdot
B_{r}(\operatorname{Id}))^{k}$. This means

$\displaystyle{\ell(\varphi)\leq\left(\max_{f\in\mathcal{F}}\ell(f)+r\right)\cdot
k},$

for all $\varphi\in G$.

######  Definition 2.4.

A group is locally CB if it has a CB neighborhood of the identity.

When discussing the (local) CB-ness of a given set, it is essential to keep
track of the ambient group since it is possible that $H\leq G$ might be CB in
$G$, but $H$ might not be CB as a subset of itself. For instance, consider the
following example.

######  Example 2.5.

Let $\Sigma$ be a surface for which $\operatorname{Map}(\Sigma)$ is CB, and
let $\gamma$ be a non-separating simple closed curve in $\Sigma$. Consider the
group generated by a Dehn twist about $\gamma$, namely $\langle
T_{\gamma}\rangle\leq\operatorname{Map}(\Sigma)$. Because
$\operatorname{Map}(\Sigma)$ is assumed to be CB, any subset
$A\subseteq\operatorname{Map}(\Sigma)$, and in particular $\langle
T_{\gamma}\rangle$ must also be CB in $\operatorname{Map}(\Sigma)$. Thus,
$\langle T_{\gamma}\rangle$ is CB in $\operatorname{Map}(\Sigma)$, but
$\langle T_{\gamma}\rangle$ is not CB as a subset of itself since $\langle
T_{\gamma}\rangle\cong\mathbb{Z}$ and the continuous isometric action of
$\mathbb{Z}$ on $\mathbb{R}$ by translations has unbounded orbits.

When $G$ is a Polish group and $H\leq G$ is a _finite index_ open Polish
subgroup, this distinction can be blurred in light of the following fact (see
[DHK23, Proposition 2.20] for a proof).

######  Fact 2.6 ([Ros22, Proposition 5.67]).

Let $G$ be a Polish group and $H\leq G$ a finite index open Polish subgroup.
Then $H$ is CB if and only if $G$ is CB. Furthermore, $H$ is coarsely embedded
in $G$; that is to say, for $A\subset H$, $A$ is CB in $H$ if and only if $A$
is CB in $G$.

######  Definition 2.7.

A group is CB-generated if it admits a CB generating set.

The relationship between CB, locally CB, and CB-generated Polish groups can be
summarized in the diagram of implications below:

${\operatorname{CB}}$${\operatorname{CB-generated}}$locally
CB$\scriptstyle{(3)}$$\scriptstyle{(1)}$$\scriptstyle{(2)}$

When a group is CB, it is CB-generated since the whole group can be taken as
the CB-generating set, so implication (1) holds. Because CB-generated groups
have a well-defined quasi-isometry type by [Ros22], this means in particular
that globally CB groups have trivial quasi-isometry type. Implication (2) is a
result of Rosendal that every CB-generated Polish group is locally CB [Ros22,
Theorem 1.2].

Of course, combining (1) and (2) results in implication (3); however, (3) can
also be seen directly. If a group is globally CB, then it is locally CB since
the whole group can be taken as the CB neighborhood of the identity.

###  2.3. CB Classification of $\operatorname{Map}(\Sigma)$

This section will briefly summarize some of the results and tools of
Mann–Rafi’s CB classification of full mapping class groups [MR22].

######  Theorem 2.8 ([MR22, Theorem 1.7]).

Suppose that $\Sigma$ is either tame or has a countable end space. Then
$\operatorname{Map}(\Sigma)$ is CB if and only if $\Sigma$ has infinite or
zero genus and $E(\Sigma)$ is self-similar or telescoping.

An end space $E(\Sigma)$ is self-similar if for any decomposition
$E(\Sigma)=E_{1}\sqcup E_{2}\sqcup\dots\sqcup E_{n}$ there is a clopen subset
$D\subset E_{i}$ for some $i$, such that $(D,D\cap
E_{G}(\Sigma))\cong(E(\Sigma),E_{G}(\Sigma))$. Telescoping is a variation of
the self-similar condition (see [MR22, Definition 3.3]).

######  Remark 2.9.

A telescoping surface necessarily has infinitely many ends (see [MR22,
Proposition 3.7]). Likewise, self-similar end spaces contain either a single
end or infinitely many ends.

The definition of tameness space mentioned above in 2.8 is a technical
condition about the ‘local self-similarity’ of certain ends of a surface. We
avoid discussing this condition since our main focus will be on surfaces with
a finite end space, and tameness will always be satisfied for those surfaces.

######  Remark 2.10.

If $|E(\Sigma)|<\infty$, then $\Sigma$ is tame.

######  Theorem 2.11 ([MR22, Theorem 5.7] ).

$\operatorname{Map}(\Sigma)$ is locally CB if and only if there is a finite-
type subsurface $K\subset\Sigma$ (possibly empty) such that the connected
components of $\Sigma\setminus K$ have infinite-type and either 0 or infinite
genus, and partition $E(\Sigma)$ into finitely many clopen sets

$E(\Sigma)=\left(\bigsqcup_{A\in\mathcal{A}}A\right)\sqcup\left(\bigsqcup_{P\in\mathcal{P}}P\right)$

such that:

  1. (1)

Each $A\in\mathcal{A}$ is self-similar, $\mathcal{M}(A)\subset\mathcal{M}(E)$,
and $\mathcal{M}(E)=\bigsqcup_{A\in\mathcal{A}}\mathcal{M}(A)$.

  2. (2)

Each $P\in\mathcal{P}$ is homeomorphic to a clopen subset of some
$A\in\mathcal{A}$.

  3. (3)

For any $x_{A}\in\mathcal{M}(A)$, and any neighborhood $V$ of the end $x_{A}$
in $\Sigma$, there is $f_{V}\in\operatorname{Map}(\Sigma)$ so that $f_{V}(V)$
contains the complementary component to $K$ with end space $A$.

Moreover, the set of mapping classes restricting to the identity on $K$,
denoted by $\mathcal{V}_{K}$, is a CB neighborhood of the identity.
Additionally, $K$ can always be taken to have genus 0 if $\Sigma$ has infinite
genus and genus equal to $\Sigma$ otherwise. If $\Sigma$ has finitely many
isolated planar ends, these may be taken as punctures of $K$.

######  Example 2.12.

The ladder surface is an example of a surface with a locally CB, but not a
globally CB mapping class group. Because it has finitely many ends, the ladder
is a tame surface by 2.10. Since it has only two ends, the ladder surface is
not self-similar or telescoping by 2.9. However, the ladder surface is locally
CB since it satisfies the modified telescoping condition in 2.11 (3).

######  Theorem 2.13 ([MR22, Theorem 1.6] ).

For a tame surface with a locally (but not globally) CB mapping class group,
$\operatorname{Map}(\Sigma)$ is CB generated if and only if $E(\Sigma)$ is
finite-rank and not of limit type.

See [MR22, Definition 6.2 and 6.5] for the definitions of finite rank and
limit type, respectively. For our purpose, it is relevant that both the
infinite rank and limit type conditions require the space to have an infinite
number of ends.

###  2.4. Nondisplaceable Subsurfaces

For two subsurfaces $S$ and $S^{\prime}$, we say $S\cap
S^{\prime}\neq\emptyset$ if every subsurface homotopic to $S$ intersects every
subsurface homotopic to $S^{\prime}$.

######  Definition 2.14.

A connected finite-type subsurface $S\subset\Sigma$ is called nondisplaceable
with respect to the group $G\leq\operatorname{Map}(\Sigma)$ if $g(S)\cap
S\neq\emptyset$ for all $g\in G$.

Using the presence of a nondisplaceable subsurface in $\Sigma$ to conclude
that $\operatorname{Map}(\Sigma)$ is not CB is one of the key tools of
Mann–Rafi. This is also important in disproving the CB-ness of
$\operatorname{PMap}(\Sigma)$ below.

######  Proposition 2.15 (cf. [MR22, Theorem 1.9]).

If $\Sigma$ has a nondisplaceable subsurface with respect to
$\operatorname{PMap}(\Sigma)$, then $\operatorname{PMap}(\Sigma)$ is not
globally CB.

The proof of 2.15 follows exactly as in Mann–Rafi, but with
$\operatorname{PMap}(\Sigma)$ instead of $\operatorname{Map}(\Sigma)$. Below,
we will give a proof sketch of their argument and justify the replacement of
$\operatorname{Map}(\Sigma)$ with $\operatorname{PMap}(\Sigma)$. The proof
aims to exploit the unboundedness of the action of pseudo-Anosov elements on
the curve complex of a finite-type subsurface proven by Masur–Minsky [MM99].
Curve graphs and subsurface projections will play a role in the proof.

The curve graph of a surface $\Sigma$, denoted by $\mathcal{C}(\Sigma)$ is the
graph whose vertices correspond with the isotopy classes of simple closed
curves in $\Sigma$ where two vertices $\alpha$ and $\beta$ are connected by an
edge if $\alpha$ and $\beta$ can be realized disjointly. The curve graph can
be equipped with the path metric $d_{\Sigma}$ induced by giving each edge
length 1.

Let $S\subset\Sigma$ be a finite-type subsurface. For a curve
$\gamma\in\mathcal{C}(\Sigma)$, we define the subsurface projection of
$\gamma$ onto $S$ as

$\pi_{S}(\gamma)\mathrel{\mathop{:}}=\begin{cases}\\\ \\\ \\\ \end{cases}$
$\emptyset$ if there is a representative of $\gamma$ disjoint from $S$,
$\gamma$ if there is a representative of $\gamma$ contained in $S$, surger
$\gamma$ along $\partial S$ otherwise.

By surgering $\gamma$ along $\partial S$, we mean the following. If
$\gamma\cap S\neq\emptyset$ but $\gamma\not\subset S$ then $\gamma\cap S$ is a
union of arcs $\set{\alpha_{i}}$ based in $\partial S$. Consider the boundary
of a regular neighborhood of an arc $\alpha_{i}$. The result is two arcs that
we surger with the boundary of a regular neighborhood of the boundary
component(s) of $S$ that $\alpha_{i}$ is based at. The projection of
$\pi_{S}(\gamma)$ is the union of all curves obtained from this surgery. See
Figure 2 for an example of this surgery.

\begin{overpic}[width=256.0748pt]{subsurfproj.jpeg}
\put(11.0,12.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\gamma$}
\put(24.5,5.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}
$S$}
\put(46.0,11.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
$\pi_{S}(\gamma)$} \end{overpic}

Figure 2. An example of the surgery used to define the subsurface projection.
Here, $\pi_{S}(\gamma)$ is the disjoint union of the blue curves.

Similarly, if $\mu\subset\mathcal{C}(\Sigma)$ is a multicurve, we define
$\pi_{S}(\mu)\mathrel{\mathop{:}}=\bigsqcup_{\gamma\in\mu}\pi_{S}(\gamma)$.
For a subsurface $R\subset\Sigma$, we define
$\pi_{S}(R)\mathrel{\mathop{:}}=\pi_{S}(\partial R)$.

The path metric $d_{S}$ on the curve complex $\mathcal{C}(S)$ of a finite-type
subsurface $S\subset\Sigma$ can be extended to curves
$\gamma_{1},\gamma_{2}\in\mathcal{C}(\Sigma)$ by defining

$d_{S}(\gamma_{1},\gamma_{2})\mathrel{\mathop{:}}=\max_{\alpha_{i}\in\pi_{S}(\gamma_{i})}d_{S}(\alpha_{1},\alpha_{2})$

and similarly for multicurves $\mu_{1},\mu_{2}\subset\mathcal{C}(\Sigma)$,
$d_{S}(\mu_{1},\mu_{2})\mathrel{\mathop{:}}=\displaystyle{\max_{\gamma_{i}\in\pi_{S}(\mu_{i})}d_{S}(\gamma_{1},\gamma_{2})}.$

_Proof of 2.15._ The idea of Mann–Rafi’s argument is to construct an unbounded
length function $\ell\colon\operatorname{Map}(\Sigma)\to\mathbb{Z}$, which in
light of 2.3 is enough to show that $\operatorname{Map}(\Sigma)$ is not CB.

We now show that their argument goes through with
$\operatorname{PMap}(\Sigma)$ in place of $\operatorname{Map}(\Sigma)$. Let
$S\subset\Sigma$ be a nondisplaceable subsurface and let
$\mathcal{I}=\set{f(S)\mid f\in\operatorname{PMap}(\Sigma)}$. Let $\mu_{S}$ be
a multi-curve that fills $S$. Because $S$ is nondisplaceable, $R\cap
S\neq\emptyset$ for all $R\in\mathcal{I}$, so
$\mu_{R}\mathrel{\mathop{:}}=\pi_{R}(\mu_{S})$ is nonempty for all
$R\in\mathcal{I}$. Define a length function

$\ell\colon\operatorname{PMap}(\Sigma)\to\mathbb{Z}^{+}\qquad\text{given
by}\qquad\ell(\varphi)\mathrel{\mathop{:}}=\max_{R\in\mathcal{I}}d_{\varphi(R)}(\varphi(\mu_{R}),\mu_{\varphi(R)})$

where $d_{\varphi(R)}$ is the path metric in the curve complex
$C(\varphi(R))$. Intuitively, the function $\ell$ is measuring the maximum
distance in the curve complex of $\varphi(R)$ between curves in $\mu_{R}$
transformed by $\varphi$ and curves in the intersection of $S$ and
$\varphi(R)$.

Mann–Rafi prove $\ell$ is a continuous length function and demonstrate that
$\ell$ is unbounded by taking a $\varphi\in\operatorname{PMap}(\Sigma)$ that
preserves $S$ and restricts to a pseudo-Anosov on $S$. Then

$\ell(\varphi^{n})\geq
d_{\varphi^{n}(S)}(\varphi^{n}(\mu_{S}),\mu_{\varphi^{n}(S)})=d_{S}(\varphi^{n}(\mu_{S}),\mu_{S})\geq
d_{S}(\varphi^{n}(\gamma),\gamma)$

where $\gamma$ is any essential curve in $S$, and the middle equality holds
because $\varphi$ preserves $S$. By the work of Masur–Minsky [MM99]
$d_{S}(\varphi^{n}(\gamma),\gamma)\to\infty$ as $n\to\infty$ for such a map
$\varphi$, so $\ell$ is unbounded. $\square$

##  3\. CB Classification of $\operatorname{PMap}(\Sigma)$

###  3.1. Globally CB Classification

We show below that one of the striking differences between
$\operatorname{PMap}(\Sigma)$ and $\operatorname{Map}(\Sigma)$ is that
surfaces with at least three ends contain a nondisplaceable subsurface with
respect to $\operatorname{PMap}(\Sigma)$. In contrast, there are many examples
of surfaces with infinitely many ends that don’t have a nondisplaceable
subsurface with respect to $\operatorname{Map}(\Sigma)$. For example, the
Cantor tree surface pictured in Figure 1 has a globally CB mapping class
group, and therefore, it does not have any nondisplaceable subsurfaces with
respect to the (full) mapping class group.

######  1.2.

Let $\Sigma$ be an infinite-type surface, possibly with an infinite end space.
If $|E(\Sigma)|\geq 3$ then $\Sigma$ contains a nondisplaceable subsurface
with respect to $\operatorname{PMap}(\Sigma).$

###### Proof.

First, we claim that there is a compact subsurface $K\subset\Sigma$ whose
complementary components partition $E(\Sigma)$ into three nonempty disjoint
subsets. To see why, suppose that $x_{1},x_{2},x_{3}\in E(\Sigma)$ are
distinct ends. Recall that by definition, $x_{1}$ is an equivalence class of
nested connected subsurfaces with compact boundary $x_{1}=[S_{1}^{0}\supset
S_{1}^{1}\supset S_{1}^{2}\supset\cdots]$. Because the ends
$x_{1},x_{2},x_{3}$ are distinct, eventually there is a subsurface, say
$S_{1}^{i}$, in the sequence $S_{1}^{0}\supset S_{1}^{1}\supset
S_{1}^{2}\supset\cdots$ that does not contain $x_{2}$ or $x_{3}$. By
definition, $S_{1}^{i}$ is a connected surface with compact boundary.

\begin{overpic}[width=142.26378pt]{cptBdry.jpeg}
\put(60.0,45.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
$\gamma_{1}$} \put(35.0,7.0){$S_{1}$} \put(70.0,-5.0){$S_{1}^{i}$}
\put(0.0,25.0){$\cdots$}
\put(100.0,45.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
$\partial S_{1}^{i}$} \end{overpic} Figure 3. Since $S_{1}^{i}$ is connected
and has compact boundary, it can be visualized with the boundary components
lined up as above, up to homeomorphism. The separating curve $\gamma_{1}$ is
chosen to cut off only the boundary components of $S_{1}^{i}$; in particular,
$E(S_{1}^{i})=E(S_{1})$.

As pictured in Figure 3, we can find a separating simple closed curve
$\gamma_{1}$, which separates all the boundary components of $S_{1}^{i}$ from
the rest of the subsurface $S_{1}^{i}$. Cutting the original surface $\Sigma$
along $\gamma_{1}$, we obtain two new subsurface, each with a single boundary
component $\gamma_{1}$: one containing the boundary of $S_{1}^{i}$ call it
$\Sigma^{\prime}$, and another subsurface, call it $S_{1}$, that has
$E(S_{1})=E(S_{1}^{i})$. Thus the curve $\gamma_{1}$ partitions the end space
of $\Sigma$ into two subsets $E(\Sigma)=E(S_{1})\sqcup E(\Sigma^{\prime})$,
where $x_{1}\in E(S_{1})$ and $x_{2},x_{3}\in E(\Sigma^{\prime})$.

We can apply the same strategy to the end $x_{2}\in E(\Sigma^{\prime})$ and
find a subsurface $S_{2}\subset\Sigma^{\prime}$ that has a single boundary
component $\gamma_{2}$, and $x_{2}\in E(S_{2})$ but $x_{3}\not\in E(S_{2})$.
To simplify notation, let
$\Sigma^{\prime\prime}\mathrel{\mathop{:}}=\Sigma\setminus\operatorname{int}(S_{1}\cup
S_{2})$. By construction $\Sigma^{\prime\prime}$ is a connected subsurface
with two boundary components $\gamma_{1}$ and $\gamma_{2}$, and $x_{3}\in
E(\Sigma^{\prime\prime})$. Finally, let $\gamma_{3}$ be a separating simple
closed curve that cuts $\Sigma^{\prime\prime}$ into two components $K$ and
$S_{3}$, where $K$ is a compact subsurface with three boundary components
$\gamma_{1},\gamma_{2},$ and $\gamma_{3}$, and $S_{3}$ is a subsurface with
$E(S_{3})=E(\Sigma^{\prime\prime})$. Figure 4 illustrates this setup.

\begin{overpic}[width=199.16928pt]{cptSubsurf.jpeg} \put(18.0,28.0){$S_{1}$}
\put(73.0,5.0){$S_{2}$} \put(73.0,80.0){$S_{3}$}
\put(20.0,40.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
$\gamma_{1}$}
\put(52.0,10.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
$\gamma_{2}$}
\put(52.0,73.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
$\gamma_{3}$}
\put(38.0,48.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
$\partial S_{1}^{i}$} \put(25.0,5.0){$\underbrace{\hskip 142.26378pt}$}
\put(53.0,-2.0){$\Sigma^{\prime}$}
\put(65.0,45.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
$K$}
\put(86.0,57.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\Sigma^{\prime\prime}$}
\put(81.0,57.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
$\begin{rcases*}{}\\\ {}\\\ {}\\\ {}\\\ {}\\\ {}\\\ {}\\\ {}\\\ {}\\\ {}\\\
{}\\\ {}\\\ {}\\\ {}\\\ {}\end{rcases*}$} \end{overpic} Figure 4. The surface
$\Sigma$ partitioned into subsurfaces $S_{1},S_{2},S_{3}$ and $K$. The
subsurface $\Sigma^{\prime\prime}=\Sigma\setminus\operatorname{int}(S_{1}\cup
S_{2})$ is shaded in red.

In summary, we have constructed a compact subsurface $K$ such that

$\Sigma\setminus\operatorname{int}(K)=S_{1}\sqcup S_{2}\sqcup S_{3}$

with $E(S_{i})\neq\emptyset$ for each $i$, and $E(\Sigma)=E(S_{1})\sqcup
E(S_{2})\sqcup E(S_{3})$.

We now aim to prove that $K$ is nondisplaceable with respect to
$\operatorname{PMap}(\Sigma)$. Let $f\in\operatorname{PMap}(\Sigma)$, and then
$\Sigma\setminus\operatorname{int}(f(K))=S_{1}^{\prime}\sqcup
S_{2}^{\prime}\sqcup S_{3}^{\prime}$ where up to relabeling
$S_{i}^{\prime}=f(S_{i})$. We claim that

(1) $E(S_{i}^{\prime})=E(S_{i}).$

By the relabeling, $E(S_{i}^{\prime})=E(f(S_{i}))$. Since homeomorphisms
$S_{i}\to S_{i}^{\prime}$ induce homeomorphisms of the end spaces $E(S_{i})\to
E(S_{i}^{\prime})$, it follows $E(f(S_{i}))=f(E(S_{i})).$ Finally, since $f$
is pure, $f$ fixes the ends point-wise, so $f(E(S_{i}))=E(S_{i}).$ Putting
this all together,

$E(S_{i}^{\prime})=E(f(S_{i}))=f(E(S_{i}))=E(S_{i})$

which proves Equation 1.

Now, suppose that $K$ were displaceable. Then $f(K)\cap K=\emptyset$, so
$f(K)$ must be contained in one of the complementary components $S_{i}$.
Without loss of generality, assume $f(K)\subset S_{1}$. Because $S_{2},S_{3},$
and $K$ are all disjoint from $f(K)$ and $S_{2}\cup S_{3}\cup K$ is connected,
the subsurface $S_{2}\cup S_{3}\cup K$ must be contained in one of the
complementary components of $f(K)$. But this is not possible, since for
example if $S_{2}\cup S_{3}\cup K\subset S_{3}^{\prime}$, then
$E(S_{2})\subset E(S_{3}^{\prime})$, but by Equation 1,
$E(S_{2})=E(S_{2}^{\prime})$ contradicting $E(S_{2}^{\prime})\cap
E(S_{3}^{\prime})=\emptyset$. Similar arguments show $S_{2}\cup S_{3}\cup
K\not\subset S_{j}^{\prime}$ for $j=1,2.$ ∎

Equipped with 2.15, we are poised to begin the proof of 1.1.

######  1.1 (a).

Let $\Sigma$ be an infinite-type surface. Then $\operatorname{PMap}(\Sigma)$
is globally CB if and only if $\Sigma$ is the Loch Ness monster surface.

Proof. By 2.15 and 1.2, if $|E(\Sigma)|\geq 3$, then $\Sigma$ has a
nondisplaceable subsurface and therefore $\operatorname{PMap}(\Sigma)$ is not
CB. There are three infinite-type surfaces with $|E(\Sigma)|\leq 2$, namely
the ladder surface, the once-punctured Loch Ness monster surface, and the Loch
Ness monster surface, which we will consider case by case.

  * •

Case 1: $\Sigma$ is the ladder surface. For surfaces with $|E_{G}(\Sigma)|\geq
2$, Aramayona, Patel and Vlamis [APV20] define a _flux map_ , which is a
continuous surjection
$\operatorname{PMap}(\Sigma)\twoheadrightarrow\mathbb{Z}$. Because the action
of $\mathbb{Z}$ on $\mathbb{R}$ by translations is continuous and by
isometries, but the orbits of $x\in\mathbb{R}$ are unbounded, $\mathbb{Z}$ is
not CB. It follows that $\operatorname{PMap}(\Sigma)$ is not CB either, by
pulling back the action of $\mathbb{Z}$ on $\mathbb{R}$ to an action of
$\operatorname{PMap}(\Sigma)$ on $\mathbb{R}$.

  * •

Case 2: $\Sigma$ is the once-punctured Loch Ness monster surface. A subsurface
containing the isolated puncture is nondisplaceable with respect to
$\operatorname{Map}(\Sigma)$, and since $\Sigma$ has two distinct ends of
different type, $\operatorname{Map}(\Sigma)=\operatorname{PMap}(\Sigma)$ in
this case. Hence, $\Sigma$ is not CB by 2.15.

  * •

Case 3: $\Sigma$ is the Loch Ness monster surface. Because the Loch Ness
monster has only one end,
$\operatorname{PMap}(\Sigma)=\operatorname{Map}(\Sigma)$ and, as before, the
CB classification of $\operatorname{Map}(\Sigma)$ by Mann–Rafi applies.
Specifically, $|E(\Sigma)|=1$ means that the end space is trivially self-
similar, so by 2.8, $\operatorname{Map}(\Sigma)$ is CB. ∎

1.1 (a) is consistent with the results of Mann–Rafi. In particular, when
$|E(\Sigma)|<\infty$, $\operatorname{PMap}(\Sigma)$ has finite index in
$\operatorname{Map}(\Sigma)$. Because $\operatorname{Map}(\Sigma)$ is a Polish
group and $\operatorname{PMap}(\Sigma)$ is a clopen Polish subgroup, 2.6
applies, and we can import Mann–Rafi’s CB classification of
$\operatorname{Map}(\Sigma)$ in 2.13 to $\operatorname{PMap}(\Sigma)$. As
pointed out in 2.9, the telescoping condition is only satisfied for surfaces
with infinitely many ends, and self-similarity only occurs when there are
infinitely many ends or exactly one end. Therefore when $|E(\Sigma)|<\infty$,
the only surface with globally CB $\operatorname{Map}(\Sigma)$ is the unique
infinite-type surface with one end, namely the Loch Ness Monster surface.

###  3.2. Locally CB Classification

We now turn our attention to classifying the surfaces for which
$\operatorname{PMap}(\Sigma)$ is locally CB. For a finite subsurface
$K\subset\Sigma$, let

$\mathcal{V}_{K}\mathrel{\mathop{:}}=\set{f\in\operatorname{PMap}(S)\mid
f_{{}{K}}=\operatorname{Id}}{.}$

If $K=\emptyset$, then $\mathcal{V}_{K}=\operatorname{PMap}(\Sigma)$.

The proof of the proposition below follows the same argument as a result of
Mann–Rafi for $\operatorname{Map}(\Sigma)$.

######  Proposition 3.1 (cf. [MR22, Lemma 5.2]).

Let $K\subset\Sigma$ be a finite-type (possibly empty) subsurface such that
each component of $\Sigma\setminus K$ is infinite-type. If $\Sigma\setminus K$
contains a nondisplaceable subsurface with respect to
$\operatorname{PMap}(\Sigma)$, then $\mathcal{V}_{K}$ is not CB in
$\operatorname{PMap}(\Sigma)$. If $\mathcal{V}_{K}$ is not CB in
$\operatorname{PMap}(\Sigma)$ for all such $K$, then
$\operatorname{PMap}(\Sigma)$ is not locally CB.

###### Proof.

First, let $S\subset\Sigma\setminus K$ be a nondisplaceable subsurface with
respect to $\operatorname{PMap}(\Sigma)$. Recall in 2.15 that
$\operatorname{PMap}(\Sigma)$ was shown to be non-CB when $S\subset\Sigma$ is
nondisplaceable by constructing an unbounded length function on
$\operatorname{PMap}(\Sigma)$. The same argument applies to construct an
unbounded length function on $\mathcal{V}_{K}$. Specifically, consider a
psuedo-Anosov $f$ supported on $S\subset\Sigma\setminus K$ and extend $f$ to
an element of $\operatorname{PMap}(\Sigma)$ by extending via the identity on
$\Sigma\setminus S$. Then $f\in\mathcal{V}_{K}$ and as in 2.15 the length
function is unbounded on $\mathcal{V}_{K}$.

To see why $\operatorname{PMap}(\Sigma)$ is not locally CB if all
$\mathcal{V}_{K}$ are not CB in $\operatorname{PMap}(\Sigma)$, first recall
that $\set{\mathcal{V}_{C}\mid C\text{ compact}}$ is a neighborhood basis of
the identity in the compact open topology. For each compact subsurface $C$,
there is a finite type subsurface $K$ that contains $C$ and whose
complementary components $\Sigma\setminus K$ are infinite-type. Specifically,
take $K$ to be the union of $C$ and its finite-type complementary components.
So $\mathcal{V}_{K}\subset\mathcal{V}_{C}$. It follows from the definition of
CB that if $\mathcal{V}_{K}$ is not CB in $\operatorname{PMap}(\Sigma)$, then
$\mathcal{V}_{C}$ is not CB in $\operatorname{PMap}(\Sigma)$. If all sets
$\mathcal{V}_{K}$ are not CB in $\operatorname{PMap}(\Sigma)$, then neither
are any of the sets $\mathcal{V}_{C}$ in the neighborhood basis of the
identity, and $\operatorname{PMap}(\Sigma)$ has no CB neighborhood of the
identity; in other words, $\operatorname{PMap}(\Sigma)$ is not locally CB. ∎

We will see that 3.1 effectively narrows down the possible surfaces with a
locally CB pure mapping class group to those with finitely many ends.

######  1.1 (b).

Let $\Sigma$ be an infinite-type surface. Then $\operatorname{PMap}(\Sigma)$
is locally CB if and only if $|E(\Sigma)|<\infty$ and either

  * _(i)_

$|E_{G}(\Sigma)|=1$ and $|E(\Sigma)\setminus E_{G}(\Sigma)|=0$ or,

  * _(ii)_

$|E_{G}(\Sigma)|>1$ and $|E(\Sigma)\setminus E_{G}(\Sigma)|\geq 0$;

in other words, $\Sigma$ is not a Loch Ness monster surface with (nonzero)
punctures.

###### Proof.

First, consider when $|E(\Sigma)|=\infty$. Let $K\subset\Sigma$ be a finite-
type subsurface (possibly empty) so that each connected component of
$\Sigma\setminus K$ has infinite-type. If $K=\emptyset$, then
$\mathcal{V}_{K}=\operatorname{PMap}(\Sigma)$ is not CB by 1.1 (a). For any
$K\neq\emptyset$, at least one of the connected components of $\Sigma\setminus
K$ must contain infinitely many ends, and by 1.2 it must contain a
nondisplaceable subsurface. So, $\mathcal{V}_{K}$ is not CB in
$\operatorname{PMap}(\Sigma)$ for all possible $K$ and by 3.1,
$\operatorname{PMap}(\Sigma)$ is not locally CB.

Before proceeding, we remark that when $|E(\Sigma)|<\infty$, we can determine
if $\operatorname{PMap}(\Sigma)$ is locally CB solely by establishing whether
$\operatorname{Map}(\Sigma)$ is locally CB. To see why, let $\mathcal{V}$ be a
CB neighborhood of the identity in $\operatorname{Map}(\Sigma)$. The
restriction of $\mathcal{V}$ to $\operatorname{PMap}(\Sigma)$ is CB in
$\operatorname{PMap}(\Sigma)$ by 2.6 since $|E(\Sigma)|<\infty$. Furthermore,
$\mathcal{V}$ restricts to an identity neighborhood in
$\operatorname{PMap}(\Sigma)$, so $\operatorname{PMap}(\Sigma)$ is locally CB.

Now when $|E_{G}(\Sigma)|=1$, and $|E(\Sigma)\setminus E_{G}(\Sigma)|=0$, the
surface $\Sigma$ is the Loch Ness monster surface which is globally CB by 1.1
(a) and hence locally CB. On the other hand, when $|E_{G}(\Sigma)|=1$ and
$\Sigma$ has nonzero, but finitely many punctures, we can show that
$\operatorname{Map}(\Sigma)$ is not locally CB by 2.11. Up to isotopy, there
is only subsurface $K$ satisfying the hypothesis of 2.11, namely the
subsurface of genus 0 containing all finitely many punctures pictured in
Figure 5.

\begin{overpic}[width=142.26378pt]{LNMonsterWithPunctures.jpeg}
\put(20.0,30.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
$K$} \end{overpic} Figure 5. The unique subsurface $K$ of the finitely
punctured Loch Ness monster surface satisfying the hypothesis of 2.11 up to
isotopy.

Let $V$ be a neighborhood of the end accumulated by genus. In general,
$\Sigma\setminus V$ has some nonzero finite genus, while
$\Sigma\setminus(\Sigma\setminus K)=K$ has zero genus. So by the
classification of surfaces, there is no $f\in\operatorname{Map}(\Sigma)$ such
that $f(V)\supset\Sigma\setminus K$, and so $\operatorname{Map}(\Sigma)$ is
not locally CB. Hence, when $|E_{G}(\Sigma)|=1$ and $|E(\Sigma)\setminus
E_{G}(\Sigma)|\neq 0$, $\operatorname{PMap}(\Sigma)$ is not locally CB.

Finally, we will show $\operatorname{PMap}(\Sigma)$ is locally CB when
$|E(\Sigma)|<\infty$ and $|E_{G}(\Sigma)|>1$, by showing that
$\operatorname{Map}(\Sigma)$ is locally CB in this case. Because $\Sigma$ has
finitely many ends, it has, at most, a finite number of punctures. Let
$K\subset\Sigma$ be a finite-type subsurface containing all punctures of
$\Sigma$ (if any), such that all components of $\Sigma\setminus K$ contain a
single end accumulated by genus. These self-similar singleton sets represent
the $A$ sets in 2.11. There are no $P$ sets for this surface. See Figure 6.

\begin{overpic}[width=199.16928pt]{locallyCB_K.jpeg}
\put(47.0,38.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
$K$} \end{overpic} Figure 6. A choice of $K$ that satisfies the hypothesis of
2.11.

We now check condition (3) in the statement of 2.11. For any end accumulated
by genus $x\in E_{G}(\Sigma)$ and any neighborhood $V$ of $x$, an iterate of a
handle shift $h\in\operatorname{Map}(\Sigma)$ between $x$ and some other end
$y\neq x\in E_{G}(\Sigma)$ will shift all of the genus in the component of
$\Sigma\setminus K$ not in $V$ into the component of $\Sigma\setminus K$
containing $y$. Thus, $h$ maps the neighborhood $V$ to the entire
complementary component of $\Sigma\setminus K$ containing the end $x$, and by
2.11 $\operatorname{Map}(\Sigma)$ is locally CB, hence
$\operatorname{PMap}(\Sigma)$ is as well. ∎

Finally, we show that the surfaces with CB-generated pure mapping class group
coincide with those with locally CB pure mapping class group, finishing the
proof of 1.1.

######  1.1 (c).

Let $\Sigma$ be an infinite-type surface. Then $\operatorname{PMap}(\Sigma)$
is CB-generated if and only if $\operatorname{PMap}(\Sigma)$ is locally CB.

###### Proof.

Recall that every CB-generated Polish group is locally CB [Ros22, Theorem
1.2], so we need only prove that if $\operatorname{PMap}(\Sigma)$ is locally
CB, then it is CB-generated. In the case where $\Sigma$ is the Loch Ness
Monster, 1.1(a) implies that $\operatorname{PMap}(\Sigma)$ is globally CB and
hence CB-generated. Thus, it remains to consider the case where $\Sigma$ has
finitely many ends and $|E_{G}(\Sigma)|>1$.

Since $\operatorname{PMap}(\Sigma)$ is locally CB by 1.1 (b), we know that for
$K\subset\Sigma$ as in 2.11, $\mathcal{V}_{K}$ is a CB neighborhood of the
identity in $\operatorname{PMap}(\Sigma)$. Now, let $D$ be a Lickorish
generating set for the finite-type subsurface $K$, and $H$ a finite collection
of pairwise commuting handle-shifts and chosen by Mann–Rafi in the proof of
2.13. As both $D$ and $H$ are finite, the set $X=\mathcal{V}_{K}\cup D\cup H$
is CB in $\operatorname{PMap}(\Sigma)$. We claim that $X$ generates
$\operatorname{PMap}(S)$. In [MR22, Lemma 6.21], they show that the set $X$
together with a collection of “generalized” shift maps generates a group $G$
that contains $\operatorname{PMap}(\Sigma)$. However, in our case,
$|E(\Sigma)|<\infty$, so there are no generalized shift maps. Since all
elements of $X$ are pure, we conclude that the CB set $X$ generates
$\operatorname{PMap}(\Sigma)$. ∎

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