# Phase transition catalyzed by primordial black holes

Zhen-Min Zeng1,2<EMAIL_ADDRESS>Zong-Kuan Guo1,2,3<EMAIL_ADDRESS>1CAS
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China  2School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
3School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute
for Advanced Study, University of Chinese Academy of Sciences, Hangzhou
310024, China

###### Abstract

We investigate the first-order phase transition catalyzed by primordial black
holes (PBHs) in the early Universe. We find that super-horizon curvature
perturbations generated in this scenario lead to the production of
gravitational waves when the scalar modes re-enter the horizon. If PBHs with
masses about $10^{-13}M_{\odot}$ constitute all dark matter, the first-order
electroweak phase transition catalyzed by PBHs can explain the gravitational
wave signal observed by pulsar timing array collaborations without the
overproduction of PBHs.

_Introduction._ First-order phase transitions (PTs) are predicted by new
physics beyond the standard model [1, 2, 3, 4]. This process can generate
gravitational waves (GWs) through bubble collisions [5, 6], sound waves [7, 8,
9] and fluid turbulence [10, 11, 12, 13, 14]. Recent observations of the
stochastic gravitational-wave backgrounds (SGWBs) from pulsar timing array
collaborations implies a supercooling quantum chromodynamics PT at low energy
scales if the signal is generated by bubble collisions [15, 16, 17], which
infers a finite baryon chemical potentials.

These PTs are assumed to take place in a flat space-time. The effect of local
strong gravitational fields (such as PBHs) are not considered in the evolution
of the first-order PT. Recent studies indicate that PBHs can significantly
enhance the vacuum decay rate [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32]. The enhancement effect includes two parts, the change of the
Bekenstein entropy and the decrement of the action that depends on the
Euclidean bubble wall trajectory. The former dominates when PBH masses are of
order of the plank mass. It is under debate that the thermal effect of such
small PBHs is so strong that may stabilize the fields in the symmetric phase
[33]. The latter dominates when PBHs are large, especially when the horizon
radius of PBHs is comparable to the bubble radius [18, 34]. In this case the
bubble nucleation rate is strongly enhanced due to the strong gravitational
effect.

In this Letter, we consider the catalyzing effect of PBHs on a thermal first-
order PT. We assume that the vacuum energy density is vanishingly small in the
true vacuum at low temperature. Due to the catalyzing effect of PBHs, the
tunneling process occurs before the temperature decreases to the PT
temperature, $T_{n}$, which is determined by $\Gamma(T_{n})\sim H^{4}(T_{n})$
[35, 36, 37]. It reveals a tunneling process from a de Sitter vacuum to a de
Sitter vacuum with a relatively small vacuum energy. This differs from the
thermal PT, where the scalar field tunnels from a de Sitter vaccum to a
Minkovski vacuum.

Assuming PBHs are sparse enough that the average number of PBHs in a Hubble-
size region during PT is less than unity, which indicates there are some
Hubble-size regions that contain a PBH while the others don’t. In an
identification Hubble-sized region that contains a PBH, slightly after the
degenerate vacuum forms at the critical temperature, $T_{c}$, PT is triggered
by the PBH and itself as a nucleation site. However, the Hubble-size regions
without PBHs stay in the false vacuum until the nucleation rate $\Gamma(T)$
gets close to $H^{4}(T)$, and then the nucleation of thermal true vacuum
bubbles starts. Such an asynchronism of vacuum decay leads to the different
evolution of the probability of the false vacuum and then affects the averaged
equation of state in the different Hubble horizons during the PT, finally
induces curvature perturbations at super-horizon scales. Since the
asynchronism originates from the distribution of PBHs as catalysts, induced
density perturbations inherit the distribution of PBHs at super-horizon
scales.

We analytically calculate the power spectrum of curvature perturbations by
deriving the number density of PBHs. We find that at scales larger than the
average separation of PBHs, the probability distribution of density
perturbations is close to a finite-width Gaussian distribution. The existence
of the upper limit of density perturbations indicates that this mechanism can
generate a strong SGWB signal and avoid the overproduction of PBHs. We apply
our formalism to the electroweak phase transition (EWPT). If PBHs with masses
$M\sim 10^{-13}M_{\odot}$ constitute all dark matter as catalysts [38], the
predicted SGWB can explain the observed signal from recent pulsar timing array
collaborations [16, 17, 15] and avoid the overproduction of PBHs.

_First-order PT catalyzed by PBHs._ We regard PBHs as the nucleation sites of
true vacuum bubbles during a thermal first-order PT. By using the technique
developed in Refs. [18, 28], the upper limit of the mass of PBHs that can
catalyze the first-order PT is estimated to be $m_{pl}^{3}/\sqrt{V_{fv}}$ ,
where $m_{pl}\equiv 1/\sqrt{8\pi G}$ is the plank mass and $V_{fv}$ is the
energy density of the false vacuum. This estimation ignores the change of the
Bekenstein entropy, which only consider the pure gravitational effect. For the
EWPT, $V_{fv}\sim 10^{7}GeV^{4}$, which suggests that PBHs with masses less
than $10^{-4}M_{\odot}$ can catalyze the PT. Here we have assumed that the
tunneling process catalyzed by PBHs occurs when the energy difference of the
true and false vacua satisfies $\Delta V(T_{b})\ll V_{fv}$, where $T_{b}$ is
the formation temperature of bubbles catalyzed by PBHs. It is equivalent to
assume that $T_{b}\approx T_{c}$. This assumption is non-trivial because
usually we should compare the Euclidean bounce action between catalyzed PT and
thermal PT which is determined by a specific case [30]. However, we can
generalize this assumption in a slow PT, where the catalyzing effect is more
significant. It is exactly the case on which we mainly focus. Further, the
discount of the reduction of the Bekenstein entropy infers that our estimate
is conservative.

In a general first-order PT, the bubble nucleation rate of the true vacuum can
be generally expressed in the exponential form [39, 40, 36]

$\Gamma(t)=\Gamma_{0}e^{\beta t},$ (1)

where $\Gamma_{0}$ and $\beta$ are approximately constant. The parameter
$\beta$ represents the increasing rate of vacuum bubbles. In the case of
$\beta\gg 1$ , $\beta^{-1}$ is approximately the PT duration. The spatial
fraction of the false vacuum in the thermal PT is [41]

$F(t)=\exp{\left[-\frac{4\pi}{3}\int_{t_{i}}^{t}dt^{\prime}\Gamma(t^{\prime})a^{3}(t^{\prime})r^{3}(t^{\prime},t)\right]},$
(2)

where
$r(t^{\prime},t)\equiv\int_{t^{\prime}}^{t}\frac{1}{a(t^{\prime\prime})}dt^{\prime\prime}$
denotes the comoving distance of true vacuum bubbles nucleated at
$t^{\prime}$, and $t_{i}$ is the nucleation time of the first bubble.

However, in the case of the catalyzed PT, the formula (2) for $F(t)$ is no
longer valid. The true vacuum bubbles first nucleate around the PBH at $T_{b}$
which is slightly lower than the critical temperature $T_{c}$. When the
temperature decreases to the PT temperature $T_{n}$ determined by
$\Gamma(T_{n})\sim H^{4}(T_{n})$, the thermal bubbles start to nucleate. At
this moment the false vacuum fraction is no longer 1 due to the expansion of
the catalyzed bubbles. The expression of $F(t)$ is modified to

$\frac{dF(t)}{dt}=-F(t)\int_{t_{i}}^{t}\Gamma(t^{\prime})a^{3}(t^{\prime})dt^{\prime}\frac{d\bar{V}_{tb}}{dt}-\frac{d\bar{V}_{cb}}{\bar{V}dt},$
(3)

where
$\bar{V}_{tb}=\frac{4\pi}{3}(\int_{t_{n}}^{t}\frac{v_{w}(\tilde{t})}{a(\tilde{t})}d\tilde{t})^{3}$
and
$\bar{V}_{cb}=\frac{4\pi}{3}(\int_{t_{c}}^{t}\frac{v_{w}(\tilde{t})}{a(\tilde{t})}d\tilde{t})^{3}$
are the comoving volumes of bubbles for the thermal PT and catalyzed PT,
respectively, $\bar{V}=\frac{4\pi}{3}(\frac{1}{\mathcal{H}(t)})^{3}$ is the
comoving Hubble volume, and $\mathcal{H}$ is the comoving Hubble parameter.
Here we have used the approximation $T_{b}\approx T_{c}>T_{n}$, which is
easily accomplished in a slow PT. Before $t_{c}$, the field stays in the
global minimum. After the degenerate vacua appear, catalyzed bubbles form and
expand due to the increasing pressure between the vacua, leading to the
decrease of $F(t)$. When the temperature decrease to $T_{n}$, thermal true
vacuum bubbles start to nucleate and the decreasing of $F(t)$ is accelarated.
As $F(t)$ decreases, the vacuum energy is transferred to bubble walls and
plasma. The Friedmann equation and equations of motion read [42, 43]

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{\rho_{r}+\rho_{w}+\rho_{v}}{3},$ (4)
$\displaystyle\rho_{v}$ $\displaystyle=$ $\displaystyle F(t)\Delta V,$ (5)
$\displaystyle\frac{d(\rho_{r}+\rho_{w})}{dt}$ $\displaystyle+$ $\displaystyle
4H(\rho_{r}+\rho_{w})=-\frac{d\rho_{v}}{dt},$ (6)

where $H$ is the Hubble parameter, $\rho_{r}$, $\rho_{w}$ and $\rho_{v}$ are
the energy densities of background radiation, bubble walls and the false
vacuum, respectively. $\Delta V$ is the energy density difference of the true
and false vacua at the tunneling time. Here we have assumed that the velocity
of bubble walls is closed to 1, which means that bubble walls are regarded as
ultra-relativistic matter that share the same equation of motion parameter of
background plasma. The left-hand side of (6) refers to the evolution of ultra-
relativistic matter in an expanding Universe and the right-hand side
represents the decay of the false vacuum.

Figure 1: Energy density ratio $\gamma$ as a function of the PT parameters
$\alpha$ and $\beta/H_{*}$ in the case of $T_{b}\approx T_{c}>T_{n}$. The
green line corresponds to $\gamma=0.53$.

PTs take place earlier in the Hubble-size regions with a PBH than in the
Hubble-size regions without PBHs. Such an asynchronism leads to the different
evolution of the total energy density in the different Hubble-size regions.
This scenario is firstly seen as the ‘late-blooming’ mechanism [42, 44], which
leaves the delayed vacuum decay regions higher energy density after the PT. On
the contrary, the advanced vacuum decay results in the lower energy density,
which implies that the ratio of the energy density for the catalyzed PT to
that for the thermal PT, $\gamma=\rho_{cpt}/\rho_{tpt}$, is less than 1, as
shown in Fig. 1. The values of $\rho_{cpt}$ and $\rho_{tpt}$ are
straightforward calculated in a specific first-order PT model that is
described by $T_{*}$, $\alpha$ and $\beta/H_{*}$. Given $\alpha$ and
$\beta/H_{*}$, we can get the value of $\gamma$. The information of the energy
scale of the PT is eliminated due to the ratio of these two parameters.
Subsequently, we shall derive the statistical properties of curvature
perturbations from the distribution of PBHs.

_Curvature perturbations from the catalyzed PT._ In our scenario of the PT
catalyzed by PBHs, we assume that seed PBHs are monochromatic and form earlier
than the PT. The comoving number density of seed PBHs is
$n_{\mathrm{PBH},c}(M)$. The average number of PBHs with masses $M$ in a
Hubble-size region at time $t$ is

$p(M,t)=n_{\mathrm{PBH},c}(M)\frac{4\pi}{3}\left(\frac{1}{\mathcal{H}(t)}\right)^{3}.$
(7)

For monochromatic PBHs with masses $M$, $p(M,t)$ increase as the Universe
expands (we assume that PBHs form in the radiation-dominant era). We are
interested in the case of $p(M,t)<1$ during the PT. In this case, PBHs
randomly distribute in space and satisfy the Poisson distribution in each
comoving Hubble volume. The probability distributions of density perturbations
at the end of the PT inherit the distributions of PBHs. Now we investigate the
statistical properties of curvature perturbations from the distribution of
PBHs.

Consider a comoving volume of $(2L)^{3}$, where $L=n/\mathcal{H}(t_{e})$,
$n\gg 1$ and $t_{e}$ is the end time of the PT. To study the statistical
properties of PBHs, $L$ should be much larger than the comoving mean
separation, $S=n_{\mathrm{PBH},c}^{1/3}(M)$, so that enough PBHs are contained
in the volume. Let $p$ denote the probability that a PBH presents in a Hubble
volume and $X_{i}$ denote the number of PBHs contained in the $i$th Hubble
volume, where $i\leq n^{3}$ and $X_{i}=0$ or 1 by definition. The expectation
value and the varience of the random variable $X_{i}$ are respectively
$E(X_{i})=p$ and $D(X_{i})=p(1-p)$. Since the PBH number in the volume
$V_{L}=(2L)^{3}$ is much larger than one, according to the central limit
theorem, the total PBH number $X=\Sigma_{1}^{n^{3}}X_{i}$ in the comoving
volume $(2L)^{3}$ is subject to the Gaussian distribution with the expectation
value $E(X)=n^{3}E(X_{i})$ and the variance $D(X)=n^{3}D(X_{i})$. Since
$\rho_{cpt}$ and $\rho_{tpt}$ are the energy densities of the causality
disconnect Hubble volumes that contain or do not contain a PHB after the PT,
the energy density of a comoving volume $V_{L}$ is
$\rho_{X}=[X\rho_{cpt}+(n^{3}-X)\rho_{tpt}]/n^{3}=\rho_{tpt}+\frac{X}{n^{3}}(\rho_{cpt}-\rho_{tpt})$.
$\rho_{X}$ is a stochastic variable that satisfies the Gaussian distribution.
Its expectation value and variance are
$E(\rho_{X})=\rho_{tpt}+p(\rho_{cpt}-\rho_{tpt})$ and
$D(\rho_{X})=(\rho_{cpt}-\rho_{tpt})^{2}p(1-p)/n^{3}$. Then the density
perturbation as a stochastic variable is written as

$\delta_{X}\equiv\frac{\rho_{X}-E(\rho_{X})}{E(\rho_{X})}=\frac{(\frac{X}{n^{3}}-p)(\gamma-1)}{1+p(\gamma-1)},$
(8)

where $\gamma<1$ by definition. We easily see that the density perturbation is
still a Gaussian stochastic variable and it is valid in the range of
$\delta_{X}\in\left[\frac{(p-1)(1-\gamma)}{1+p(\gamma-1)},\frac{p(1-\gamma)}{1+p(\gamma-1)}\right]$,
as shown in Fig. 2. With the increasing of $n$, due to the central limit
theorem, the probability distribution of $\delta_{X}$ is getting closer and
closer to the Gaussian distribution in that certain range. It is known that
when density perturbations exceed a threshold value $\delta_{c}\sim 0.4$ [45,
46, 47, 38], the over-dense region collapse into a black hole. The number
density of black holes that collapse from Gaussian density perturbations are
determined by the Gaussian tail of the probability distribution function.
However, we emphasize that in our case, the amplitude of density perturbations
induced by catalyzed PT has an upper limit, which is determined by $p$ and
$\gamma$. If this upper limit is less than the threshold $\delta_{c}$, no
black holes are produced.

Figure 2: Probability distribution function of the stochastic variable
$\delta_{X}$. It is valid within the range by definition. The black dashed
line is the threshold of the primordial black hole formation $\delta_{c}\sim
0.4$.

Following the Gaussian distribution of density perturbations, we get the
variance of the density contrast

$\sigma_{\delta}^{2}(L)=\frac{p(1-p)}{(1+\frac{1}{1-\gamma}-p)^{2}\mathcal{H}^{3}(t_{e})L^{3}}.$
(9)

The variance is related to the power spectrum of curvature perturbations by
[48]

$\sigma^{2}_{\delta}(L)=\frac{16}{81}\int
d\ln{k}\,\mathcal{P}_{\mathcal{R}}(k)\exp{(-k^{2}L^{2})}.$ (10)

Here we have used the super-horizon relation
$\mathcal{P}_{\delta}(k)=\frac{16}{81}\mathcal{P}_{\mathcal{R}}(k)$ because
the length scale of induced perturbations is larger than the Hubble radius. It
is proved that the power spectrum of density perturbations from the Poisson-
type source scales as $k^{3}$ [49]. We parameterize
$\mathcal{P}_{\mathcal{R}}(k)$ in the form
$\mathcal{P}_{\mathcal{R}}(k)=A_{b}(k/k_{cut})^{3}$ where $k_{cut}$ is a cut-
off scale arising from the requirement of the central limit theorem $L>S$.
Since at smaller scales, the distribution of PBHs becomes non-Gaussian and
$\mathcal{P}_{\mathcal{R}}(k)$ decreases rapidly, we simply adopt the
approximation $k_{cut}=S^{-1}$ and $\mathcal{P}_{\mathcal{R}}(k)=0$ for
$k>k_{cut}$. Then (10) is rewritten in the form

$\displaystyle\sigma_{\delta}^{2}(L)$ $\displaystyle=$
$\displaystyle\frac{16A_{b}}{81(k_{cut}L)^{3}}\int_{0}^{k_{cut}}d\left(kL\right)\exp{\left(-k^{2}L^{2}\right)}\left(kL\right)^{2}$
(11) $\displaystyle=$
$\displaystyle\frac{4\sqrt{\pi}A_{b}}{81(k_{cut}L)^{3}}\,,$

where

$A_{b}=\frac{81}{4\sqrt{\pi}}\frac{p^{2}(1-p)}{(1+\frac{1}{1-\gamma}-p)^{2}}.$
(12)

The power spectrum of curvature perturbations induced by the catalyzed PT is
given by

$\mathcal{P}_{\mathcal{R}}=\left\\{\begin{aligned}
&\frac{81}{4\sqrt{\pi}}\frac{p^{2}(1-p)}{(1+\frac{1}{1-\gamma}-p)^{2}}\left(\frac{k}{k_{cut}}\right)^{3}\quad&\mathrm{for}\;k\leq
k_{cut}\,,\\\ &0\quad&\mathrm{for}\;k>k_{cut}\,,\end{aligned}\right.$ (13)

where $k_{cut}=n_{\mathrm{PBH}}^{\frac{1}{3}}(M)$ and $p$ is evaluated at the
end of the PT, which is related to the comoving number density of seed PBHs
and the energy scale of the first-order PT. The effects of $p$ and $\gamma$ on
$\mathcal{P}_{\mathcal{R}}(k)$ are quite clear. If $\gamma$ approaches 1,
which means that the catalyzing effect is negligible, then
$\mathcal{P}_{\mathcal{R}}(k)$ is suppressed. When $p$ approaches $0$ or $1$,
the energy density differences are diluted at the scale $L$, so that the power
spectrum of curvature perturbations is suppressed. However, the latter may
generate a new phenomenon, where PBHs catalyze a super-cooling PT ($\gamma\ll
1$). When $p\rightarrow 1$ and $\gamma\ll 1$, since density perturbations can
exceed the threshold of the PBH formation, the formation mechanism of PBHs is
controlled by the Gaussian tail of density perturbations. This implies that
seed PBHs lead to the generation of PBHs [27]. Leaving these extreme case
behind, we will see that there is a large parameter space in which a catalyzed
PT may generate large curvature perturbations and even generate a detectable
SGWB. The energy spectrum of the SGWB is roughly estimated by
$\Omega_{GW}(k)\sim\mathcal{P}_{\mathcal{R}}^{2}(k)$. In this case the peak
frequency of GWs is determined by the comoving number density of seed PBHs,
and the amplitude is related to $A_{b}$ in (12), which is both related to the
abundance of seed PBHs and a specific PT model that characterized by $T_{*}$,
$\alpha$ and $\beta/H_{*}$.

_Specific example._ As a primary case of study, we focus on the first-order
EWPT model. To provide an example, we need to first come up with a question:
which kind of PBHs play a role as catalysts and generate curvature
perturbations during the EWPT? To answer this question, we need two key
quantities: the PBH masses and the PBH abundance.

PBHs form in the early Universe. It is known that PBHs form due to the
collapse of the overdense region during the radiation-dominated epoch and are
the candidate of dark matter [50, 51, 38, 47]. Taking the assumption that PBHs
are monochromatic, it is reported that PBHs with masses around
$10^{-15}M_{\odot}\sim 10^{-10}M_{\odot}$ can constitute all dark matter.
Because such a mass range of PBHs lies within the critical mass, they work as
catalysts during the EWPT. Their number density can be estimated by

$n_{\mathrm{PBH}}(M,t_{0})\approx 3.285\times
10^{10}\left(\frac{M_{\odot}}{M}\right)\rm{Mpc^{-1}},$ (14)

where $t_{0}$ is the present time. The average number of PBHs with masses $M$
in a Hubble-size region, $p(M,t)$, is derived in (7), which is increasing over
time due to the expansion of the Hubble horizon. As mentioned above, we are
interested in the case of $p(M,T_{e})<1$, where $T_{e}$ is the energy scale at
the end of the EWPT. We take PBHs with $M=10^{-13}M_{\odot}$ as an example,
the average number in a Hubble-size region is less than unity until the
temperature decreases to $68\,$GeV. We show PBHs that can be catalysts during
the EWPT in Fig. 3, whose masses lie in the mass range of our interest. If
these PBHs form before the EWPT, the catalyzed EWPT is expected to generate
large curvature perturbations at super-horizon scales.

Figure 3: Average number of PBHs in a Hubble volume under the assumption that
PBHs constitute all dark matter. The vertical dashed line corresponds to
$T=100\,$GeV.

It is an exciting coincidence that if the mass of PBHs is $M\sim
10^{-13}M_{\odot}$ which constitute all dark matter, the SGWB from the EWPT
catalyzed by PBHs can explain the observed signal from the NANOGrav and
European pulsar timing array. By setting the energy scale at the end of the
EWPT, we calculate the average number of PBHs as catalysts in the Hubble
volume by (7). Fig. 4 shows the energy spectrum of GWs (blue line) predicted
by the catalyzed EWPT with $T_{e}=100\,\rm{GeV}$, $M=10^{-13}M_{\odot}$ and
$\gamma=0.53$. The green line in Fig. 1 corresponds to $\gamma=0.53$ in the
parameter space of $\alpha$ and $\beta/H_{*}$. Further more, we find that the
upper limit of density perturbations are $\delta_{up}\sim 0.176$, which does
not exceed the threshold of the PBH formation. Therefore, curvature
perturbations induce the SGWB to explain the pulsar timing array data [16, 17,
15, 52] and naturally avoid the overproduction of PBHs.

If the EWPT ends at a lower energy scale, the average number of PBHs in the
Hubble volume will increase. If $p<1$ is still valid, which corresponds to
$T_{e}>68\,\rm{GeV}$, a large value of $\gamma$ is required to explain the
observation result. It implies a relatively weaker EWPT compared to the EWPT
model we consider in this Letter.

Figure 4: Energy spectrum of GWs (blue line) predicted by the catalyzed EWPT
with the parameters $T_{e}=100\,\rm{GeV}$, $M=10^{-13}M_{\odot}$ and
$\gamma=0.53$. The green and orange violins correspond to the SGWB
observations from the NANOGrav and European pulsar timing array
collaborations.

_Conclusions and discussions._ We have investigated the first-order PT
catalyzed by PBHs. Due to the enhancement of the vacuum decay rate around a
PBH, the PT occurs earlier in the regions containing PBHs than in the regions
without PBHs. Such an asynchronism leads to super-horizon curvature
perturbations which induce the SGWB when re-entering the horizon in the
radiation-dominated era. If all dark matter is composed of PBHs with masses
about $10^{-13}M_{\odot}$, we find that the first-order EWPT catalyzed by PBHs
provides an explanation for the origin of the SGWB signal recently reported by
pulsar timing array collaborations. Usually PBHs are overproduced if the
observed SGWB is induced by large curvature perturbations that follow the
Gaussian distribution. In our scenario, the upper limit of the amplitude of
curvature perturbations is less than the threshold of the PBH formation. Thus
our scenario is free from the PBH overproduction problem.

PBHs in the mass range of $10^{-15}M_{\odot}\sim 10^{-10}M_{\odot}$ can
account for all dark matter. In our analysis we only consider PBHs with masses
of $10^{-13}M_{\odot}$ as catalysts. The increase of the PBH mass leads to a
lower peak frequency of the energy spectrum of the SGWB because the peak
frequency of GWs is determined by the PBH number density that is directly
related to the PBH mass. If PBHs constitute a part of dark matter, the
fraction of dark matter in PBHs affects the peak frequency of GWs. As the
fraction decreases, the peak frequency becomes low for a fixed PBH mass.

Our conclusion is independent of the details of the first-order PT. In
principle, a SGWB might be generated by bubble collisions, sound waves, and
fluid turbulence during the PT. The GW wavelengths of such a SGWB are shorter
than the horizon at the PT time. After the PT the SGWB is induced by curvature
perturbations from the catalyzed PT. Therefore, the peak frequency of the SGWB
from the PT is higher than that of the SGWB from the catalyzed PT.

In our work we consider PBHs as catalysts. In principle, our formalism can be
applied to other cases of impurity-catalyzed first-order PTs as long as
impurities are so sparse that their average number within a horizon volume is
less than unity.

_Acknowledgements_. This work is supported in part by the National Key
Research and Development Program of China Grant No. 2020YFC2201501, in part by
the National Natural Science Foundation of China under Grant No. 12075297 and
No. 12235019.

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# Non-Roelcke precompactness of groups of surface homeomorphisms

J. de la Nuez González<EMAIL_ADDRESS>Korean Institute of Advanced
Study (KIAS)

###### Abstract.

We prove that no subgroup of the group of boundary-fixing homeomorphisms of a
compact surface whose action on the interior of the surface is sufficiently
transitive can be Roelcke precompact with the topology inherited from the
compact-open topology.

Work supported by Samsung Science and Technology Foundation under Project
Number SSTF-BA1301-51.

##  1\. Introduction

Roelcke precompactness is the topological group counterpart of the model
theoretic notion of $\omega$-categoricity in classical and continuous logic.
As such, it has received a lot of attention in the literature from a variety
of angles.

######  Definition 1.1.

We say that a topological group $G$ is Roelcke precompact if for every
neighbourhood of the identity $U$ there exists a finite subset
$g_{1},g_{2},\dots g_{k}$ such that $G=\bigcup_{j=1}^{k}Ug_{j}U$.

It is a result of Culler and Rosendal [2] that the homeomorphism group of a
manifold of dimension $\geq 2$ (and more generally, any group of
homeomorphisms containing a specific class of elements) is not Roelcke
precompact. In this note we elaborate on the general idea of their proof to
show that the same property holds for a large class of groups of
homeomorphisms of compact surfaces.

Given a surface with (possibly empty) boundary $S$, from now on simply ´a
surface’, and some tuple $p_{1},p_{2},\dots p_{r}$ of points in the interior
of $S$, $int(S)$, we denote by $S_{p_{1},p_{2},\dots p_{r}}$ the result of
removing $p_{1},\dots p_{r}$ (the punctures) from $S$. We write simply
$S_{r}$, $r>0$ when only interested in the homeomorphism type.

The pure mapping class group of a punctured surface $S_{p_{1},\dots p_{r}}$,
or $PMod(S_{p_{1},\dots p_{r}})$ is the quotient of the pointwise stabilizer
of the set of punctures in $Homeo_{\partial S}(S)$ (whose elements we can
regard alternatively as homeomorphisms of $S_{p_{1},p_{2}\dots p_{m}}$ fixing
the boundary) by identifying any two elements which are homotopic via a
homotopy fixing the punctures. Given $g\in S_{p_{1},\dots p_{m}}$ we will
denote the corresponding mapping class by $[g]$.

Accordingly, there is a natural quotient map $PMod(S_{p_{1},\dots p_{r}})\to
PMod(S_{p_{1},\dots p_{r-1}})$. The kernel is given by the image in
$PMod(S_{p_{1},p_{2},\dots p_{r}})$ of the point pushing map
$\mathcal{P}:\pi(S_{p_{1},p_{2},\dots p_{r-1}},p_{r})\to
PMod(S_{p_{1},p_{2},\dots p_{r}})$, which plays a certain role in this paper
and about which we say a few more words at the beginning of section 1.

For simplicity we assume $S$ is compact, so that the compact-open topology on
$Homeo_{\partial S}(S)$ coincides with the uniform convergence topology. We
will work with some fixed hyperbolic or flat Riemannian metric $d$ on $S$
compatible with its topology with respect to which the boundary is geodesic. A
system of symmetric neighbourhoods of the identity is thus given by
$\\{U_{\epsilon}\\}_{\epsilon>0}$ where $U_{\epsilon}=V_{\epsilon}\cap
V_{\epsilon}^{-1}$ and $V_{\epsilon}=\\{g\in G\,|\,\forall x\in
S\,d(x,gx)<\epsilon\\}$.

######  Definition 1.2.

Let $\zeta(S)$ the minimum positive integer $m$ such that the point pushing
map $\mathcal{P}:\pi_{1}(S_{m-1})\to PMod(S_{m})$ is an embedding (a fortiori,
of a torsion-free group), that is $m$ is equal to:

  * •

$4$ if $S$ is a sphere

  * •

$3$ if $S$ is a projective plane

  * •

$2$ if $S$ is a torus or a disk

  * •

$1$ otherwise

######  Theorem 1.3.

Let $S$ be a compact surface possibly with boundary and $m=\zeta(S)$. If
$H\leq Homeo_{\partial S}(S)$ is $m$-transitive, then $H$ with the topology
inherited from the compact-open topology is not Roelcke precompact.

For the remainder of the paper we will use the notation $G=Homeo_{\partial
S}(S)$. Given $H\leq G$ we write $H_{p_{1},p_{2},\dots p_{r}}$ to denote the
pointwise stabilizer of $\\{p_{1},\dots p_{r}\\}$.

In view of one of the open questions at the end it might be worthwhile to be
somewhat more precise.

######  Definition 1.4.

Let $H\leq Homeo_{\partial S}(S)$ and $\hat{p}$ a tuple of points in the
interior of $S$ (possibly empty). Denote by $\mathcal{O}_{\hat{p}}(H)$ be the
collection of orbits of points of $int(S_{\hat{p}})$ by $H_{\hat{p}}$.

We say that $H$ is _sufficiently transitive_ if there exists a tuple of
$\zeta(S)-1$ points $\hat{p}$ in $S$ and a non null-homotopic simple closed
curve $\alpha$ in $S_{\hat{p}}$ such that $im(\alpha)$ is contained in some
$O\in\mathcal{O}_{\hat{p}}(H)$.

Notice that if $H$ is $\zeta(S)$-transitive then it is sufficiently
transitive, so the following clearly implies 1.3.

######  Theorem 1.5.

Let $S$ be a compact surface. If $H\leq G$ is sufficiently transitive, then
$H$ with the topology inherited from the compact-open topology is not Roelcke
precompact.

### Point pushing maps

Fix some tuple $\hat{p}$ of points in $int(S)$. For any path $\alpha:J\to
S_{\hat{p}}$ parametrized by an interval $J$ and any open $U\subseteq
S_{\hat{p}}$ containing $im(\alpha)$ we choose a map
$\mathcal{P}^{U}_{\alpha}\in Homeo_{\partial S}(S)$ supported in $U$ pushing
$\alpha(0)$ to $\alpha(1)$ along $\alpha$. Very concretely: if
$\alpha(0)\neq\alpha(1)$ and $\alpha$ is simple this just means a map
supported in some open disk $U^{\prime}\subseteq U$ which maps $\alpha(0)$ to
$\alpha(1)$ and is supported in $U^{\prime}$. If
$\alpha=\alpha_{1}*\alpha_{2}\dots*\alpha_{q}$, where the $\alpha_{i}$ are
simple paths, then we can take as
$\mathcal{P}_{\alpha}^{U}=\mathcal{P}^{U}_{\alpha_{q}}\circ\mathcal{P}^{U}_{\alpha_{q-1}}\dots\circ\mathcal{P}^{U}_{\alpha_{1}}$.
When $N_{\epsilon}(im(\alpha))\subseteq S_{\hat{p}}$ we use the expression
$\mathcal{P}^{\epsilon}_{\alpha}$ to denote
$\mathcal{P}^{N_{\epsilon}(im(\alpha))}_{\alpha}$, where
$N_{\epsilon}(X)=\\{x\in S\,|\,d(x,X)<\epsilon\\}$.

We recall the following fundamental fact:

######  Fact 1.6 (Alexander lemma).

The mapping class group of a once-punctured disk is trivial.

From it one can deduce the following (see Chapter $4$ of [1]):

######  Fact 1.7.

Let $\alpha,\beta$ be paths in $S_{\hat{p}}$ from a point $q$ to a point
$q^{\prime}$ and $U,V\subseteq S_{\hat{p}}$ open sets containing $im(\alpha)$
and $im(\beta)$ respectively.

If $\alpha$ and $\beta$ are homotopic in $S_{\hat{p}}$ then
$(\mathcal{P}_{\beta}^{V})^{-1}\mathcal{P}_{\alpha}^{U}$ and
$\mathcal{P}_{\beta}^{V}(\mathcal{P}_{\alpha}^{U})^{-1}$ represent the
identity in $PMod(S_{\hat{p},q})$ and $PMod(S_{\hat{p},q^{\prime}})$
respectively regardless of the choices made. If $\alpha,\beta$ are paths in
$S_{\hat{p}}$ with $\alpha(1)=\beta(0)$ and $U,V$ neighbourhoods of
$im(\alpha)$ and $im(\beta)$ respectively, then
$\mathcal{P}^{V}_{\beta}\circ\mathcal{P}^{U}_{\alpha}=\mathcal{P}^{U\cup
V}_{\alpha*\beta}\circ h$ for some $h\in G_{\hat{p},\alpha(0)}$ trivial in
$PMod(S_{\hat{p},\alpha(0)})$.

The induced map $\mathcal{P}:\pi_{1}(S_{\hat{p}},q)\to PMod(S_{\hat{p},q})$ is
a group homomorphism and an a non-trivial group embedding of a torsion free
group under the conditions described in definition 1.2.

We will occasionally omit the superscript from $\mathcal{P}_{\alpha}$ in
contexts where in light of the above the choice of neighbourhood and
representative do not affect the resulting expression.

Given a curve $\alpha$, for any $0\leq s\leq t\leq 1$ let $\alpha_{s,t}$ be
the restriction of $\alpha$ to $[s,t]$ reparametrized to the domain is
$[0,1]$. If $s>t$ we let $\alpha_{s,t}=\alpha_{t,s}^{-1}$. We extend the
parametrization of a curve to $\mathbb{R}$ cyclically when needed.

We write $H_{p_{1},p_{2},\dots p_{r},p\to q}$ to denote the collection of
elements in $H_{p_{1},\dots p_{r}}$ sending $p$ to $q$. Since for any $g\in
Homeo_{\partial S}(S)_{\hat{p},p\to q}$ the inner automorphism $f\mapsto
f^{g}$ induces an isomorphism between $PMod(S_{\hat{p},q})$ and
$PMod(S_{\hat{p},p})$ for which we will use the same expression $(-)^{g}$.

######  Lemma 1.8.

Let $\hat{p}=(p_{1},\dots p_{m-1})$ be an $(m-1)$-tuple of points in $int(S)$
and $\alpha$ an essential simple closed curve in $S_{\hat{p}}$ with
$im(\alpha)\subseteq O\in\mathcal{O}_{\hat{p}}(H)$. Let $p_{m}=\alpha(0)$.
Assume that to any point $q\in im(\alpha)$ we have assigned some subgroup
$F_{q}\leq PMod(S_{\hat{p},q})$ such that the following holds:

  1. (a)

For any $q,q^{\prime}\in Im(\alpha)$ and any $h\in H_{\hat{p},q\to
q^{\prime}}$ we have $F_{q^{\prime}}^{h}=F_{q}$.

  2. (b)

For any $t\in[0,1]$ there exists some $\delta_{t}>0$ such that for all
$s\in(t-\delta_{t},t+\delta_{t})$ and sufficiently small $\epsilon$ the set
$(\mathcal{P}_{\alpha_{s,t}}^{\epsilon})^{-1}H_{\hat{p},\alpha(t)\to\alpha(s)}$
maps to a subset of the set $F_{\alpha(t)}$

Then there exists some $f\in F_{p_{m}}$ and some $h\in H_{p_{1},\dots p_{m}}$
such that $[h]=\mathcal{P}([\alpha])f\in PMod(S_{p_{1},\dots p_{m}})$ .

###### Proof.

By compactness there is a finite tuple $0\leq t_{0}<t_{1}<\dots<t_{r}\leq 1$
such that
$[0,1]\subseteq\bigcup_{i=0}^{r}(t_{i}-\delta_{i},t_{i}+\delta_{i})$. We may
assume that this set is minimal for the property so that one can choose
$q_{i}\in(t_{i-1},t_{i})$ for $1\leq i\leq r$ so that
$q_{i}\in(t_{i-1},t_{i-1}+\delta_{i})\cap(t_{i}-\delta_{i},t_{i})$. Let also
$q_{0}=0$ and $q_{r+1}=1$.

Pick $h_{0}\in H_{\hat{p},\alpha(t_{0})\to\alpha(0)}$, for each $0\leq i\leq
r$ some $g_{i}\in H_{\hat{p},\alpha(t_{i})\to\alpha(q_{i})}$ and for $1\leq
i\leq r+1$ some $h_{i}\in H_{\hat{p},\alpha(t_{i-1})\to\alpha(q_{i})}$.
Consider the group element $h=h_{r+1}g^{-1}_{r}\dots
g_{1}^{-1}h_{1}g_{0}^{-1}$. One can check that $h\in H_{p_{1},p_{2},\dots
p_{m}}$.

By assumption we can write
$g_{i}=\mathcal{P}^{\epsilon}_{\alpha_{t_{i},q_{i}}}c_{i}$ and
$h_{i}=\mathcal{P}^{\epsilon}_{\alpha_{t_{i},q_{i+1}}}d_{i}$, where
$c_{i},d_{i}\in G_{\hat{p},\alpha(t_{i})}$ and $[c_{i}],[d_{i}]\in
F_{\alpha(t_{i})}$ and any fixed $\epsilon<d(im(\alpha),\hat{p})$.
Alternatively, by property (a) we can write
$g_{i}^{-1}=(\mathcal{P}^{\epsilon}_{\alpha_{t_{i},q_{i}}})^{-1}e_{i}$, where
$e_{i}\in F_{q}$.

Now we can write

$\displaystyle[h]=[h_{r+1}g^{-1}_{r}\dots
g_{1}^{-1}\mathcal{P}_{\alpha_{t_{0},q_{1}}}d_{1}\underbracket{(\mathcal{P}_{\alpha_{t_{0},0}})^{-1}e_{0}}_{=g_{0}^{-1}\in
H_{\hat{p},\alpha(0)\to\alpha(t_{0})}}]=$
$\displaystyle=[h_{r+1}g^{-1}_{r}\dots(\mathcal{P}_{\alpha_{t_{1},q_{1}}})^{-1}e_{1}\mathcal{P}_{\alpha_{t_{0},q_{1}}}(\mathcal{P}_{\alpha_{t_{0},0}})^{-1}e_{0}]\underbracket{[d_{1}^{g_{0}^{-1}}]}_{\in
F_{\alpha(0)}}$
$\displaystyle=[h_{r+1}g^{-1}_{r}\dots(\mathcal{P}_{\alpha_{t_{1},q_{1}}})^{-1}\mathcal{P}_{\alpha_{t_{0},q_{1}}}(\mathcal{P}_{\alpha_{t_{0},0}})^{-1}[e_{0}][d_{1}^{g_{0}^{-1}}][e_{1}^{h_{1}g_{0}^{-1}}]$
$\displaystyle\dots=[\mathcal{P}_{\alpha_{t_{n},1}}(\mathcal{P}_{\alpha_{t_{n},q_{n}}})^{-1}\dots\mathcal{P}_{\alpha_{t_{0},q_{1}}}(\mathcal{P}_{\alpha_{t_{0},0}})^{-1}][h^{\prime}]$

where $[h^{\prime}]$ is a product of elements in $F_{\alpha(0)}=F_{p_{m}}$ and
thus itself in $F_{p_{m}}$. Notice that it is the same kind of manipulation
that allows us to ignore the choice of particular point pushing maps in the
expression above. In view of 1.7 we have
$[h]\in\mathcal{P}([\alpha])F_{p_{m}}$. ∎

######  Lemma 1.9.

Let $\hat{p}=(p_{1},\dots p_{m-1})$, $\bar{p}=(\hat{p},p_{m})$, $H$ and
$\alpha$ be as in Lemma 1.8 and assume that $m=\eta(S)$. Then given any
$\delta>0$ there is a sequence $((f_{n},b_{n},w_{n}))_{n\in\mathbb{N}}$ of
triples of elements of $G$ such that:

  1. (1)

$f_{n}\in H_{\hat{p}}$ and $w_{n}\in U_{\delta}\cap H_{\hat{p}}$

  2. (2)

$b_{n}=w_{n}^{-1}f_{n}\in G_{\bar{p}}$

  3. (3)

$\\{[b_{n}]\\}_{\in\mathbb{N}}\subseteq PMod(S_{\bar{p}})$ is an infinite set

###### Proof.

Notice that any element supported in $B(p_{m},\frac{1}{n})$ belongs to
$U_{\frac{2}{n}}$.

For any $q\in im(\alpha)$ and any $n>0$ let

$A_{q,n}=\\{[wh]\,|\,h\in H_{\hat{p}},w\in U_{\frac{1}{n}}\cap
G_{\hat{p}},wh(q)=q\\}\subseteq PMod(S_{\hat{p}})$

We distinguish two mutually exclusive cases:

  1. (1)

$A_{q,n}$ is infinite for all $n>0$

  2. (2)

$A_{q,n}$ is finite for some $n>0$

Continuity of inner automorphisms and transitivity of $H_{\hat{p}}$ on
$im(\alpha)$ implies that the same case holds for any $q\in im(\alpha)$. The
first clearly implies the condition of the statement is satisfied, so from now
on assume the second is always the case.

Define $F_{q}=\bigcap_{n>0}A_{q,n}$. This is the intersection of a descending
chain of non-empty sets which are eventually finite. Hence, it is a finite
non-empty subset of $PMod(S_{\hat{p},q})$. Continuity of inner automorphisms
implies that for any $q,q^{\prime}\in im(\alpha)$ and any $h\in
H_{\hat{p},q\to q^{\prime}}$ conjugation by $h$ induces an isomorphisms
between $F_{q^{\prime}}$ and $F_{q}$.

######  Lemma 1.10.

For any $q\in im(\alpha)$ the set $F_{q}$ is closed under multiplication. It
is therefore a finite subgroup of $PMod(S_{\hat{p},q})$.

###### Proof.

Consider any $\eta_{1},\eta_{2}\in F_{q}$ and $n>0$. Then pick some $h_{1}\in
H_{\hat{p}}$ that can be written as $h_{1}=u_{1}a_{1}$, where $u_{1}\in
G_{\frac{1}{2n}}$ and $a_{1}\in G_{\hat{p},q}$ maps to $\eta_{1}$. By
continuity of the group operations there exists some $n^{\prime}>0$ such that
$(U_{\frac{1}{n^{\prime}}})^{a^{-1}}\subseteq U_{\frac{1}{2n}}$. Choose
$h_{2}\in H_{\hat{p}}$ of the form $u_{2}a_{2}$, where $u_{2}\in
G_{\frac{1}{n^{\prime}}}$ and $a_{2}\in G_{\hat{p},q}$ maps to $\eta_{2}$.
Then $h_{1}h_{2}=u_{1}u_{2}^{a_{1}^{-1}}a_{1}a_{2}\in
U_{\frac{1}{n}}a_{1}a_{2}$ and $\eta_{1}\eta_{2}\in A_{q,n}$. Since $n$ is
arbitrary, we are done. ∎

We are now in the position to apply Lemma 1.8 to the family $\\{F_{q}\\}_{q\in
im(\alpha)}$. This yields some element $h\in H_{p_{1},\dots p_{m}}$ (so that
$[h]\in F_{p_{m}}$) whose image in $PMod(S_{p_{1},\dots p_{m}})$ is of the
form $f\mathcal{P}(\alpha)$, where $f\in F_{p_{m}}$. Since $F_{p_{m}}$ is a
group we get that $\mathcal{P}(\alpha)\in F_{p_{m}}$, contradicting the
finiteness of $F_{p_{m}}$ and the fact that by the choice of $m$ the map
$\mathcal{P}:\pi_{1}(S_{\hat{p}},p_{m})\to PMod(S_{\hat{p},p_{m}})$ is an
embedding of a torsion-free group. ∎

##  2\. Discriminating between mapping classes

Given $\alpha,\alpha^{\prime}$ arcs between the same pair of boundary points
of $\partial S$ or simple closed curves in $S$ and given some subset
$F\subseteq S$ we write $\alpha\simeq_{F}\alpha^{\prime}$ to indicate that $F$
does not intersect $im(\alpha)\cup im(\alpha^{\prime})$ and $\alpha$ is
homotopic to $\alpha^{\prime}$ in $S\setminus F$ (rel $\partial I$ in the arc
case).

######  Definition 2.1.

Let $\delta_{e}$ be small enough that for any $x,y$ at distance less than
$\delta_{e}$ the point $y$ is in the domain of injectivity of the exponential
map at $x$, so that there is a unique geodesic segment from $x$ to $y$ of
minimal length in $B(x,\delta_{e})\cap B(y,\delta_{e})$, which we denote by
$[x,y]$.

As it is well known, given two maps $f,f^{\prime}$ from a space $X$ into $S$
such that $d(f(x),f^{\prime}(x))<\delta_{e}$ for any $x\in X$ a uniform
parametrizataion of the geodesic $[f(x),f^{\prime}(x)]$ yields a homotopy in
$S$ between the two maps $f$ and $f^{\prime}$.

An immediate consequence of this is the following observation:

######  Observation 2.2.

Let $\bar{p}=(p_{1},p_{2}\dots p_{m})$ a tuple of points in $int(S)$ and let
$u\in U_{\delta_{e}}\cap G_{\bar{p}}$. Then $u$ represents the identity in
$PMod(S_{\bar{p}})$.

Another one is the Lemma below, which for future reference we state in a
degree of generality not needed here. Its content is rather standard but we
include a proof for the sake of completeness.

######  Definition 2.3.

We say that a path connected subset $U\subseteq X$ is $\epsilon$-branching if
there exist $x_{1},x_{2},x_{3}\in U$ and paths $\gamma_{ij}$ between $x_{i}$
and $x_{j}$ for distinct $i,j$ such that $\\{i,j,k\\}=\\{1,2,3\\}$ the path
$\gamma_{ij}$ does not contain points at distance $<\epsilon$ from $x_{k}$.
Clearly, any path connected open set is $\epsilon$-branching for some
$\epsilon$.

######  Lemma 2.4.

Let $D_{1},D_{2},\dots D_{m}\subseteq S$ be $\epsilon$-branching disjoint
closed subdisks of $S$. Then there is some constant $\delta$ depending only on
the metric such that the following holds. If $\alpha,\beta:I\to S$ are two
loops in $S\setminus\bigcup^{m}_{i=1}D_{i}$ with common endpoints and
$d(\alpha(t),\beta(t))<\min\\{\delta_{e},\epsilon\\}$ for all $t\in I$, then
$\alpha_{\simeq_{\bigcup_{i=1}^{m}}D_{i}}\beta$.

###### Proof.

For $1\leq i\leq m$ let the non-linearity of $D_{i}$ be witnessed by a triple
$(x_{j}^{i})_{1\leq j\leq 3}$ of points in $D_{i}$ and paths
$(\gamma_{jk}^{i})_{1\leq j<k\leq 3}$ in $D_{i}$ between the points in the
triple.

Let $H:I^{2}\to S$ be the homotopy between $\alpha$ and $\beta$ where
$s\mapsto H(t,s)$ is a constant parametrization of the geodesic segment
$[\alpha(t),\beta(t)]$.

The following claim is easy to verify using rudiments of differential
topology.

######  Claim 2.5.

After performing a small perturbation of $\alpha$, $\beta$, $H$ and
$\gamma^{i}_{jk}$ we may assume that $\alpha,\beta$ and $H$ are
differentiable, that $H$ is transverse to $\partial D_{i}$, $\gamma^{i}_{jk}$
and the points $x^{i}_{j}$, while preserving the property that
$d(H(t,s_{1}),H(t,s_{2}))<\delta$ for any $t,s_{1},s_{2}\in I$. After
modifying the $\gamma^{i}_{jk}$ further and shrinking $D_{i}$ we may
additionally assume that $\partial D_{i}=\bigcup_{1\leq j<k\leq
3}\gamma^{i}_{jk}$. In particular $\gamma^{i}_{jk}$ meet only at the endpoints
and they do so smoothly.

######  Lemma 2.6.

For any connected component $C$ of $H^{-1}(D_{i})$ the map
$H_{\restriction}:\partial C\to\partial D_{i}$ is nullhomotopic.

###### Proof.

For convenience we will drop the superindex $i$ for the remainder of this
proof. For $1\leq j,k\leq 3,j\neq k$ let $Y_{j}=H^{-1}(\\{x_{j}\\})$ and
$Y_{jk}=H^{-1}(im(\gamma_{jk}))$. Let also $Y=\bigcup_{j=1}^{3}Y_{j}$. By
transversality, we have the following:

  * •

$Y_{j}$ is finite for $1\leq j\leq 3$

  * •

$Y_{jk}$ is a union of finitely many arcs between points in $Y_{j}\cup Y_{k}$
(potentially arcs between two points in $Y_{j}$ or in $Y_{k}$)

  * •

$H^{-1}(\partial C)=\bigcup_{j\neq k}Y_{jk}$ is the image of a simple closed
curve

  * •

$y\in Y_{j}$ is the endpoint of exactly one arc in $H^{-1}(\gamma_{kj})$ for
$k\neq j$ (say $\gamma_{jk}=\gamma_{kj}$ ).

This makes $\partial C$ into a simplicial complex with
$Y=\coprod_{j=1}^{3}Y_{j}$ and the components of $Y_{jk}$ as edges. Let $\pi$
be the projection of $I\times I$ onto the second coordinate.

Notice that if $\\{i,j,k\\}=\\{1,2,3\\}$ and $x,y,z\in U$, then we cannot have
$\pi(x)\in[\pi(y),\pi(z)]$ for $x\in Y_{i}$ and $[\pi(y),\pi(z)]\subseteq
Y_{jk}$, since this would imply the existence of $w\in im(\gamma_{jk})$ such
that $d(w,x_{i})<\delta$.

We show that $H_{\restriction}:\partial C\to\partial D$ is null-homotopic by
induction on the number of vertices in $Y$.

Number the vertices of $Y$ cyclically as $y_{1},y_{2},\dots y_{N}$. Let
$\tau_{i}=\pi(y_{i})$ and let $c_{i}\in\\{1,2,3\\}$ be such that $y_{i}\in
Y_{c_{i}}$ and $d_{i}\in\\{12,23,31\\}$ such that $[y_{i},y_{i+1}]\subseteq
Y_{d_{i}}$.

Let $i$ be such that $\tau_{i}$ is minimal. Without loss of generality we may
assume $i=2$ and $c_{2}=1$. The case $\\{c_{1},c_{3}\\}=\\{2,3\\}$ is excluded
by our condition on $f$.

Consider first the case in which $\\{c_{1},c_{3}\\}=\\{1,k\\}$, with $k\neq
1$. Without loss of generality we can assume that $k=2=c_{1}$. We cannot have
$\tau_{1}\leq\tau_{3}$ since then $\tau_{1}\in[\tau_{2},\tau_{3}]$. But
$c_{1}=2$ and $d_{1}=12$, which forces $d_{2}=31$, resulting in a collision.
So $\tau_{1}>\tau_{3}$. We may assume $c_{1}=1$. Necessarily $d_{1}=13$ which
in turn implies $d_{0}=12$.

At this point we can remove vertices $y_{1}$ and $y_{2}$ connect $y_{0}$ and
$y_{3}$ directly with an edge of type $12$ (so compatible with the other
incident edges at the endpoints). The non-collision condition continues to be
satisfied, since the image of the new edge is contained in the image of two
edges of the same type in the old configuration:
$[\tau_{0},\tau_{3}]\subseteq[\tau_{0},\tau_{1}]\cup[\tau_{2},\tau_{3}]$, as
$\tau_{1}\in(\tau_{2},\tau_{3})$.

We have replaced a path of type $2_{12}1_{13}1_{12}2$ (resp
$1_{12}1_{13}1_{12}2$) with one of type $2_{12}2$ (resp. $1_{12}2$), both of
which represent homotopic paths between $x_{2}$ and itself (resp. $x_{1}$ and
$x_{2}$) on the circle $\partial D$. Since the new configuration corresponds
to a null-homotopic path by the induction hypothesis, so does the original.

The only case left is that in which $\\{c_{1},c_{3}\\}=\\{1\\}$. We may assume
$d_{1}=13$ and thus $d_{2}=12$, as well as $\tau_{1}<\tau_{3}$ and we can
apply an argument identical to the one of the previous case.

∎

Given $H$ satisfying the conclusion of the sublemma, we can redefine $H$ on
each component $C$ as above by extending $H_{\partial C}$ to a map from $C$ to
$\partial D_{i}$. Postcomposing with a homeomorphism that slightly enlarges
each $D_{i}$ while fixing $\alpha$ and $\beta$ yields a homotopy witnessing
$\alpha\simeq_{\bigcup_{i=1}^{m}D_{i}}\beta$.

∎

In the proof of the theorem below we will use the following (see [1],
Proposition 2.8)

######  Fact 2.7.

The action of the homeomorphism group descends to a well-defined action of the
mapping class group on free homotopy classes of curves in the surface and
homotopy classes relative to the boundary of arcs between points in the
boundary. The kernel of the action of the mapping class group on the union of
the two families is trivial.

See 1.5

###### Proof.

Let $m=\zeta(S)$. Let $\hat{p}=(p_{1},\dots p_{m-1})$ and
$\bar{p}=(\hat{p},p_{m})$ be tuples of distinct points as in Lemma 1.9.

$\delta_{0}=\frac{1}{7}\min\\{\delta_{e},\min\\{d(p_{i},\partial
S),d(p_{i},p_{j})\,|\,1\leq i<j<m\\}\\}.$

where $\delta_{e}$ is as in definition 1.2

Roelcke precompactness of $H$ implies the existence of a finite collection
$h_{1},\dots h_{M}\in H$ such that
$H\subseteq\bigcup_{j=1}^{M}U_{\delta_{0}}h_{j}U_{\delta_{0}}$.

Let $f_{n}\in H_{p_{1},p_{2},\dots p_{m-1}}$, $b_{n}\in G_{\bar{p}}$,
$w_{n}\in U_{\delta_{1}}\cap G_{\hat{p}}$ for $n\in\mathbb{N}$ be as given by
Lemma 1.9 applied to $H$ with constant $\delta=\delta_{0}$. In view of the
above, for any $n$ there exists $j_{n}\in\\{1,2,\dots M\\}$ and
$u_{n},v_{n}\in U_{\delta_{0}}$ such that $u_{n}f_{n}=h_{j_{n}}v_{n}$. Notice
that for any $1\leq i\leq m$ we have $v_{n}(p_{i})\in
B(p_{i},\delta_{0})\subseteq v_{n}(B(p_{i},2\delta_{0}))$ (for the last
inclusion, notice that $v_{n}^{-1}\in U_{\delta_{0}}$). If follows that

$u_{n}f_{n}p_{i}=h_{j_{n}}v_{n}p_{i}\in
h_{j_{{}_{n}}}(B(p_{i},\delta_{0}))\subseteq
h_{j_{n}}v_{j_{n}}B(p_{i},2\delta_{0})=u_{n}f_{n}B(p_{i},2\delta_{0})$

Up to passing to a subsequence again we may assume that $j_{n}$ equals some
constant $j_{0}$ for all $n$. For $1\leq i\leq m$ let
$K_{i}=h_{j_{0}}\bar{B}(p_{i},\delta_{0})$.

For $n>0$ write $q_{n}^{i}=u_{n}f_{n}p_{i}$. We know that
$d(f_{n}p_{i},p_{i})<\delta\leq\delta_{0}$ and that
$d(f_{n}p_{i},q_{n}^{i})<\delta_{0}$ ($w_{n}\in U_{\delta}$) so that
$d(q_{n}^{i},p_{m}^{i})<2\delta_{0}$. We choose a series of paths between
these points as follows. For $n\in\mathbb{N}$, $1\leq i\leq m$ take any path
$\nu_{n}^{i}$ in $B(p_{m},2\delta_{0})$ from $p_{i}$ to $q^{i}_{n}$.

######  Lemma 2.8.

After passing to a subsequence we can choose for $1\leq i\leq m$ and
$n,n^{\prime}\geq 0$ paths $\mu_{n^{\prime},n}^{i}$ from $q_{n}^{i}$ to
$q_{n^{\prime}}^{i}$ and $\epsilon\in(0,\delta_{0})$ with the following
properties.

  * •

$N_{\epsilon}(im(\mu_{n^{\prime},n}^{i}))\subseteq
h_{j_{0}}(B(p_{i},\delta_{0}))\setminus\\{q_{n}^{i}\,|\,n\in\mathbb{N},1\leq
l\leq m,l\neq i\\}$

  * •

Neither of the points $q_{n^{\prime}}^{j}$ or $q_{n}^{j}$ belongs to
$W_{n^{\prime},n}^{i}$ for $1\leq j\leq m$, $j\neq m$.

  * •

Let

$P_{n^{\prime},n}^{i}=\prod_{1\leq
l<i}\mathcal{P}^{\delta_{0}}_{\nu^{l}_{n^{\prime}}}\prod_{i<l\leq
m}\mathcal{P}^{\delta_{0}}_{\nu^{l}_{n}}\in G$

and notice that it fixes the ball $B(p_{i},\delta_{0})$. Consider also the
loops at $p_{i}$:

$\displaystyle\sigma_{n^{\prime},n}^{i}:=$
$\displaystyle\nu_{n^{\prime}}^{i}*\mu_{n^{\prime},n}^{i}*(\nu_{n}^{i})^{-1}$
$\displaystyle\rho_{n^{\prime},n}^{i}:=$
$\displaystyle(P_{n^{\prime},n}^{i})^{-1}\circ\sigma_{n^{\prime},n}^{i}=\nu_{n^{\prime}}^{i}*((P^{i}_{n^{\prime},n})^{-1}\circ\mu_{n^{\prime},n}^{i})*(\nu_{n}^{i})^{-1}$

Then for any $1\leq i\leq m$ the image $\mathcal{R}^{i}$ of the set
$\\{\rho_{n^{\prime},n}^{i}\\}_{n,n^{\prime}\in\mathbb{N}}$ in
$\pi_{1}(S_{\bar{p}\setminus\\{p_{i}\\}},p_{i})$ is finite.

###### Proof.

After passing to a subsequence we may assume that the points $q_{n}^{i}$
converge to a point $q_{\infty}^{i}$ for all $1\leq i\leq m$. If we let
$\mathcal{Q}^{i}=\\{q_{n}^{i},q_{\infty}^{i}\,|\,n\in\mathbb{N}\\}$, then each
of the sets $h_{j_{0}}(B(p_{i}))\setminus\bigcup_{l\neq i}\mathcal{Q}^{i}$ are
connected.

We start with the following observation:

######  Observation 2.9.

Let $W$ be the interior of an embedded disk, $x\in S$, $U,U^{\prime}$ two open
sets in $S$ containing $x$ with $\bar{U^{\prime}}\subseteq U$ and
$\mathcal{C}$ some countable closed sets of points. Then there exits some
finite family $\mathcal{L}$ of arcs between points $W\setminus\mathcal{C}$
such that for any two points $x,y\in(W\cap U^{\prime})\setminus\mathcal{C}$
there exists some $\lambda_{x,y}\in\mathcal{L}$ and arcs $\beta_{x,y}$ and
$\gamma_{x,y}$ in $(W\cap U)\setminus\mathcal{C}$ such that

$\beta_{x,y}(0)=x,\,\,\beta_{x,y}(1)=\lambda_{x,y}(0),\,\,\lambda_{x,y}(1)=\gamma_{x,y}(0),\,\,\gamma_{x,y}(1)=y.$

###### Proof.

This follows from a standard argument. By compactness, there are finitely many
connected components $U_{1},\dots U_{q}$ of $U\cap\bar{W}$ such that
$\bar{W}\cap\bar{U}^{\prime}\subseteq\bigcup_{i=1}^{q}U_{i}$. Choose some
point $y_{i}\in U_{i}\cap(W\setminus\mathcal{C})$ for $1\leq i\leq q$. Now
$W\setminus\mathcal{C}$ is connected and as $\mathcal{L}$ it suffices to
choose some finite collection of arcs $\lambda_{i,j}$ in
$W\setminus\mathcal{C}$ between $y_{i}$ and $y_{j}$ for every pair of distinct
$i,j\in\\{1,2,\dots q\\}$. ∎

For fixed $1\leq i\leq m$ let $\lambda_{-},\beta_{-},\gamma_{-}$ be the
families that result from applying Observation 2.9 with
$W=h_{j_{0}}(B(p_{i},\delta_{0}))\setminus\bigcup_{l\neq i}\mathcal{Q}^{l}$,
$x=p_{i}$, $U=B(p_{i},3\delta_{0})$ and $U^{\prime}=B(p_{i},2\delta_{0})$ and
$\mathcal{C}=\mathcal{Q}$.

For simplicity, write
$\lambda^{i}_{n,n^{\prime}}=\lambda_{q_{n}^{i},q_{n^{\prime}}^{i}}$ and so
forth. Let
$\mu_{n,n^{\prime}}^{i}=\beta_{n,n^{\prime}}^{i}*\lambda_{n,n^{\prime}}^{i}*\gamma_{n,n^{\prime}}^{i}$.
We can assume that $\lambda^{i}_{n,n^{\prime}}$ and hence
$\mu_{n,n^{\prime}}^{i}$ and $\sigma_{n,n^{\prime}}^{i}$ intersects
$\bar{B}(p_{l},2\delta_{0})$ in a finite collection of arcs for $1\leq l\leq
m$, $l\neq m$.

We first note that it follows from the finiteness of $\mathcal{L}$ that the
image $\mathcal{S}^{i}$ of the collection $\\{\sigma_{n,n^{\prime}}^{i}\\}$ in
$\pi_{1}(S_{\bar{p}\setminus\\{p_{i}\\}},p_{i})$ in the statement is finite.

Indeed, if $\lambda_{n_{0},n_{1}}^{i}=\lambda_{n_{2},n_{3}}^{i}=:\lambda$ then
we can rewrite $(\sigma_{n_{0},n_{1}}^{i})^{-1}\sigma_{n_{2},n_{3}}^{i}$ as

$\nu_{n_{1}}^{i}*(\gamma_{n_{0},n_{1}}^{i})^{-1}*\lambda^{-1}*(\beta_{n_{0},n_{1}}^{i})^{-1}*(\nu_{n_{0}}^{i})^{-1}*\nu_{n_{2}}^{i}*\beta_{n_{0},n_{1}}^{i}*\lambda*\gamma_{n_{2},n_{3}}^{i}*(\nu_{n_{3}}^{i})^{-1}$

Since the simple closed paths
$(\beta_{n_{0},n_{1}}^{i})^{-1}*(\nu_{n_{0}}^{i})^{-1}*\nu_{n_{2}}^{i}*\beta_{n_{0},n_{1}}^{i}$
and
$\nu_{n_{1}}^{i}*(\gamma_{n_{0},n_{1}}^{i})^{-1}*(\gamma_{n_{2},n_{3}}^{i})*(\nu_{n_{3}}^{i})^{-1}$
are contained in a disk in $S_{\bar{p}\setminus\\{p_{i}\\}}$ they must be
homotopic to the identity there. Therefore so is
$(\sigma_{n_{0},n_{1}}^{i})^{-1}\sigma_{n_{2},n_{3}}^{i}$.

In order to ensure the finiteness of the homotopy classes of the
$\rho^{i}_{n^{\prime},n}$ we will need the following two standard
observations, whose proof we include for the sake of completeness:

######  Observation 2.10.

Let $T$ be a punctured surface, $U\subseteq T$ an open disk , $p\in U$ and
$\alpha$ a curve based at some $*\notin U$. Then given any $q\notin
im(\alpha)$ and any $g\in G$ supported in $U$ sending $q$ to $p$, the homotopy
class of $g(\alpha)$ in $T_{p}$ depends only on the connected component of
$U\setminus im(\alpha)$ to which the point $q$ belongs.

###### Proof.

Assume that $q$ and $q^{\prime}$ are two points in the same connected
component $C$ of $U\setminus im(\alpha)$ and $g,g^{\prime}$ supported in $U$
map $q$ and $q^{\prime}$ respectively to $p$. If
$g(\alpha)\nsimeq_{p}g^{\prime}(\alpha)$, then
$\alpha\nsimeq_{q}g^{-1}g^{\prime}(\alpha)$. Take some $g^{\prime\prime}$
supported in $C$ mapping $q$ to $q^{\prime}$, then
$\alpha=g^{\prime\prime}\alpha\nsimeq_{q^{\prime}}g^{\prime\prime}g^{-1}g\alpha$.
But by Fact 1.6 the map $g^{\prime\prime}g^{-1}g$ must be the identity in
$PMod(T_{q^{\prime}})$: a contradiction. ∎

The following follows from a standard application of transversality.

######  Observation 2.11.

For any any embedded disk $D$ contained in some open set $W$ and any finite
collection $\mathcal{M}$ of disjoint simple arcs between points outside of $W$
there exists another closed disk $D^{\prime}$ with $D\subseteq
D^{\prime}\subseteq W$ such that each $\mu\in\mathcal{M}$ intersects
$D^{\prime}$ in a finite collection of arcs.

Consider the loop
$\rho_{n^{\prime},n}^{i}:=(P_{n^{\prime},n}^{i})^{-1}\circ\sigma_{n^{\prime},n}^{i}=\nu_{n^{\prime}}^{i}*((P^{i}_{n^{\prime},n})^{-1}\circ\mu_{n^{\prime},n}^{i})*(\nu_{n}^{i})^{-1}$.
This is a loop in $S_{p_{1},\dots p_{i-1},p_{i+1},\dots p_{m}}$, since im
$q_{n}^{l},q_{n^{\prime}}^{l}\notin im(\alpha)$ for $l\neq i$.

For $1\leq l\leq m$, $l\neq i$ let $F_{l}$ be a closed embedded disk with
$B(p_{l},2\delta_{0})\subseteq F_{l}\subseteq B(p_{l},3\delta_{0})$ and such
that $\lambda_{n,n^{\prime}}^{i}$ (and hence $\sigma_{n,n^{\prime}}^{i}$)
intersects $F_{l}$ in a finite collection of arcs.

By iteratively applying Observation 2.10, we conclude that the homotopy class
of $\rho_{n^{\prime},n}^{i}$ depends only on the connected component of
$F_{l}\setminus im(\mu_{n^{\prime},n}^{i})$ in which $q_{n^{\prime}}^{l}$ lies
for $1\leq l<i$ as well as the connected component of $F_{l}\setminus
im(\mu_{n^{\prime},n}^{i})$ in which one $q_{n}^{l}$ lies for $i<l\leq m$.
Since in each case there are only finitely many possibilities the result
follows.

The existence of $\epsilon$ such that
$N_{\epsilon}(im(\mu^{i}_{n^{\prime},n}))$ does not intersect
$\mathcal{Q}^{l}$ for $1\leq l\leq m$, $l\neq i$ is clear. ∎

Choose $\eta>0$ such that $h_{j_{0}}B(p_{i},\delta_{0})$ is $\eta$-branching
and let $\delta_{1}=\min\\{\eta,\delta_{0}\\}$.

By another application of Roelcke precompactness, after passing to a
subsequence again we may assume that for all distinct $n<n^{\prime}\leq N$
there exists $u^{\prime}_{n,n^{\prime}},u^{\prime\prime}_{n,n^{\prime}}\in
U_{\delta_{1}}$ such that
$u^{\prime\prime}u_{n^{\prime}}f_{n^{\prime}}=hu_{n}f_{n}u^{\prime}$. Fix
$n^{\prime}=0$ and for the time being also $n>0$ and let
$u^{\prime}=u^{\prime}_{0,n},u^{\prime\prime}=u^{\prime\prime}_{0,n}$.

Let also $\alpha$ be any essential simple closed curve or simple arc between
boundary points $\alpha$ that does not intersect any of the disks
$\bar{B}(p_{i},4\delta_{0})$. It follows that $u^{\prime}\circ\alpha$ does not
intersect $B(p_{i},2\delta_{0})$ for any $1\leq i\leq m$. Therefore
$u_{n}f_{n}u^{\prime}\circ\alpha$ does not intersect
$h_{j_{0}}B(p_{i},\delta_{0})\subseteq u_{n}f_{n}B(p_{i},2\delta_{0})$ for any
$1\leq i\leq m$ and the same holds for $u_{0}f_{0}\circ\alpha$.

On the other hand, since $u^{\prime\prime}\in U_{\delta}$ from the choice of
$\delta_{1}$ and $\delta_{0}$ it follows that

$d((u_{0}f_{0}\circ\alpha)(t),(u_{n}f_{n}u^{\prime}\circ\alpha)(t))<\min\\{\eta,\delta_{e}\\}.$

Lemma 2.4 yields that
$u_{0}f_{0}\circ\alpha\simeq_{K}u_{n}f_{n}u^{\prime}\circ\alpha$, where
$K=\bigcup_{i=1}^{m}K_{i}$.

Now, $\alpha$ and $u^{\prime}\circ\alpha$ can be homotoped to each other
through a homotopy with values in
$S\setminus\bigcup_{i=1}^{m}B(p_{i},2\delta_{0})$, it follows that
$u_{n}f_{n}u^{\prime}\circ\alpha\simeq_{K}u_{n}f_{n}\circ\alpha$

Consider the commuting products
$\mathcal{P}_{\mu_{0,n}}=\prod_{i=1}^{m}\mathcal{P}_{\mu^{i}_{0,n}}^{\epsilon}$
and
$\mathcal{P}_{\nu_{n}}=\prod_{i=1}^{m}\mathcal{P}_{\nu^{i}_{n}}^{\delta_{0}}$.
We may assume
$(\mathcal{P}^{\delta_{0}}_{\nu^{i}_{0,n}})^{-1}=\mathcal{P}^{\delta_{0}}_{(\nu^{i}_{0,n})^{-1}}$.
Since $\mathcal{P}_{\mu_{0,n}}$ is supported in $K$, for any $\alpha$ as above
we have
$\mathcal{P}_{\mu_{0,n}}u_{0}f_{0}\circ\alpha\simeq_{K}u_{0}f_{0}\circ\alpha$.
It follows that

$\mathcal{P}_{\mu_{0,n}}u_{0}f_{0}\circ\alpha\simeq_{K}u_{n}f_{n}$

Since $q^{i}_{n}\in K$ and $\mathcal{P}_{\nu_{n}}^{-1}(q^{i}_{n})=p_{i}$ we
have

$\mathcal{P}_{\nu_{n}}^{-1}\mathcal{P}_{\mu_{0,n}}u_{0}f_{0}\circ\alpha\simeq_{\bar{p}}\mathcal{P}_{\nu_{n}^{-1}}u_{n}f_{n}\circ\alpha.$

Since both $\mathcal{P}_{\nu_{n}}^{-1}\mathcal{P}_{\mu_{0,n}}u_{0}f_{0}$ and
$\mathcal{P}_{\nu_{n}}^{-1}u_{n}f_{n}$ belong to $H_{\bar{p}}$ and $\alpha$
ranges over a set of curves/arcs containing representatives of all homotopy
classes in $S_{\bar{p}}$ we conclude that the images of these two elements in
$PMod(S_{\bar{p}})$ are equal. Expanding $f_{n}$ as $w_{n}b_{n}$ with
$b_{n}\in G_{\bar{p}}$ we obtain

$[\mathcal{P}_{\nu_{n}}^{-1}\mathcal{P}_{\mu_{0,n}}\mathcal{P}_{\nu_{0}}][\mathcal{P}_{\nu_{0}}^{-1}u_{0}w_{0}][b_{0}]=[\mathcal{P}_{\nu_{n}}^{-1}u_{n}w_{n}][b_{n}].$

Notice that $\mathcal{P}_{\nu_{n}}^{-1}u_{n}w_{n}\in G_{\bar{p}}$. On the
other hand, we have $\mathcal{P}_{\nu^{i}_{n}}^{\delta_{0}}\in
U_{6\delta_{0}}$, since it is supported on
$N_{\delta_{0}}(im(\nu_{n}))\subseteq B(p_{i},3\delta_{0})$. The latter also
implies that the supports of $\mathcal{P}_{\nu^{i}_{n}}^{\delta_{0}}$ for
different values of $i$ are disjoint and hence $\mathcal{P}_{\nu_{n}}\in
U_{6\delta_{0}}$. It follows that $\mathcal{P}_{\nu_{n}}^{-1}u_{n}w_{n}\in
U_{6\delta_{0}+\delta_{0}+\delta_{1}}\subseteq U_{\delta_{e}}$. By Observation
2.2 we then have $[\mathcal{P}_{\nu_{n}}^{-1}u_{n}w_{n}]=1\in
PMod(S_{\bar{p}})$ and like-wise $[\mathcal{P}_{\nu_{0}}^{-1}u_{0}w_{0}]=1$.

We now claim that
$[\mathcal{P}_{\nu_{n}}^{-1}\mathcal{P}_{\mu_{0,n}}\mathcal{P}_{\nu_{0}}]\in\mathcal{P}(\mathcal{R}^{1})\cdot\mathcal{P}(\mathcal{R}^{2})\dots\mathcal{P}(\mathcal{R}^{m})$,
where $\mathcal{R}^{i}$ is the finite set in Lemma 2.8. This concludes the
proof, since letting $n>0$ range we obtain that the set
$\\{[b_{n}]\\}_{n\in\mathbb{N}}$ has to be finite, contradicting the choice of
$b_{n}$.

The claim follows from a straightforward algebraic manipulation, which we
sketch below, justifiably ignoring the choice of push representatives:

$\begin{gathered}=[(\mathcal{P}_{\nu_{n}^{1}}^{-1}\prod_{i=2}^{m}\mathcal{P}_{\nu_{n}^{i}}^{-1})(\mathcal{P}_{\mu_{0,n}^{1}}\prod_{i=2}^{m}\mathcal{P}_{\mu_{0,n}^{i}})(\mathcal{P}_{\nu_{0}^{1}}\prod_{i=2}^{m}\mathcal{P}_{\nu_{0}^{i}})]\\\
=[\mathcal{P}_{\nu_{n}^{1}}^{-1}\mathcal{P}_{\mu_{0,n}^{1}}^{\prod_{i=2}^{m}\mathcal{P}_{\nu_{n}^{i}}}\mathcal{P}_{\nu_{0}^{1}}(\prod_{i=2}^{m}\mathcal{P}_{\nu_{n}^{i}}^{-1}\prod_{i=2}^{m}\mathcal{P}_{\mu_{0,n}^{i}})^{\mathcal{P}_{\nu_{0}^{1}}}\prod_{i=2}^{m}\mathcal{P}_{\nu_{0}^{i}}]\\\
=\mathcal{P}([\rho^{1}_{0,n}])[(\prod_{i=2}^{m}\mathcal{P}_{\nu_{n}^{i}}^{-1}\prod_{i=2}^{m}\mathcal{P}_{\mu_{0,n}^{i}}\prod_{i=2}^{m}\mathcal{P}_{\nu_{0}^{i}})^{\mathcal{P}_{\nu_{0}^{1}}}]\\\
=\mathcal{P}([\rho^{1}_{0,n}])\mathcal{P}([\rho^{2}_{0,n}])\dots\mathcal{P}([\rho^{m}_{0,n}])\end{gathered}$

∎

##  3\. Questions

The torus admits a $1$-transitive Roelcke precompact subgroup of
homeomorphisms, namely the one given by the diagonal action of
$Homeo(S^{1})\times Homeo(S^{1})$ on $S^{1}\times S^{1}$.

######  Question 3.1.

Is the bound $\zeta(S)$ on the degree of transitivity on $S$ of a Roelcke
precompact subgroup of $Homeo_{\partial S}(S)$ sharp in those cases in which
$\zeta(S)>1$ and $S$ is not a torus?

######  Question 3.2.

Is the subgroup of $Homeo(S^{1}\times S^{1})$ given above the only
$1$-transitive Roelcke precompact subgroup on $S^{1}\times S^{1}$ up to
conjugacy?

######  Question 3.3.

Can a transitive (resp. $\omega$-transitive) group of homeomorphisms of an
$m$-manifold, $m\geq 3$ be Roelcke precompact?

## References

  * [1] Benson Farb and Dan Margalit.  A primer on mapping class groups (pms-49).  Princeton university press, 2011.
  * [2] Christian Rosendal.  Global and local boundedness of polish groups.  Indiana University Mathematics Journal, pages 1621–1678, 2013.