Higher flow harmonics of strange hadrons in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV and Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
with HYDJET++ model

Gauri Devi1, Arpit Singh1, Saraswati Pandey1, B. K. Singh1,2*,

1Department of Physics, Institute of Science, Banaras Hindu University (BHU),
Varanasi, 221005, India
2Discipline of Natural Sciences, PDPM Indian Institute of Information
Technology Design & Manufacturing, Jabalpur, 482005, India

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###### Abstract

Using the HYDJET++ model, we measure the higher-order flow harmonics
$v_{n}$(n=2,3,4) of (multi-) strange hadrons in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV and Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV.
We have compared our model results and experimental data at RHIC and LHC
energies. The model reproduces the higher flow harmonics of (multi-) strange
hadrons as a function of $p_{T}$, centrality, and quark content. We have
studied and discussed mass ordering of flows $v_{n}$(n=2,3,4) among
$\pi^{+}+\pi^{-}$, $K^{-}+K^{+}$, $K_{s}^{0}$, $p+\overline{p}$,
$\Lambda+\overline{\Lambda}$, $\Xi^{-}+\overline{\Xi}^{+}$,
$\Omega^{-}+\overline{\Omega}^{+}$ at low $p_{T}$ and the baryon-meson
grouping at intermediate $p_{T}$. NCQ scaling using the HYDJET++ model and its
comparison with experimental results at RHIC and LHC energies have also been
discussed. Study of flow harmonics behaviour in RHIC and LHC energy regimes
will provide an additional insight into the dynamics of anisotropic flow and
the effect of radial flow expansion in the system.

##  1 Introduction

The relativistic heavy-ion collision experiments aim to explore the Quark-
Gluon Plasma (QGP) formed at high densities and temperatures [1, 2, 3, 4, 5,
6, 7]. Among various signals [8] suggested to probe the QGP matter,
anisotropic flow [9] serves as a powerful tool in non-central heavy-ion
collisions to study the properties and dynamics of QGP. The anisotropic flow
arises in the final hadronic distribution from the spatial asymmetry in the
initial geometry of the collision transformed into the momentum anisotropy
combined with the initial state inhomogeneities of the system’s energy
density. The anisotropic flow is calculated in the form of Fourier series
expansion with respect to the event plane as:

$\frac{dN}{d\phi}=1+2\sum_{n=1}^{\infty}v_{n}\cos[n(\phi-\psi_{n})]$ (1)

The $n^{th}$ order flow harmonics $v_{n}$ are represented by the Fourier
coefficients as :

$v_{n}=\langle\langle\cos[n(\phi-\psi_{n})]\rangle\rangle$ (2)

where $\phi$ is the azimuthal angle of the produced particles, n is the
harmonic order of flow coefficients and $\psi_{n}$ is the azimuthal angle of
the reaction plane. The averaging is performed over all the particles in a
single event and over all the events. The elliptic flow $v_{2}$ arises from
the almond-shaped geometry of the interaction volume being, the largest
contribution to the asymmetry of non-central collisions. The triangular flow
$v_{3}$ arises from fluctuations in the initial distribution of particles [10,
11, 12, 13]. The quadrangular flow $v_{4}$ arises from both, the asymmetry of
non-central collisions and from fluctuations. The higher flow harmonics
$v_{n}(n>2)$ arise from the event-by-event fluctuations in the initial
distribution of nucleons in overlap region and non-viscous effects [14], [15].
They are more sensitive to transport coefficients ($\eta/s$) than elliptic
flow [4, 12].

In the past few decades, different facets of (multi-) strange hadrons have
been explored theoretically [16, 17, 5, 18] and experimentally [1, 4, 19, 20,
21, 5, 22, 23, 24] in heavy-ion collisions. The main motivation of our
presented work is to study anisotropic flow $v_{n}$(n=2,3,4) of (multi-)
strange hadrons. Strange hadrons, especially having more than one valence
strange quark, are considered as a good probe to study the collective
properties of the medium created in high energy heavy-ion collisions [24, 18].
Anisotropic flow harmonics are transverse momentum ($p_{T}$) dependent and
produce a hump structure [25, 26, 27]. It increases monotonically with $p_{T}$
up to 2-3 GeV/c where a maximum is achieved and then decreases. The position
of this maxima depends on collision centrality and particle species [1, 21,
25]. Flow harmonics also show mass ordering at low $p_{T}$ i.e.,
$v_{n}^{mesons}>v_{n}^{baryons}$ [28, 29]. A characteristic mass dependence of
$p_{T}$\- differential $v_{n}$ occurs as a consequence of the interplay
between the azimuthally symmetric radial flow and anisotropic flow, $v_{n}$
(n$\geq$ 2). Mass ordering in anisotropic flow at RHIC (the Relativistic Heavy
Ion Collider) [30, 31], and LHC (the Large Hadron Collider) [32, 33] has been
observed at $p_{T}<$ 2-3 GeV/c [21, 1, 34]. At high $p_{T}$, particle-type
dependence was observed i.e., $v_{n}^{baryons}>v_{n}^{mesons}$ [5, 20, 21, 1,
4, 35].

The collective behaviour of (multi-) strange hadrons is weaker than that of
the light strange and non-strange hadrons. Since the hadronic interaction
cross-section of multi-strange hadrons with their large mass is expected to be
small, their freeze-out temperatures are close to the quark-hadron transition
temperature as predicted by lattice QCD [4, 24, 19]. Other studies on the
behaviour of strange hadrons based on temperature effects have been performed
using the hydrodynamic-inspired models [36]. It shows that (multi-) strange
baryons thermally freeze-out close to the point where chemical freeze-out
occurs with $T_{ch}$=160-165 MeV [22], which at these collision energies
coincides with the critical temperature $T_{c}$. In addition, according to the
Quark-Coalesence mechanism [37], [38], strange quarks may have a smaller flow
than light quarks at high $p_{T}$ because heavy strange quarks are expected to
lose less energy in nuclear medium [38]. Thus, the flow harmonics for strange
and non-strange hadrons have same qualitative behaviour for u, d, and s
quarks; however, the strength could be comparable [17, 20]. Moreover, compared
to non-strange hadrons, flow for (multi-) strange hadrons provide more
information about the partonic stage than the hadronic stage due to the small
hadronic cross-section of strange hadrons. This can be visualized by studying
NCQ scaling [19, 20, 21] of anisotropic flow.

In recent times, studies have been performed on the flow coefficients of all
charged (both strange and non-strange) and identified hadrons at RHIC [4, 19,
24, 20, 39] and LHC [1, 21, 25, 29, 35, 40, 32] using different models such as
AMPT [4, 19], [11], VISHNU [34, 41, 42], ideal Hydrodynamic model [4, 43], and
HYDJET++ [26, 39, 45, 35, 46, 44, 47] models using by several analysis methods
like reaction-plane method $v_{n}\\{RP\\}$, sub-event plane method
$v_{n}^{sub}\\{EP\\}$, multi-particle cumulant method $v_{n}\\{2,3,4\\}$, and
LYZ method $v_{n}\\{LYZ\\}$ [17, 11, 34, 41, 48, 12, 7].

In this article, we have used the Monte Carlo HYDJET++ model to analyze higher
flow harmonics ($v_{2}-v_{4}$) as a function of transverse momentum and
centrality for (multi-) strange hadrons at RHIC and LHC energies. A brief
description of HYDJET++ model is discussed in section 2. In section 3, results
obtained by HYDJET++ model on anisotropic flow coefficients are discussed
along with its mass ordering and NCQ scaling ($n_{q}$ is the number of
constituent quarks). At last, we have presented the summary of our study in
Section 4.

##  2 Model

HYDJET++ model is a Monte Carlo event generator that simulates relativistic
heavy-ion collisions in an event-by-event basis [49, 50, 51]. A detailed
description of the used physics and the simulation procedure is provided in
the HYDJET++ manual [52, 50]. HYDJET++ is primarily composed of two
independent components which simulate soft (low $p_{T}$ physics) and hard
processes (high $p_{T}$ physics) [51]. The soft part provides the
hydrodynamical evolution of the system while the hard part describes
multiparton fragmentation within the formed medium. The soft state in HYDJET++
is a thermal hadronic state generated on the chemical and thermal freezeout
hypersurfaces obtained from a parameterization of relativistic hydrodynamics
with preset freezeout conditions provided by the FASTMC [53, 54] generator.
The soft part includes the generation of the 4-momentum of hadrons, fluid flow
4-velocity, and the two- and three-body decays of resonances with branching
ratios taken from the SHARE [55] particle decay table. The hard part of the
model consists of PYTHIA [56] and PYQUEN [49] event generators. These
generators simulate initial parton-parton collisions, the radiative energy
loss of partons, and parton hadronization. The model has several free
parameters for the soft component and can be fixed by fitting multi-parameters
at the preset freezeout surface and calculating various observations at RHIC
and LHC.

Figure 1:  The first row ((a)-(c)) represent the higher harmonic coefficients
$v_{n}$ of charged hadrons at mid-rapidity ($|y|<$ 1.0) as a function of
transverse momentum $p_{T}$ at $\sqrt{s_{NN}}$= 200 GeV in Au+Au collisions
and the second row ((a)-(c)) represent $v_{n}$ of charged hadrons at mid-
rapidity ($|y|<$ 0.5) with respect to $p_{T}$ at $\sqrt{s_{NN}}$= 2.76 TeV in
Pb+Pb collisions. The solid markers represent the HYDJET++ model results while
the open markers represent experimental data from the PHENIX [57] and ATLAS
[58] collaborations.

The hard part includes the generation of multi-jets according to the binomial
distribution in PYQUEN. PYQUEN includes generating initial parton spectra with
PYTHIA and production vertex at a given impact parameter followed by the
rescattering-by-rescattering simulation of the parton path in a dense zone
where the partons experience radiative and collisional energy losses. Then,
the final hadronization of hard partons and in-medium emitted gluons takes
place in accordance with the Lund string model. The average number of jets
produced in an AA event collision is calculated as a product of the number of
binary NN collisions at a given impact parameter per the integral cross-
section of the hard process in NN collisions with the minimum transverse
momentum transfer $p_{T}^{min}$.

Within the soft part of HYDJET++, anisotropic flow arises from the
corresponding spatial eccentricities of the overlap zone of the fireball. The
flow fluctuations in HYDJET++ arise from fluctuations in particle
multiplicity, decays of resonances, and the production of (mini-)jets
[26],[15]. The second harmonic coefficient $v_{2}$ is calculated using the
initial state coordinate anisotropy($\epsilon_{0}(b)$) parameters and momentum
anisotropy ($\delta(b)$) parameters in the model. The third harmonic
coefficient $v_{3}$ is calculated by the spatial triangularity parameter
$\epsilon_{3}(b)$. These parameters can be treated either independently for
each centrality or related to each other through the dependency on initial
ellipticity $\epsilon_{0}(b)=b/2R_{A}$, where $R_{A}$ is the nucleus radius.
During the evolution, the initial system coordinates anisotropy
$\epsilon_{0}(b)$ is converted into the momentum anisotropy $\delta(b)$ in the
transverse plane at the freezeout surface [51]. The anisotropy eccentricity
parameters are related to the transverse radius of the fireball by the
equations :

$R_{ell}(b,\phi)=R_{eff}(b)\frac{\sqrt{1-\epsilon^{2}(b)}}{1+\epsilon(b)\cos(2\phi)}$
(3)

and,

$R_{trian}(b,\phi)=R_{ell}(b,\phi)[1+\epsilon_{3}(b)\cos[3(\phi-\psi_{3})]]$
(4)

respectively. Experiments have shown that there is no correlation between the
event planes of the second and third harmonics. All produced particles are
emitted from a thermal expanding source with a maximum transverse flow
velocity $\rho_{u}$ whose parametrization is presented in reference [52].

$\rho_{u}^{max}=\rho_{u}^{max}(0)[1+\rho_{2u}(b)\cos(2\phi)+\rho_{3u}(b)\cos(3(\phi-\psi_{3})+\rho_{4u}(b)\cos(4\phi)]$
(5)

This also permits us to find the higher order anisotropic flow coefficients
with respect to the reaction plane i.e., $v_{2,4}(\psi_{2}^{RP}=0)$ and
$v_{3}(\psi_{3}^{RP})$. Here, we perform the modulation of the transverse
velocity profile in all freeze-out hypersurfaces. The new parameters
$\rho_{3u}(b)$ and $\rho_{4u}(b)$ can also be treated either centrality
independent or dependent on initial ellipticity. We have used the former way
for our simulation and chosen a fixed value of these parameters for Au+Au
collisions at $\sqrt{s_{NN}}$ = 200 GeV and Pb+Pb collisions at
$\sqrt{s_{NN}}$ = 2.76 TeV, respectively [26].

Figure 2: The transverse momentum $p_{T}$ dependence of $v_{2}$ (a), $v_{3}$
(b) and $v_{4}$ (c) of the (multi-) strange hadrons for $(0-80)\%$ centrality
in Au+Au collisions at $\sqrt{s_{NN}}$= 200 GeV. The solid markers represent
the HYDJET++ model results while the open markers represent experimental data
from the STAR collaboration [20, 23].

Figure 3: The second harmonic coefficient $v_{2}$ of (multi-) strange hadrons
as a function of transverse momentum $p_{T}$ for different centrality
intervals at $\sqrt{s_{NN}}$= 200 GeV in Au+Au collisions, for
(a)$K^{-}+K^{+}$, (b) $K_{s}^{0}$, (c) $\Lambda+\overline{\Lambda}$, (d)
$\Xi^{-}+\overline{\Xi}^{+}$, and (e) $\Omega^{-}+\overline{\Omega}^{+}$. The
solid markers represent the HYDJET++ model results while the open markers
represent experimental data from the PHENIX [31], STAR [19],[23]
collaborations. Figure 4: The elliptic flow $v_{2}$ of (multi-) strange
hadrons as a function of transverse momentum $p_{T}$ for different centrality
intervals at $\sqrt{s_{NN}}$= 2.76 TeV in Pb+Pb collisions, for
(a)$K^{-}+K^{+}$, (b) $K_{s}^{0}$, (c) $\Lambda+\overline{\Lambda}$, (d)
$\Xi^{-}+\overline{\Xi}^{+}$, and (e) $\Omega^{-}+\overline{\Omega}^{+}$. The
solid markers represent the HYDJET++ model results while the open markers
represent experimental data from the ALICE collaboration [21].

##  3 Results and Discussions

###  3.1 Higher harmonics flow as a function $p_{T}$ at different centrality
interval

The top ((a)-(c)) row of Figure 1 shows $v_{n}$ (n=2,3,4) as a function of
$p_{T}$ for primary charged hadrons in Au+Au collisions at $\sqrt{s_{NN}}$ =
200 GeV. The results obtained by the HYDJET++ model are presented for three
different centrality intervals i.e, $(0-10)\%$, $(20-30)\%$, and $(40-50)\%$.
The model results are compared with the available experimental data from
PHENIX collaboration [57]. We observe that the $v_{n}$ for charged hadrons
rises linearly with $p_{T}$ initially in low $p_{T}$ region i.e., $p_{T}\leq$
2-3 GeV/c, where a maximum is matched. With further increase in $p_{T}$,
$v_{n}$ starts decreasing for all the centrality intervals in Au+Au
collisions. The model calculations provide a good estimate of the experimental
data [57] towards low $p_{T}$ region for $v_{2}$, $v_{3}$, and $v_{4}$ in all
the centrality intervals.

Similarly the bottom ((a)-(c)) row of Figure 1 shows the variation of $v_{n}$
as a function of $p_{T}$ for charged hadrons in Pb+Pb collisions at
$\sqrt{s_{NN}}$ = 2.76 TeV. The results are shown for three centrality
intervals i.e, $(0-5)\%$, $(20-30)\%$, and $(40-50)\%$. The model results are
compared with experimental data from ATLAS collaboration [58]. It is observed
that the model results for $v_{2}$ and $v_{3}$ suitably match with the
experimental data [58] towards low $p_{T}$ in all the centrality intervals. At
high $p_{T}$, the model results underestimate the experimental data. The model
results of $v_{4}$ underestimate the experimental data in central collisions
because higher flow coefficients are produced by the nonlinear contributions
of $v_{2}$ and $v_{3}$ thereby dominating over the intrinsic momentum
anisotropy $v_{n}$. However, it provides a good estimate of the experimental
data for $(20-30)\%$, and $(40-50)\%$ centrality intervals [59]. These results
are well matched with the results obtained by HYDJET++ of $p_{T}$-differential
$v_{3}$ and $v_{4}$ for charged hadrons in Pb+Pb collisions at $\sqrt{s_{NN}}$
= 2.76 TeV [26]. One particular observation in both Au+Au and Pb+Pb collisions
is that in the intermediate $p_{T}$ region the model overestimates the
experimental data of charged hadrons. This observation is discussed in detail
in the article [26], where authors suggested that it can be improved by
incorporating mini-jet production or some other mechanisms.

Figure 5: The third harmonic coefficient $v_{3}$ of (multi-) strange hadrons
as a function of $p_{T}$, for different centralities in Au+Au collisions at
$\sqrt{s_{NN}}$= 200 GeV. The solid markers represent the HYDJET++ model
results while the open markers represent experimental data from the PHENIX
collaboration [31].

Figure 6: The third harmonic coefficient $v_{3}$ of (multi-) strange hadrons
as a function of $p_{T}$, for different centralities in Pb+Pb collisions at
$\sqrt{s_{NN}}$= 2.76 TeV. The solid markers represent the HYDJET++ model
results while the open markers represent experimental data from ALICE
collaboration [12].

The characteristic behaviour of $v_{3}$ and $v_{4}$ is quite similar to that
of $v_{2}$ as a function of $p_{T}$ with lower magnitude. It shows that
elliptic flow is a dominant source of anisotropic flow. Using the same model
and with the same tuned parameter values, we study the anisotropic flow of
(multi-) strange hadrons in Au+Au and Pb+Pb collision at RHIC and LHC
energies, respectively.

Figure 7: The fourth harmonic coefficient $v_{4}$ of (multi-) strange hadrons
as a function of $p_{T}$, for different centralities in Au+Au collisions at
$\sqrt{s_{NN}}$= 200 GeV. The solid markers represent the HYDJET++ model
results while the open markers represent experimental data from PHENIX
collaboration [31].

Figure 8: The fourth harmonic coefficient $v_{4}$ of (multi-) strange hadrons
as a function of $p_{T}$ for different centralities for Pb+Pb collisions at
$\sqrt{s_{NN}}$= 2.76 TeV. The solid markers represent the HYDJET++ model
results while the open markers represent experimental data from ALICE
collaboration [12].

Figure 9: The higher harmonic coefficients $v_{n}$ of (multi-) strange hadrons
as a function of transverse momentum $p_{T}$ for $(0-10)\%$ centrality
interval at $\sqrt{s_{NN}}$= 200 GeV in Au+Au collisions.The solid markers
represent the HYDJET++ model results while the open markers represent
experimental data from the STAR, PHENIX collaborations[31], [23].

Figure 10: The higher harmonic coefficients $v_{n}$ of (multi-) strange
hadrons as a function of transverse momentum $p_{T}$ for central $(0-5)\%$ at
$\sqrt{s_{NN}}$= 2.76 TeV in Pb + Pb collisions. The solid markers represent
the HYDJET++ model results while the open markers represent experimental data
from ALICE [21].

Figure 3 shows the variation of $v_{n}$ as a function of $p_{T}$ for (multi-)
strange hadrons in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV obtained by
HYDJET++ model. The model results are compared with the recently published
experimental data [20] from STAR collaboration. The results are shown for
$(0-80)\%$ centrality interval. Figure 3(a) shows that $v_{2}$ results are
well matched with the experimental data [23] upto $p_{T}\leq$ 2 GeV/c and then
overpredicts the experimental data with the increasing $p_{T}$. Figure 3(b)
presents $v_{3}$ results for different strange hadron species. It is observed
that the model provides a good description of experimental data [20] of
$K^{-}+K^{+}$ and $\Lambda+\overline{\Lambda}$ upto $p_{T}\leq$ 2.5 GeV/c
thereafter it overpredicts the experimental data at higher $p_{T}$. Due to non
availability of experimental data for $\Xi^{-}+\overline{\Xi}^{+}$ and
$\Omega^{-}+\overline{\Omega}^{+}$ in $(0-80)\%$ centrality interval, the
$\Xi^{-}+\overline{\Xi}^{+}$ and $\Omega^{-}+\overline{\Omega}^{+}$ results
are the model predictions. We expect that it should follow the trends of
$K^{-}+K^{+}$ and $\Lambda+\overline{\Lambda}$. Figure 3(c) shows the results
of $v_{4}$ for strange hadrons. It is observed that the model overpredicts the
experimental data of $K^{-}+K^{+}$. The model prediction for $v_{4}$ of
$K_{s}^{0}$, $\Lambda+\overline{\Lambda}$, $\Xi^{-}+\overline{\Xi}^{+}$ and
$\Omega^{-}+\overline{\Omega}^{+}$ are also presented. It is observed that the
model describes the experimental result of a rapid increase in $v_{n}$ of
mesons ($K^{-}+K^{+}$ and $K_{s}^{0}$) in comparison to baryons at low $p_{T}$
i.e, $p_{T}\leq$ 1 GeV/c. It is because of the fact that the probability of
formation of $K^{-}$ mesons by a combination of a strange quark and an up or a
down quark is much higher than the baryon formation at low $p_{T}$.

Figures 3 and 4 show elliptic flow $v_{2}(p_{T})$ results obtained from
HYDJET++ model for $K^{-}+K^{+}$, $K_{s}^{0}$, $\Lambda+\overline{\Lambda}$,
$\Xi^{-}+\overline{\Xi}^{+}$ and $\Omega^{-}+\overline{\Omega}^{+}$ in various
centrality intervals for Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV and
Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV, respectively. The model
results are compared with the corresponding experimental data from STAR,
PHENIX, and ALICE collaborations [19, 31, 23, 21]. It is observed that the
model reproduces the experimental data of (multi-) strange hadrons. However,
towards peripheral collisions the match between the model results and
experimental data is limited to much lower $p_{T}$ regions. This may be due to
the fact that the model parameters in the present study responsible for the
evaluation of anisotropic flow coefficient is same for all the centralities.
Fine-tuning of these parameters for different centrality intervals may lead to
better agreement between the model results and experimental data. It is also
observed that moving from central to peripheral collisions, maxima of $v_{2}$
increases and shifts towards lower $p_{T}$. It is because of the smaller
radial flow generated during fireball expansion in peripheral collisions
compared to that produced in central collisions [21].

In the same way, Figures 6 and 6 show variation of $v_{3}$ as a function of
$p_{T}$ for (multi-) strange hadrons obtained by HYDJET++ model for different
centrality intervals in Au+Au and Pb+Pb collisions, respectively. The model
results are compared with the available experimental data from PHENIX and
ALICE experiments [31, 12]. The Au+Au results for $v_{3}$ are similar to those
described in Figure 3 and Pb+Pb results show same behaviour as $v_{2}$ in
Figure 4 but with different magnitude.

Likewise, Figures 8 and 8 show the variation of $v_{4}$ as a function of
$p_{T}$ in Au+Au and Pb+Pb collisions, respectively, for different centrality
intervals. We observe a much better agreement between $v_{4}$ results obtained
from the model and the experimental data towards central collisions in
comparison to $v_{3}$ results.

Figure 11: $p_{T}$ integrated flow $v_{2}$, $v_{3}$, and $v_{4}$ of (multi-)
strange hadrons as a function of centrality in Au+Au collisions at
$\sqrt{s_{NN}}$= 200 GeV and $\sqrt{s_{NN}}$= 2.76 TeV in Pb+Pb collisions.
The solid markers represent the HYDJET++ model results while the open markers
represent experimental data from STAR [20] .

Figures 10 and 10 show a comparison among $v_{2}$, $v_{3}$, and $v_{4}$ of
(multi-) strange hadrons in most central collision i.e, $(0-10)\%$ centrality
interval for Au+Au and $(0-5)\%$ centrality interval for Pb+Pb collisions,
respectively. It is observed that HYDJET++ elliptic flow $v_{2}$ has slightly
higher magnitude than $v_{3}$ at low $p_{T}$ i.e, $p_{T}\leq$ 2 GeV/c in Au+Au
collisions for all the strange hadron species. At higher $p_{T}$, both $v_{2}$
and $v_{3}$ have same magnitude. However, in Pb+Pb collisions, $v_{2}$ is
slightly higher in magnitude than $v_{3}$ up to $p_{T}\leq$ 1.5 GeV/c, beyond
which $v_{3}$ is larger than $v_{2}$. It indicates that event-by-event
fluctuations contribute more to the development of flow in most central
collisions [31, 23, 21, 12, 60] which gets reflected in the higher value of
$v_{3}$ than $v_{2}$ at intermediate $p_{T}$. It can be seen that $v_{4}$ for
$K^{-}+K^{+}$ matches well with the experimental data in Au+Au collisions. For
Pb+Pb collisions, the model underpredicts the experimental data of $v_{4}$ for
$K^{-}+K^{+}$. This observation is in accordance with the results obtained in
ref. [26] for Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV.

###  3.2  Higher harmonics flow as a function of centrality

Figure 11 shows $p_{T}$ integrated $v_{n}$ as a function of centrality for
(multi-) strange hadrons in Au+Au collisions at $\sqrt{s}_{NN}$ =200 GeV and
Pb+Pb collisions at $\sqrt{s}_{NN}$ =2.76 TeV, respectively. For comparision,
the available experimental data of $K^{+}+K^{-}$ from STAR collaboration [20]
is also shown in top row of Figure 11. The model approximately reproduces the
experimental $\left<v_{2}\right>$ data for $K^{+}+K^{-}$ but it underestimates
the $\left<v_{3}\right>$ data for all the centrality intervals. It is observed
that $\left<v_{2}\right>$, $\left<v_{3}\right>$ and $\left<v_{4}\right>$
increase with increasing centrality. $\left<v_{3}\right>$ initially increases
with increasing centrality but towards peripheral collision, a saturation is
observed in $\left<v_{3}\right>$ value. It may happen because event-by-event
fluctuations dominate in $v_{3}$ in central collisions. Towards peripheral
collisions, the contribution of fluctuations is small and shows weak
dependence on centrality since odd-order flow harmonics are produced only by
fluctuations [26, 11].

Figure 12: The scaled elliptic flow $v_{2}/n_{q}$ versus the scaled transverse
momentum $p_{T}/n_{q}$ of (multi-) strange hadrons for $(0-10)\%$, and
$(0-80)\%$ centrality intervals in Au+Au collisions at $\sqrt{s_{NN}}$= 200
GeV. The solid markers represent the HYDJET++ model results while the open
markers represent experimental data from STAR and PHENIX collaborations [20,
23, 31].

From top row of Figure 11, baryon-meson grouping is clearly observed in
$\left<v_{n}\right>$ for Au+Au collisions at $\sqrt{s_{NN}}$ =200 GeV, but it
is absent in Pb+Pb collisions at $\sqrt{s_{NN}}$ =2.76 TeV in bottom row of
Figure 11. This is because of the large contribution of jets at LHC energy
regime. It is the main motivation to test NCQ scaling at both RHIC and LHC
energies.

Figure 13: The scaled elliptic flow $v_{2}/n_{q}$ versus the scaled transverse
momentum $p_{T}/n_{q}$ of (multi-) strange hadrons for $(0-5)\%$, $(20-30)\%$
and $(40-50)\%$ centrality intervals in Pb+Pb collisions at $\sqrt{s_{NN}}$=
2.76 TeV [21]. Figure 14: The scaled triangular $v_{3}/n_{q}$ and quadrangular
$v_{4}/n_{q}$ flows versus the scaled transverse momentum $p_{T}/n_{q}$ of
(multi-) strange hadrons for $(0-10)\%$ and $(0-80)\%$ centrality intervals in
Au+Au collisions at $\sqrt{s_{NN}}$= 200 GeV. Figure 15: The scaled triangular
$v_{3}/n_{q}$ and quadrangular $v_{4}/n_{q}$ flows versus the scaled
transverse momentum $p_{T}/n_{q}$ of (multi-) strange hadrons for $(0-5)\%$,
$(20-30)\%$, and $(40-50)\%$ centrality intervals in Pb+Pb collisions at
$\sqrt{s_{NN}}$= 2.76 TeV.

###  3.3 Test of NCQ scaling

Figure 12 shows NCQ scaling results in Au+Au collisions at $\sqrt{s_{NN}}$
=200 GeV obtained from HYDJET++ model and compared with corresponding results
from STAR collaboration [20, 23, 31]. STAR data is shown in Figure 12(a) for
$(0-10)\%$ centrality interval and in Figure 12(d) for $(0-80)\%$ interval.
HYDJET++ results on NCQ scaling are presented in Figure 12(b) and Figure 12(e)
for $(0-10)\%$ and $(0-80)\%$ centrality intervals, respectively.

Figure 16: $v_{n}$ of (non-) strange hadrons as a function $p_{T}$ for
$(0-80)\%$ and $(20-30)\%$ centrality intervals in Au+Au collisions at
$\sqrt{s_{NN}}$= 200 GeV and in Pb+Pb collisions at $\sqrt{s_{NN}}$= 2.76 TeV,
respectively. Experimental results from STAR [23, 19, 20] and ALICE [34, 21]
collaborations for elliptic flow $v_{2}$ are shown in panel (a), while
HYDJET++ results for all $v_{n}$ are shown in (b), (c) and (d).

The NCQ scaling is studied by the variation of $v_{2}/n_{q}$ as a function of
$p_{T}/n_{q}$, where $n_{q}$ is the number of constituent quark. It is
observed that the NCQ scaling remains valid for strange mesons throughout the
$p_{T}$ range in $(0-10)\%$ centrality. For strange baryons, the NCQ scaling
is valid up to $p_{T}\approx$ 1.5 GeV. Thereafter, because of large
fluctuation in model calculation for strange baryons, the NCQ scaling can not
be described properly. However, increasing the event statistics may reduce the
large fluctuation in strange baryon results. Comparing the HYDJET++ results
with the experimental data in Figure 12(a) and (b), it is observed that the
model calculations reproduce the shape of the experimental data for all the
particle species but with higher magnitude. Similar observations are made in
Figure 12(d) and (e). In Figure 12(c), for non-central collisions, it is
observed that the strange baryon results are consistently below the strange
meson results. It is because of the transition between thermal and
thermal+shower recombination. For peripheral collisions, the transition can
happen at lower $p_{T}/n_{q}$ while in central collisions, this transition
occurs at higher $p_{T}/n_{q}$ value [61].

Likewise, Figure 13 shows the results of NCQ scaling in Pb+Pb collisions at
$\sqrt{s_{NN}}=$ 2.76 TeV. The experimental data and the model results are
shown separately for $(0-5)\%$, $(20-30)\%$, and $(40-50)\%$ centrality
intervals, respectively. The observations are similar to that of NCQ scaling
in Au+Au collisions. However, in non-central collisions a clear separation
between strange meson and baryon spectra is observed in both the model
calculations and experimental data. It indicates the violation of NCQ scaling
in non-central collisions for Pb+Pb collisions at LHC [28].

Figures 14 and 15 show NCQ scaling of $v_{3}$ and $v_{4}$ in HYDJET++ for
Au+Au and Pb+Pb collisions, respectively over different centrality intervals.
Here also, it can be seen that the model clearly shows the NCQ scaling for all
the hadrons in Au+Au collisions up to $p_{T}\approx$ 1.5 GeV, while for Pb+Pb
collisions a violation in the NCQ scaling is observed towards peripheral
collisions due to stronger influence of jet hadrons at LHC [47, 28].

###  3.4  Mass Ordering effect in flow harmonics of (non-)strange hadrons

Figure 16 presents the mass ordering of anisotropic flow among
$\pi^{+}+\pi^{-}$, $K^{+}+K^{-}$, $K_{s}^{0}$, $p+\overline{p}$,
$\Lambda+\overline{\Lambda}$, $\Xi^{-}+\overline{\Xi}^{+}$, and
$\Omega^{-}+\overline{\Omega}^{+}$) in Au+Au collisions at $\sqrt{s}_{NN}$
=200 GeV and Pb+Pb collisions at $\sqrt{s}_{NN}$ =2.76 TeV, respectively. The
available experimental data and the model results for $v_{2}$ are shown
separately. The results are shown for $(0-80)\%$ centrality interval in Au+Au
collisions and for $(20-30)\%$ in Pb+Pb collisions. It can be seen that the
model reproduces the experimental data of $v_{2}$ up to $p_{T}\approx$ 2.0
GeV/c implying that the model provides a good description of mass ordering
among all the hadron species at low $p_{T}$. This is because of the interplay
between radial and elliptic flows generated during hadronic evolution. The
radial flow pushes the heavier particles towards higher $p_{T}$ which gets
manifested to $p_{T}$-dependent mass ordering of elliptic flow at low $p_{T}$.
This tends to decrease the elliptic flow with increase of hadron mass. Towards
higher $p_{T}$, baryon-meson grouping (particle-type dependence) is observed
in the model calculation and in the experimental data. From the figures, we
also observe that the separation between mass ordering of $\pi^{+}+\pi^{-}$
and $K^{+}+K^{-}$ is less visible in HYDJET++ model in comparison to ALICE
experimental data. However, experimental results show a clear separation
between the mass ordering for all the hadrons species. This may be due to the
strangeness enhancement factor $\gamma_{s}$ used in soft particle production
component of the model. Similar observations are made for $v_{3}$ and $v_{4}$
in Au+Au and Pb+Pb collisions.

##  4 Conclusion

We have presented $p_{T}$-differential flow harmonics $v_{n}(n=2,3,4)$ of
(multi-) strange hadrons at RHIC and LHC energies using HYDJET++ model. We
have compared our findings with the available experimental data at both
energies. HYDJET++ model suitably describes the experimental data at low
$p_{T}$. At intermediate $p_{T}$, the model overpredicts the experimental data
for $v_{n}$. The difference between the model calculations and the
experimental data increases as we move toward peripheral collisions. This may
be because the mechanism involved to address the interaction between the
medium and jet partons is absent in the HYDJET++ model framework at the
present scenario. We observe a clear baryon-meson grouping in
$p_{T}$-integrated flow coefficients $v_{n}$ in Au+Au collisions at
$\sqrt{s}_{NN}$= 200 GeV. The meson-baryon grouping is absent in Pb+Pb
collisions at $\sqrt{s}_{NN}$= 2.76 TeV.

We observed that the model reproduces the experimental results on constituent
quark scaling at both RHIC and LHC energies. Moreover, similar to the
experimental results, a violation of NCQ scaling is also observed in the model
calculations at LHC energy. Further, we also observed mass ordering in the
anisotropic flow of strange and non-strange hadrons at low $p_{T}$ in both
Au+Au and Pb+Pb collisions, where mesons have higher magnitude of flow than
the baryons. At intermediate $p_{T}$, the mass order reverses, and heavier
baryons acquire higher flow in comparison to mesons.

Thus, our study highlighted the contribution of flow of (multi-) strange
hadrons to the collective properties of QGP medium occurring in the partonic
phase of Au+Au and Pb+Pb collision systems at RHIC and LHC energy regimes,
respectively. Comparison of the $v_{2}$($p_{T}$), $v_{3}$($p_{T}$), and
$v_{4}$($p_{T}$) values for different particle species in these two different
energy regimes might be helpful in providing additional insight into the
dynamics of anisotropic flow and the effect of radial expansion of the system.

##  5 Acknowledgements

BKS gratefully acknowledges the financial support provided by the BHU
Institutions of Eminence (IoE) Grant No. 6031, Govt. of India. AS would like
to thank CSIR, India for providing Senior Research Fellowship. SP and GD
acknowledge the financial support obtained from UGC under the research
fellowship scheme in central universities.

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#  Asymptotic resemblance relations on Groups

Sh. Kalantari  Department of Basic Sciences, Babol Noshirvani University of
Technology, Shariati Ave.,Babol, Iran, Post Code:47148-71167.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study properties of asymptotic resemblance relations induced
by compatible coarse structures on groups. We generalize the notion of
asymptotic dimensiongrad for groups with compatible coarse structures and show
this notion is coarse invariant. We end by defining the notion of set
theoretic coupling for groups with compatible coarse structures and showing
this notion is the generalization of the notion of topological coupling for
finitely generated groups. We show if two groups with compatible coarse
structures admit a set theoretic coupling then they are asymptotic equivalent.

###### Key words and phrases:

asymptotic dimensiongrad, asymptotic resemblance, coarse structure, set
theoretic coupling, topological coupling

###### 2020 Mathematics Subject Classification:

51F99, 53C23, 54F45, 54D35

##  1\. Introduction

The Gromov’s approach of investigating finitely generated groups with large
scale properties of their Cayley graphs ([3]) has been generalized rapidly for
studying more general objects. The key tool for this purpose is the notion of
_coarse structure_ introduced by Roe ([14]). A coarse structure on a set $X$
is a family of subsets of $X\times X$ with some additional properties. Coarse
structures have opened a new perspective for investigating more general forms
of groups, for instance, _topological groups_. There have been some attempts
for this goal. For example in [12], the authors defined the large scale
counterpart of topological groups. In fact in [12] they defined compatible
_ball structures_ on groups and for this reason they used the notion of _group
ideals_. Ball structures can be considered as alternative ways for defining
coarse structures on sets. In [10] a way of defining coarse structures on
general groups has been introduced. They used the notion of _generating
family_ on a group $G$ for defining compatible coarse structures with the
group structure of $G$. The notion of generating family and the notion of
group ideal are closely related and they are equivalent in some ways. There
are also other works in this area, one can see [4, 11, 13] and the relatively
new monograph [15]. We survey elementary properties of coarse structures on
nonempty sets and compatible coarse structures on groups in §2.1. As another
way of defining large scale structures on a given set, the authors of [6]
introduced the notion of _asymptotic resemblance relations_ (AS.R). An
asymptotic resemblance relation on a set $X$ is an equivalence relation on the
family of all subsets of $X$ with two additional properties. It can be shown
that every coarse structure on a set $X$ can induce an asymptotic resemblance
relation on $X$ and almost all concepts in coarse geometry can be carried on
to asymptotic resemblance spaces. In §2.2 we discuss asymptotic resemblance
relations and their basic properties briefly.
In Section 3 we describe asymptotic resemblance relations on groups induced by
compatible coarse structures. In this section, we drive special meanings of
concepts in AS.R spaces for the AS.R relations induced by compatible coarse
structures on groups. In particular, we show what are the notions of
_asymptotic disjointness_ and _compatible AS.R with the topology_ equivalent
to.
Section 4 is devoted to generalize the concept of _asymptotic dimensiongrad_
for groups with compatible coarse structures. The notion of asymptotic
dimensiongrad for metric spaces has been introduced in [1] as an inductive way
for finding dimensions in large scale. In this section, we describe the
meaning of an _asymptotic cut_ between two asymptotic disjoint subsets of a
coarse space and define the asymptotic dimensiongrad inductively by using this
notion. In addition, we show asymptotic dimensiongrad is invariant under
coarse equivalences and we study the relation between asymptotic cut and
_large scale separator_. Even though we define the asymptotic dimensiongrad
for all coarse spaces, our main subject in this section is the asymptotic
dimensiongrad of groups with compatible coarse structures.
The notion of _topological coupling_ has first appeared in [3]. A topological
coupling for two finitely generated groups $G$ and $H$ is a locally compact
topological space $X$ such that $G$ and $H$ act cocompactly and properly on
$X$ from left and right respectively and these actions commute. It can be
shown that two finitely generated groups are coarsely equivalent if and only
if they admit a topological coupling (§0.2.$C_{2}^{\prime}$ of [3]). In
Section 5 we show how the notion of topological coupling can be generalized to
the concept of _set theoretic coupling_ for groups with AS.R relations induced
by compatible coarse structures. We show that if two groups admit a set
theoretic coupling then they are asymptotic equivalent. This section contains
some examples of set theoretic couplings for some important compatible coarse
structures on groups.

##  2\. preliminaries

###  2.1. Coarse Structures

We start by recalling that a _coarse structure_ $\mathcal{E}$ on a nonempty
set $X$ is a family of subsets of $X\times X$ such that each subset of an
element of $\mathcal{E}$ is an element of $\mathcal{E}$ and for each
$E,F\in\mathcal{E}$ the subsets $E\bigcup F$, $E^{-1}$ and $E\circ F$ belong
to $\mathcal{E}$, where $E^{-1}=\\{(y,x)\in X\times X\mid(x,y)\in E\\}$ and
$E\circ F$ denotes the set of all $(x,y)\in X\times X$ such that $(x,z)\in F$
and $(z,y)\in E$ for some $z\in X$ ([14]). If $\mathcal{E}$ is a coarse
structure on the set $X$, then the pair $(X,\mathcal{E})$ is called a coarse
space. The coarse structure $\mathcal{E}$ on the set $X$ is called unitary if
it contains the diagonal $\Delta=\\{(x,x)\mid x\in X\\}$ and it is called
connected if for each $x,y\in X$, $\\{(x,y)\\}\in\mathcal{E}$. In the rest of
this paper by a coarse structure we mean a unitary and connected coarse
structure. Before going further we should recall that a subset $B$ of a coarse
space $(X,\mathcal{E})$ is called bounded if $B\times B\in\mathcal{E}$.
Suppose that $(X,\mathcal{E})$ is a coarse space and let $Y\subseteq X$ be a
nonempty set. Define $\mathcal{E}_{Y}=\\{E\bigcap(Y\times Y)\mid
E\in\mathcal{E}\\}$. Then $\mathcal{E}_{Y}$ is a coarse structure on $Y$,
which is called the coarse structure induced on $Y$ by $\mathcal{E}$.

######  Example 2.1.

The metric coarse structure $\mathcal{E}_{d}$ on a metric space $(X,d)$ is the
family of all subsets $E$ of $X\times X$ such that for some $k>0$, $d(x,y)\leq
k$ for all $(x,y)\in E$.

In the category of coarse spaces morphisms are called _coarse maps_.

######  Definition 2.2.

Let $(X,\mathcal{E})$ and $(Y,\mathcal{E}^{\prime})$ be two coarse spaces. A
map $f:X\rightarrow Y$ is called a _coarse map_ if for each $E\in\mathcal{E}$,
$(f\times f)(E)\in\mathcal{E}^{\prime}$ and the inverse image of each bounded
subset of $Y$ is a bounded subset of $X$. A coarse map $f:X\rightarrow Y$ is
called a _coarse equivalence_ if there exists a coarse map $g:Y\rightarrow X$
such that $\\{(g(f(x)),x)\mid x\in X\\}\in\mathcal{E}$ and $\\{(f(g(y)),y)\mid
y\in Y\\}\in\mathcal{E}^{\prime}$. In this case we call $g$ a _coarse inverse_
of $f$.

The following definition is from [10].

######  Definition 2.3.

Let $G$ be a group. For subsets $A$ and $B$ of $G$ suppose that
$A^{-1}=\\{g^{-1}\mid g\in G\\}$ and $AB=\\{gh\mid g\in A,h\in B\\}$. By a
_generating family_ $\mathcal{F}$ on the group $G$ we mean a family
$\mathcal{F}$ of subsets of $G$ such that $\mathcal{F}$ contains a nonempty
subset of $G$ and for each $A,B\in\mathcal{F}$, the subsets $A\bigcup B$, $AB$
and $A^{-1}$ belongs to $\mathcal{F}$.

######  Example 2.4.

Let $G$ be a group and let $\mathcal{F}$ be a generating family on $G$. Let
$\mathcal{E}_{\mathcal{F}}$ denotes the family of all subsets $E$ of $X\times
X$ such that $E\subseteq G(F\times F)$ for some $F\in\mathcal{F}$, where

$G(F\times F)=\\{(ga,gb)\mid g\in G,a,b\in F\\}$

It can be shown that the family $\mathcal{E}_{\mathcal{F}}$ is a coarse
structure on $G$ and we call it the coarse structure associated to the
generating family $\mathcal{F}$ on $G$ (Proposition 2.4 of [10]). This coarse
structure is not connected in general. In fact the coarse structure
$\mathcal{E}_{\mathcal{F}}$ is connected if and only if
$G=\bigcup_{F\in\mathcal{F}}F$ (Proposition 2.21 of [10]). From now on we
assume all generating sets $\mathcal{F}$ with the additional assumption
$G=\bigcup_{F\in\mathcal{F}}F$.

Let $G$ be a group and let $\mathcal{F}$ be a generating family on $G$. We
denote by $\hat{\mathcal{F}}$ the family of all $A\subseteq G$ such that
$A\subseteq B$ for some $B\in\mathcal{F}$. It is easy to see that
$\hat{\mathcal{F}}$ is a generating family on $G$ and
$\mathcal{E}_{\mathcal{F}}=\mathcal{E}_{\hat{\mathcal{F}}}$. Another thing
that we should mention here is that if the coarse structure
$\mathcal{E}_{\mathcal{F}}$ is connected then the family of all bounded
subsets coincide with the family $\hat{\mathcal{F}}$. Let us end this section
with two examples of generating sets on groups (for more examples see [10]).

######  Example 2.5.

Let $G$ be a group and suppose that $\mathcal{F}$ denotes the family of all
finite subsets of $G$. Then $\mathcal{F}$ is a generating family on $G$.

######  Example 2.6.

Let $G$ be a (Hausdorff) topological group. Let $\mathcal{K}$ denotes the
family of all subsets of $G$ with compact closure. Then $\mathcal{K}$ is a
generating family on $G$. We call the associated coarse structure
$\mathcal{E}_{\mathcal{K}}$ the _group compact coarse structure_ ([10]).

######  Example 2.7.

Let $G$ be a group and let $d$ be a left invariant metric on $G$. Assume that
$\mathcal{F}$ denotes the family of all bounded subsets of the metric space
$(G,d)$. The family $\mathcal{F}$ is a generating family on $G$.

Let $X$ be a topological space. A coarse structure $\mathcal{E}$ on $X$ is
called to be compatible with the topology of $X$ if there exists some
$E\in\mathcal{E}$ which is open in the product topology of $X\times X$ and $E$
contains the diagonal. The coarse structure $\mathcal{E}$ is called to be
proper if each bounded subset of the coarse space $(X,\mathcal{E})$ has a
compact closure. Note that if there exists a compatible and proper coarse
structure on a topological space $X$ then $X$ is locally compact. Now let
$\mathcal{E}$ be a compatible and proper coarse structure on the topological
space $X$. A bounded and continuous function $f:X\rightarrow\mathbb{C}$ is
called a _Higson function_ if for each $E\in\mathcal{E}$ the restriction of
the function $g(x,y)=\mid f(x)-f(y)\mid$ to $E$ vanishes to infinity. If we
denote the family of all Higson functions by $C_{h}(X)$ then there exists a
compactification $hX$ of $X$ such that $C_{h}(X)=C(hX)$ ([14] section 2.3),
where $C(hX)$ denotes the family of all continuous functions
$f:hX\rightarrow\mathbb{C}$. The compactification $hX$ of $X$ is called the
_Higson compactification_ of $X$. The boundary $\nu X=hX\setminus X$ is called
the _Higson corona_ of $X$. For a subset $A$ of $X$ suppose that $\bar{A}$
denotes the closure of $A$ in $hX$. We denote the boundary $\bar{A}\cap\nu X$
by $\nu A$.

###  2.2. Asymptotic Resemblance

Let $X$ be a nonempty set by an _asymptotic resemblance relation_ (an AS.R) on
$X$ we mean an equivalence relation $\lambda$ on the family of all subsets of
$X$ with the following two additional properties,
1) If $A_{1},A_{2},B_{1},B_{2}\subseteq X$ and $A_{1}\lambda B_{1}$ and
$A_{2}\lambda B_{2}$ then $(A_{1}\cup A_{2})\lambda(B_{1}\cup B_{2})$.
2) Suppose that $A_{1},A_{2},B\subseteq X$ and $A_{1},A_{2}\neq\emptyset$. If
$(A_{1}\cup A_{2})\lambda B$ then there exists nonempty subsets $B_{1}$ and
$B_{2}$ of $X$ such that $B=B_{1}\cup B_{2}$ and $A_{i}\lambda B_{i}$ for
$i\in\\{1,2\\}$.
If $\lambda$ denotes an AS.R on the set $X$ then we call the pair
$(X,\lambda)$ an AS.R space. If $A\lambda B$ for two subsets $A$ and $B$ of
the AS.R space $(X,\lambda)$, we say $A$ and $B$ are asymptotically alike. One
can see [6] for more details about AS.R spaces. Let $Y$ be a nonempty subset
of the AS.R space $(X,\lambda)$. For two subsets $A$ and $B$ of $Y$ define
$A\lambda_{Y}B$ if $A$ and $B$ are two asymptotically alike subsets of $X$.
The relation $\lambda_{Y}$ is an AS.R on $Y$, which we call the subspace AS.R
induced on $Y$ from $X$.

######  Example 2.8.

Let $(X,d)$ be a metric space. For two subsets $A$ and $B$ define
$A\lambda_{d}B$ if $d_{H}(A,B)<+\infty$, where $d_{H}$ denotes the Hausdorff
distance between $A$ and $B$. Then $\lambda_{d}$ is an AS.R on $X$, which we
call the metric AS.R associated to the metric $d$ on $X$.

The following example illustrates how the notions of coarse structures and
asymptotic resemblance relations are related.

######  Example 2.9.

Let $\mathcal{E}$ be a coarse structure on the set $X$. Suppose that for
$E\in\mathcal{E}$ and $x\in X$, $E(x)=\\{y\in X\mid(y,x)\in E\\}$ and for
$A\subseteq X$, $E(A)=\bigcup_{x\in A}E(x)$. For two subsets $A$ and $B$ of
$X$ define $A\lambda_{\mathcal{E}}B$ if $A\subseteq E(B)$ and $B\subseteq
E(A)$ for some $E\in\mathcal{E}$. The relation $\lambda_{\mathcal{E}}$ is an
AS.R on $X$ and we call it the AS.R associated to the coarse structure
$\mathcal{E}$ on $X$. It is worth mentioning that two different coarse
structures on a set $X$ may have the same associated AS.R (Example 3.1 of
[6]).

We call the subset $A$ of the AS.R space $(X,\lambda)$ bounded if
$A=\emptyset$ or $A\lambda\\{x\\}$ for some $x\in X$. If $\mathcal{E}$ is a
coarse structure on the set $X$ then $A\subseteq X$ is bounded in the coarse
space $(X,\mathcal{E})$ if and only if $A$ is bounded in the AS.R space
$(X,\lambda_{\mathcal{E}})$.
Suppose that $(X,\lambda)$ and $(Y,\lambda^{\prime})$ are two AS.R spaces. A
map $f:X\rightarrow Y$ is called an _AS.R mapping_ if the inverse image of
each bounded subset of $Y$ is a bounded subset of $X$ and if $A\lambda B$ then
$f(A)\lambda^{\prime}f(B)$, for all $A,B\subseteq X$. An AS.R mapping
$f:X\rightarrow Y$ is called an _asymptotic equivalence_ if there exists an
AS.R mapping $\tilde{f}:Y\rightarrow X$ such that $\tilde{f}(f(A))\lambda A$
and $f(\tilde{f}(B))\lambda^{\prime}B$ for all $A\subseteq X$ and $B\subseteq
Y$. In this case $\tilde{f}$ is called the asymptotic inverse of $f$ and $X$
and $Y$ are called asymptotically equivalent.

######  Definition 2.10.

We say two subsets $A$ and $B$ of the AS.R space $(X,\lambda)$ are
_asymptotically disjoint_ if $L_{1}\lambda L_{2}$ for $L_{1}\subseteq A$ and
$L_{2}\subseteq B$ then $L_{1}$ and $L_{2}$ are bounded. In other words two
subsets of an AS.R space are asymptotically disjoint if they do not have any
unbounded asymptotically alike subsets. An AS.R space $(X,\lambda)$ is called
_asymptotically normal_ if $A$ and $B$ are two asymptotically disjoint subsets
of $X$ then $X=X_{1}\cup X_{2}$ such that $X_{1}$ and $X_{2}$ are
asymptotically disjoint from $A$ respectively.

######  Proposition 2.11.

Let $d$ be a metric on the set $X$. Then the AS.R space $(X,\lambda_{d})$ is
asymptotically normal.

###### Proof.

See Proposition 4.5 of [6]. ∎

Let $\lambda$ be an AS.R on the topological space $X$. We say $\lambda$ is
compatible with the topology of $X$ if for each subset $A$ of $X$ there exists
an open subset $A\subseteq U$ such that $A\lambda U$ and in addition
$\bar{A}\lambda A$. The AS.R $\lambda$ is called to be proper if every bounded
subset has compact closure. It can be shown that each AS.R associated to a
compatible and proper coarse structure on a topological space is compatible
and proper (Proposition 4.2 of [6]). Let $\lambda$ be a compatible and proper
AS.R on the normal topological space $X$. Assume that $(X,\lambda)$ is
asymptotically normal. Define $A\delta B$ if $A$ and $B$ are not
asymptotically disjoint or $\bar{A}\cap\bar{B}\neq\emptyset$. The relation
$\delta$ defines a compatible proximity on $X$ and we call the associated
Smirnov compactification the asymptotic compactification of $X$ (for proximity
spaces and their Smirnov compactifications see [9]). It can be shown that if
$\mathcal{E}$ is a compatible and proper coarse structure on the normal
topological space $X$, such that $\lambda_{\mathcal{E}}$ is asymptotically
normal then the Higson compactification of $X$ and the asymptotic
compactification of $X$ are equal up to a homeomorphism ([6]).

##  3\. asymptotic resemblance relations on groups

######  Definition 3.1.

Let $\mathcal{F}$ be a generating family on a group $G$. Define
$A\lambda_{\mathcal{F}}B$ if there exists some $H\in\mathcal{F}$ such that
$A\subseteq BH$ and $B\subseteq AH$.

######  Proposition 3.2.

Let $\mathcal{F}$ be a generating family on the group $G$. Then
$\lambda_{\mathcal{F}}$ is the AS.R associated to the coarse structure
$\mathcal{E}_{\mathcal{F}}$ on $G$.

###### Proof.

Assume that $\lambda$ denotes the AS.R associated to
$\mathcal{E}_{\mathcal{F}}$. For two subsets $A$ and $B$ of $X$ if $A\lambda
B$, then there exists some $H\in\mathcal{F}$ and $E\subseteq G(H\times H)$
such that $A\subseteq E(B)$ and $B\subseteq E(A)$. Let $K=H^{-1}H$. So
$K\in\mathcal{F}$. If $a\in A$, there is some $b\in B$ such that $(a,b)\in E$.
Thus there are $h_{1},h_{2}\in H$ and $g\in G$ such that
$(a,b)=(gh_{1},gh_{2})$. So $b^{-1}a=h_{2}^{-1}h_{1}\in K$ and it shows that
$a\in BK$ and therefore $A\subseteq BK$. Similarly one can show that
$B\subseteq AK$ and this leads to $A\lambda_{\mathcal{F}}B$. Now suppose that
$A\lambda_{\mathcal{F}}B$. There exists some $H\in\mathcal{F}$ such that
$A\subseteq BH$ and $B\subseteq AH$. Let $K=H\cup HH^{-1}$ and $E=G(K\times
K)$. Clearly $E\in\mathcal{F}$. If $a\in A$, then there exists some $b\in B$
and $h\in H$ such that $a=bh$. Since $e\in HH^{-1}\subseteq K$, we have
$(a,b)=b(h,e)\in E$ ($e$ denotes the neutral element of $G$). So $A\subseteq
E(B)$ and similarly it can be shown that $B\subseteq E(A)$. Therefore
$A\lambda B$. ∎

######  Proposition 3.3.

Let $\mathcal{F}$ be a generating family on the group G. Then two subsets $A$
and $B$ of $G$ are asymptotically disjoint in the AS.R space
$(G,\lambda_{\mathcal{F}})$ if and only if $A\cap BH$ is bounded for each
$H\in\mathcal{F}$.

###### Proof.

Assume that $A$ and $B$ are two asymptotically disjoint subsets of
$(G,\lambda_{\mathcal{F}})$. Suppose that $L_{1}=A\cap BH$ is unbounded for
some $H\in\mathcal{F}$. Let $L_{2}=B\cap AH^{-1}$ and $K=H\cup H^{-1}$. It is
straightforward to show that $L_{1}\subseteq L_{2}K$ and $L_{2}\subseteq
L_{1}K$, thus $L_{1}\lambda_{\mathcal{F}}L_{2}$. It shows that $A$ and $B$ are
not asymptotically disjoint, a contradiction. To prove the converse suppose
that $A$ and $B$ are two subsets of $G$ and they are not asymptotically
disjoint. So there are unbounded subsets $L_{1}\subseteq A$ and
$L_{2}\subseteq B$ and some $H\in\mathcal{F}$ such that $L_{1}\subseteq
L_{2}H$ and $L_{2}\subseteq L_{1}H$. Thus $L_{1}\subseteq A\cap BH$, which
shows that $A\cap BH$ is unbounded. ∎

Since in this paper we consider each coarse structure to be connected so as we
mentioned before if $\mathcal{F}$ is a generating family on the group $G$,
then the family of all bounded subsets of $(G,\lambda_{\mathcal{F}})$ coincide
with $\hat{\mathcal{F}}$. Thus we have the following corollary.

######  Corollary 3.4.

Let $\mathcal{F}$ be a generating family on the group $G$. Two subsets $A$ and
$B$ of $G$ are asymptotically disjoint if and only if $A\cap
BH\in\hat{\mathcal{F}}$, for all $H\in\mathcal{F}$.

######  Corollary 3.5.

Let $G$ be a topological group. Then two subsets $A$ and $B$ of $G$ are
asymptotically disjoint with respect to the AS.R $\lambda_{\mathcal{K}}$ if
and only if for all compact subset $K$ of $G$, $A\cap BK$ has compact closure
($\mathcal{K}$ denotes the family of all compact subsets of $G$).

###### Proof.

It is evident from Corollary 3.4. ∎

######  Proposition 3.6.

Let $\mathcal{F}$ be a generating family on the topological group $G$. Then
the coarse structure $\mathcal{E}_{\mathcal{F}}$ is compatible with the
topology of $G$ if and only if there exists an open set $U$ containing the
identity such that $U\in\hat{\mathcal{F}}$.

###### Proof.

Let $e$ denotes the identity of $G$. Suppose that $\mathcal{E}_{\mathcal{F}}$
is compatible with the topology of $G$ and let $E$ be an open entourage that
contains the diagonal. Then $e\in E(e)$ is an open set and
$E(e)\in\hat{\mathcal{F}}$. To prove the converse suppose that $U$ is an open
neighbourhood of the identity and $U\in\hat{\mathcal{F}}$. Then clearly
$E=G(U\times U)$ is an open entourage containing the diagonal. ∎

######  Proposition 3.7.

Let $\mathcal{F}$ be a generating family on the topological group $G$. Then
the coarse structure $\mathcal{E}_{\mathcal{F}}$ is proper if and only if each
element of $\mathcal{F}$ has compact closure.

###### Proof.

It is straightforward. ∎

On a topological group $G$, we call the AS.R $\lambda_{\mathcal{K}}$ the
_group compact AS.R_ on $G$ ($\mathcal{K}$ denotes the family of all subsets
of $G$ with compact closure as mentioned before). By Proposition 3.7 it is
evident that the group compact AS.R is proper. The following corollary is a
direct consequence of Proposition 3.6.

######  Corollary 3.8.

The group compact AS.R is compatible with the topology of the topological
group $G$ if and only if $G$ is locally compact.

Let $\lambda_{\mathcal{F}}$ denotes the AS.R on the topological group $G$
associated to the generating set $\mathcal{F}$ on $G$. For two subsets $A$ and
$B$ of $G$ define $A\delta_{\mathcal{F}}B$ if
$\bar{A}\cap\bar{B}\neq\emptyset$ or there exists some $K\in\mathcal{F}$ such
that $A\cap BK$ is not in $\hat{F}$ (i.e they are not asymptotically
disjoint). If $\lambda_{\mathcal{F}}$ is an asymptotically normal, proper and
compatible AS.R on the normal topological group $G$ then
$\delta_{\mathcal{F}}$ is a compatible proximity on $G$. Thus the Smirnov
compactification of the proximity space $(G,\delta_{\mathcal{F}})$ is
homeomorphic with the Higson compactification of
$(G,\mathcal{E}_{\mathcal{F}})$.
Unfortunately the AS.R $\lambda_{\mathcal{F}}$ fails to be asymptotically
normal in general. In the following we give an example of a locally compact
normal topological group such that the group compact AS.R is not
asymptotically normal.

######  Example 3.9.

Suppose the additive group $\mathbb{R}$ with the discrete topology. Note that
in this topological group the family $\mathcal{K}$ is equal to the family of
all finite subsets of $\mathbb{R}$. We claim that AS.R $\lambda_{\mathcal{K}}$
on this topological group is not asymptotically normal. To prove this it
suffices two show that there are two subsets $A$ and $B$ of $\mathbb{R}$ such
that they are asymptotically disjoint but for each asymptotically disjoint
subset $X$ from $A$, the subset $\mathbb{R}\setminus X$ is not asymptotically
disjoint from $B$. Let $A=[0,+\infty)$ and $B=\\{-n\mid n\in\mathbb{N}\\}$.
Two subsets $A$ and $B$ of $\mathbb{R}$ are asymptotically disjoint. Suppose
that $X\subseteq\mathbb{R}$ is asymptotically disjoint from $A$. So for each
$n\in\mathbb{N}$ the set $A_{n}=\\{\epsilon>0\mid-n+\epsilon\in X\\}$ is
finite. So $Y=\bigcup_{n\in\mathbb{N}}A_{n}$ is countable. Thus there exists
some $\epsilon>0$ such that for each $n\in\mathbb{N}$, $-n+\epsilon\notin X$.
This means $\mathbb{R}\setminus X$ is not asymptotically disjoint from $B$.

Let us recall that a Hausdorff topological space $X$ is called _hemicompact_
if there exists a sequence $K_{1}\subseteq K_{2}\subseteq...$ of compact
subsets of $X$ such that $X=\bigcup_{i\in\mathbb{N}}K_{i}$ and each compact
subset of $X$ contains in $K_{i}$ for some $i\in\mathbb{N}$. It can be shown
that the group compact coarse structure on a topological group $G$ coincides
with the metric coarse structure for some left invariant metric on $G$ if and
only if $G$ is hemicompact (It is a straightforward consequence of Theorem
2.17 of [10]). Since each metrizable AS.R is asymptotically normal
(Proposition 2.11) we have the following result.

######  Proposition 3.10.

Let $G$ be a hemicompact topological group. Then the group compact AS.R is
asymptotically normal.

##  4\. Asymptotic dimensiongrad

Suppose that $(X,d)$ is a metric space and $r>0$. Let us recall that a
$r$-chain from $x\in X$ to $y\in X$ is a finite sequence $x=x_{0},...,x_{n}=y$
such that $d(x_{i},x_{i+1})\leq r$ for all $i\in\\{0,...,n-1\\}$.

######  Definition 4.1.

Let $(X,\mathcal{E})$ be a coarse space and $E\in\mathcal{E}$. A $E$-chain in
$(X,\mathcal{E})$ from $a\in X$ to $b\in X$ is a finite sequence
$a=x_{0},...,b=x_{n}$ such that $(x_{i},x_{i+1})\in E$, for all
$i\in\\{0,...,n-1\\}$.

######  Definition 4.2.

Suppose that $G$ is a group and $\mathcal{F}$ is a generating family on $G$.
Let $X\subseteq G$ be a nonempty set. For $K\in\mathcal{F}$ by a $K$-chain in
$X$ from $x\in X$ to $y\in X$ we mean the finite sequence
$x=g_{0},...,g_{n}=y$, such that for each $i\in\\{1,...,n\\}$, $g_{i}\in X$
and $g_{i}\in g_{i-1}K$.

The following proposition is a direct consequence of Definition 4.1 and
Definition 4.2.

######  Proposition 4.3.

Let $\mathcal{F}$ be a generating family on the group $G$. Then
$a=g_{0},...,g_{n}=b$ is a $E$-chain in $(X,\mathcal{E}_{\mathcal{F}})$ for
some $E\in\mathcal{E}_{\mathcal{F}}$ if and only if it is a $K$-chain for some
$K\in\mathcal{F}$.

######  Proposition 4.4.

Suppose that $d$ is a left invariant metric on the group $G$. Let
$\mathcal{F}$ denote the family of all bounded subsets of $(G,d)$ and let $X$
be a nonempty subset of $G$. Then the sequence $x=g_{0},...,g_{n}=y$ is a
$r$-chain in $X$ for some $r>0$ if and only if there exists some
$K\in\mathcal{F}$ such that $x=g_{0},...,g_{n}=y$ is a $K$-chain between $x$
and $y$ in $X$.

###### Proof.

It is straightforward. ∎

######  Definition 4.5.

Let $(X,\mathcal{E})$ be a coarse space. Suppose that $A$ and $B$ are two
asymptotically disjoint subsets of $X$. We call $C\subseteq X$ an _asymptotic
cut_ between $A$ and $B$ if $C$ is asymptotically disjoint from both $A$ and
$B$ and for each $E\in\mathcal{E}$ there exists some $F\in\mathcal{E}$ such
that each $E$-chain from an element of $A$ to an element of $B$ meets $F(C)$.

######  Proposition 4.6.

Let $G$ be a group and let $\mathcal{F}$ denote a generating family on $G$.
Assume that $X\subseteq G$ is nonempty and consider $X$ with the subspace
coarse structure. For two asymptotically disjoint subsets $A$ and $B$ of $X$,
the subset $C$ of $X$ is an asymptotic cut between $A$ and $B$ in $X$ if and
only if $C$ is asymptotically disjoint from both $A$ and $B$ and for each
$H\in\mathcal{F}$ there exists some $K\in\mathcal{F}$ such that each $H$-chain
in $X$ from an element of $A$ to an element of $B$ in $X$ has a member in
$CK$.

###### Proof.

It is straightforward. ∎

By using Proposition 4.4 and Proposition 4.6 it is evident that Definition 4.5
is a natural generalization of the well known definition of the asymptotic cut
in finitely generated groups ([1]).
We recall the following definition from [7].

######  Definition 4.7.

Assume that $(X,\lambda)$ is an AS.R space. A subset $C$ of $X$ is called a
_large scale separator_ between two asymptotically disjoint subsets
$A,B\subseteq X$, if it is asymptotically disjoint from both $A$ and $B$ and
we have $X=X_{1}\cup X_{2}$ such that $X_{1}$ and $X_{2}$ are asymptotically
disjoint from $A$ and $B$ respectively and if $L_{1}\lambda L_{2}$ for
$L_{1}\subseteq X_{1}$ and $L_{2}\subseteq X_{2}$, then there is a subset $L$
of $C$ such that $L_{1}\lambda L$.

We recall that a closed subset $C$ of a topological space $X$ is called a
separator between closed and disjoint subsets $A$ and $B$ of $X$ is
$X\setminus C=U\cup V$, such that $U$ and $V$ are two open and disjoint
subsets of $X$ and they contain $A$ and $B$ respectively.
The following proposition is a direct consequence of Proposition 3.2 of [7].

######  Proposition 4.8.

Let $\mathcal{F}$ be a generating family on the normal topological group $G$.
Assume that $\lambda_{\mathcal{F}}$ is an asymptotically normal AS.R on $G$
and $\mathcal{E}_{\mathcal{F}}$ is a proper and compatible coarse structure on
$G$. Then for two asymptotically disjoint subsets $A$ and $B$ of $G$ if $C$ is
a large scale separator between $A$ and $B$ then $\nu C$ is a separator
between $\nu A$ and $\nu B$ in $\nu G$.

######  Corollary 4.9.

Let $G$ be a normal and locally compact topological group such that
$\lambda_{\mathcal{K}}$ is asymptotically normal. Then if $C\subseteq G$ is a
large scale separator between asymptotically disjoint subsets $A$ and $B$ of
$G$, then $\nu C$ is a separator between $\nu A$ and $\nu B$ in $\nu G$.

###### Proof.

It is a direct consequence of Corollary 3.8 and Proposition 4.8. ∎

######  Proposition 4.10.

Let $(X,\mathcal{E})$ be a coarse space. Assume that $A$ and $B$ are two
asymptotically disjoint subsets of the AS.R space $(G,\lambda_{\mathcal{E}})$.
Then each large scale separator between $A$ and $B$ in
$(X,\lambda_{\mathcal{E}})$ is an asymptotic cut between $A$ and $B$ in
$(X,\mathcal{E})$.

###### Proof.

Suppose that $C$ is a large scale separator between $A$ and $B$ and
$X=X_{1}\cup X_{2}$ such that $X_{1}$ and $X_{2}$ satisfy the properties of
Definition 4.7. Since $A$ and $X_{1}$ are asymptotically disjoint so $A\cap
X_{1}$ is bounded. Similarly $B\cap X_{2}$ is bounded. So, without loss of
generality, we can assume that $A\subseteq X_{2}$ and $B\subseteq X_{1}$.
Suppose that $E\in\mathcal{E}$. Let $L_{1}$ denote the set of all $x\in X_{1}$
such that for some $E$-chain $x_{0},...,x_{n}$ from $A$ to $B$ there exists
some $i\in\\{1,...,n\\}$ such that $x=x_{i}$ and $x_{i-1}\in X_{2}$. Similarly
let $L_{2}$ denotes the set of all $x\in X_{2}$ such that for some $E$-chain
$x_{0},...,x_{n}$ joining $A$ and $B$ and some $i\in\\{0,...,n-1\\}$,
$x=x_{i}$ and $x_{i+1}\in X_{1}$. Clearly $L_{1}\subseteq F(L_{2})$ and
$L_{2}\subseteq F(L_{1})$, where $F=E\cup E^{-1}$. So
$L_{1}\lambda_{\mathcal{E}}L_{2}$. Thus there exists some $L\subseteq C$ such
that $L_{1}\lambda_{\mathcal{E}}L$. It means that $L_{1}\subseteq O(L)$ for
some $O\in\mathcal{E}$, which shows each $E$-chain joining $A$ and $B$ meets
$O(C)$. ∎

######  Definition 4.11.

Let $G$ be a group and let $\mathcal{F}$ denote a generating family on $G$. We
say $G$ is $H$-_convex_ for some $H\in\mathcal{F}$, if $H$ generates $G$ and
for each $K\in\mathcal{F}$ there exists some $n\in\mathbb{N}$ such that
$K\subseteq H^{n}$.

######  Example 4.12.

Let $G$ be a finitely generated group and let $\mathcal{F}$ denote the family
of all finite subsets of $G$. Assume that the finite subset $K$ generates $G$.
Then clearly $G$ is $K$-convex. Recall that the coarse structure associated to
the generating family $\mathcal{F}$ is equal to the bounded coarse structure
associated to the metric $d_{K}$.

######  Proposition 4.13.

Let $G$ be a compactly generated locally compact group. Consider $G$ with the
AS.R $\lambda_{\mathcal{K}}$, where $\mathcal{K}$ denotes the family of all
compact subsets of $G$. Then $G$ is $H$-convex for some compact subset
$H\subseteq G$.

###### Proof.

Let $H$ be a compact set of generators for $G$. Clearly
$G=\bigcup_{i\in\mathbb{N}}H^{i}$. By using the Baire category theorem there
exists some $n\in\mathbb{N}$ such that $H^{n}$ has nonempty interior. Choose
the open neighbourhood $U$ of the identity such that for some $a\in H^{n}$,
$aU\subseteq H^{n}$. Assume that $K\subseteq G$ is a compact subset. So there
exists $a_{1},...,a_{l}$ such that $K\subseteq\bigcup_{i=1}^{l}a_{i}U$. Let
$m\in\mathbb{N}$ be such that $a_{1}a^{-1},...,a_{l}a^{-1}\in H^{m}$. Now for
$x\in K$ choose $i\in\\{1,...,l\\}$ such that $x\in a_{i}U$. Thus $x\in
a_{i}a^{-1}aU\subseteq H^{m+n}$. It shows $K\subseteq H^{m+n}$, which
completes our proof. ∎

######  Proposition 4.14.

Assume that $G$ is a group and $\mathcal{F}$ is a generating set on $G$. If
$G$ is $H$-convex for some $H\in\mathcal{F}$ then each asymptotic cut between
asymptotic disjoint subsets of $G$ is a large scale separator between them.

###### Proof.

First it is straightforward to show that if the group $G$ is $H$-convex for
some $H$ in $\mathcal{F}$ then it is also $H\cup H^{-1}$-convex. Thus without
loss of generality we can assume that $H=H^{-1}$. Let $A$ and $B$ be two
asymptotically disjoint subsets of $X$ and let $C$ be an asymptotic cut
between $A$ and $B$. By the definition of asymptotic cut there exists some
$K\in\mathcal{F}$ such that each $H$-chain connecting $A$ and $B$ meets $CK$.
Since $A$ and $B$ are asymptotically disjoint from $C$, $A\cap CK$ and $B\cap
CK$ are bounded so we can assume, without loss of generality, $A,B\subseteq
G\setminus CK$. Let $Y=G\setminus CK$. For $a\in Y$ assume that $[a]_{H}^{Y}$
denotes all points $b\in Y$ such that there exists a $H$-chain in $Y$
connecting $a$ and $b$. Let $X_{1}=\cup_{a\in A}[a]_{H}^{Y}$ and
$X_{2}=X\setminus X_{1}$. Since each $H$-chain connecting $A$ and $B$ meets
$CK$, $A\subseteq X_{1}$ and $B\subseteq X_{2}$. Suppose that $L_{1}\subseteq
X_{1}$ and $L_{2}\subseteq X_{2}$ and $L_{1}\lambda_{\mathcal{F}}L_{2}$. So
there is some $O\in\mathcal{F}$ such that $L_{1}\subseteq L_{2}O$ and
$L_{2}\subseteq L_{1}O$. Since we assumed that $G$ is $H$-convex, there exists
some $n\in\mathbb{N}$ such that $O\subseteq H^{n}$. So $L_{1}\subseteq
L_{2}H^{n}$. It shows that for each $c\in L_{1}$ there exists some $d\in
L_{2}$ and a $H$-chain $x_{0}=c,...,x_{n}=d$. Since $L_{2}\subseteq X_{2}$ and
$[c]_{H}^{Y}\subseteq X_{1}$, $d\notin[c]_{H}^{Y}$. Thus there is some
$i\in\\{1,..,n\\}$ such that $x_{i}\in CK$. Thus there is some $k\in K$ such
that $x_{i}k^{-1}\in C$. Let $\tilde{H}=H\cup...\cup H^{n}$ and let
$F=\tilde{H}(K^{-1}\cup K)$. Clearly $F=F^{-1}$. Since $x_{i}\in c\tilde{H}$
so $x_{i}k^{-1}\in cF$. Thus for each $c\in L_{1}$ we found some $g\in C$ such
that $c\in gF$. It shows that $L_{1}\subseteq CF$. If we assume $L=C\cap
L_{1}F$. Clearly $L_{1}\subseteq LF$ and $L\subseteq L_{1}F$, which shows
$L\lambda_{\mathcal{F}}L_{1}$. ∎

######  Remark 4.15.

One can shows that if $G$ is a group and $\mathcal{F}$ is a generating family
on $G$ such that $G$ is $H$-convex for some $H\in\mathcal{F}$, then the AS.R
space $(G,\lambda_{\mathcal{F}})$ is metrizable and it is $r$-convex for some
$r>0$. Thus Proposition 4.14 is a consequence of Proposition 4.9 of [7].

The following example shows that the inverse of Proposition 4.10 is not true
in general and in addition it shows being $H$-convex is an important condition
for Proposition 4.14.

######  Example 4.16.

Assume that $G$ is the additive group $\mathbb{Q}\times\mathbb{Q}$ and
consider $G$ with the discrete topology. We assume $G$ with the AS.R
$\lambda_{\mathcal{K}}$. Since each compact subset of $G$ is finite and $G$ is
not finitely generated so $G$ is not $H$-convex for each compact subset $H$ of
$G$. Remark that $G$ is hemicompact. Now suppose that
$(p_{n})_{n\in\mathbb{N}}$ denotes the sequence of all prime numbers and
$p_{1}<p_{2}<...$. Let $A=\\{(n+\frac{1}{p_{n}},0)\mid n\in\mathbb{N}\\}$ and
$B=\\{(0,n)\mid n\in\mathbb{N}\\}$. It is easy to verify that for each finite
subset $F$ of $G$, $(A+F)\bigcap B$ is finite. So $A$ and $B$ are
asymptotically disjoint. Assume $F=\\{(a_{1},b_{1}),...,(a_{k},b_{k})\\}$ is a
finite subset of $G$. For each $i\in\\{1,..,k\\}$ choose $m_{i}\in\mathbb{Z}$
and $n_{i}\in\mathbb{N}$ such that $a_{i}=\frac{m_{i}}{n_{i}}$. Suppose that
$r\in\mathbb{N}$ is such that $p_{r}$ is strictly greater than each prime
factor of $n_{1},...,n_{k}$. For $l_{1},...,l_{k}\in\mathbb{N}$, if
$n+\frac{1}{p_{r}}+l_{1}a_{1}+...+l_{k}a_{k}=0$ then

$np_{r}n_{1}...n_{k}+n_{1}...n_{k}+l_{1}m_{1}p_{r}n_{2}...n_{k}+l_{2}m_{2}p_{r}n_{1}n_{3}...n_{k}+...+l_{k}m_{k}p_{r}n_{1}...n_{k-1}=0$

The last equality shows that there are integers $\alpha_{0},...,\alpha_{k}$
such that

$\alpha_{0}p_{r}+n_{1}...n_{k}+\alpha_{1}p_{r}+...+\alpha_{k}p_{r}=0$

So

$-p_{r}(\alpha_{0}+...+\alpha_{k})=n_{1}...n_{k}$

This means $p_{r}$ is a prime factor of $n_{1}...n_{k}$ which contradicts our
chose of $p_{r}$. This argument shows that the set of all $i\in\mathbb{N}$
such that there exists a $F$-chain from $(0,i+\frac{1}{p_{i}})$ to an element
of $B$ is finite. Therefore the set $C=\\{(0,0)\\}$ is an asymptotic cut
between $A$ and $B$ ($C$ can be any finite subset of $G$). We are going to
show that $C$ is not a large scale separator between $A$ and $B$. Suppose the
set $X_{1}\subseteq X$ is such that $X_{2}=X\setminus X_{1}$ is asymptotically
disjoint from $B$. So for each $\alpha\in\mathbb{Q}$ the set
$A_{\alpha}=\\{n\in\mathbb{N}\mid(\alpha,n)\in X_{2}\\}$ is finite. For each
$\alpha\in\mathbb{Q}$ if $A_{\alpha}\neq\emptyset$, let $M_{\alpha}=\max
A_{\alpha}$. Thus $(\alpha,M_{\alpha})\in X_{2}$ and $(\alpha,M_{\alpha}+1)\in
X_{1}$. If we assume that $X_{2}$ is asymptotically disjoint from $X_{1}$,
then we find out there exists some $N\in\mathbb{N}$ such that for each
$\alpha>N$, $A_{\alpha}=\emptyset$. It shows that
$L_{1}=\\{(n+\frac{1}{p_{n}},1)\mid n\geq N\\}\subseteq X_{1}$. Let
$L_{2}=\\{(n+\frac{1}{p_{n}},0)\mid n\geq N\\}$. Clearly $L_{2}\subseteq A$
and $L_{1}\lambda L_{2}$. Since $L_{1}$ and $L_{2}$ are unbounded (they are
infinite), $X_{1}$ and $A$ are not asymptotically disjoint. This argument
shows that it is impossible to have $X=X_{1}\bigcup X_{2}$ such that $X_{1}$
and $X_{2}$ are asymptotically disjoint and they are asymptotically disjoint
from $A$ and $B$ respectively. Therefore $C$ can not be a large scale
separator between $A$ and $B$.

Let us recall that if $Y$ is a nonempty subset of the coarse space
$(X,\mathcal{E})$ we denote the induced coarse structure on $Y$ by
$\mathcal{E}_{Y}$.

######  Definition 4.17.

Let $X$ be a set and let $\mathcal{E}$ be a coarse structure on $X$. We denote
the _asymptotic dimensiongrad_ of $(X,\mathcal{E})$ by
$\operatorname{asDg}_{\mathcal{E}}(X)$ and define it inductively as follows.
We say $\operatorname{asDg}_{\mathcal{E}}(X)=-1$ if and only if $X$ is
bounded. For $n\in\mathbb{N}$ we say $\operatorname{asDg}_{\mathcal{E}}(X)\leq
n$ if for each two asymptotically disjoint subsets $A$ and $B$ of
$(X,\lambda_{\mathcal{E}})$ there exists an asymptotic cut $C$ between $A$ and
$B$ such that $\operatorname{asDg}_{\mathcal{E}_{C}}(C)\leq n-1$. For
$n\in\mathbb{N}\bigcup\\{0\\}$ we say $\operatorname{asDg}_{\mathcal{E}}(X)=n$
if $\operatorname{asDg}_{\mathcal{E}}(X)\leq n$ and
$\operatorname{asDg}_{\mathcal{E}}(X)\leq n-1$ is not true. We say
$(X,\mathcal{E})$ has infinite asymptotic dimensiongrad if
$\operatorname{asDg}_{\mathcal{E}}(X)\leq n$ is not true for each
$n\in\mathbb{N}$.

######  Proposition 4.18.

Assume that $(X,\mathcal{E})$ and $(Y,\mathcal{F})$ are two coarsely
equivalent coarse spaces. Then
$\operatorname{asDg}_{\mathcal{E}}X=\operatorname{asDg}_{\mathcal{F}}Y$.

###### Proof.

Let us denote $\lambda_{\mathcal{E}}$ by $\lambda_{1}$ and
$\lambda_{\mathcal{F}}$ by $\lambda_{2}$. We first show
$\operatorname{asDg}_{\mathcal{E}}X\leq\operatorname{asDg}_{\mathcal{F}}Y$. We
proceed by induction on $\operatorname{asDg}_{\mathcal{F}}Y$. If
$\operatorname{asDg}_{\mathcal{F}}Y=-1$ then $Y$ is bounded and clearly $X$ is
also bounded, so $\operatorname{asDg}_{\mathcal{E}}X=-1$. Now assume that the
inequality holds when $\operatorname{asDg_{\mathcal{F}}}Y\leq n$. Suppose that
$\operatorname{asDg_{\mathcal{F}}}Y=n+1$. Also assume that $\phi:X\rightarrow
Y$ is a coarse equivalence between coarse spaces $(X,\mathcal{E})$ and
$(Y,\mathcal{F})$ and $\psi:Y\rightarrow X$ is a coarse inverse of $\phi$. Let
$A$ and $B$ be two asymptotically disjoint subsets of $X$. By Proposition 2.3
of [5], $\phi(A)$ and $\phi(B)$ are two asymptotically disjoint subsets of
$Y$. Since $\operatorname{asDg}_{\mathcal{F}}Y=n+1$ there exists an asymptotic
cut $C$ between $\phi(A)$ and $\phi(B)$ in $Y$ such that
$\operatorname{asDg}_{\mathcal{F}_{C}}C\leq n$. It is easy to show that
$(\psi(C),\mathcal{E}_{\psi(C)})$ is coarse equivalent to
$(C,\mathcal{F}_{C})$, so by our assumption of induction
$\operatorname{asDg}_{\mathcal{E}_{\psi(C)}}\psi(C)\leq n$. Now it remains to
show that $\psi(C)$ is an asymptotic cut between $A$ and $B$. Again by using
Proposition 2.3 of [5], $\psi(C)$ is asymptotically disjoint from both
$\psi(\phi(A))$ and $\psi(\phi(B))$. Since $\psi(\phi(A))\lambda_{1}A$ and
$\psi(\phi(B))\lambda_{2}B$, $\psi(C)$ is asymptotically disjoint from both
$A$ and $B$. Now assume that $E\in\mathcal{E}$. Since $\phi$ is a coarse map,
$F=(\phi\times\phi)(E)\in\mathcal{F}$. Since $C$ is an asymptotic cut between
$\phi(A)$ and $\phi(B)$, there exists some $O\in\mathcal{F}$ such that each
$F$-chain joining an element of $\phi(A)$ to an element of $\phi(B)$ meets
$O(C)$. Since $\psi$ is a coarse map, $L=(\psi\times\psi)(O)\in\mathcal{E}$.
In addition since $\psi$ is a coarse equivalence and $\phi$ is its coarse
inverse, there exists some symmetric $L^{\prime}\in\mathcal{E}$ such that for
each $x\in X$, $(\psi(\phi(x)),x)\in L^{\prime}$. Let $M=L\circ L^{\prime}$.
Assume that $a=x_{0},...,x_{n}=b$ is a $E$-chain from $a\in A$ to $b\in B$. So
$\phi(x_{0}),...,\phi(x_{n})$ is a $F$-chain from $\phi(a)\in\phi(A)$ to
$\phi(b)\in\phi(B)$ and hence it meets $O(C)$. It means there exists some
$i\in\\{0,...,n\\}$ such that $(\phi(x_{i}),c)\in O$, for some $c\in C$. Thus
$(\psi(\phi(x_{i})),\psi(c))\in L$ and hence $(x_{i},\psi(c))\in M$. It shows
that each $E$-chain from $A$ to $B$ meets $M^{-1}(\psi(C))$ and hence
$\psi(C)$ is an asymptotic cut between $A$ and $B$. Thus
$\operatorname{asDg}_{\mathcal{E}}X\leq
n+1=\operatorname{asDg}_{\mathcal{F}}Y$. Similarly one can show that
$\operatorname{asDg}_{\mathcal{F}}Y\leq\operatorname{asDg}_{\mathcal{E}}X$. ∎

######  Corollary 4.19.

Let $\mathcal{F}$ be a generating set on the group $G$ and let $\mathcal{I}$
be a generating set on the group $H$. If the coarse spaces
$(G,\mathcal{E}_{\mathcal{F}})$ and $(H,\mathcal{E}_{\mathcal{I}})$ are
coarsely equivalent then
$\operatorname{asDg_{\mathcal{F}}}G=\operatorname{asDg_{\mathcal{I}}}H$.

It is worth mentioning that if we replace the word “asymptotic cut” with the
word “large scale separator” in Definition 4.17 then we have the definition of
_large scale inductive dimension_. Proposition 4.10 shows that the asymptotic
dimensiongrad is always less than or equal to the large scale inductive
dimension.

##  5\. Set Theoretic Coupling

######  Definition 5.1.

Let $G$ and $H$ be two groups. Assume that $\mathcal{F}$ and $\mathcal{L}$ are
two generating sets on $G$ and $H$ respectively. We call the nonempty set $X$
a _set theoretic coupling_ for AS.R spaces $(G,\lambda_{\mathcal{F}})$ and
$(H,\lambda_{\mathcal{L}})$ if $G$ and $H$ act on $X$ from right and left
respectively, such that these actions commute with each other and there exists
a subset $O\subseteq X$ such that the following properties hold,
i) $GO=OH=X$,
ii) For each $F\in\hat{\mathcal{F}}$ and $K\in\hat{\mathcal{L}}$,

$F_{G}(K)=\\{g\in G\mid gOK\cap OK\neq\emptyset\\}\in\hat{\mathcal{F}}$
$F_{H}(F)=\\{h\in H\mid FOh\cap FO\neq\emptyset\\}\in\hat{\mathcal{L}}$

iii) For each $F\in\mathcal{F}$ there exists some $E_{H}(F)\in\mathcal{L}$
such that

$FO\subseteq OE_{H}(F)$

and for each $K\in\mathcal{L}$ there exists some $E_{G}(K)\in\mathcal{F}$ such
that

$OK\subseteq E_{G}(K)O$

The following proposition shows that Definition 5.1 is equivalent to
Definition 5.5.1 of [8] if we just deal with the finitely generated groups.

######  Proposition 5.2.

Let $G$ and $H$ be two groups and let $\mathcal{F}$ and $\mathcal{F}^{\prime}$
denote the family of all finite subsets of $G$ and $H$ respectively. Suppose
that $G$ and $H$ act from left and right on the set $X$ respectively and these
actions commute. Then $X$ is a set theoretic coupling for
$(G,\lambda_{\mathcal{F}})$ and $(H,\lambda_{\mathcal{F}^{\prime}})$ if and
only if there exists a subset $O\subseteq X$ such that the following
properties hold,
i) $GO=OH=X$,
ii) The subsets $F_{G}=\\{g\in G\mid gO\cap O\neq\emptyset\\}$ and
$F_{H}=\\{h\in H\mid Oh\cap O\neq\emptyset\\}$ are finite subsets of $G$ and
$H$ respectively.
iii) For all $g\in G$ and $h\in H$ there exist finite subsets
$F_{H}(g)\subseteq H$ and $F_{G}(h)\subseteq G$ such that $gO\subseteq
OF_{H}(g)$ and $Oh\subseteq F_{G}(h)O$.

###### Proof.

The ‘only if’ part of Proposition is a straightforward consequence of the fact
$F_{G}=F_{G}(\\{e_{H}\\})$ and $F_{H}=F_{H}(\\{e_{G}\\})$, where $e_{G}$ and
$e_{H}$ are neutral elements of $G$ and $H$ respectively. To prove the
converse suppose that $O\subseteq X$ is such that $GO=OH=X$ and it satisfies
the properties i), ii) and iii) of this proposition. Assume that $L\subseteq
H$ is finite and $L=\\{h_{1},...,h_{n}\\}$. By the property iii) of this
proposition there are finite subsets $F_{G}(h_{i})$ such that $Oh_{i}\subseteq
F_{G}(h_{i})O$, for $i\in\\{1,...,n\\}$. Let $D=\cup_{i=1}^{n}F_{G}(h_{i})$.
Now suppose that $g\in F_{G}(L)$. Since $OL\subseteq DO$, $gDO\cap
DO\neq\emptyset$. It shows that there are $d_{1},d_{2}\in D$ and
$o_{1},o_{2}\in O$ such that $gd_{1}o_{1}=d_{2}o_{2}$. So
$d_{2}^{-1}gd_{1}O\cap O\neq\emptyset$, which shows $d_{2}^{-1}gd_{1}\in
F_{G}$. Therefore $g\in DF_{G}D^{-1}$ and this shows $F_{G}(L)\subseteq
DF_{G}D^{-1}$. Since $DF_{G}D^{-1}$ is finite, $F_{G}(L)$ is also finite.
Similarly one can show for each finite subset $K$ of $G$, $F_{H}(K)$ is
finite. The property iii) of Definition 5.1 is an immediate consequence of the
property iii) of this proposition. ∎

######  Proposition 5.3.

Suppose that $G$ and $H$ are two groups and $\mathcal{F}$ and $\mathcal{L}$
are two generating families on $G$ and $H$ respectively. If
$(G,\lambda_{\mathcal{F}})$ and $(H,\lambda_{\mathcal{L}})$ admit a set
theoretic coupling then they are asymptotically equivalent.

###### Proof.

Suppose that $X$ is a set theoretic coupling for $(G,\lambda_{\mathcal{F}})$
and $(H,\lambda_{\mathcal{L}})$. Our notation here have similar meanings to
the notation in Definition 5.1. Let $x\in O$ and define a map $f:G\rightarrow
H$ such that for each $g\in G$, $g^{-1}x\in O(f(g))^{-1}$. Suppose that
$D\in\hat{\mathcal{L}}$. If $g\in f^{-1}(D)$, $g^{-1}x\in OD^{-1}$. Let
$L=D^{-1}\cup\\{e_{H}\\}$ ($e_{H}$ is the neutral element of $H$). So $x\in
gOL$ and $x\in OL$. Thus $gOL\cap OL\neq\emptyset$, and it shows $g\in
F_{G}(L)$. Therefore $f^{-1}(D)\subseteq F_{G}(L)$ and it shows $f^{-1}(D)$ is
a bounded subset of $G$. Now suppose that $A,B\subseteq G$ and
$A\lambda_{\mathcal{F}}B$. So there exists some $F\in\mathcal{F}$ such that
$A\subseteq BF$ and $B\subseteq AF$. Without loss of generality we can assume
$F=F^{-1}$. So for each $a\in A$ there exists some $b\in B$ such that
$b^{-1}a\in F$ and $a^{-1}b\in F$. We have $a^{-1}xf(b)=a^{-1}bb^{-1}xf(b)\in
a^{-1}bO\subseteq FO$. Besides $FO\subseteq OE_{H}(F)$ and $a^{-1}xf(b)\in
O(f(a))^{-1}f(b)$. Let $K=E_{H}(F)\cup\\{e_{H}\\}$. So $a^{-1}xf(b)\in
OK(f(a))^{-1}f(b)$ and $a^{-1}xf(b)\in OK$, therefore $(f(a))^{-1}f(b)\in
F_{H}(K)$. If we let $K^{\prime}=F_{H}(K)\cup(F_{H}(K))^{-1}$ we have
$f(A)\subseteq f(B)K^{\prime}$. Similarly one can show that $f(B)\subseteq
f(A)K^{\prime}$, thus $f(A)\lambda_{\mathcal{L}}f(B)$. Thus $f$ is an AS.R
mapping. For each $h\in H$ choose $\tilde{f}(h)\in G$ such that
$xh\in\tilde{f}(h)O$. Similar argument can show that $\tilde{f}$ is an AS.R
mapping. Now suppose that $A\subseteq G$. Let $a\in A$. We have $a^{-1}x\in
O(f(a))^{-1}$ so $a^{-1}xf(a)\in O$. Besides $xf(a)\in\tilde{f}(f(a))O$ so
$a^{-1}xf(a)\in a^{-1}(\tilde{f}(f(a)))O$. This shows that
$a^{-1}\tilde{f}(f(a))\in F_{G}(e_{H})=P$. We showed $\tilde{f}(f(a))\in aP$
and $a\in\tilde{f}(f(a))P^{-1}$, for each $a\in A$. Thus
$A\subseteq\tilde{f}(f(A))P^{-1}$ and $\tilde{f}(f(A))\subseteq AP$. We know
from the definition of set theoretic coupling $M=P\cup P^{-1}\in\mathcal{F}$.
Thus $A\lambda_{\mathcal{F}}\tilde{f}(f(A))$ for all $A\subseteq G$. One can
show that $f(\tilde{f}(B))\lambda_{\mathcal{L}}B$ for all $B\subseteq H$ in
similar way. Thus $f$ is an asymptotic equivalence and $\tilde{f}$ is its
coarse inverse. ∎

Let $G$ be a (Hausdorff) topological group that acts on a locally compact
topological space on the right (or on the left). This action is called to be
continuous if the map $(g,x)\rightarrow gx$ from $G\times X$ to $X$ is
continuous and it is called cocompact if there exists a compact subset $K$ of
$X$ such that $GK=X$. The action of $G$ on $X$ is called proper if the set
$\\{g\in G\mid gD\bigcap D\neq\emptyset\\}$ has compact closure for all
compact subsets $D$ of $X$.

######  Definition 5.4.

Suppose that $G$ and $H$ are two (Hausdorff) topological groups and $G$ and
$H$ act continuously, properly and cocompactly on the locally compact
topological space $X$ from left and right respectively. If the actions of $G$
and $H$ on $X$ commute, then $X$ is called a topological coupling for $G$ and
$H$.

######  Proposition 5.5.

Let $G$ and $H$ be two (Hausdorff) topological groups and let $X$ be a locally
compact topological space. Then if $X$ is a topological coupling for $G$ and
$H$ then it is a set theoretic coupling for $(G,\lambda_{\mathcal{K}})$ and
$(H,\lambda_{\mathcal{K}}^{\prime})$, where $\lambda_{\mathcal{K}}$ and
$\lambda_{\mathcal{K}}^{\prime}$ denote the group compact AS.Rs on $G$ and $H$
respectively.

###### Proof.

Since the actions of $G$ and $H$ are cocompact so clearly there exists a
compact subset $K$ of $X$ such that $GK=KH=X$. For each point $x\in K$ suppose
that $U_{x}$ denotes an open subset of $X$ with compact closure such that
$x\in U_{x}$. Since $K$ is compact there are $x_{1},...,x_{n}\in K$ such that
$K\subseteq\bigcup_{i=1}^{n}U_{x_{i}}$. So $W=\bigcup_{i=1}^{n}U_{x_{i}}$ is
an open subset of $X$ such that $GW=WH=X$ and $\overline{W}$ is compact. Let
$O=\overline{W}$. Clearly $O$ satisfies the property i) of 5.1. Now suppose
that $D\subseteq G$ is compact. Since these actions are continuous,
$DO\subseteq X$ is compact and the properness of the action of $H$ leads to
$F_{H}(D)$ is relatively compact. Similar argument shows that
$F_{G}(D^{\prime})$ is relatively compact for each compact subset $D^{\prime}$
of $H$. This proves the property ii) of 5.1. For proving the property iii) of
5.1 suppose that $D\subseteq G$ is compact. Since $WH=X$ and $DO$ is compact
there are $h_{1},...,h_{n}\in H$ such that
$DO\subseteq\bigcup_{i=1}^{n}Wh_{i}$. Clearly the set
$E_{H}(D)=\\{h_{1},...,h_{n}\\}$ is the desired set for the property iii) of
5.1. Similarly for each compact $D^{\prime}\subseteq H$ one can find a finite
set $E_{G}(D^{\prime})$ such that $OD^{\prime}\subseteq E_{G}(D^{\prime})O$. ∎

######  Example 5.6.

Let $G$ be a locally compact Hausdorff topological group and suppose that $H$
and $K$ are two closed normal subgroups of $G$. If $G/H$ and $G/K$ are two
compact topological groups then $G$ is a topological coupling for $H$ and $K$
with their group compact AS.Rs as topological groups. To see this it can be
shown that there exists a compact subset $O$ of $G$ such that $\pi_{H}(O)=G/H$
and $\pi_{K}(O)=G/K$, where $\pi_{H}:G\rightarrow G/H$ and
$\pi_{K}:G\rightarrow G/K$ are quotient maps (for example see Lemma 7.2.5 of
[2]). Clearly $HO=OK=G$, so the trivial actions of $H$ and $K$ on $G$ is
cocompact. Let $D\subseteq G$ be a compact subset. Since $L=\\{h\in H\mid
hD\bigcap D\neq\emptyset\\}=DD^{-1}\bigcap H$ and $DD^{-1}$ is a compact
subset of $G$, $L$ has a compact closure and it shows that $H$ acts properly
on $G$. Similarly one can shows that $K$ acts properly on $G$ too. Therefore
Proposition 5.3 shows that two topological groups $H$ and $K$ are
asymptotically equivalent.

######  Example 5.7.

Recall that a discrete subgroup $H$ of a locally compact group $G$ is called a
_uniform lattice_ if there exists a compact subset $K$ of $G$ such that $HK=G$
($KH=G$) (§5.5.1 of [8]). It is evident from Example 5.6 and Proposition 5.3
that two uniform lattices in a locally compact topological group are
asymptotically equivalent.

## ACKNOWLEDGMENTS

The author wishes to express his gratitude to the anonymous referee for his
(or her) several useful comments. The author was supported by Babol Noshirvani
University of Technology with Grant Program No. BNUT/393072/99.

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