Fang-Zhou Yan

*Zhi-Fu Gao. Xinjiang Astronomical Observatory, CAS,150, Science 1-Street, Urumqi, Xinjiang, 830011, China.

# Explaining high-braking indice of magnetars SGR 0501$+$4516 and 1E
2259$+$586 using the double magnetic-dipole model

Fang-Zhou Yan  Zhi-Fu Gao  Wen-Shen Yang  Ai-Jun Dong  Xinjiang Astronomical
Observatory, CAS,150, Science 1-Street, Urumqi, Xinjiang, 830011, China  Key
Laboratory of Radio Astronomy, Chinese Academy of Sciences, West Beijing Road,
Nanjing, 210008, China  University of Chinese Academy of Sciences, No.19,
Yuquan Road, Beijing, 100049, China  Shchool of Physics and Electronic
Science, Guizhou Normal University, Guiyang, Guizhou, 550001, China
<EMAIL_ADDRESS>
(20 Nov. 2020)

###### Abstract

In this paper, we attribute high braking indices $n>3$ of two magnetars SGR
0501$+$4516 and 1E 2259$+$586 to the decrease in their inclination angles
using the double magnetic-dipole model proposed by [Hamil . (2016)]. In this
model, there are two magnetic moments inside a neutron star, one is generated
by the rotation effect of a charged sphere, $M_{1}$, and the other is
generated by the magnetization of ferromagnetically ordered material, $M_{2}$.
Our calculations indicate that the magnetic moment $M_{2}$ would evolve
towards alignment with the spin axis of the two magnetars, and cause their
magnetic inclination angles to decrease. We also define a ratio
$\eta=M_{2}/M_{1}$, which reflects the magnetization degree, and find that the
values of $\eta$ of the two magnetars are about two-orders of magnitude higher
than that of rotationally powered pulsar PSR J1640-4631 with $n=3.15(3)$,
assuming that they have the same rate of decrease in their inclination angles.

###### keywords:

Magetar braking index– Double magnetic-dipole model–SGR 0501$+$4516–1E
2259$+$586

††articletype: Article Type

##  1 Introduction

The spin-down evolution of pulsars has become an important research hot spot
in the field of compact objects (Dang ., 2020; Kou ., 2018; Magalhaes ., 2012,
2016; Oliveira ., 2018; Wen, Chen ., 2020; Yan, Manchester, van Straten .,
2011; Yan, Manchester, Hobbs ., 2011; Yan ., 2018; Zhao, 2019, 2020). Allen
Horvath (1997) proposed that the long-term slowdown of a pulsar’s rotation
follows a power-law form $\tau_{\rm ext}\propto\Omega^{n}$ with $\tau_{\rm
ext}$ being the external torque acting on the crust, $\Omega$ the rotational
angular velocity, and $n$ the braking index, and gave an expression

$\dot{\Omega}=-K\Omega^{n},$ (1)

where $\dot{\Omega}$ is the derivative of $\Omega$, and $K$ is a positive
proportional parameter. The braking index of a pulsar is defined as

$n=\frac{\Omega\ddot{\Omega}}{\dot{\Omega}^{2}}=\frac{\nu\ddot{\nu}}{\dot{\nu}^{2}}=2-\frac{P\ddot{P}}{\dot{P}^{2}},$
(2)

where $\ddot{\Omega}$ is the second derivative of $\Omega$, $\nu=\Omega/2\pi$
the spin frequency, and $P=1/\nu$ the spin period. A pure magneto-dipole
radiation (MDR) model predicts the braking index $n=3$. However, the observed
braking indices of pulsars always deviate from three predicted by the pure MDR
model. Michel Goldwire (1970) proposed the origin of the alignment and slowing
down torque with the corresponding equations of motion

$I\dot{\alpha}=-\frac{2M^{2}\sin\alpha\cos\alpha\Omega^{2}}{3c^{3}}$ (3)

when pulsars are slowed down, where $I$ is the star’s moment of inertia,
$\alpha$ is the inclination angle between the rotation and magnetic axes,
$\dot{\alpha}$ is the derivative of $\alpha$, $M=B_{d}R^{3}$ is the magnetic
moment ($B_{d}$ is the surface dipole magnetic field strength at the equator,
and $R$ the radius of the star), and $c$ is the speed of light in vacuum. In
addition, a similar model of the alignment or counteralignment of the magnetic
axis of a pulsar with its spin axis was proposed by Macy (1974). A change in
the inclination angle $\alpha$ of a pulsar is usually companied with a change
in pulse profiles (Wen ., 20161, 20162; Yan ., 2019, 2020; Yuan ., 2017; Wen,
Yan Yuan, 2020). A dipole rotating in vacuum is subject to the radiation field
torque (Davis Goldstein, 1970)

$I\dot{\Omega}=-\frac{2M^{2}\sin^{2}\alpha\Omega^{3}}{3c^{3}}$ (4)

The starting point of this work comes from Michel Goldwire (1970), Macy (1974)
and Allen Horvath (1997). The so-called “vacuum model” includes the alignment
torque and radiation field torque, described by Eq. (3) and Eq. (4),
respectively, and allows a change in $\alpha$ and/or a change in the magnetic
momentum $M$ of a pulsar. From Eq (4), we get the second derivative of
$\Omega$,

$\displaystyle\ddot{\Omega}=-\frac{2}{3c^{3}}(3M^{2}\sin^{2}\alpha\Omega^{2}\dot{\Omega}+$
$\displaystyle~{}~{}~{}~{}2M\dot{M}\sin^{2}\alpha\Omega^{3}+2M^{2}\sin\alpha\cos\alpha\dot{\alpha}\Omega^{3}).$
(5)

Substituting Eqs.(3),(4) and (5) into Eq. (2), the observed braking index
$n_{\rm obs}$ in vacuum model is expressed as

$n_{\rm
obs}=3+\frac{2\Omega}{\dot{\Omega}}(\frac{\dot{\alpha}}{\tan\alpha}+\frac{\dot{M}}{M}).$
(6)

If we ignore the change in $M$, the above equation becomes

$n_{\rm
obs}=3+\frac{2\Omega}{\dot{\Omega}}\frac{\dot{\alpha}}{\tan\alpha}=3+\frac{2\nu}{\dot{\nu}}\frac{\dot{\alpha}}{\tan\alpha}.$
(7)

The rotationally powered pulsar PSR J1640$-$4631 has $\nu$=4.843 Hz and
$\dot{\nu}=-2.28\times 10^{-11}$ Hz s-1 (Archibald et al. 2016), implying a
characteristic age of $\tau_{c}=-\nu/2\dot{\nu}$=3370 yr and surface dipole
field $B_{d}=1.43\times 10^{13}I^{1/2}_{45}$ G, where $I_{45}$ is the moment
of inertia in units of $10^{45}$ g cm2 (Gotthelf ., 2014). It is the first to
have a braking index greater than 3, $n=3.15(3)$ (the number in parentheses
denotes the error) (Archibald ., 2016), measured with high precision. Ekşi .
(2016) explained the braking index of PSR J1640-4631 in plasma-filled model
and inferred the change in the inclination angle to be at the rate
$\dot{\alpha}=-0.23(5)^{\circ}$/century, about 2.5 times smaller in absolute
value than the rate $\dot{\alpha}=-0.56^{\circ}$/century of the Crab pulsar
through 45 years of observations (Lyne ., 2015).

Because of lack of long-term pulsed emission in quiescence and strong timing
noise, it is impossible to directly measure the braking index of a magnetar,
powered by superhigh multiple magnetic fields. Based on the estimated ages of
their potentially associated supernova remnants (SNR) and the timing
parameter,Gao . (2016) measured the mean braking indices of $n=13(4)$,
6.3(1.7) and 32(10) for 1E 1841-045, SGR 0501$+$4516 and 1E 2259$+$586,
respectively, using the expression $n=1+P/(\dot{P}t_{SNR})$, where $t_{SNR}$
is the SNR age. Since the uncertainty in estimating the SNR age of 1E 1841-045
is too large, 1E 1841-045 will not be considered in this work. The high
braking indices of pulsars have been explained by different models (see Gao .
(2017) for a brief review). We attribute high braking indices $n>3$ of SGR
0501$+$4516 and 1E 2259$+$586 to the decrease in their inclination angles, and
estimate their initial magnetic moments, initial inclination angles by using a
novel double magnetic-dipole model proposed by Hamil . (2016) (hereafter the
HSS model).

##  2 The HSS model

The pulsar’s magnetic field was thought to originate from the following
mechanisms: the dynamo effect and the magnetization of a ferromagnetic core in
the liquid interior of neutron stars (NSs) (Sliverstein, 1969). Since there is
no internal source of energy to generate convection and activate dynamo, it is
very difficult to produce a continuous dynamo action in the pulsar. In the HSS
model, the pulsar has two dipoles with magnetic moments $M_{1}$ and $M_{2}$,
and the distance between two dipole centers is a constant $r$. The second
magnetic moment $M_{2}$ is thought to be off-centered dipole, and the
inclination angle between the $M_{1}$ and the dipole-dipole axis is
$\theta_{1}$. Because the NS interior is in superfluidity phase, the friction
received by the rotational region supporting $M_{2}$ is neglected. Due to the
interaction between two dipoles, the change rate of $\theta_{2}$ between the
$M_{2}$ and the dipole-dipole axis is

$\dot{\theta_{2}}=\sqrt{\frac{2M_{1}M_{2}}{I_{2}r^{3}}F(\Theta)},$ (8)

where
$F(\Theta)=(sin\theta_{1i}\sin\theta_{2i}-2\cos\theta_{1i}\cos\theta_{2i})-(\sin\theta_{1f}\sin\theta_{2f}-2\cos\theta_{1f}\cos\theta_{2f})$,
$I_{2}$ is the moment of inertia of the rotating region, $i$ and $f$ represent
the initial and final values, respectively.

During the motion of the $M_{2}$, its potential energy would convert into the
kinetic energy. By taking $C=0$ in Eq. (10) of Hamil . (2016), we can obtain
the direction of $M_{2}$ when its potential energy is minimum,

$\theta_{2}^{\rm min}=-\arctan(\frac{\tan\theta_{1}}{2}).$ (9)

From, Eq.(9), it is obvious that $\theta_{2}^{\rm min}$ should be negative,
meaning that the minimum energy position should locate the clockwise direction
of the dipole-dipole axis. For an arbitrary initial $\theta_{2}$, the physical
law indicates that the $M_{2}$ will freely rotate around the minimum energy
position, which is similar to a simple pendulum.

According to the geometric relationship, the inclination angle and its change
rate are determined by

$~{}~{}~{}~{}~{}\alpha=\theta_{1}-\theta_{2},~{}~{}~{}{\rm
and}~{}~{}\dot{\alpha}=-\dot{\theta}_{2},$ (10)

respectively, where $\theta_{1}$ keeps constant, positive (negative) angles
$\theta_{1}$ and $\theta_{2}$ are measured from the dipole-dipole axis
counterclockwise (clockwise), the rotation axis is taken as $\alpha=0$, an
increase (decrease) in $\alpha$ and a consequent movement away (toward) the
axis of rotation. In this work, for simplicity, we take range of
$-90^{\circ}<\theta_{2}<0$ in our simulation.

##  3 Applied to two high-$n$ magnetars

###  3.1 Relation between $\dot{\alpha}$ and $\alpha$

SGR 0501$+$4516 was first discovered by the Swift $\gamma$ -ray observatory,
and is associated with SNR HB9 with an age of $t_{\rm SNR}\sim 1.85-5.85$ kyr.
It has the spin period and the period derivative $\nu$=0.1735 Hz and
$\dot{\nu}=-1.7\times 10^{-13}$ Hz s-1, implying a surface dipole magnetic
field $B_{d}=1.9\times 10^{14}I^{1/2}_{45}$ G (Camero ., 2014). Its braking
index is constrained as $6.3(1.7)$. AXP 1E2259$+$586 lies within the geometric
centre of CTB 109 with an age of $t_{SNR}\sim 10-20$ kyr, and has $\nu=0.143$
Hz, and $\dot{\nu}=0.967\times 10^{-14}$ Hz s-1, implying a surface dipole
magnetic field $B_{d}=0.59\times 10^{14}I^{1/2}_{45}$ G (Olausen Kaspi, 2014).
Its constrained value of braking index is $n\sim 32(10)$ (Gao ., 2016).
Regularly monitored with RXTE from 1996 to 2018, influenced by frequent
glitches and outbursts, SGR 0501$+$4516 and 1E 2259$+$586 exhibited large
fluctuations in spin-down parameters $\nu$ and $\dot{\nu}$. These large
decreases in $\dot{\nu}$ were explained as the effects of timing-noise torque
and outbursts, instead of as a secular dipole magnetic field decay (e.g.Lin .
(2020). Their outbursts were explained as resulting from internal multiple
magnetic field decay (Duncan Thompson, 1992).

According to the HSS model, when $M_{2}$ is rotating in the vicinity of the
equilibrium position with the counter clockwise direction,
$\dot{\theta}_{2}>0$, hence $\dot{\alpha}<0$, we obtain a braking index
greater than three from Eq. (7), which can naturally explain high braking
indices of SGR 0501$+$4516 and 1E 2259$+$586\. Noted that, if
$\dot{\theta}_{2}<0$, it will result in a braking index smaller than three
however, this case is not the subject of this work. From Eq. (7), the change
rate of the inclination angel, $\dot{\alpha}$, is estimated from

$\dot{\alpha}=(n_{\rm obs}-3)\frac{\dot{\Omega}}{2\Omega}\tan\alpha=(n_{\rm
obs}-3)\frac{\dot{\nu}}{2\nu}\tan\alpha.$ (11)

Inserting the values of $\nu$ and $\dot{\nu}$ into Eq. (11), we make the plots
of $-d\alpha/dt$ versus $\alpha$ for the two high-$n$ magnetars. In Fig. 1, we
take $n_{\rm obs}\approx\bar{n}=6.3$ and 32 for SGR 0501$+$4516 and 1E
2259$+$586, respectively. Obviously, $-d\alpha/dt$ increases with the increase
in $\alpha$ when $n_{\rm obs}$ is given.

Figure 1: Relations of $\dot{\alpha}$ and $\alpha$ of two magnetars

To date, due to the lack of polarization observations, we cannot obtain the
actual value of $\alpha$ of a magnetar and its change rate $\dot{\alpha}$
observationally. In this work we assume that SGR 0501$+$4516 and 1E 2259$+$586
are experiencing a decrease in $\alpha$, it is possible that the actual values
of $\dot{\alpha}$ of the two magnetars are near to that of PSR J1640-4631
(Ekşi ., 2016) or to that of the Crab pulsar (Lyne ., 2015). We can obtain the
information related to the inclination angles, their decrease rates and
magnetic moments of the two magnetars by numerically simulating.

###  3.2 SGR 0501 and 1E 2259

If the current decrease rate of $\alpha$ of SGR 0501$+$4516 is same to that of
PSR J1640-4631, $\dot{\alpha}=-0.23^{\circ}$/century (Ekşi ., 2016), since its
average age is $t=t_{\rm SNR}=3.85$ kyr, we take a mean decrease rate
$\bar{\dot{\alpha}}=-0.115^{\circ}$/century, the total decrease of $\alpha$ is
then estimated as $\Delta\alpha=38.5\times(-0.115^{\circ})\approx 4^{\circ}$.
Inserting $\nu$ and $\dot{\nu}$ into Eq. (7). we get the current value of
$\alpha$, $\alpha_{f}\approx 37^{\circ}$, and the initial value of $\alpha$,
$\alpha_{i}=41^{\circ}$. In order to obtain a negative $\dot{\alpha}$,
$\theta_{2}$ should be less than $\theta_{2}^{\rm min}$, and $\theta_{1}$
should be less than a critical angle $\theta_{1}^{\rm crit}$, which is
determined by

$\theta_{1}^{\rm crit}\leq\alpha_{f}-\arctan(\frac{\tan(\theta_{1}^{\rm
crit})}{2}).$ (12)

From Eq. (12), we get $\theta_{1}^{\rm crit}=24.4^{\circ}$ and
$\alpha_{f}=37^{\circ}$. If the current decrease rate of $\alpha$ of the star
is same to that of the Crab, we obtain $\Delta\alpha\approx 11^{\circ}$,
$\alpha_{f}\approx 61^{\circ}$, and $\alpha_{i}\approx 72^{\circ}$ by taking a
mean decrease rate $\bar{\dot{\alpha}}=-0.28^{\circ}$/century. In the same
way, we obtain $\theta_{1}^{\rm crit}=39.1^{\circ}$, corresponding to
$\alpha_{f}=61^{\circ}$. By choosing particular values of $\theta_{1}$, we
calculate the values of $F(\Theta)$ and $\theta_{2}^{\rm min}$. In Table 1,
Case A and Case B correspond to $\bar{\dot{\alpha}}=-0.115^{\circ}$/century
and $-0.28^{\circ}$/century, respectively, which are same to those of PSR
J1640-4631 and the Crab, respectively. It’s obvious that $F(\Theta)$ decreases
with the increase in $\theta_{1}$. Comparing $\theta_{2}$ with
$\theta_{2}^{\rm min}$, the $M_{2}$ should rotate along the counter clockwise,
and the angle $\alpha$ decreases.

We choose a medium-mass NS with $m=1.45M_{\odot}$ and $R=11.5$ km,
corresponding to the moment of inertia $I=1.534(1)\times 10^{45}$ g cm2 in the
BSK26 EoS. The moment of inertia of $M_{1}$ is $I_{1}\approx I$. Taking
$I_{2}$ as $1\%$ of the total $I$ and $r=0.5R$, the magnetic moment $M_{2}$ of
SGR 0501+4516 is calculated as $M_{2}=B_{d}R^{3}\approx 3.58\times 10^{32}$ G
cm3. If $\dot{\theta_{2}}=-\dot{\alpha}=0.23^{\circ}$/century, taking
$\theta_{i}=0^{\circ},10^{\circ},20^{\circ}$ and $24^{\circ}$, respectively,
and using Eq.(8), we calculate the values of $M_{1}$ and $M_{2}/M_{1}$ and
list them in Table 2. If
$\dot{\theta_{2}}=-\dot{\alpha}=0.56^{\circ}$/century, taking
$\theta_{i}=0^{\circ},10^{\circ},20^{\circ},30^{\circ}$ and $38^{\circ}$,
respectively, we also calculate the values of $M_{1}$ and $M_{2}/M_{1}$ and
list them in Table 2.

Table 1: Main angle parameters of SGR 0501+4516 in the HSS model

$\alpha_{f}$ |  $\theta_{1i}$ |  $\theta_{1f}$ |  $\theta_{2i}$ |  $\theta_{2f}$ |  $\theta_{2}^{\rm min}$ | $F(\Theta)$
---|---|---|---|---|---|---
|  |  |  Case A |  |  |
$37^{\circ}$ |  $0^{\circ}$ |  $0^{\circ}$ |  $-41^{\circ}$ |  $-37^{\circ}$ |  $0^{\circ}$ | $0.0878$
$37^{\circ}$ |  $5^{\circ}$ |  $5^{\circ}$ |  $-36^{\circ}$ |  $-32^{\circ}$ |  $-2.5^{\circ}$ | $0.0727$
$37^{\circ}$ |  $10^{\circ}$ |  $10^{\circ}$ |  $-31^{\circ}$ |  $-27^{\circ}$ |  $-5.0^{\circ}$ | $0.0561$
$37^{\circ}$ |  $15^{\circ}$ |  $15^{\circ}$ |  $-26^{\circ}$ |  $-22^{\circ}$ |  $-7.6^{\circ}$ | $0.0383$
$37^{\circ}$ |  $20^{\circ}$ |  $20^{\circ}$ |  $-21^{\circ}$ |  $-17^{\circ}$ |  $-10.3^{\circ}$ | $0.0201$
$37^{\circ}$ |  $24^{\circ}$ |  $24^{\circ}_{\ast}$ |  $-17^{\circ}$ |  $-13^{\circ}$ |  $-12.6^{\circ}$ | $0.0056$
|  |  |  Case B |  |  |
$61^{\circ}$ |  $0^{\circ}$ |  $0^{\circ}$ |  $-72^{\circ}$ |  $-61^{\circ}$ |  $0^{\circ}$ | $0.3516$
$61^{\circ}$ |  $5^{\circ}$ |  $5^{\circ}$ |  $-67^{\circ}$ |  $-56^{\circ}$ |  $-2.5^{\circ}$ | $0.3277$
$61^{\circ}$ |  $10^{\circ}$ |  $10^{\circ}$ |  $-62^{\circ}$ |  $-51^{\circ}$ |  $-5.0^{\circ}$ | $0.2965$
$61^{\circ}$ |  $15^{\circ}$ |  $15^{\circ}$ |  $-57^{\circ}$ |  $-46^{\circ}$ |  $-7.6^{\circ}$ | $0.2589$
$61^{\circ}$ |  $20^{\circ}$ |  $20^{\circ}$ |  $-52^{\circ}$ |  $-41^{\circ}$ |  $-10.3^{\circ}$ | $0.2162$
$61^{\circ}$ |  $25^{\circ}$ |  $25^{\circ}$ |  $-47^{\circ}$ |  $-36^{\circ}$ |  $-13.1^{\circ}$ | $0.1692$
$61^{\circ}$ |  $30^{\circ}$ |  $30^{\circ}$ |  $-42^{\circ}$ |  $-31^{\circ}$ |  $-16.1^{\circ}$ | $0.1204$
$61^{\circ}$ |  $35^{\circ}$ |  $35^{\circ}$ |  $-37^{\circ}$ |  $-26^{\circ}$ |  $-19.3^{\circ}$ | $0.0703$
$61^{\circ}$ |  $38^{\circ}$ |  $38^{\circ}$ |  $-34^{\circ}$ |  $-23^{\circ}$ |  $-21.3^{\circ}$ | $0.0404$

Table 2: Partial values of $\alpha_{f}$, $F(\Theta)$, $M_{1}$ and $M_{2}$ and
$\eta$ for SGR 0501+4516

$\alpha_{f}$ |  $F(\Theta)$ |  $M_{2}$ |  $M_{1}$ |  $\eta$
---|---|---|---|---
|  |  G cm3 |  G cm3 |
$37^{\circ}$ |  0.0878 |  $3.58\times 10^{32}$ |  $7.52\times 10^{4}$ |  $4.76\times 10^{27}$
$37^{\circ}$ |  0.0561 |  $3.58\times 10^{32}$ |  $1.18\times 10^{5}$ |  $3.04\times 10^{27}$
$37^{\circ}$ |  0.0201 |  $3.58\times 10^{32}$ |  $3.28\times 10^{5}$ |  $1.09\times 10^{27}$
$37^{\circ}$ |  0.0056 |  $3.58\times 10^{32}$ |  $1.18\times 10^{6}$ |  $3.04\times 10^{26}$
$61^{\circ}$ |  0.3516 |  $3.58\times 10^{32}$ |  $1.11\times 10^{5}$ |  $3.22\times 10^{27}$
$61^{\circ}$ |  0.2965 |  $3.58\times 10^{32}$ |  $1.32\times 10^{5}$ |  $2.71\times 10^{27}$
$61^{\circ}$ |  0.2162 |  $3.58\times 10^{32}$ |  $1.81\times 10^{5}$ |  $1.98\times 10^{27}$
$61^{\circ}$ |  0.1204 |  $3.58\times 10^{32}$ |  $3.25\times 10^{5}$ |  $1.10\times 10^{27}$
$61^{\circ}$ |  0.0404 |  $3.58\times 10^{32}$ |  $9.68\times 10^{5}$ |  $3.69\times 10^{26}$

In order to explore the information of paramagnetic magnetization of a NS in
its early ferromagnetic phase, we define a model-dependent ratio,
$\eta=M_{2}/M_{1}$, which reflects the magnetization degree and depends on
angle parameters, e.g., $\dot{\alpha}$ and $F(\Theta)$. From Table 2, we find
that the values of $\eta$ are about $10^{26}-10^{27}$ for SGR 0501+4516. Fig.
2 denotes $n$ as a function of $\alpha$ for SGR 0501+4516, the red and blue
solid lines are for $\dot{\alpha}=-0.23^{\circ}$/century,
$\dot{\alpha}=-0.56^{\circ}$/century, respectively, and the horizontal blue
dotted line and the surrounding shaded region denote, respectively, the
constrained value of $n=13$ and its possible range given by the uncertainty 4
of SGR 0501$+$4516\. From Fig. 2, it is obvious that $n$ decreases with the
increase in $\alpha$ when the values of $\dot{\alpha}$, $\nu$ and $\dot{\nu}$
are given. If the change rate of $\alpha$ of SGR 0501$+$4516 is similar to the
Crab, it probably has a larger inclination angle.

Figure 2: Relation of $n$ and $\alpha$ for SGR 0501+4516.

If the current change rate of $\alpha$ for 1E 2259$+$586 is same to that of
PSR J1640-4631, $\dot{\alpha}=-0.23^{\circ}$/century (Ekşi ., 2016). Since its
SNR age is $t\approx t_{\rm SNR}$=15 kyr, assuming a mean change rate
$\bar{\dot{\alpha}}=-0.115^{\circ}$/century ,the total change of $\alpha$ is
$\Delta\alpha=-150\times 0.115^{\circ}/\approx 17^{\circ}$. Inserting $\nu$
and $\dot{\nu}$ into Eq.(7), we obtains the current value $\alpha_{f}\approx
51^{\circ}$, and the initial value of $\alpha$, $\alpha_{i}=68^{\circ}$. From
Eq.(12),the critical angle $\theta_{1}^{crit}=33.1^{\circ}$. If the current
change rate of $\alpha$ is same to that of the Crab,
$\dot{\alpha}=-0.56^{\circ}$/century, in the same method, the current value of
$\alpha_{f}=72^{\circ}$, but the initial angle $\alpha_{i}>90^{\circ}$, which
is unreasonable. Thus, the change rate of $\alpha$ for 1E 2259$+$586 could be
smaller than that of the Crab pulsar in the HSS model.

Assuming $\bar{\dot{\alpha}}=-0.115^{\circ}$/century and choosing several
particular angles of $\theta_{1}$, we calculate the values of $F(\Theta)$ and
$\theta_{2}^{\rm min}$ for 1E 2259$+$586\. From Table 3, $F(\Theta)$ decreases
with the increase in $\theta_{1}$. Comparing $\theta_{2}$ with
$\theta_{2}^{\rm min}$, the $M_{2}$ of the star should also rotate along the
counter clockwise. Using the same EoS, the magnetic moment $M_{2}$ of 1E
2259$+$586 is calculated as $M_{2}=B_{d}R^{3}\approx 1.12\times 10^{32}$ G
cm3. We calculate the values of $M_{1}$, $M_{2}$ and $\eta$ and list them in
Table 4. It is found that both $F(\Theta)$ and $\eta$ decrease with the
increase in $\theta_{1i}$, and the values of $\eta$ are also in the range of
$10^{26}-10^{27}$. Fig. 3 denotes $n$ as a function of $\alpha$ for 1E
2259$+$586 assuming $\dot{\alpha}=-0.23^{\circ}$/century. As a comparison,
using the same method, we calculate the values of $\dot{\theta}_{2}$,
$F(\Theta)$, $M_{1}$ and $M_{2}$ and $\eta$ for PSR J1640$-$4631 and list them
in Table 5. We find that the values of $M_{2}$ of PSR J1640$-$4631 are about
4-10 times smaller than those of 1E 2259$+$586 and SGR 0501$+$4516, and the
values of $\eta$ of PSR J1640$-$4631 are 1-2 orders of magnitude less than
those of the two magnetars. Through PSR J1640$-$4631 is a rotationally powered
X/$\gamma$-ray pulsar, it posses a higher dipole magnetic field $\sim 10^{13}$
G and lacks of radio emission. These features are close to magnetars. However,
for a common radio pulsar with typical dipole field $B_{d}^{{}^{\prime}}\sim
10^{12}$ G, its ratio of $\eta^{{}^{\prime}}$ could be 3-5 orders of magnitude
less than that of a canonical magnetar with $B_{d}\sim 10^{14}-10^{15}$ G.

Table 3: Main angle parameters of 1E 2259+586

$\alpha_{f}$ | $\theta_{1i}$ | $\theta_{1f}$ | $\theta_{2i}$ | $\theta_{2f}$ | $\theta_{2}^{\rm min}$ | $F(\Theta)$
---|---|---|---|---|---|---
$51^{\circ}$ | $0^{\circ}$ | $0^{\circ}$ | $-68^{\circ}$ | $-51^{\circ}$ | $0^{\circ}$ | $0.5094$
$51^{\circ}$ | $5^{\circ}$ | $5^{\circ}$ | $-63^{\circ}$ | $-46^{\circ}$ | $-2.5^{\circ}$ | $0.4645$
$51^{\circ}$ | $10^{\circ}$ | $10^{\circ}$ | $-58^{\circ}$ | $-41^{\circ}$ | $-5.0^{\circ}$ | $0.4096$
$51^{\circ}$ | $15^{\circ}$ | $15^{\circ}$ | $-53^{\circ}$ | $-36^{\circ}$ | $-7.6^{\circ}$ | $0.3457$
$51^{\circ}$ | $20^{\circ}$ | $20^{\circ}$ | $-48^{\circ}$ | $-31^{\circ}$ | $-10.3^{\circ}$ | $0.2754$
$51^{\circ}$ | $25^{\circ}$ | $25^{\circ}$ | $-43^{\circ}$ | $-26^{\circ}$ | $-13.1^{\circ}$ | $0.2005$
$51^{\circ}$ | $32^{\circ}$ | $32^{\circ}$ | $-36^{\circ}$ | $-19^{\circ}$ | $-17.4^{\circ}$ | $0.0925$

Table 4:  Partial values of $\alpha_{f}$, $F(\Theta)$, $M_{1}$, $M_{2}$ and
$\eta$ for 1E 2259+586

$\alpha_{f}$ |  $F(\Theta)$ |  $M_{2}$ |  $M_{1}$ |  $\eta$
---|---|---|---|---
|  |  G cm3 |  G cm3 |
$51^{\circ}$ |  0.5094 |  $1.12\times 10^{32}$ |  $4.14\times 10^{4}$ |  $2.70\times 10^{27}$
$51^{\circ}$ |  0.4645 |  $1.12\times 10^{32}$ |  $4.54\times 10^{4}$ |  $2.47\times 10^{27}$
$51^{\circ}$ |  0.4096 |  $1.12\times 10^{32}$ |  $5.15\times 10^{4}$ |  $2.17\times 10^{27}$
$51^{\circ}$ |  0.3457 |  $1.12\times 10^{32}$ |  $4.71\times 10^{4}$ |  $3.09\times 10^{27}$
$51^{\circ}$ |  0.2754 |  $1.12\times 10^{32}$ |  $6.10\times 10^{4}$ |  $1.83\times 10^{27}$
$51^{\circ}$ |  0.2005 |  $1.12\times 10^{32}$ |  $1.05\times 10^{5}$ |  $1.06\times 10^{27}$
$51^{\circ}$ |  0.0925 |  $1.12\times 10^{32}$ |  $2.28\times 10^{5}$ |  $4.91\times 10^{26}$

Figure 3: Relation of $n$ and $\alpha$ for 1E 2259+586.

A simple estimation is given as follows. Firstly, from Eq. (8), we get
$M_{2}/M_{1}=2M^{2}_{2}F(\Theta)/(\dot{\alpha}^{2}I_{2}r^{3})$ in the HSS
model. The differences in EoSs of a common radio pulsar and a magnetar may be
very small, and can be ignored. Assuming a same decrease rate $\dot{\alpha}$,
we give an approximation expression

$\frac{\eta}{\eta^{{}^{\prime}}}\approx\frac{M^{2}_{2}F(\Theta)}{M^{{}^{\prime}2}_{2}F^{{}^{\prime}}(\Theta)}\approx\frac{B^{2}_{d}F(\Theta)}{B^{{}^{\prime}2}_{d}F^{{}^{\prime}}(\Theta)},$
(13)

where $\eta$ and $\eta^{{}^{\prime}}$ are for a magnetar and a common radio
pulsar, respectively.

Table 5:  Partial values of $\dot{\theta}_{2}$, $F(\Theta)$, $M_{1}$ and
$M_{2}$ and $\eta$ for PSR J1640-4631

$\dot{\theta}_{2}$ |  $F(\Theta)$ |  $M_{2}$ |  $M_{1}$ |  $\eta$
---|---|---|---|---
∘/cent |  |  G$\cdot$cm3 |  G$\cdot$cm3 |
0.23 |  0.122 |  $2.7\times 10^{31}$ |  $7.2\times 10^{5}$ |  $3.7\times 10^{25}$
0.23 |  0.097 |  $2.7\times 10^{31}$ |  $9.1\times 10^{5}$ |  $2.9\times 10^{25}$
0.56 |  0.657 |  $2.7\times 10^{31}$ |  $7.9\times 10^{5}$ |  $3.4\times 10^{25}$
0.56 |  0.919 |  $2.7\times 10^{31}$ |  $5.7\times 10^{5}$ |  $4.8\times 10^{25}$

From Eq. (13), it is obvious that the degree of magnetization of a magnetar is
far higher that that of a common radio pulsar. The above equation implies that
there is a difference in the magnetic field formation mechanism (including
paramagnetic magnetization) between rotationally powered pulsars and magnetic-
field powered magnetars during their early ferromagnetic phases. However, the
cause of such a difference is not yet known to date. I hope that this work
will be helpful in understanding the nature of magnatars.

##  4 Summary

We have applied a toy two-dipole model to the two high-$n$ magnetars
SGR0501$+$4516 and 1E 2259$+$586, and ignored other factors influencing their
braking indices, which are attributed to the decrease in inclination angles.
For these two sources, their initial magnetic moments, initial inclination
angles and average decrease rates of inclination angles are estimated and
comparisons with rotationally powered pulsars are presented. Our results could
be useful in understanding the nature of magnetars, and will be tested by
future observations.

### Acknowledgements

This work was supported by the NSFC (11573009, U1831120, U1731238), the
Foundation of Science and Technology of Guizhou Provice([2019]1241,[2017]7349,
[2016]4008, (2017) 5726-37)), Outstanding Postdoctoral Foundation of XinJiang
and the Foudation of Guizhou Provincial Education Department ((2020)003) and
Doctor Starting Up Foundation of Guizhou Normal University (0516134).

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# Isoperimetric Profiles and Regular Embeddings of locally compact groups

Juan Paucar  Institut de Mathématiques de Jussieu-Paris Rive Gauche,
Université de Paris-Cité, 75205 Paris Cedex 13, France<EMAIL_ADDRESS>
###### Abstract.

In this article we extend the notion of $L^{p}$-measured subgroups couplings;
a quantitative asymmetric version of measure equivalence that was introduced
by Delabie, Koivisto, Le Maître and Tessera for finitely generated groups, to
the setting of locally compact compactly generated unimodular groups. As an
example of these couplings; using ideas from Bader and Rosendal, we prove a
”dynamical criteria” for the existence of regular embeddings between amenable
locally compact compactly generated unimodular groups, namely the existence of
an $L^{\infty}$-measured subgroup coupling that is coarsely $m$ to $1$. We
then proceed to prove that the existence of an $L^{p}$-measured subgroup that
is coarsely $m$ to $1$ implies the monotonicity of the $L^{p}$-isoperimetric
profile. We conclude then that the $L^{p}$-isoperimetric profile is monotonous
under regular embeddings, as well as coarse embeddings, between amenable
unimodular locally compact compactly generated groups.

###### Key words and phrases:

Measured Group Theory, Isoperimetric Profile, Regular Embeddings, Locally
compact groups, Polish spaces

##  1\. Introduction

###  1.1. Regular Embeddings and Measured Subgroup Couplings

Let’s consider $\Gamma$ and $\Lambda$ two finitely generated groups with fixed
word metrics. In his seminal article [Gromov], Gromov proved the following
characterization of the existence of quasi-isometries between finitely
generated groups:

######  Proposition 1.1 (Gromov’s Dynamical Criteria).

The following are equivalent:

  1. (1)

$\Gamma$ and $\Lambda$ are quasi-isometric.

  2. (2)

There exists a locally compact topological space $\Omega$ with a continuous
$\Gamma$-action and a continuous $\Lambda$-action that commute with each
other, such that both actions are proper and cocompact.

We will say that such a topological space $\Omega$ is a Topological
Equivalence coupling between $\Gamma$ and $\Lambda$. This proposition
motivated Gromov to introduce the following definition:

######  Definition 1.2.

We say that a standard measured space $(\Omega,\mu)$ is a  Measured
Equivalence coupling between $\Gamma$ and $\Lambda$ if there exists measure
preserving commuting $\Gamma,\Lambda$-actions such that both actions are free
and have fundamental domains of finite measure. We say as well that $\Gamma$
and $\Lambda$ are Measured Equivalent if there exists a Measured Equivalence
coupling between them.

This marked the starting point of Measured Group Theory. For a comprehensive
survey of this vast subject and the links with other areas, see the references
[Gab10] and [Fur].

Furthermore, Proposition 1.1 has also an asymmetric version as well, to be
more precise, let’s first define:

######  Definition 1.3.

We say that $\varphi\colon\Gamma\to\Lambda$ is a coarse embedding from
$\Gamma$ to $\Lambda$ if $\varphi$ is $L$-Lipschitz for some $L>0$ and for any
$\lambda\in\Lambda$ and any $R>0$ there exists $C(R)>0$ such that:

$\text{diam}_{\Gamma}(\varphi^{-1}(B_{\Lambda}(\lambda,R)))\leq C(R)$

Note first, that this definition is equivalent to the definition given in
[Sha04] of uniform embeddings, moreover we have the following characterization
of the existence of coarse embeddings between finitely generated groups:

######  Proposition 1.4 (Theorem 2.1.2 in [Sha04]).

The following are equivalent:

  1. (1)

There exists a coarse embedding from $\Gamma$ to $\Lambda$.

  2. (2)

There exists a locally compact topological space $\Omega$ with a continuous
$\Gamma$-action and a continuous $\Lambda$-action that commute with each
other, such that both actions are proper actions, and the $\Lambda$-action is
cocompact.

We will say then that such a topological space $\Omega$ will be a Topological
Subgroup coupling from $\Gamma$ to $\Lambda$. See [Sau06] for other
consequences of this characterization.

It is then natural to ask ourselves if, following the spirit of Measured Group
Theory, there is a measured analog of this proposition. This turns out to be
the case, as seen in [DKLMT], in this article the authors began an study of
quantitative asymmetric versions of measured equivalence by demanding less
conditions on the actions over the coupling space, which they call measured
subgroup couplings and measured subquotient couplings. Moreover, they also
study quantitative versions of this concept, which includes, for example
$L^{p}$-Measure Equivalence and $L^{p}$-measured subgroups. They also consider
regular embeddings; a notion introduced in [BST12] for graphs and studied for
non-discrete spaces in [HMT20], as the measured analogue of coarse embeddings.

######  Definition 1.5.

We say that $\varphi\colon\Gamma\to\Lambda$ is a regular embedding from
$\Gamma$ to $\Lambda$ if $\varphi$ is $L$-Lipschitz for some $L>0$ and there
exists $m>0$ such that for any $\lambda$ in $\Lambda$:

$\\#\\{\varphi^{-1}(\lambda)\\}\leq m.$

Note that this definition gives us uniform control of the preimages of balls
in a measured sense and not in a metric sense as in the case of a coarse
embedding. For example, the application $\varphi\colon\mathbb{Z}\to\mathbb{Z}$
given by $\varphi(n)=|n|$ is a regular embedding but not a coarse embedding.

In the Theorem 5.4 of [DKLMT], the authors proved the following theorem that
can be considered as a measured version of Proposition 1.4:

######  Theorem 1.6.

Given $\Gamma$ amenable. The following are equivalent:

  1. (1)

There exists a regular embedding from $\Gamma$ to $\Lambda$.

  2. (2)

There exists $(\Omega,\mu)$ a $L^{\infty}$-measured subgroup coupling from
$\Gamma$ to $\Lambda$ that is $m$ to $1$ for some $m>0$.

On the other hand, Bader and Rosendal proved in [BR18], the analog of
Proposition 1.1 for locally compact compactly generated groups. Using their
ideas as inspiration, we extend then the Theorem 1.6 to the locally compact
setting, thus obtaining the first main contribution of this article:

######  Theorem A.

Given $H,G$ two locally compact compactly generated unimodular groups, with
$H$ being amenable, we have the following equivalence:

  1. (1)

There exists a regular embedding from $H$ to $G$.

  2. (2)

There exists a $L^{\infty}$-measured subgroup coupling from $H$ to $G$ that is
coarsely $m$ to $1$ for some $m>0$.

Note that the approach given in [DKLMT], doesn’t translate well to the locally
compact setting, since they use that $\Gamma$ can be well ordered; however, in
the locally compact setting this is not possible to do in a Borel way. It is
also not clear how to obtain an injective map out of a regular embedding
between locally compact groups. This is where the ideas coming from [BR18]
come into play, note however that applying directly their ideas would only
gives us a topological subgroup coupling out of a coarse embedding between
locally compact groups; so in order to obtain results in the ”measured” world,
some extra work is needed, the main difficulty being the proof of Claim 6 that
will use Lemma 2.2 and Proposition 3.2 as its main ingredients.

###  1.2. Monotonicity of the isoperimetric Profile

On the other hand, the $l^{p}$-isoperimetric profile of a group for $p\geq 1$
is a well studied invariant in the discrete setting that measures how
”amenable” is our group. More precisely, given $\Gamma$ a finitely generated
group with fixed generating set $S_{\Gamma}$, for any $f\in l^{p}(\Gamma)$ of
finite support we define the $l^{p}$-gradient of $f$ as:

$||\nabla_{\Gamma}f||^{p}_{p}=\sum_{s\in S_{\Gamma}}||f-s*f||_{p}^{p}$

Let’s define then the $l^{p}$-isoperimetric profile as:

$j_{p,\Gamma}(n)=\sup_{|\textrm{supp}f|\leq
n}\dfrac{||f||_{p}}{||\nabla_{\Gamma}f||_{p}}$

We will be interested in the asymptotic behaviour of this function as $n$ goes
to infinity. To be more precise, given two monotonous real-valued functions
$f$ and $g$ we say that $f$ is asymptotically less than $g$ and write
$f\preccurlyeq g$ if there exists a positive constant $C$ such that
$f(n)=O(g(Cn))$ as $n\to\infty$. We say that $f$ and $g$ are asymptotically
equivalent and write $f\approx g$ if $f\preccurlyeq g$ and $g\preccurlyeq f$.
The asymptotic behavior of $f$ is its equivalence class modulo $\approx$.

Note that the asymptotic behaviour of $j_{p,\Gamma}$ doesn’t depend on the
choice of the generating set $S_{\Gamma}$. Moreover we have that the
$j_{p,\Gamma}$ is unbounded if and only if $\Gamma$ is amenable. In that sense
$j_{p,\Gamma}$ measures how amenable $\Gamma$ is.

Furthermore, $j_{1,\Gamma}$ has the following geometric
interpretation[Coul00]:

$j_{1,\Gamma}(n)=\sup_{|A|\leq n}\dfrac{|A|}{|\partial A|}$

In the theorem 4.3 of [DKLMT], the authors proved the following monotonicity
result:

######  Theorem 1.7.

If there exists a $L^{p}$-measured subgroup coupling from $\Lambda$ to
$\Gamma$ that is $m$ to $1$, we then have that:

$j_{p,\Gamma}\preccurlyeq j_{p,\Lambda}$

On the other hand, in [Tes08], the $L^{p}$-isoperimetric profile of unimodular
amenable locally compact compactly generated groups is introduced. It is
natural then to ask about an analogue of Theorem 1.7 in the locally compact
setting. We prove the analog statement in the locally compact setting, thus
obtaining the second main contribution of the article. Let’s consider in all
this section $H,G$ two unimodular locally compact compactly generated groups,
we then have that:

######  Theorem B.

Suppose there exists an $L^{p}$-measured subgroup coupling from $H$ to $G$,
that is coarsely $m$ to $1$, we then have that:

$j_{p,G}\preccurlyeq j_{p,H}$

It is now clear that we obtain the following corollary:

######  Corollary 1.

If there exists a regular embedding from $H$ to $G$ and $H,G$ are amenable,
then for any $p\geq 1$ we know that:

$j_{p,G}\preccurlyeq j_{p,H}$

On the other hand, it is known that the $L^{2}$-isoperimetric profile and the
return probability to the origin are related under some mild assumptions, this
was seen in the discrete case in [Coul00] and in the locally compact case in
[Tes08]. More precisely we have the following:

######  Proposition 1.8 (Theorem 9.2 in [Tes08]).

Let $(X,d,\mu)$ be a metric measure space and let $P=(P_{x})_{x\in X}$ be a
symmetric viewpoint at scale $h$ on $X$. Consider $\gamma$ defined by:

$t=\int^{1/\gamma(t)}(j_{2,X}(v))^{2}\dfrac{dv}{v}.$

where $j_{2,X}$ is the isoperimetric profile of $X$ with the gradient
$|\cdot|_{P^{2},2}$. If the logarithmic derivative of $\gamma$ has at most
polynomial growth, then there exists a constant such that:

$\gamma(Cn)\leq\sup_{x\in X}p_{x}^{2n}(x)\leq\gamma(n).$

Let’s denote then by $p_{H}^{2n}$ and $p_{G}^{2n}$ the return probability to
the origin with symmetric viewpoints given by Example 3.4 in [Tes08]. Now as
consequence of this theorem we obtain the following corollary:

######  Corollary 2.

Given $H,G$ amenable that satisfy the condition of Proposition 1.8, if there
exists a regular embedding from $H$ to $G$, we then have that:

$p_{G}^{2n}\preccurlyeq p_{H}^{2n}$

Furthermore, we also have the following:

######  Corollary 3.

Given $G$ amenable and unimodular, if there exists a regular embedding from
$G$ to $GL(d,k)$ where $k$ is a finite product of local fields, we obtain
that:

$\log t\preccurlyeq j_{p,G}(t)$

###### Proof.

Since $T(d,k)$ the group of upper triangular matrices in $GL(d,k)$ is
cocompact in $GL(d,k)$, we can suppose that there exists a regular embedding
from $G$ to $T(d,k)$, however $T(d,k)$ is not unimodular. In order to obtain a
regular embedding to an unimodular amenable group, let’s note that $T(d,k)$ is
a semidirect product of $A(d,k)$ and $N(d,k)$, where $A(d,k)$ is abelian and
$N(d,k)$ is nilpotent. Let’s denote by $\sigma\colon
A(d,k)\to\text{Aut}(N(d,k))$ the morphism coming from this semi-direct
product. If we consider $S(d,k)=A(d,k)\ltimes(N(d,k)\times N(d,k))$; given by
$\overline{\sigma}\colon A(d,k)\to\text{Aut}(N(d,k)\times N(d,k))$,
$\overline{\sigma}(a)=(\sigma(a),\sigma(a^{-1}))$, then we obtain that there
exists a regular embedding from $G$ to $S(d,k)$, with $S(d,k)$ being an
unimodular amenable group. We can now apply the theorem B to obtain the
desired corollary. ∎

Moreover, we have the following known proposition that is an important source
of coarse embeddings:

######  Proposition 1.9.

Let’s consider $G$ acting properly and by isometries on a compact pseudo-
riemannian manifold $M$ of signature $(p,q)$, we then have that there exists a
coarse embedding from $G$ to $O(p,q)$.

###### Proof.

This is easy to see using the Proposition 1.4, that is we will provide a
topological subgroup coupling $\Omega$ given by the frame bundle
$\text{Fr}(M)$ of orthonormal basis under the pseudo-riemannian metric on $M$.
It is clear that the $G$-action on $\text{Fr}(M)$ is proper and the
$O(p,q)$-action on $\text{Fr}(M)$ is proper and cocompact, and that both
actions commute with each other. This will provide us with a coarse embedding
from $G$ to $O(p,q)$. ∎

As a consequence we have that:

######  Corollary 4.

Given $G$ an amenable unimodular group that acts properly and by isometries on
a compact pseudo-riemannian manifold, we then have that:

$\log t\preccurlyeq j_{p,G}(t)$

##  2\. Preliminaries

###  2.1. Locally compact second countable group actions

In this subsection we will introduce the necessary background to understand
Borel actions, as well as fix notations needed in the article. In particular,
we say that a Polish topological space is a separable completely metrizable
topological space, and a standard Borel space is a measurable space
$(X,\mathcal{S})$ such that there exists a Polish topology on $X$ such that
$\mathcal{S}$ is the $\sigma$-algebra generated by this topology.

We will be interested on smooth actions, with this we mean the following:

######  Definition 2.1.

Let’s consider $G$ a locally compact second countable group and $X$ a standard
Borel space. We say that a Borel action $G\curvearrowright X$ is  smooth  if
there exists a Borel subset $\mathcal{F}\subset X$ that intersects each orbit
in exactly one point, moreover $\mathcal{F}$ will be called a  fundamental
domain  of this action.

The following theorem due to Effros will be our main tool to prove that a
given action is smooth

######  Lemma 2.2 (Theorem 2.1.14 on [Zim]).

Given $G$ be a locally compact second countable group, $X$ a Polish
topological space and $G\curvearrowright X$ a continuous action. The following
assertions are equivalent:

  1. (1)

For every $x$ in $X$, the map $R_{x}\colon G/G_{x}\to G\cdot x$ is an
homeomorphism.

  2. (2)

The Borel action $G\curvearrowright X$ is smooth.

  3. (3)

All orbits are locally closed; this is, they are the intersection of a open
and a closed subset.

We have as well the following lemma that can be considered as a weaker
”topological” version of the previous lemma:

######  Lemma 2.3.

Given $\Omega$ a locally compact second countable space, let’s consider a
proper cocompact continuous action $G\curvearrowright\Omega$; this is an
action such that $p\colon G\times\Omega\to\Omega\times\Omega$ given by
$p(g,x)=(x,g\cdot x)$ is proper and such that there exists a compact subset
$K$ of $\Omega$ such that $G\cdot K=\Omega$, we then have that this action is
smooth and that $\Omega/G$ is compact.

###### Proof.

Let’s consider any $x$ in $\Omega$, then since $p\colon
G\times\Omega\to\Omega\times\Omega$ is proper and $\Omega$ is locally compact,
then it is closed, which implies that $p(G\times\\{x\\})=\\{x\\}\times G\cdot
x$ is closed, so the orbits are closed which by the lemma 2.2 implies that the
action is smooth. Moreover since the action is proper, we have that $\Omega/G$
is a Haussdorff space and if we denote by $\pi\colon\Omega\to\Omega/G$ the
continuous projection, we have that $\pi(K)=\Omega/G$ which implies that
$\Omega/G$ is a compact Hausdorff space. ∎

Now given a smooth Borel action $G\curvearrowright\Omega$ and a fixed
fundamental domain $\mathcal{F}_{G}$, there is a natural identification that
will be denoted by $i_{G}\colon G\times\mathcal{F}_{G}\to\Omega$ given by
$i_{G}(g,x)=g\cdot x$. Note that $i_{G}$ is an isomorphism of measurable
spaces.

We recall now the following useful lemma from [KKR21]:

######  Lemma 2.4 (Lemma 2.14 on [KKR21]).

Given $G$ be a locally compact second countable group, $\lambda_{G}$ a fixed
left-invariant Haar measure and $\mathcal{F}$ a standard Borel space, let’s
consider the natural action of $G$ on $G\times\mathcal{F}$. We then have the
following:

  1. (1)

If $[\eta]$ is a probability measure class on $\mathcal{F}$ and $[\mu]$ is a
$G$-invariant class of a $\sigma$-finite measure on $G\times\mathcal{F}$ such
that it projects to $\eta$ under the projection $p_{2}\colon
G\times\mathcal{F}\to\mathcal{F}$, we then have that
$[\mu]=[\lambda_{G}\otimes\ \eta]$

  2. (2)

If $\eta$ is a probability on $\mathcal{F}$ and $\mu$ is a $G$-invariant
$\sigma$-finite measure on $G\times\mathcal{F}$ such that
$[(p_{2})_{*}\mu]=[\eta]$, this then implies that, there exists a measurable
function $b\colon\mathcal{F}\to[0,\infty]$ such that $\mu=\lambda_{G}\otimes
b\eta$

It is easy to see that this implies the following lemma:

######  Lemma 2.5.

Given $G$ a locally compact second countable group, $\Omega$ a Polish space, a
free smooth action of $G$ on $\Omega$ with fundamental domain $\mathcal{F}$
and $\mu$ a $G$-invariant $\sigma$-finite measure on $\Omega$, we then have
that there exists a measure $\nu$ on $\mathcal{F}$ such that if we denote
$i_{G}\colon G\times\mathcal{F}\to\Omega$ the natural map coming from the
action, we have that:

$(i_{G})_{*}(\lambda_{G}\otimes\nu)=\mu$

###### Proof.

Consider $\eta$ a probability in the same measure class as
$(p_{2})_{*}(i_{G})^{*}\mu$, and define $\nu$ as $b.\eta$ where $b$ is the
function coming from the lemma 2.4.

∎

To finish this subsection, we cite the following result from [AS93], needed to
obtain free actions out of non-free actions:

######  Lemma 2.6 (Proposition 5.3 on [AS93]).

Given a locally compact second countable group $G$, there exists a compact
metrizable topological space $X$ and $G\curvearrowright X$ a continuous free
action.

###  2.2. Quantitative Measured Couplings of locally compact groups

In this subsection we will extend the notion of quantitative measured subgroup
coupling given in [DKLMT] to the locally compact setting. Moreover, we will
always consider $H,G$ two locally compact compactly generated groups with
$S_{H},S_{G}$ two symmetric compact generating subsets of $H,G$ respectively.
Let’s denote by $|\cdot|_{H},|\cdot|_{G}$ the word length on $H,G$ with
respect to $S_{H},S_{G}$ respectively. This induces compatible metrics
$d_{H},d_{G}$ on $H,G$ respectively that are left-invariant, proper and quasi-
isometric to the word length in each respective group.

######  Convention.

In this subsection, we will denote the smooth actions by $*$ and other actions
by $\cdot$

######  Definition 2.7.

We say that $(\Omega,\mu)$ is a measured subgroup coupling from $H$ to $G$ if
we have two smooth commuting measure preserving actions from $H,G$ on
$(\Omega,\mu)$ with fundamental domains $\mathcal{F}_{H},\mathcal{F}_{G}$
respectively, such that:

  1. (1)

The $G$-action is free;

  2. (2)

The $H$-action is free;

  3. (3)

There is a $\sigma$-finite measure $\nu_{H}$ on $\mathcal{F}_{H}$ and a finite
measure $\nu_{G}$ on $\mathcal{F}_{G}$ such that, the following maps are
isomorphisms of measured spaces

$\displaystyle i_{G}$
$\displaystyle\colon(G\times\mathcal{F}_{G},\mu_{G}\otimes\nu_{G})\to(\Omega,\mu)$
$\displaystyle i_{H}$
$\displaystyle\colon(H\times\mathcal{F}_{H},\mu_{H}\otimes\nu_{G})\to(\Omega,\mu).$

If there exists a measured subgroup coupling from $H$ to $G$ we say that $H$
is a measured subgroup of $G$.

Let’s define as well the induced action of $G$ on $\mathcal{F}_{H}$ as the
natural action of $G$ over $\Omega/H$ viewed under the identification
$s_{H}\colon\Omega/H\to\mathcal{F}_{H}$. In a similar manner, we define as
well the induced action of $H$ on $\mathcal{F}_{G}$, and given any $g$ in $G$
and any $x$ in $\mathcal{F}_{H}$ we will denote by $g\cdot x$ this induced
action. On the other hand the $G,H$-actions on $\Omega$ will be denoted by
$g*x,h*x$ for $g$ in $G$,$h$ in $H$ and $x$ in $\Omega$.

Let’s define the cocycle $c_{G}\colon H\times\mathcal{F}_{G}\to G$ by the
following:

$h*x=c_{G}(h,x)*(h\cdot x)$

Now, the definition of quantitative measured subgroups(see [DKLMT], section
2.4, pag. 16) becomes natural to extend, which we define as:

######  Definition 2.8.

For any $p$ in $[1,\infty]$, we say that $(\Omega,\mu)$ is an $L^{p}$-measured
subgroup coupling from $H$ to $G$ if $(\Omega,\mu)$ is a measured subgroup
coupling from $H$ to $G$ and also we have that:

$\sup_{h\in S_{H}}\lVert|c_{G}(h,\cdot)|_{G}\rVert_{p}<\infty$

where $|c_{G}(h,\cdot)|_{G}$ belongs to $L^{0}(\mathcal{F}_{G},\nu_{G})$,
since $\nu_{G}$ is a finite measure, we have that $L^{p}$ implies $L^{q}$ for
every $p>q$.

###  2.3. Regular Embeddings and Discretizations

In order to pass from the discrete setting to the non-discrete one, it is
usually helpful to define the following notions:

######  Definition 2.9.

Given $(X,d)$ a metric space and a parameter $s>0$, we say that $Y\subset X$
is $s$-discrete if $d(x,y)\geq s$ for all $x,y$ in $Y$ such that $x\neq y$. We
say that $Y\subset X$ is $s$-dense if for all $x$ in $X$, there exists $y\in
Y$ such that $y\in B(x,s)$. We say that $Y\subset X$ is a $s$-discretization
if it is a maximal $s$-discrete subset.

By the maximality condition in the definition of $s$-discretization, we can
see that an $s$-discretization is always $s$-dense as well. More precisely we
have that:

######  Lemma 2.10.

Given any set $A$ in a metric space $(X,d)$, let’s consider $C$ a maximal
$r$-discrete subset of $A$. We then have that:

$A\subset[C]_{r}=\bigcup_{c\in C}B(c,r)$

###### Proof.

Let’s suppose that there is an $a$ in $A$ not belonging to $[C]_{r}$, this
implies that $d(a,c)\geq r$ for all $c$ in $C$, we can then consider $C\cup a$
as a $r$-discret subset of $A$ that contains $A$ contradicting the maximality
of $A$. ∎

######  Definition 2.11.

Given a metric space $(X,d)$ we say it is balanced if for any two
$r_{1}>r_{2}>0$, there exists $N$; depending only on $r_{1},r_{2}$, such that
we can cover any $r_{1}$-ball by at most $N$ $r_{2}$-balls.

Let’s consider the following generalization of regular embeddings of graphs to
non-discrete spaces given in [HMT20]:

######  Definition 2.12 (Subsection 1.1, p.4 in [HMT20]).

Consider $X,Y$ two balanced metric spaces, a map $f\colon X\to Y$ is called a
regular embedding if there exists scales $r,s>0$, $L>0$ and $N\in\mathbb{N}$
such that:

  * •

$d(f(x),f(x^{\prime}))\leq Ld(x,x^{\prime})+L$ for all $x,x^{\prime}$ in $X$
and

  * •

the preimage of each $r$-ball in $Y$ can be covered by at most $N$ $s$-balls
on $X$

It is clear from the definition that quasi-isometries are regular embeddings.
Moreover since we are working with balanced metric spaces, there is no
importance on the scale we use, that is, if a map $f$ is regular at scales
$(r,s)$ it is regular at all scales. It is also clear that the composition of
regular embeddings is also regular.

Note as well that for $G$,$H$ locally compact compactly generated groups,
$(G,d_{G})$ and $(H,d_{H})$ are balanced metric spaces; for that reason we can
fix the scales used in each group, we will consider $s$-discretizations on $H$
and $3$-discretizations on $G$. Now that we have fixed the scales on $G,H$ we
can define the analog of Definition 4.1 in [DKLMT]:

######  Definition 2.13.

Given $(\Omega,\mu)$ a measured subgroup coupling from $H$ to $G$, we say that
$(\Omega,\mu)$ is coarsely $m$ to $1$ if for any $x$ in $\mathcal{F}_{G}$, the
map $c_{x}\colon H\to G$ given by $c_{x}(h)=c_{G}(h^{-1},x)$ satisfies that
the preimage under this map of any ball of radius $3$ in $G$ can be covered
but at most $m$ balls of radius $s$ in $H$.

We will be interested on the following definition as well:

######  Definition 2.14.

We say that $A\subset(X,d)$ is $r$-thick if it’s the union of balls of radius
at least $r$. Since each ball of radius at least $r$ can be covered by balls
of radius $r$, this is the same as saying that there exists $C$ such that
$A=[C]_{r}$.

##  3\. Main Theorem

Let’s first prove the following proposition:

######  Proposition 3.1.

Let $f\colon H\to G$ be a regular embedding between unimodular locally compact
compactly generated groups, if $H$ is amenable, we then have that there exists
$L^{\infty}$-measured subgroup coupling $(\Omega,\mu)$ from $H$ to $G$ that is
coarsely $m$ to $1$ for some $m$.

###### Proof.

Let’s fix $Y$ a $s$-discretization of $G$ and $Z$ a $3$-discretization of $H$,
let’s denote by $\pi\colon G\to Z$ a retraction of $G$ to $Z$, that takes
every element $g$ in $G$ to some $\pi(g)$ in $Z$ with $d(g,\pi(g))\leq 3$. Up
to considering $\pi\circ f$ instead of $f$ we can suppose that $f(Y)\subset
Z$.

Since $f$ is regular, it’s clear as well that for every $z$ in $Z$, the number
of elements of $\\{y\in Y:f(y)=z\\}$ is uniformly bounded by $T$, for some
$T>0$.

Now as in [BR18] we will obtain a $C$-Lipschitz map $\eta\colon
H\to\Delta(G)\subset L^{1}(G,\lambda_{G})$ where $\Delta(G)$ will be a large-
scale analog of $G\times\Delta$ where $\Delta$ is a finite dimensional simplex
and we will build our coupling space $(\Omega,\mu)$ with the help of this map.

More specifically, we can find $(\beta_{y}\colon H\to[0,1])_{y\in Y}$ a family
of $M$-Lipschitz functions for some $M>0$ such that
$\textrm{supp}\beta_{y}=\overline{B}(y,s+1)$ for all $y$ in $Y$ and such that
for all $h$ in $H$:

$\sum_{y}\beta_{y}(h)=1$

This is done in [BR18], by considering for all $y$ in $Y$, $\theta_{y}\colon
H\to[0,s+1]$ given by $\theta_{y}(h)=\max\\{0,s+1-d_{H}(y,h)\\}$, it is clear
that $\theta_{y}$ is $1$-Lipschitz and $\theta_{y}\geq 1$ on $B(y,s)$ and
$\theta_{y}=0$ outside of $B(y,s+1)$. It follows that $\Theta(h)=\sum_{y\in
Y}\theta_{y}(h)$ is a bounded Lipschitz function with $\Theta\geq 1$. We can
get then $\beta_{y}$ by defining as $\beta_{y}=\theta_{y}/\Theta$ for all $y$
in $Y$.

Let’s define then the family of functions $(\alpha_{z})_{z\in Z}\colon
H\to[0,1]$ by:

$\alpha_{z}(h)=\sum_{f(y)=z}\beta_{y}(h)$

Since for every $z$ in $Z$, the number of elements of $\\{y\in Y:f(y)=z\\}$ is
uniformly bounded by $T$, we have that $\alpha_{z}$ is $TM$-Lipschitz for all
$z$ in $Z$. For all $h$ in $H$, we denote by $Y_{h}=\\{y\in
Y:\beta_{y}(h)>0\\}$ and $Z_{h}=\\{z\in Z:\alpha_{y}(h)>0\\}$, we clearly have
that $Z_{h}\subset f(Y_{h})$ which implies that the diameter of $Z_{h}$ is
bounded by $L(\text{diam}(B(y,s+1)))+L=L(2s+2)+L=R$. Now consider $N$ to be
the maximum number of a $3$-discrete subset of $G$ of diameter bounded by $R$.

Let’s define then $\eta\colon H\to L^{1}(G,\lambda_{G})$ given by:

$\eta(h)=\sum_{z\in Z}\alpha_{z}(h)\chi_{zB}$

where $B$ is the unitary closed ball in $G$, and define $\Delta(G)\subset
L^{1}(G,\lambda_{G})$ as:

$\Delta(G)=\left\\{\sum_{i=1}^{m}\alpha_{i}\chi_{z_{i}B}:\\{z_{1},\ldots,z_{m}\\}\text{
is }\text{3-discrete}\text{ of diameter}\leq R,\sum\alpha_{i}=1,\alpha_{i}\geq
0\right\\}$

Since $Z_{h}$ has diameter bounded by $R$ it is clear then that $\eta(h)$
belongs to $\Delta(G)$ for all $h$ in $H$. Furthermore; in Claim 3,p.3 in
[BR18], the authors proved the following claim:

######  Claim 1.

$\Delta(G)$ is locally compact in the $L^{1}$-topology in $L^{1}(G,\mu_{G})$.
In fact:

$[K,\epsilon]=\\{\xi\in\Delta(G):\langle\xi|\chi_{K}\rangle\geq\epsilon\\}$

is compact for every compact set $K\subset G$ and $\epsilon>0$. Moreover,
every compact set is contained in some $[K,\epsilon]$.

Let’s consider the space of maps $\Delta(G)^{H}$ equipped with the product
topology, endowed with two natural commuting actions of $G$ and $H$ given by:

$\displaystyle(g*\xi)(h)$ $\displaystyle=\lambda(g)\xi(h)$
$\displaystyle(h_{1}*\xi)(h)$ $\displaystyle=\xi(h_{1}^{-1}h).$

where $g$ belongs to $G$, $h,h_{1}$ belong to $H$, $\xi$ belongs to
$\Delta(G)^{H}$ and $\lambda$ is the left regular representation of $G$ on
$L^{1}(G,\lambda_{G})$. We then set $\Omega_{0}$ as the subset of
$\Delta(G)^{H}$ given by $\overline{(G\times H)*\zeta}$.

Note that $\zeta$ is a Lipschitz function, to be more precise:

$\displaystyle\left\|\zeta(h)-\zeta(h^{\prime})\right\|_{1}$
$\displaystyle=\left\|\sum_{z\in Z_{h}\cup
Z_{h^{\prime}}}(\alpha_{z}(h)-\alpha(h^{\prime}))\cdot\chi_{zB}\right\|_{1}$
$\displaystyle\leq\sum_{z\in Z_{h}\cup
Z_{h^{\prime}}}|\alpha_{z}(h)-\alpha(h^{\prime})|\cdot\lVert\chi_{zB}\rVert_{1}$
$\displaystyle\leq 2NTMd_{H}(h,h^{\prime}).$

Since $d_{H}$ is left-invariant, we have that any $\xi$ in $(G\times H)*\zeta$
is also $2NTM$-Lipschitz. This then implies that any $\xi$ in $\Omega_{0}$ is
also $2NTM$-Lipschitz.

Note as well, that for any $h,h^{\prime}$ in $H$, we have the following:

$\displaystyle
d_{G}(\textrm{supp}(\zeta(h)),\textrm{supp}(\zeta(h^{\prime})))$
$\displaystyle\leq 2+d_{G}(Z_{h},Z_{h^{\prime}})$ $\displaystyle\leq
2+L(d_{H}(Y_{h},Y_{h^{\prime}}))+L$ $\displaystyle\leq
2+L(d_{H}(B(h,s+1),B(h^{\prime},s+1)))+L$ $\displaystyle\leq
2+L(2s+2+d_{H}(h,h^{\prime}))+L.$

It can be seen as well that the same is true for any $\eta$ in $\Omega_{0}$.

We will consider then $\Omega=X\times\Omega_{0}$, where $X$ is the compact
topological space equipped with a free continuous $G\times H$-action given by
the Lemma 2.6. This makes that there exists two natural commuting
$G,H$-actions on $\Omega$ that are now free. By an abuse of notation, we will
still by $*$ these two actions on $\Omega$. We claim then that:

######  Claim 2.

$\Omega$ is a locally compact space.

###### Proof.

It is clear that we need only to prove that $\Omega_{0}$ is a locally compact
topological space. This follows from the same arguments as [BR18], to be more
precise, given any $\xi$ in $\Omega_{0}$, if we denote by $K$ the support of
$\xi(1_{H})$ we then have that:

$K_{\xi}=\left\\{\eta\in\Omega_{0}:\eta(1_{H})\in[K,1/2]\right\\}$

is a compact neighbourhood of $\xi$, since for any $\eta$ in $K_{\xi}$ we have
that $\textrm{supp}(\eta(1_{H}))\cap K\neq\emptyset$, which then implies that
for any $h$ in $H$ we have that
$\textrm{supp}(\eta(h))\subset[K]_{3(R+2)+L(d_{H}(1,h))}$ and that
$\eta(h)\in[[K]_{3(R+2)+L(d_{H}(1,h))},1]$. ∎

######  Claim 3.

$\Omega$ is a second countable topological space.

###### Proof.

This follows from the same arguments as [KKR21], since $\Delta(G)\subset
L^{1}(G,\lambda_{G})$ is a separable space and $\Omega_{0}\subset
C(H,\Delta(G))$ is locally compact and cosmic(see lemma 6.18 and its proof on
[KKR21] for the definition), we have that $\Omega_{0}$ is a locally compact
second countable Hausdorff topological space. This implies then that $\Omega$
is second countable. ∎

######  Claim 4.

The action $H\curvearrowright\Omega$ is continuous.

###### Proof.

It is only needed to prove that $H\curvearrowright\Omega_{0}$ is continuous,
this follows from the same arguments as in [BR18]. ∎

######  Claim 5.

The action $G\curvearrowright\Omega$ is continuous, proper and cocompact.

###### Proof.

In order to prove that $G\curvearrowright\Omega$ is continuous, we need only
to prove that $G\curvearrowright\Omega_{0}$ is continuous; this follows from
the same arguments as in [BR18].

In order to prove that $G\curvearrowright\Omega$ is proper, we need to prove
that $G\curvearrowright\Omega_{0}$ is proper, let’s consider $\tilde{K}$ a
compact subset of $\Omega_{0}\subset\Delta(G)^{H}$, since the projection onto
the $1_{H}$-coordinate is continuous, we can suppose there exists $K$ a
compact subset of $G$ and $\epsilon>0$ such that:

$\tilde{K}\subset\\{\xi\in\Omega_{0}:\xi(1_{H})\in[K,\epsilon]\\}$

Now, for any $g$ in $\\{g\in G:g\tilde{K}\cap\tilde{K}\neq\emptyset\\}$ we
have that there exists $\xi$ in $\Omega_{0}$ such that
$\xi(1_{H})\in[K,\epsilon]\cap[gK,\epsilon]$. We thus have that
$\textrm{supp}\ \xi(1_{H})\cap K\neq\emptyset$ and $\textrm{supp}\
\xi(1_{H})\cap gK\neq\emptyset$, which allows us to prove that $d_{G}(1,g)\leq
2\text{diam}(K)+\text{diam}(\textrm{supp}(\xi(1_{H})))\leq
2\text{diam}(K)+R+2$. So we conclude that $G\curvearrowright\Omega$ is proper.

In order to prove that $G\curvearrowright\Omega$ is cocompact, let’s consider
$B(1_{H},2R)$, given any $\xi$ in $\Omega_{0}$ since $\textrm{supp}\
\xi(1_{H})$ has diameter bounded by $R+2$, there exists $g$ in $G$ such that
$\textrm{supp}(g*\xi)(1_{H})\subset B(1_{H},R+2)$, so if we define:

$C=\\{\xi\in\Omega_{0}:\xi(1_{H})\in[B(1_{G},R+2),1]\\}$

we have that $G*C=\Omega_{0}$ and that $C$ is a compact neighbourhood by Claim
$2$. This then implies that $G*(X\times C)=\Omega$

∎

######  Claim 6.

The action $H\curvearrowright\Omega$ is a smooth action.

###### Proof.

In order to prove this claim, we will need to prove that all $\xi$ in
$\Omega_{0}$ will be proper maps, more specifically we have that:

######  Proposition 3.2.

For all $\xi$ in $\Omega_{0}$ we have that for any $K$ of diameter bounded by
$r$, there exists $N(r)$ only depending on $r$ such that
$\xi^{-1}([K,\epsilon])$ can be covered by at most $N(r)$ balls of radius $s$.

###### Proof.

Let’s suppose that $\xi_{n}$ in $(G\times H)*\zeta$ converges to $\xi$ in
$\Omega_{0}$, it is possible to see that for each $r>0$ there is a $N(r)$ such
that all $\xi_{n}^{-1}([K,\epsilon/2])$ can be covered by at most $N(r)$ balls
of radius $s/2$. Let’s consider then $C\subset H$ a $s$-discrete maximal
subset of $\xi^{-1}([K,\epsilon])$, suppose that $C$ has cardinality at least
$N(r)+1$ and let’s denote $F\subset C$ some finite subset of cardinality
$N(r)+1$. We then have that since $\langle\chi_{K}|\xi\rangle\geq\epsilon$ and
$\xi_{n}(c)\to\xi(c)$ for all $c\in F$, there exists $n_{c}$ such that for all
$n\geq n_{c}$ we have that $\langle\chi_{K}|\xi_{n}(c)\rangle\geq\epsilon/2$.
If we consider $m=\max_{F}{n_{c}}$ we get that
$c\in\xi_{m}^{-1}([K,\epsilon/2])$ for all $c\in F$. Since $F$ is s-discrete
we have that $|F|\leq N(r)$ which is a contradiction. Therefore, $C$ is finite
and moreover $|C|\leq N(r)$ which implies that $\xi^{-1}([K,\epsilon])$ can be
covered by at most $N(r)$ balls of radius $s$ by the Lemma 2.10. ∎

Moreover, since $[K,\epsilon]$ is a basis of $\Delta(G)$, Proposition 3
implies as well that any $\xi$ in $\Omega_{0}$ is proper when we see it as a
map from $H$ to $\Delta(G)$. Furthermore, for any $(x,\xi)$ in $\Omega$ we
have that the map:

$R_{(x,\xi)}\colon H\to H*(x,\xi)$

given by $R_{(x,\xi)}(h)=(h\cdot x,h*\xi)$; is a proper map; this is so
because any $\tilde{K}$ compact set in $\Omega$ is included on $X\times V$,
with $V=\\{\xi\in\Omega:\xi(1_{H})\in[K,\epsilon]\\}$ for some $K$ compact
subset of $G$ and $\epsilon>0$. Then we have that $R_{(x,\xi)}^{-1}(X\times
V)=\\{h\in
H:\xi(h^{-1})\in[K,\epsilon]\\}=\left(\xi^{-1}([K,\epsilon])\right)^{-1}$
which is compact. This implies that $R_{(x,\xi)}$ is an homeomorphism (see
Lemma 2.2 in [Houd]). Since by the claims $3,4$ we have that $\Omega$ is a
Polish topological space, we can then use the Lemma 2.2 to deduce that
$H\curvearrowright\Omega$ is a smooth action. ∎

Now by the lemma 2.3, we have that the $G$ action is smooth, so let’s fix then
$\mathcal{F}_{G}$ and $\mathcal{F}_{H}$ fundamental domains of the respective
$G$-action and $H$-action respectively. Now in order to construct the measure
$\mu$ on $\Omega$, let’s consider the continuous induced action
$H\curvearrowright\Omega/G$, since the $G$-action is proper cocompact, by
lemma 2.3 we have that $\Omega/G$ is compact, which since $H$ is amenable,
implies that there exists $\nu$ probability measure on $\Omega/G$ that is
$H$-invariant, we then define $\nu_{G}$ as the probability on
$\mathcal{F}_{G}$ given by the identification
$s_{G}\colon\Omega/G\to\mathcal{F}_{G}$, and then define the measure $\mu$ as
$(i_{G})_{*}(\lambda_{G}\otimes\nu_{G})$. We then have that:

######  Claim 7.

$\mu$ is $H$-invariant

###### Proof.

Given any $h$ in $H$, and any measurable $B\subset G,F\subset\mathcal{F}_{G}$,
we need to prove:

$\mu(h^{-1}*{i_{G}(B\times F)})=\mu(i_{G}(B\times F))$

Which is equivalent to:

$\displaystyle\mu(h^{-1}*{i_{G}(B\times F)})$
$\displaystyle=\int_{\Omega}\chi_{h^{-1}*{i_{G}(B\times F)}}d\mu$
$\displaystyle=\int_{G\times\mathcal{F}_{G}}\chi_{B\times
F}(i_{G}^{-1}(h*g*x))d\lambda_{G}(g),d\nu_{G}(x)$
$\displaystyle=\int_{G\times\mathcal{F}_{G}}\chi_{B\times F}(g\cdot
c_{G}(h,x),h\cdot x)d\lambda_{G}(g)d\nu_{G}(x)$
$\displaystyle=\int_{\mathcal{F}_{G}}\lambda_{G}(B)\chi_{F}(h\cdot
x)d\nu_{G}(x)$ $\displaystyle=\lambda_{G}(B)\nu_{G}(F).$

Which implies that $\mu$ is $H$-invariant. ∎

Now since $\mu$ is $H$-invariant, by Lemma 2.5, there exists $\nu_{H}$ a
measure on $\mathcal{F}_{H}$ such that:

$(i_{H})_{*}(\lambda_{H}\otimes\nu_{H})=\mu$

This then implies that $(\Omega,\mu)$ is a measured subgroup coupling from $H$
to $G$. Moreover, we have that:

######  Claim 8.

$(\Omega,\mu)$ is a $L^{\infty}$-measured subgroup coupling.

###### Proof.

Since the $G$-action is cocompact, we can assume that $\mathcal{F}_{G}$ is
precompact. Now, since $S_{H}$ is precompact, we have that
$S_{H}*\mathcal{F}_{G}$ is precompact as well since the $H$-action is
continuous. Furthermore, as the $G$-action is proper we have that:

$\\{g\in G:g*\mathcal{F}_{G}\cap S_{H}*\mathcal{F}_{G}\neq\emptyset\\}$

is precompact, let’s say that it is included in $B_{G}(1_{G},L)$ for some
$L>0$. Now for any $(x,\xi)$ in $\mathcal{F}_{G}$ and any $h$ in $S_{H}$, if
we denote $g=c_{G}(h,x,\xi)$, we have that $h*(x,\xi)=g*(h\cdot(x,\xi))$ which
implies that $g$ belongs to $B_{G}(1_{G},L)$, since $d_{G}$ and $|\cdot|_{G}$
are quasi-isometric, this implies that there exists $C>0$ such that
$|c_{G}(h,x,\xi)|_{G}\leq C$ for all $(x,\xi)$ in $\mathcal{F}_{G}$, which
implies that $(\Omega,\mu)$ is an $L^{\infty}$-measured subgroup coupling.

∎

######  Claim 9.

$(\Omega,\mu)$ is a coarsely $m$ to $1$ measured subgroup coupling for some
$m>0$.

###### Proof.

Since $\mathcal{F}_{G}$ is precompact, the projection of $\mathcal{F}_{G}$
onto the coordinate $1_{H}$ is precompact as well, so there exists
$[B(1,d),\epsilon]$ such that $\mathcal{F}_{G}\subset
X\times\\{\xi\in\Omega_{0}:\xi(1_{H})\in[B(1,d),\epsilon]\\}$. Now given any
$(x,\xi)$ in $\mathcal{F}_{G}$ and any $g$ in $G$, let’s consider $S$ to be
$c_{G}(\cdot,x,\xi)^{-1}(B(g,3))$. Now for any $h$ in $S$, there exists some
$\overline{g}$ in $B_{G}(g,3)$ such that:

$h*(x,\xi)=\overline{g}*(h\cdot(x,\xi))$

This then implies that
$\xi(h^{-1})\in\lambda(\overline{g})([B_{G}(1_{G},d),\epsilon])\subset[B_{G}(g,d+3),\epsilon]$
which by proposition 3.2 implies that there exists $m$ such that $S^{-1}$ can
be covered by at most $m$ balls of radius $s$. This then implies that
$(\Omega,\mu)$ is coarsely $m$ to $1$.

∎

∎

We can now prove the announced theorem at the beginning of the article:

######  Theorem 3.3.

Given $H$,$G$ two locally compact compactly generated unimodular groups, with
$H$ being amenable, the following are equivalent:

  1. (1)

There exists a regular embedding from $H$ to $G$.

  2. (2)

There exists a $L^{\infty}$-measured subgroup coupling $(\Omega,\mu)$ from $H$
to $G$ that is coarsely $m$ to $1$.

###### Proof.

The fact that $(1)$ implies $(2)$ comes from the last proposition, so we only
need to prove the converse. Suppose then that we have $(\Omega,\mu)$ a
$L^{\infty}$-measured subgroup coupling from $H$ to $G$ that is coarsely $m$
to $1$. Since this implies that for any $x$ in $\mathcal{F}_{G}$ the map
$c_{x}\colon H\to G$ given by $c_{x}(h)=c_{G}(h^{-1},x)$ satisfies the second
condition on the definition of a regular embedding, all we need to prove is
that this map satisfies the first condition, now since the coupling is
$L^{\infty}$ we know there exists some $C>0$ such that for any $h$ in $S_{H}$
and any $x$ in $\mathcal{F}_{G}$:

$||c(h,x)||_{G}\leq C$

Using this and the cocycle condition, for any $h$ in $H$, we then have that:

$||c_{G}(h,x)||_{G}\leq C||h||_{H}$

Then for any $h_{1},h_{2}$ in $H$, we then have that $||c_{x}(h_{1})||_{G}\leq
C||h_{1}||_{H}$ and that $||c_{x}(h_{2})||_{G}\leq C||h_{2}||_{H}$, which then
implies that:

$||c_{x}(h_{1})^{-1}c_{x}(h_{2})||_{G}\leq C||h_{1}||_{H}+C||h_{2}||_{H}$

Since $d_{G}$ is quasi-isometric to $|\cdot|_{G}$ and $d_{H}$ is quasi-
isometric to $|\cdot|_{H}$ we then have that there exists $L$ such that:

$d_{G}(c_{x}(h_{1}),c_{x}(h_{2}))\leq Ld_{H}(h_{1},h_{2})+L$

∎

##  4\. Monotonocity of Isoperimetric Profile

###  4.1. Isoperimetric profile of locally compact groups

Given $G$ a locally compact unimodular group with compact generating symmetric
subset $S_{G}$, with fixed Haar measure $\lambda_{G}$; acting by isometries on
a $L^{p}$-space $E$, we define the $p$-gradient of a function $f$ in $E$ as:

$||\nabla_{G}f||_{p}=\sup_{s\in S_{G}}||f-s*f||_{p}$

The following lemma will be useful on the next subsection, so we prove it
here:

######  Lemma 4.1.

Let’s consider $f$ in $E$, some $L^{p}$ space, for any $g$ in $G$, we have the
following:

$||f-g\ast f||_{p}\leq|g|_{G}||\nabla_{G}f||_{p}$

###### Proof.

By the definition of $|g|_{G}=n$, we have that $g=s_{1}\cdot s_{2}\cdots
s_{n}$ where each $s_{i}$ belongs to $S_{G}$ for $i$ in $\\{1,\cdots,n\\}$. It
is clear then that:

$\displaystyle||f-g\ast f||_{p}$ $\displaystyle=\sum_{i=0}^{n-1}||s_{1}\cdots
s_{i}\ast f-s_{1}\cdots s_{i+1}\ast f||_{p}$
$\displaystyle=\sum_{i=0}^{n-1}||f-s_{i+1}\ast f||_{p}$ $\displaystyle\leq
n||\nabla_{G}f||_{p}=|g|_{G}||\nabla_{G}f||_{p}$

∎

Now, in order to define the $L^{p}$-isoperimetric profile of $G$, we will
consider either the gradient coming from the left regular representation
$\lambda:G\curvearrowright L^{p}(G)$, which will be denoted by
$||\nabla_{G}^{l}f||_{p}$ or the right regular representation $\rho\colon
G\curvearrowright L^{p}(G)$ which will be denoted $||\nabla_{G}^{r}f||_{p}$.

Now, given any subset $A$ of $G$ of finite measure, we define the right
$L^{p}$-isoperimetric profile of $A$ as:

$J_{p,G}^{r}(A)=\sup_{f\in
L^{p}(A)}\dfrac{||f||_{p}}{||\nabla^{r}_{G}f||_{p}}$

and we define the right $L^{p}$-isoperimetric profile of $G$ as:

$j_{p,G}^{r}(v)=\sup_{\lambda_{G}(A)\leq v}J_{p,G}^{r}(A)$

We define in a similar manner the left isoperimetric profile. Note that, since
$G$ is unimodular, the right and left isoperimetric profiles coincide.
Moreover, it can be seen as well, that in order to compute $J^{r}_{p,G}(A)$,
we can restrict ourselves to functions in $L^{\infty}(A)$. More precisely:

######  Proposition 4.2.

If we define:

$\overline{J}_{p,G}(A)=\sup_{f\in
L^{\infty}(A)}\dfrac{||f||_{p}}{||\nabla_{G}^{r}f||_{p}}$

we obtain that $\overline{J}_{p,G}(A)=J_{p,G}(A)$.

###### Proof.

It is clear that $J_{p,G}(A)\geq\overline{J}_{p,G}(A)$. In order to prove the
inverse inequality, let’s consider any $f$ in $L^{p}(A)$, there exists a
sequence $(f_{n})_{n\in\mathbb{N}}$ in $L^{\infty}(A)$ such that $f_{n}\to f$
in $L^{p}(A)$. For any $s$ in $S_{G}$, we have that:

$\big{|}||f_{n}-\rho(s)f_{n}||_{p}-||f-\rho(s)f||_{p}\big{|}\leq
2||f_{n}-f||_{p}$

So, for any $\epsilon>0$ there exists $n_{0}$, such that for all $n\geq n_{0}$
and all $s\in S_{G}$, we have that:

$||f-\rho(s)f||_{p}-\epsilon\leq||f_{n}-\rho(s)f_{n}||_{p}\leq||f-\rho(s)f||_{p}+\epsilon$

Which then implies that for all $n\geq n_{0}$ we have that:

$||\nabla_{G}f||_{p}-\epsilon\leq||\nabla_{G}f_{n}||_{p}\leq||\nabla_{G}f||_{p}+\epsilon$

So, $||\nabla_{G}f_{n}||_{p}\to||\nabla_{G}f||_{p}$ when $n\to\infty$. Then,
it’s clear to see that:

$||f_{n}||_{p}\leq\overline{J}_{p,G}(A)||\nabla_{G}f_{n}||_{p}.$

Which then implies that:

$||f||_{p}\leq\overline{J}_{p,G}(A)||\nabla_{G}f||_{p}.$

For all $f$ in $L^{p}(A)$, so $J_{p,G}(A)\leq\overline{J}_{p,G}(A)$. ∎

Furthermore, we will need to restrict ourselves to thick subsets as well, in
order to do that, we will use the other two natural notions of gradients
introduced in [Tes08]. More precisely, for any $h>0$, any subset $A$ of finite
measure and any $f\in L^{\infty}(A)$ we obtain the following 3 notions of
gradients:

$\displaystyle||\nabla^{1}_{h}f||^{p}_{p}$ $\displaystyle=\int_{G}\sup_{s\in
B(1,h)}\left|f(gs)-f(g)\right|^{p}d\lambda_{G}(g)$
$\displaystyle||\nabla_{h}f||_{p}^{p}$ $\displaystyle=\sup_{s\in
B(1,h)}\int_{G}|f(gs)-f(g)|^{p}d\lambda_{G}(g)$
$\displaystyle||\nabla^{2}_{h}f||_{p}^{p}$ $\displaystyle=\int_{G}\fint_{s\in
B(1,h)}\left|f(gs)-f(g)\right|^{p}d\lambda_{G}(s)d\lambda_{G}(g).$

The gradient $||\nabla^{1}||_{p}$ was introduced in [Tes08], p.7, as $|\nabla
f|_{h}$; while the gradient $||\nabla^{2}||_{p}$ was introduced in the same
article as $|\nabla_{P,p}|$ for
$dP_{x}=\frac{1}{\lambda_{G}(B(1,h))}1_{B(x,h)}d\lambda_{G}$. Now, it is easy
to see that we have the following inequalities:

$||\nabla^{1}_{h}f||_{p}\geq||\nabla_{h}f||_{p}\geq||\nabla_{h}^{2}f||_{p}$

Now, it is proven in the Proposition 7.2 in [Tes08] that the asymptotic
behaviour of the respective isoperimetric profiles under $\nabla^{1}$ and
$\nabla^{2}$ coincide, and in the Proposition 10.1 in [Tes08] that the
isoperimetric profile under $\nabla^{1}$ doesn’t depend on the parameter $h$,
provided $h$ is large enough. Moreover, the following lemma is proven in the
same article:

######  Lemma 4.3 (Proposition 8.3 in [Tes08]).

There exists $C>0$ such that for any $f$ in $L^{\infty}(A)$, there is a
function $\tilde{f}$ in $L^{\infty}(\tilde{A})$ such that $\tilde{A}$ is
$h$-thick and we have that:

$\displaystyle\lambda_{G}(\tilde{A})$ $\displaystyle\leq\lambda_{G}(A)+C$
$\displaystyle\dfrac{||f||_{p}}{||\nabla^{1}_{2h}f||_{p}}$ $\displaystyle\leq
C\dfrac{||\tilde{f}||_{p}}{||\nabla_{h}^{1}\tilde{f}||_{p}}$

Let’s consider $h$ large enough that the isoperimetric profile under
$\nabla^{1}$ doesn’t depend on $h$, and such that $S_{G}\subset B(1,h)$. We
can then restrict ourselves to $h$-thick subsets, more precisely:

######  Lemma 4.4.

Let’s define:

$\tilde{j}_{h,G}(v)=\sup_{\lambda_{G}(A)\leq v,\textrm{A is
}h-\textrm{thick}}\overline{J}_{p,G}(A).$

We then have that $\tilde{j}_{h,G}\approx j^{r}_{G}$.

###### Proof.

Let’s denote by $j^{1}_{h}$ and $j^{2}_{h}$ the isoperimetric profiles with
respect to $\nabla_{h}^{1}$ and $\nabla_{h}^{2}$ respectively. It’s clear that
$\tilde{j}_{h,G}\leq j^{r}_{p,G}$, so in order to prove the converse
inequality, consider $f$ in $L^{\infty}(A)$ with $\lambda_{G}(A)\leq v$. By
the Lemma 4.3 and since
$||\nabla_{h}^{1}||_{p}\geq||\nabla_{h}||_{p}\geq||\nabla_{S_{G}}||_{p}$ we
have that:

$\dfrac{||f||_{p}}{||\nabla_{2h}^{1}f||_{p}}\leq C\tilde{j}_{h,G}(v+C)$

Which then implies that $j^{1}_{2h}\preccurlyeq\tilde{j}_{h,G}$. Since
$j^{1}_{2h}\approx j^{1}_{h}\approx j^{r}_{p,G}$ we obtain the desired result.
∎

###  4.2. Monotonicity of Isoperimetric Profile

In all of this subsection, we consider $G,H$ being locally compact unimodular
groups with fixed compact generating symmetric subsets $S_{G},S_{H}$
respectively. We will consider $(\Omega,\mu)$ a coarsely $m$ to $1$,
$L^{p}$-measured subgroup coupling from $H$ to $G$ with the same notation from
the Definition 2.7.

######  Lemma 4.5.

For any $f$ in $L^{p}(G,\lambda_{G})$ with $\textrm{supp}f$ of finite measure,
let’s consider $\tilde{f}$ in $L^{p}(\Omega,\mu)$ defined by:

$\tilde{f}(g\ast x)=f(g)$

for all $g$ in $G$ and $x$ in $\mathcal{F}_{G}$. We then have that
$||\tilde{f}||_{p}^{p}=\nu_{G}(\mathcal{F}_{G})\cdot||f||_{p}^{p}$ and that:

$||\nabla_{H}\tilde{f}||^{p}_{p}\leq C||\nabla_{G}^{r}f||^{p}_{p}$

where $C=\sup_{s\in S_{H}}\int_{\mathcal{F}_{G}}|c(s,x)|_{G}^{p}d\nu_{G}(x)$.

###### Proof.

Given any $s$ in $S_{H}$, we have that:

$\displaystyle||\tilde{f}-s^{-1}\ast\tilde{f}||_{p}^{p}$
$\displaystyle=\int_{\Omega}|\tilde{f}(\omega)-\tilde{f}(s\ast\omega)|^{p}d\mu(\omega)$
$\displaystyle=\int_{G\times\mathcal{F}_{G}}|\tilde{f}(g\ast
x)-\tilde{f}(s\ast g\ast x)|^{p}d\lambda_{G}(g)d\nu_{G}(x).$

However since both actions commute, we have that $s\ast g\ast x=g\ast s\ast
x$; this implies that $s\ast g\ast x=(g\cdot c(s,x))\ast(s\cdot x)$ with
$g\ast c(s,x)$ in $G$ and $s\cdot x$ in $\mathcal{F}_{G}$ which in turn
implies that:

$\tilde{f}(s\ast g\ast x)=f(g\cdot c(s,x)).$

This then by Lemma 4.1 implies that:

$\displaystyle||\tilde{f}-s^{-1}\ast\tilde{f}||_{p}^{p}$
$\displaystyle=\int_{G\times\mathcal{F}_{G}}|f(g)-f(g\cdot
c(s,x))|^{p}d\lambda_{G}(g)d\nu_{G}(x).$
$\displaystyle\leq\int_{\mathcal{F}_{G}}|c(s,x)|_{G}^{p}||\nabla^{r}_{G}f||^{p}_{p}d\nu_{G}(x)$
$\displaystyle\leq C||\nabla_{G}^{r}f||_{p}^{p}.$

Which implies the desired result. ∎

######  Lemma 4.6.

Let $f\colon H\to G$ be an application such that the preimage of any ball in
$G$ of radius $4h$ can be covered by at most $m$ balls of radius $s$ in $H$.
If $A$ is $h$-thick, that is, it is the union of balls of radius at least $h$,
and of finite measure; we then have that for some $C_{1}>0$:

$\lambda_{H}(f^{-1}(A))\leq C_{1}.\lambda_{G}(A)$

###### Proof.

Since $A$ is $h$-thick we can suppose that there exists $C$ such that
$A=[C]_{h}$. Let’s consider $Z$ a maximal $3h$-discrete subset of $C$, by the
lemma 2.10, we have that $C\subset[Z]_{3h}$, which then implies
$A\subset[Z]_{4h}$. Moreover, let’s note that since $Z$ is $3h$-discrete, we
also have that $[Z]_{h}\subset A$ which implies that:

$\\#Z\cdot\lambda_{G}(B(1_{G},h))\leq\lambda_{G}(A)$

By the hypothesis on $f$ we have that $\lambda_{H}(f^{-1}(A))\leq
m\\#Z\cdot\lambda_{H}(B(1_{H},s))$ which implies that for
$C_{1}=m\cdot\lambda_{H}(B(1_{H},s))/\lambda_{G}(B(1_{G},h))$, we have that:

$\lambda_{H}(f^{-1}(A))\leq C_{1}.\lambda_{G}(A)$

∎

Let’s consider then the family of functions
$(\tilde{f}_{x})_{x\in\mathcal{F}_{G}}\colon H\to\mathbb{R}$ given by
$\tilde{f}_{x}(h)=\tilde{f}(h*x)$, note that since $(\Omega,\mu)$ is a
coarsely $m$ to $1$ coupling if $\textrm{supp}f\subset A$ and $A$ is
$h$-thick, by the lemma 4.6 we have that there exists $C_{1}>0$ such that
$\lambda_{H}(c_{x}^{-1}(A))\leq C_{1}\lambda_{G}(A)$. Since
$\textrm{supp}\tilde{f}_{x}\subset(c_{x}^{-1}(A))^{-1}$ and $H$ is unimodular,
we then have that for all $x$ in $\mathcal{F}_{G}$:

$\lambda_{H}(\textrm{supp}\tilde{f}_{x})\leq C_{1}\cdot\lambda_{G}(A)$

######  Theorem 4.7.

Let $H$ be a coarsely $m$ to $1$ $L^{p}$-measure subgroup of $G$, we then have
that:

$j_{p,G}\preccurlyeq j_{p,H}$

###### Proof.

Let’s consider $\epsilon>0$ and $A$ a $h$-thick subset of $G$ of finite
measure, there exists then $f$ in $L^{p}(G,\mu_{G})$ such that
$\textrm{supp}(f)\subset A$ and:

$J_{p,G}^{r}(A)-\epsilon\leq\dfrac{||f||_{p}}{||\nabla_{G}^{r}f||_{p}}\leq
J_{p,G}^{r}(A)$

Using Lemma 4.5 we have that, there exists a constant $\overline{C}$ such
that:

$\dfrac{||\nabla_{H}\tilde{f}||_{p}}{||\tilde{f}||_{p}}\leq\overline{C}\dfrac{||\nabla_{G}^{r}f||_{p}}{||f||_{p}}.$
(1)

Furthermore, this implies that there exists $Z\subset\mathcal{F}_{H}$ with
$\nu_{H}(Z)>0$ such that:

$||\nabla_{H}^{l}\tilde{f}_{y}||_{p}<2\overline{C}\dfrac{||\nabla_{G}^{r}f||_{p}}{||f||_{p}}||\tilde{f}_{y}||_{p}$

for all $y$ in $Z$. This is true, since if that were not the case, we would
have that for all $y$ in conull subset of $\mathcal{F}_{H}$:

$||\nabla_{H}^{l}\tilde{f}_{y}||_{p}\geq
2\overline{C}\dfrac{||\nabla_{G}^{r}f||_{p}}{||f||_{p}}||\tilde{f}_{y}||_{p}$

Which by integrating over $\mathcal{F}_{H}$ would imply that:

$||\nabla_{H}\tilde{f}||\geq
2\overline{C}\dfrac{||\nabla_{G}^{r}f||_{p}}{||f||_{p}}||\tilde{f}||_{p}$

that contradicts the inequality (1). For any $y$ in $Z$, we have that
$\tilde{f}_{y}$ is nonzero. Moreover, since $(\Omega,\mu)$ is a coarsely
$m$-to-$1$ coupling, the Lemma 4.6 implies that $\textrm{supp}\tilde{f}_{y}$
has finite measure, which implies that
$||\nabla^{l}_{H}\tilde{f}_{y}||_{p}>0$. This implies that for any $y$ in $Z$
we have that:

$\dfrac{||f||_{p}}{||\nabla^{r}_{G}f||_{p}}\leq
2\overline{C}\dfrac{||\tilde{f}_{y}||_{p}}{||\nabla_{H}^{l}\tilde{f}_{y}||_{p}}$

Furthermore, we get:

$J_{p}^{r}(A)-\epsilon\leq 2\overline{C}J^{l}_{p}(C_{1}v).$

Since we can make $\epsilon$ converge to $0$, we obtain:

$J_{p}^{r}(A)\leq 2\overline{C}J^{l}_{p}(C_{1}v).$

Which then implies that:

$j_{p,G}^{r}\preccurlyeq j^{l}_{p,H}$

∎

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