# Geometric and Topological Entropies of Sphere Packing

Jack A. Logan  Department of Physics and Astronomy, Stony Brook University,
Stony Brook, NY 11794  Center for Functional Nanomaterials, Brookhaven
National Laboratory, Upton NY 11973  Alexei V. Tkachenko  Center for
Functional Nanomaterials, Brookhaven National Laboratory, Upton NY 11973

###### Abstract

We present a statistical mechanical description of randomly packed spherical
particles, where the average coordination number is treated as a macroscopic
thermodynamic variable. The overall packing entropy is shown to have two
contributions: geometric, reflecting statistical weights of individual
configurations, and topological, which corresponds to the number of
topologically distinct states. Both of them are computed in the thermodynamic
limit for isostatic packings in 2D and 3D, and the result is further expanded
to the case of “floppy” particle clusters. The theory is directly applicable
to sticky colloids, and in addition, generalizes concepts of granular and
glassy configurational entropies for the case of non-jammed systems.

The deceivingly simple task of packing spheres has been an inspiration for
multiple problems in mathematics and natural sciences since the times of
Kepler [1]. It provides insights into the physics of crystalline solids, as
well as into the world of disordered states of matter [2, 3, 4]. One such
fundamental problem is finding the entropy of a random packing of spheres. In
its simplified form, this amounts to counting the number of ways in which
distinct particle arrangements can be generated for a given ensemble. For
instance, more than two decades ago, Sam Edwards introduced the notion of
granular entropy [5, 6]. A similar concept also arises in the context of hard-
sphere glasses, as a measure of degeneracy of the locally stable
configurations [4, 7]. In both examples, the packing entropy would be a
measure of the multiplicity of the “jammed” states. Due to the non-equilibrium
nature of jamming, even a rigorous definition of the granular entropy remains
non-trivial. Nevertheless, significant progress in understanding and computing
it has been demonstrated in recent years [8, 9, 10, 11].

In this paper, we discuss the packing entropy in an equilibrium system of
spheres, where it can be properly defined as a conventional thermodynamic
quantity. The system studied is not jammed, but rather is constrained to have
a specific number of direct interparticle contacts. In other words, we use the
total number of contacts (or equivalently, mean coordination number of the
particles, $Z$) as a thermodynamic variable. This approach is immediately
relevant to packings of “sticky” spheres where each contact is associated with
a fixed binding energy. Two limits of that problem have been explored in the
past: thermodynamics of a sticky sphere liquid (e.g. Baxter model) [12, 13],
and, more recently, free energy landscapes and kinetics of mesoscopic multi-
colloidal clusters [14, 15, 16, 17, 18, 19]. In this work, we seek to bridge
the gap between these two limits, and, more importantly, use this model system
to better understand the nature of packing entropy. The latter can be
subsequently connected to the granular and/or glassy entropy by applying
external pressure to the system that would lead to its jamming. The difference
from the original contexts in which those entropies were introduced, is that
one would start with an already discrete configurational space, and select its
subset that corresponds to the local minima of the total volume.

Consider a system of $N$ hard spherical particles in $d$-dimensional space,
with average particle diameter $\bar{a}$. The packing is weakly polydisperse,
so that the width of the particle diameter distribution is much smaller than
the average: $\delta a\ll\bar{a}$. We define a pair of particles to be in
contact if the gap between them is less than some small value
$\Delta\ll\bar{a}$. The gap between particles $i$ and $j$ is defined as
$x_{ij}=\left|{\bf r}_{i}-{\bf r}_{j}\right|-(a_{i}+a_{j})/2$, where $a$’s and
${\bf r}$’s are their respective diameters and positions. For any
configuration, the topology of the packing can be specified by an adjacency
matrix $\hat{\bf C}$, with elements $C_{ij}=1$ for all particles $i\neq j$ in
contact, and $C_{ij}=0$ otherwise. As already mentioned above, the average
coordination number $Z=1/N\sum_{i<j}C_{ij}$ is treated as a macroscopic
thermodynamic variable of the system. Note that throughout the paper we set
$k_{B}T=1$.

The weak polydispersity is introduced to avoid hyperstatic (over-constrained)
configurations. These contain “accidental” contacts that could be removed,
e.g., by slight variations of particle sizes (subject to the constraint that
all other contacts are intact). Examples of such over-constrained
configurations are close packed crystals (FCC/HCP in 3D or hexagonal lattice
in 2D). If those are disqualified, any rigid packing has to be isostatic, i.e.
the number of contacts has to be equal to the total number of degrees of
freedom of $N$ spheres, $dN$, minus the number of rigid body degrees of
freedom of the packing as a whole, $d(d+1)/2$ [20, 21]. In the thermodynamic
limit, this corresponds to $Z^{*}=2d$, while for a finite isostatic packing
$Z^{*}=2d-d(d+1)/N$. Below we start by discussing the isostatic limit, and
then generalize our results to the under-constrained case $Z<Z^{*}$, in which
each “missing” bond gives rise to a single zero mode.

The packing entropy for a given $Z$ is found by performing the summation of
statistical weights of all topologically distinct realizations:

$e^{NS_{pack}(Z)}=\sum_{\bf\hat{C}}\frac{\delta(Z({\bf\hat{C}})-Z)}{N!}e^{NS_{geo}({\bf\hat{C}})}$
(1)

Here we have introduced the geometric entropy $S_{geo}({\bf\hat{C}})$ which
determines the statistical weight of a specific realization of the packing:

$e^{NS_{geo}({\bf\hat{C}})}=\int{\frac{\mathrm{d}^{d}{\bf
r}_{2}...\mathrm{d}^{d}{\bf
r}_{N}}{\Omega_{d}\bar{a}^{(N-1)d}}\prod_{i>j,C_{ij}=1}{\bar{a}\delta(x_{ij})}\prod_{i>j,C_{ij}=0}{\Theta(x_{ij})}}$
(2)

This expression assumes that all $N$ spheres belong to a single cluster.
Division by the factor $\Omega_{d}$ eliminates contributions from its rigid
body rotations (specifically, $\Omega_{2}=2\pi$ for 2D and
$\Omega_{3}=2\pi\cdot 4\pi=8\pi^{2}$ for 3D). Translation of the cluster as a
whole is not included since integration is only performed over positions on
$N-1$ out of $N$ spheres. Without loss of generality, the position of the
first particle ${\bf r}_{1}$ will be assumed to be fixed at the origin. The
factor $N!$ in Eq. (1) deserves a special clarification due to the widespread
confusion regarding its origin in statistical mechanics. As we show in
Supplementary Materials (SM), this factor does not require the particles to be
indistinguishable, and moreover, it has nothing to do with quantum mechanics
[10, 18].

For an isostatic packing, we can switch variables from the positions $({\bf
r}_{2},...,{\bf r}_{N})$ to the gaps between pairs of particles in contact,
$(x_{1},...,{x}_{NZ/2})$. The new variables should also include $d(d-1)/2$
independent rigid body rotations of the cluster:
${\bf\hat{\theta}}=\left(\theta_{1}...,\theta_{d(d-1)/2}\right)$. The Jacobian
associated with this change in variables is known as the rigidity matrix [15].
According to Eq. (2), the corresponding Jacobian determinant can be used to
find the geometric entropy:

$NS_{geo}({\bf\hat{C}})=\ln\left[\frac{\partial\left({\bf r}_{2},...,{\bf
r}_{N}\right)}{\partial\left({\bf\hat{\theta}},x_{1},...,{x}_{NZ/2}\right)}\right]$
(3)

A more conventional approach to finding the statistical weight for each
topologically distinct configuration, e.g. in the context of sticky colloids,
is to calculate the free energy that includes both phonons and rigid body
modes of the entire packing [16, 15, 14, 17, 18]. That route is practical, but
it has led to a number of seemingly paradoxical observations. In particular,
one needs to assign specific masses to all of the particles, and replace rigid
bonds with effective springs. Of course, in the non-quantum regime masses may
only give a constant contribution to the free energy. And yet, the overall
statistical weight of a cluster in this formulation depends, e.g. on its
moment of inertia [18]. The paradox may be resolved by the direct
demonstration that when the phonon and rotational partition functions are
combined, all of the masses would only give rise to a trivial multiplier,
independent of specific configurations [19]. The same is true for the spring
constants: as expected in the isostatic case, the free energy is not sensitive
to the details of the bonding potential [17]. The geometric entropy gives the
statistical weights of any configuration in the form of Eq. (3), making the
phonon-based calculation redundant.

Figure 1: (A,B) Minimal clusters (red) for $d=2$ and $d=3$. An additional
particle (purple) bonds to each particle in the minimal cluster along vectors
$\hat{n}_{i}$. (C) Three six-particle clusters with the same $S_{geo}$,
despite having different topologies.

Consider some examples of calculating the geometric entropy. We define a
minimal cluster in $d$ dimensions as $d$ particles, all in contact with each
other, such as the red particles in Fig. 1A-B. A minimal cluster is a special
case that gives $S_{geo}=0$. The proof for $d=2,3$ is presented in SM. The
next most trivial cluster is formed by adding a particle that bonds to each of
the particles in the minimal cluster, such as the purple particles shown in
the figure. In 2D, the geometric entropy associated with this additional
particle is $S_{geo}=\ln\left[\hat{n}_{1}\times\hat{n}_{2}\right]$, where
$\hat{n}_{i}$ are unit vectors in the directions of the bonds. If the
particles are arranged in a square lattice, for example, the bond directions
$\hat{n}_{1}$ and $\hat{n}_{2}$ are orthogonal, and hence $S_{geo}=0$. In the
case of a triangular lattice, however, each additional particle forms an
equilateral triangle and adds $-\mathrm{ln}(\frac{\sqrt{3}}{2})$ to the
overall geometric entropy of the minimal cluster. Similarly in 3D, adding a
particle to the minimal cluster creates a regular tetrahedron, and the triple
product of the unit vectors along the bonds adds
$-\mathrm{ln}(\frac{1}{\sqrt{2}})$ to the geometric entropy. Just like the
square lattice in 2D, a cubic lattice is a special case of an isostatic
packing with $S_{geo}=0$ because the bonds are mutually orthogonal.

We can continue to add particles to the minimal clusters that have exactly $d$
bonds each, maintaining the isostaticity of the packing. Three such examples
where four particles have been added to a 2D minimal cluster are shown in Fig.
1C. The addition of each new particle always increases the cumulative
geometric entropy of the cluster by $-\mathrm{ln}(\frac{\sqrt{3}}{2})$. For
this reason, despite the differences in their topologies, they all have the
same $S_{geo}$. Note, however, that if the distinction between particles is
ignored, the higher-symmetry triangular cluster would have a lower statistical
weight, as it corresponds to a smaller number of non-trivial particle
permutations [16, 19].

The overall packing entropy can be expressed as

$\displaystyle S_{pack}(Z)=\langle S_{geo}\rangle_{Z}+S_{topo}(Z)$ (4)

Here $\langle...\rangle$ denotes the averaging over all topological
realizations, weighted proportionally to $\exp(NS_{geo})$, with a given $Z$.
The fact that $-NS_{geo}$ acts as an effective Hamiltonian enables one to
employ a Monte-Carlo (MC) approach to generate the equilibrium ensemble of
isostatic packs and calculate, e.g. $\langle S_{geo}\rangle_{Z}$ itself, or
any other ensemble-averaged quantity. However, the generated “energy”
landscape is very rough, with lots of local minima, and making large, non-
local, moves to escape them is highly non-trivial. To resolve this
complication, we introduce a generalized effective Hamiltonian,
$H_{eff}=-\lambda NS_{geo}(\mathbf{\hat{C}})$. The parameter $\lambda$ here
allows one to tune the model from its original form (for $\lambda=1$), to one
with a completely flat “energy” landscape (in $\lambda=0$ limit). That limit
corresponds to a model in which all plausible topological arrangements with
the same coordination number $Z$ have the same statistical weight. MC
simulations are run for $\lambda=0$, and the results for $\lambda=1$ are
extrapolated from those simulations by calculating the linear response to
$\lambda$. Let $S_{geo}^{(0)}={\langle S_{geo}\rangle}$ and $S^{(0)}_{pack}$
be the geometric and the full packing entropies computed for $\lambda=0$ (note
that topological entropy in the limit $\lambda=0$ is identical to
$S^{(0)}_{pack}$). By expanding $S_{pack}$ in leading orders of $\lambda$ and
$S_{geo}$, we obtain:

$S_{pack}(\lambda,S_{geo})=\lambda
S_{geo}+S^{(0)}_{pack}-\frac{1}{2\chi}\left(S_{geo}-S^{(0)}_{geo}\right)^{2}$
(5)

Here, the linear response coefficient
$\chi=N\left\langle(S_{geo}-S^{(0)}_{geo})^{2}\right\rangle$ can be extracted
from statistics of $S_{geo}$. By maximizing Eq. (5) with respect to $S_{geo}$,
one can extrapolate geometric, topological, and the overall packing entropy to
$\lambda=1$. In particular, that gives $S_{geo}\approx S^{(0)}_{geo}+\chi$ and
$S_{topo}^{*}=S^{(0)}_{pack}-\frac{\chi}{2}$. Below we describe the numerical
procedure that was used to compute $S_{geo}^{(0)}$, $\chi$ and
$S^{(0)}_{pack}$.

In our simulations, it was ensured that the packing remains strictly isostatic
and does not have any interparticle overlaps. A typical MC move is illustrated
in Fig. 2C: a randomly chosen bond is broken, and the gap between the two
particles increases, while all the spheres are pushed along the single zero
mode associated with the lost contact. The move is complete once two,
previously unbound, particles make a contact. If $\lambda=0$, there is no
difference in statistical weights between different (isostatic)
configurations, so the only constraint is non-overlapping of the particles.

Figure 2: Example packings. (A) Packing with periodic boundary conditions in
the horizontal directions, with a closed bottom, but open upward. (B) A
typical cluster packing in free space. (C) A typical Monte Carlo move. The
bonded green particles slowly open, allowing the packing to move along a
single zero mode until the purple particles make contact. At this point, the
bond is broken between the green pair of particles and created between the
purple particles, completing the move to a new topological state. (D) A
cluster packing in equilibrium with an infinitely large square lattice. At any
moment, a free particle may “condense” onto the packing in any of the $N_{+}$
positions where it will have exactly $d$ bonds (dashed open circles).
Likewise, any of the $N_{-}$ particles in the packing with exactly $d$ bonds
(blue) may “evaporate” off of the packing and back to the lattice. Here
$N_{+}=9$ (the top-center is two closely spaced) and $N_{-}=3$.

Two different classes of isostatic systems have been studied: (i) packings in
a semi-periodic box, with periodic boundary conditions (PBC) along the x- and
y- axes, and a free boundary in the upward z-direction (only x- and z- axes in
2D), and (ii) clusters in open space. In both cases, the size of the system
was gradually increased, by adding one particle at a time, in a way that
preserved the overall isostaticity. Examples of both packing types can bee
seen in Fig. 2A-B.

By using Eq.(3), distribution of $S_{geo}$ has been computed directly, which
determined parameters $S_{geo}^{(0)}$ and $\chi$ in Eq. (5). In order to
compute $S_{pack}^{(0)}$, we imagine that the packing can exchange particles
with an infinitely large cubic (or square in 2D) lattice through “evaporation”
and “condensation” events, as shown in Fig. 2D. Technically, particles have
not been moved between the random packing and the reference lattice, but only
the probabilities of such moves have been computed. In equilibrium, chemical
potential of particles in the random packing, $\mu=-S_{pack}^{(0)}$, should
match that in the reference lattice. For any configuration, we find a number
$N_{+}$ of sites where a particle can be added to the packing, and a number
$N_{-}$ of “removable” particles that has exactly $d$ bonds. By requiring the
addition and removal processes to balance each other, one obtains
$S_{pack}^{(0)}=-\mu=\ln\left(\langle N_{+}\rangle/\langle
N_{-}\rangle\right)$.

Figure 3: Geometric and Topological entropy values per particle for 2D (top,
dashed) and 3D (bottom, solid), including PBC packings (red) and cluster
packings (blue) for all packing sizes, up to 400 particles.

Our results for the topological and geometric entropies are shown in Fig. 3 as
functions of packing size for both cases of PBC and clusters. Note an
excellent agreement between the two methods, and apparent convergence in the
limit of large $N$. The implied infinite-size values of different types of
entropies for 2D and 3D, together with the corresponding volume fractions
$\eta$, are presented in Table 1. Interestingly, the obtained geometric
entropies are quite close to our original estimates which were based on a
sequential packing procedure: $S_{geo}\approx 0.15\pm 0.01$, vs.
$-\mathrm{ln}(\frac{\sqrt{3}}{2})\approx 0.14$ in 2D, and $S_{geo}\approx
0.35\pm 0.04$ vs. $-\mathrm{ln}(\frac{1}{\sqrt{2}})\approx 0.35$ in 3D.
Topological entropies in both cases are quite substantial: $1.52\pm 0.13$ and
$3.02\pm 0.32$, respectively. It is important to keep in mind that we are
exploring the space of isostatic packs, rather than jammed ones. In other
words, they do not represent local density maxima. This explains why our
results for the topological entropy are significantly larger than those
extracted from recent computation of granular entropies: $0.5$ and $0.7$ for
2D and 3D, respectively [10, 11]. This also explains why the observed
equilibrium volume fraction in 3D is significantly lower than that of Random
Close Packing (or, Maximally Random Jammed state defined in Ref. [3]).
Curiously, the found equilibrium value $\eta\approx 0.54\pm 0.02$ is close to
the so-called random loose packing density [22], although this is likely to be
just a coincidence.

| PBC | Clusters
---|---|---
$\mathbf{d=2}$ | $\mathbf{S_{pack}}$ | 1.66 | 1.58
$\mathbf{S_{geo}}$ | 0.15 | 0.15
$\mathbf{S_{topo}}$ | 1.52 | 1.43
$\bm{\eta}$ | 0.73 | 0.78
$\mathbf{d=3}$ | $\mathbf{S_{pack}}$ | 3.37 | 3.04
$\mathbf{S_{geo}}$ | 0.35 | 0.42
$\mathbf{S_{topo}}$ | 3.02 | 2.62
$\bm{\eta}$ | 0.52 | 0.56
Table 1: The geometric, topological, and full packing entropy per particle in
2D and 3D Cluster and PBC packings with 400 particles. The average packing
fraction $\langle\eta\rangle$ is also listed.

So far, we focused on an isostatic case, where the number of contacts exactly
matched the number of frozen degrees of freedom in a rigid packing. To expand
our approach to the under-constrained case of so-called “floppy” networks, we
note that breaking any bond in an isostatic network generates a single zero
mode. Such a mode is bounded by two isostatic states, which allows one to
count the number of distinct underconstained configurations, and find the
corresponding topological entropy:

$S_{topo}(Z)\approx
S_{topo}(Z^{*})+\frac{Z^{*}-Z}{2}\ln\left[\frac{Z^{*}}{2(Z^{*}-Z)}\right]$ (6)

As for the geometric entropy, it is obtained by integration over all activated
zero modes, and the corresponding correction is simply proportional to their
number: $S_{geo}(Z)\approx S_{geo}(Z^{*})+\frac{Z^{*}-Z}{2}\ln(\xi/\bar{a})$,
where $\xi$ represents the typical range over which the interparticle gap
$x_{ij}$ may change until the new contact is formed. In the specific case of
sticky spheres, the packing entropy $S_{topo}+S_{geo}$ makes an important
contribution to the chemical potential of a disordered aggregate (i.e. a large
cluster):

$\mu(Z)=\frac{\epsilon Z}{2}-S_{geo}(Z)-S_{topo}(Z)$ (7)

Here $\epsilon=-\ln\left(\int_{0}^{\infty}\exp(-V(x))dx/\bar{a}\right)$ is the
binding free energy, determined by short-range interparticle potential $V(x)$.
Minimization of chemical potential $\mu(Z)$ predicts an exponential
suppression of zero modes with bond strength: $(Z^{*}-Z)\sim e^{-\epsilon}$.
Therefore, residual topological and geometric entropies of an aggregate are
close to their isostatic values, given in Table 1.

In summary, we proposed a statistical mechanical description of sphere
packings based on treating the coordination number $Z$ as a macroscopic
thermodynamic parameter, and identified two contributions to the packing
entropy: geometric and topological. They correspond to the statistical weight
of a particular topological configuration, and the number of non-equivalent
arrangements, respectively. The topological entropy is thus analogous, but not
equivalent, to Edwards granular entropy or to the residual entropy of hard
sphere glasses. An important difference of our approach is that it is built
entirely within the framework of equilibrium statistical mechanics, and does
not impose a requirement on the individual configurations to be jammed. Hence,
our results are directly applicable to systems with tensile short-range
forces, such as sticky spherical colloids. We developed an MC scheme to
compute the geometric and topological entropies of isostatic packings in both
2D and 3D, and further generalized the results for the case of floppy (under-
constrained) packings.

Acknowledgment. The research was carried out at the Center for Functional
Nanomaterials, which is a U.S. DOE Office of Science Facility, at Brookhaven
National Laboratory under Contract No. DE-SC0012704

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##  I Supplementary Materials

###  I.1 Origin of factor $N!$

There is considerable confusion in statistical physics literature (including
multiple textbooks) regarding the origin of the $N!$ factor that appears in
Eq. (1). Traditionally, this factor is introduced as a resolution to the Gibbs
paradox, with a justification of the particles (e.g., molecules of a gas)
being “indistinguishable”. It is often claimed that the origin of this factor
lies in quantum mechanics. Neither of these justifications is valid.
Statistical mechanics of an ideal gas is routinely applied e.g. to systems of
colloids which are neither quantum nor strictly identical. To clarify this
issue, consider a system, containing $N$ particles that can be exchanged with
a much larger system (thermal bath, TB) containing $N_{tot}-N$ particles. All
of the particles in this example are distinguishable, but we assume the free
energy of TB to only depend on the total number of particles that it contains,
not on their specific subset (e.g. the thermal bath may be an ideal gas). If
we start with all particles being in the thermal bath, there are $N_{tot}$
ways of selecting a particle to be moved to the system, then there are
$N_{tot}-1$ ways of selecting the next one, etc. When taking into account the
arbitrary order in which $N$ particles can be moved from TB to the system, we
obtain the overall statistical weight of the configuration with a given $N$:

$\displaystyle\tilde{Z}(N)=\lim_{N_{tot}\rightarrow\infty}\frac{N_{tot}!}{N!(N_{tot}-N)!}\frac{Z(N)Z_{tb}(N_{tot}-N)}{Z_{tb}(N_{tot})}=$
$\displaystyle=\frac{Z(N)e^{-\mu N}}{N!}$ (S1)

Here $\mu=-\partial\ln(Z_{tb}(N_{tot}))/\partial N_{tot}-\ln N_{tot}$ is
chemical potential of a particle in the thermal bath. Partition function of
the system $Z(N)$ is averaged over all possible selections of $N$ particles
from $N_{tot}$ that belong to the TB.

###  I.2 Geometric Entropy of Minimal Clusters

Figure S1: Initial two particles ($i$ and $j$) shown in red along the $X$-axis
in contact with particle $k$ (purple). Particle $k$ has a rotational degree of
freedom, here seen at an angle $\psi$ with the $XY$-plane, that allows it to
remain in contact with the other particles.

Consider two identical particles, $i$ and $j$, of diameter $a$. Particle $i$
is fixed as the origin of the system, and particle $j$ is placed along the
$X$-axis in contact with $i$, as shown in Fig. S1. To get the Jacobian, we
consider infinitesimal displacements $\delta\mathbf{r}$, $\delta\bm{\theta}$,
$\delta\bm{\phi}$. The effect of $\delta\mathbf{r}$ is to change the gap
$x_{ij}$ between the particles, while the other two displacements lead to
small changes in $y_{j}$ and $z_{j}$ through infinitesimal rotations.
Altogether we find

$\displaystyle\begin{bmatrix}\delta x_{j}\\\ \delta y_{j}\\\ \delta
z_{j}\end{bmatrix}=\begin{bmatrix}1&0&0\\\ 0&a&0\\\
0&0&-a\end{bmatrix}\begin{bmatrix}\delta r\\\ \delta\theta\\\
\delta\phi\end{bmatrix}.$ (S1)

The Jacobian determinant, given $a=1$, is then $|J|=1$. This serves two
purposes: it allows us to build to a 3D minimal cluster one particle at a
time, and restricting above to the $XY$-plane shows that a 2D minimal cluster
has geometric entropy of zero.

For a minimal cluster in 3D, we add a third identical particle $k$ to the
configuration above. The two previous particles are fixed along the $X$-axis,
and the third particle is placed in contact with them. We need to transform
coordinates from $(x_{k},y_{k},z_{k})$ to $(x_{ik},x_{jk},\psi)$, where $\psi$
is the angle from the $XY$-plane to particle $k$, and $x_{ik}$ and $x_{jk}$
are the gaps between the particles. Again we consider infinitesimal
displacements, this time in $\delta\mathbf{x_{k}}$, $\delta\mathbf{y_{k}}$,
and $\delta\mathbf{z_{k}}$ and write down how it affects the gap sizes
$x_{ik}$, $x_{jk}$, and the tilt angle $\psi$.

$\displaystyle\begin{bmatrix}\delta x_{ik}\\\ \delta x_{jk}\\\
\delta\psi\end{bmatrix}=\begin{bmatrix}\frac{1}{2}&0&\frac{\sqrt{3}}{2}\\\
-\frac{1}{2}&0&\frac{\sqrt{3}}{2}\\\
0&-\frac{2}{\sqrt{3}}&0\end{bmatrix}\begin{bmatrix}\delta x_{k}\\\ \delta
y_{k}\\\ \delta z_{k}\end{bmatrix}$ (S2)

We find the Jacobian determinant is again $|J|=1$. For 3D, the product of the
jacobians for each individual particle gives the complete Jacobian value. The
entropy is then found as the natural logarithm of the Jacobian,
$S_{geo}=\ln(1)=0$.

###  I.3 Packing Evolution

A packing, whether PBC or cluster, evolves in a $dN$-dimensional space that
can be represented by the positions of the $N$ spheres or the $dN$ bond gaps.
The constraints for them to be hard spheres in contact are given by
$|\vec{r}_{i}-\vec{r}_{j}|^{2}=\left(\frac{a_{i}+a_{j}}{2}\right)^{2}$. The
Jacobian of this equation is the rigidity tensor, shown in Eq. (3). At each
step of our simulations a bond is broken and a zero mode enters into the
packing. The packing then evolves in such a way that it moves orthogonal to
the constraints imposed by the rigidity tensor until contact is made between
two unbonded particles, and a new isostatic packing is realized. The rigidity
tensor $R(\vec{r})$ relates the particle displacements $\vec{u}_{i}$ to the
gaps between particles $x_{ij}$. If we consider that the bond $\alpha$ breaks
and opens by an amount $x_{\alpha}$, the displacement of any particle $i$ can
be calculated as $\vec{u}_{i}=R^{-1}_{i,\alpha}x_{\alpha}$. However, to avoid
the computationally expensive inverse function, we instead use a QR
decomposition with column pivoting [23] to solve the equation for the
displacements $\vec{u}_{i}$ of all particles caused by the opening of
$\alpha$.

Figure S2: The original $S_{geo}$ distribution for the $400$-particle PBC
packings. A clear second peak is seen that points to zero modes persisting in
the configurations after a new topology is made. Fit with a normal distrubtion
in dashed red. The black vertical lines represent $3\sigma$ of the fit.

It’s important that the trajectory of the evolution moves the packing from one
isostatic packing to another. The new bond that should be made is completely
determined by the specific bond that is broken. Sometimes, however, due to the
finite precision of the simulation, a bond other than the “true” bond will
close. When this occurs one part of the packing will be overconstrained, while
a zero mode will persist in another. This zero mode appears as anomalously
high geometric entropy in our distribution, as demonstrated by the second peak
that appears at larger values than the main peak in Fig. S2. This peak is a
collection of packings that were made when the incorrect bond was closed. In
order to maintain results that only include true isostatic packings, we must
filter out the configurations that include zero modes. We first fit our main
peak with a normal distribution, shown in red in the figure. The black
vertical lines mark three standard deviations from the mean of the fit. Any
data outside of the black lines is assumed to come from a zero mode
configuration and is discarded.

###  I.4 Simulation Procedure

We begin a PBC simulation with a fixed layer of particles at the base of the
box ($z=0$). This layer acts as a substrate on which particles can adsorb.
These substrate particles are fixed throughout the simulation, but any bonds
they share with free particles are not. For the packing to be isostatic, it
requires $dN_{free}$ bonds to match the number of translational degrees of
freedom from the free particles. We use Eq. (3) to calculate the geometric
entropy for each configuration.

To build a packing, we first identify every possible location where a new
particle could come into existence and have exactly $d$ bonds without leading
to any overlaps with the current particles. We call these positions “virtual
particles”. In general, for any configuration there exists an enormous number
of virtual particles, and we randomly choose one in a way that tends to
maximize our packing fraction by applying a weight to each virtual particle.
While building the packing to the desired number of particles, our MC move
(Fig. 2C) is used to rearrange the packing, allowing it to mix while it grows.
The results are not particularly meaningful while the number of particles is
constantly changing, so no data is collected while a packing is growing.

Figure S3: The distributions of geometric entropy for 2D clusters and PBC
packings (top) and 3D clusters and PBC packings (bottom) computed using our MC
algorithm. These distributions represent the geometric entropy for different
configurations with $\lambda=0$.

When all of the boundaries are free, as they are for a cluster, both the
translational and the rotational degrees of freedom need to be considered. For
$d=3$, for instance, Maxwell’s criterion for stiffness tells us that there
must be exactly $3N-6$ bonds. In order to do this, for any given configuration
of $N$ total particles, we fix $d$ bonded particles that are randomly chosen.
Simply fixing $d$ particles leaves the system underconstrained, so $d$ bonds
between the particles must also be fixed. For this reason, it is imperative
that we choose $d$ particles that are all bonded to each other—a minimal
cluster, as defined in the main text. For $d=3$ the minimal cluster can be any
rigid triangle of three particles with three bonds, and for $d=2$ it can be
any bonded pair of particles (Fig. 1). In this way, we find ourselves in a
similar situation to a PBC packing—the number of contacts considered exactly
matches the number of degrees of freedom. This allows us, again, to use Eq.
(3) to find the geometric entropy. In this case, however, it is broken into
two parts: the Jacobian for the $N-d$ particles that are free to move,
analogous to the PBC packing, and an additional Jacobian for the $d$ particles
in the minimal cluster that have been chosen to be temporarily fixed. The
Jacobian of the minimal cluster is calculated to be proportional to the
product of the distances between the $d$ fixed particles in contact. The full
Jacobian to be used in Eq. (3) is the product of the Jacobian of the $N-d$
free particles and that of the $d$ fixed particles.

Figure S4: The geometric entropy for all packing sizes in 2D (top) and 3D
(bottom). The values for $\lambda=0$ are drawn with solid markers, while the
extrapolated $\lambda=1$ values have open markers. PBC packings are shown in
red and cluster packings in blue.


# On equivalence relations induced by locally compact abelian Polish groups

Longyun Ding  School of Mathematical Sciences and LPMC, Nankai University,
Tianjin, 300071, P.R.China<EMAIL_ADDRESS>and  Yang Zheng
<EMAIL_ADDRESS>
###### Abstract.

Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation
$G^{\omega}/c(G)$, where $c(G)$ is the group of all convergent sequences in
$G$. The connected component of the identity of a Polish group $G$ is denoted
by $G_{0}$.

Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq_{B}E(H)$,
then there is a continuous homomorphism $S:G_{0}\rightarrow H_{0}$ such that
$\ker(S)$ is non-archimedean. The converse is also true when $G$ is connected
and compact.

For $n\in{\mathbb{N}}^{+}$, the partially ordered set $P(\omega)/\mbox{Fin}$
can be embedded into Borel equivalence relations between $E({\mathbb{R}}^{n})$
and $E({\mathbb{T}}^{n})$.

###### Key words and phrases:

Borel reduction, locally compact abelian Polish group, equivalence relation

###### 2010 Mathematics Subject Classification:

Primary 03E15, 22B05, 20K45

Research is partially supported by the National Natural Science Foundation of
China (Grant No. 11725103).

##  1\. Introduction

A topological space is Polish if it is separable and completely metrizable.
For more details in descriptive set theory, we refer to [13]. It is an
important application of descriptive set theory to study equivalence relations
by using Borel reducibility. Given two Borel equivalence relations $E$ and $F$
on Polish spaces $X$ and $Y$ respectively, recall that $E$ is Borel reducible
to $F$, denoted $E\leq_{B}F$, if there exists a Borel map $\theta:X\rightarrow
Y$ such that for all $x,y\in X$,

$xEy\Longleftrightarrow\theta(x)F\theta(y).$

We denote $E\sim_{B}F$ if both $E\leq_{B}F$ and $F\leq_{B}E$, and denote
$E<_{B}F$ if $E\leq_{B}F$ and $F\nleq_{B}E$. We refer to [7] for background on
Borel reducibility.

Polish groups are important tools in the research on Borel reducibility. A
topological group is Polish if its topology is Polish. For a Polish group $G$,
the authors [5] defined an equivalence relation $E(G)$ on $G^{\omega}$ by

$xE(G)y\iff\lim_{n}x(n)y(n)^{-1}\mbox{ converges in }G$

for $x,y\in G^{\omega}$. We say that $E(G)$ is the equivalence relation
induced by $G$. Indeed, $E(G)$ is the right coset equivalence relation
$G^{\omega}/c(G)$, where $c(G)$ is the group of all convergent sequences in
$G$.

In this article, we focus on equivalence relations induced by locally compact
abelian Polish groups. Some interesting results have been found in some
special cases. For instance, for $c_{0},e_{0},c_{1},e_{1}\in{\mathbb{N}}$,
$E({\mathbb{R}}^{c_{0}}\times{\mathbb{T}}^{e_{0}})\leq_{B}E({\mathbb{R}}^{c_{1}}\times{\mathbb{T}}^{e_{1}})$
iff $e_{0}\leq e_{1}$ and $c_{0}+e_{0}\leq c_{1}+e_{1}$ (cf. [5, Theorem
6.19]).

Given a group $G$, the identity element of $G$ is denoted by $1_{G}$. If $G$
is a topological group, the connected component of $1_{G}$ in $G$ is denoted
by $G_{0}$. Recall that a Polish group $G$ is non-archimedean if it has a
neighborhood base of $1_{G}$ consisting of open subgroups.

######  Theorem 1.1.

Let $G$ and $H$ be two locally compact abelian Polish groups. If
$E(G)\leq_{B}E(H)$, then there is a continuous homomorphism
$S:G_{0}\rightarrow H_{0}$ such that $\ker(S)$ is non-archimedean.

By restricting attention to compact connected abelian Polish groups, we prove
the following theorem.

######  Theorem 1.2 (Rigid Theorem).

Let $G$ be a compact connected abelian Polish group and $H$ a locally compact
abelian Polish group. Then $E(G)\leq_{B}E(H)$ iff there is a continuous
homomorphism $S:G\rightarrow H$ such that $\ker(S)$ is non-archimedean.

Recall that ${\mathbb{T}}$ is the multiplicative group of all complex numbers
with modulus $1$. We also consider finite dimensional compact abelian Polish
groups, and obtain the following results.

######  Theorem 1.3.

Let $G,H$ be locally compact abelian Polish groups.

  1. (1)

If $G$ is non-archimedean, then $E(G)\leq_{B}E_{0}^{\omega}$.

  2. (2)

If $G$ is not non-archimedean, then $E({\mathbb{R}})\leq_{B}E(G)$.

  3. (3)

If $G$ is not non-archimedean and $G_{0}$ is open, then
$E(G)\sim_{B}E(G_{0})$.

  4. (4)

If $n$ is a positive integer, then $E({\mathbb{T}}^{n})\leq_{B}E(G)$ iff
${\mathbb{T}}^{n}$ embeds in $G$.

  5. (5)

If $n$ is a positive integer and $G$ is compact, then $G$ is $n$-dimensional
iff $E({\mathbb{R}}^{n})<_{B}E(G)\leq_{B}E({\mathbb{T}}^{n})$.

Let $\mathcal{P}$ denote the set of all primes. For
$P,Q\in\mathcal{P}^{\omega}$, $Q\preceq P$ means there is an injection
$f:\omega\to\omega$ such that $Q(n)=P(f(n))$ for large enough $n$. We consider
$P$-adic solenoids $\Sigma_{P}$ for $P\in\mathcal{P}^{\omega}$.

######  Theorem 1.4.

Let $P$ and $Q$ be in $\mathcal{P}^{\omega}$. Then
$E(\Sigma_{P})\leq_{B}E(\Sigma_{Q})$ iff $Q\preceq P$.

The partially ordered set $P(\omega)/\mbox{Fin}$ is very complicated that
every Boolean algebra of size $\leq\omega_{1}$ embeds into it (see [2]). We
usually express that some classes of Borel equivalence relations are extremely
complicated under the order of Borel reducibility by showing that
$P(\omega)/\mbox{Fin}$ embeds into them. For instance, Louveau-Velickovic [14]
and Yin [18] showed that $P(\omega)/\mbox{Fin}$ embeds into both LV-equalities
and Borel equivalence relations between $\ell_{p}$ and $\ell_{q}$
respectively. As an application, we prove that, the partially ordered set
$P(\omega)/\mbox{Fin}$ embeds into the partially ordered set of $E(G)$ under
the ordering of Borel reducibility.

######  Theorem 1.5.

Let $n\in\mathbb{N}^{+}$. Then for $A\subseteq\omega$, there is a
$n$-dimensional compact connected abelian Polish group $G_{A}$ such that
$E({\mathbb{R}}^{n})<_{B}E(G_{A})<_{B}E({\mathbb{T}}^{n})$ and for
$A,B\subseteq\omega$, we have

$A\subseteq^{*}B\Longleftrightarrow E(G_{A})\leq_{B}E(G_{B}).$

Let $G$ be an abelian topological group, then the topological group
Hom$(G,{\mathbb{T}})$ is called the dual group of $G$, denoted by
$\widehat{G}$ (cf. [11, Definition 7.4]).

######  Theorem 1.6 (Dual Rigid Theorem).

Let $G$ be a compact connected abelian Polish group and $H$ a locally compact
abelian Polish group. Then $E(G)\leq_{B}E(H)$ iff there is a continuous
homomorphism $S^{*}:\widehat{H}\to\widehat{G}$ such that $\widehat{G}/{\rm
im}(S^{*})$ is a torsion group.

This article is organized as follows. In section 2, we prove theorems 1.1–1.3.
In section 3, we consider $P$-adic solenoids and prove theorems 1.4 and 1.5.
Finally, In section 4, we consider dual groups and prove Theorem 1.6.

##  2\. Locally compact abelian Polish groups

######  Definition 2.1 ([5, Definition 6.1]).

Let $G$ be a Polish group. We define equivalence relation $E_{*}(G)$ on
$G^{\omega}$ as: for $x,y\in G^{\omega}$,

$xE_{*}(G)y\iff\lim_{n}x(0)x(1)\dots x(n)y(n)^{-1}\dots
y(1)^{-1}y(0)^{-1}\mbox{ converges}.$

It is clear that $E(G)\sim_{B}E_{*}(G).$ In this article, we focus on abelian
Polish groups. For abelian Polish groups $G$, it is more convenient to take
$E_{*}(G)$ as research object than $E(G)$.

Some reducibility results are obtained in [5]. Since we will use them again
and again in this article, for the convenience of readers, we list them as
follows.

######  Proposition 2.2 ([5, Proposition 3.4]).

Let $G,H$ be two Polish groups. If $G$ is topologically isomorphic to a closed
subgroup of $H$, then $E(G)\leq_{B}E(H)$.

A metric $d$ on a group $G$ is called two sided invariant if
$d(ghl,gkl)=d(h,k)$ for all $g,h,k,l\in G$. We say that a Polish group $G$ is
TSI if it admits a compatible two sided invariant metric. Any abelian Polish
group is TSI (cf. [7, Exercise 2.1.4]).

######  Lemma 2.3 ([5, Theorem 6.5]).

Let $G,H,K$ be three TSI Polish groups. Suppose $\psi:G\to H$ and
$\varphi:H\to K$ are continuous homomorphisms with $\varphi(\psi(G))=K$ such
that $\ker(\varphi\circ\psi)$ is non-archimedean. If the interval $[0,1]$
embeds into $H$, then $E(G)\leq_{B}E(H)$.

######  Lemma 2.4 ([5, Theorem 6.13]).

Let $G,H$ are two TSI Polish groups such that $H$ is locally compact. It
$E(G)\leq_{B}E(H)$, then there exist open normal subgroups $G_{c}$ of $G$ and
a continuous map $S:G_{c}\to H$ with $S(1_{G})=1_{H}$ such that, for $x,y\in
G_{c}^{\omega}$, if $\lim_{n}x(n)y(n)^{-1}=1_{G}$, then

$xE_{*}(G_{c})y\iff S(x)E_{*}(H)S(y),$

where $S(x)(n)=S(x(n)),S(y)(n)=S(y(n))$ for each $n\in\omega$.

In particular, if $G=G_{c}$ and the interval $[0,1]$ embeds in $H$, then the
converse is also true.

######  Remark 2.5.

Since $G_{c}$ in the preceding lemma is an open subgroup, it is also closed.
So $G_{0}\subseteq G_{c}$ as it is connected. Since $S$ is continuous, we have
$S(G_{0})\subseteq H_{0}$. Moreover, for all $x,y\in G_{0}^{\omega}$, if
$\lim_{n}x(n)y(n)^{-1}=1_{G}$, we have

$xE_{*}(G_{0})y\Longleftrightarrow S(x)E_{*}(H_{0})S(y).$

The next lemma plays the key role in the proof of Theorem 2.7.

######  Lemma 2.6.

Let $G$ and $H$ be two abelian Polish groups such that:

  1. (1)

$H$ is locally compact,

  2. (2)

$H_{0}\subseteq{\mathbb{R}}^{\omega}\times{\mathbb{T}}^{\omega}$,

  3. (3)

there is a nonzero continuous homomorphism $f:{\mathbb{R}}^{m}\rightarrow G$
for some $m\in{\mathbb{N}}^{+}$.

If $E_{*}(G)\leq_{B}E_{*}(H)$, then there is a continuous map
$S:G_{0}\rightarrow H_{0}$ such that the map $S$ restricted on
$f({\mathbb{R}}^{m})$ is a homomorphism to $H_{0}$.

###### Proof.

First, from Remark 2.5, we can obtain a continuous map $S:G_{0}\rightarrow
H_{0}$ with $S(1_{G_{0}})=1_{H_{0}}$ such that, for $x,y\in G_{0}^{\omega}$,
if $\lim_{n}x(n)y(n)^{-1}=1_{G_{0}}$, then

$xE_{*}(G_{0})y\Longleftrightarrow S(x)E_{*}(H_{0})S(y),$

where $S(x)(n)=S(x(n)),S(y)(n)=S(y(n))$ for each $n\in\omega$.

Since $H_{0}\subseteq{\mathbb{R}}^{\omega}\times{\mathbb{T}}^{\omega}$,
without loss of generality we may assume that $h(2k)\in{\mathbb{R}}$ and
$h(2k+1)\in{\mathbb{T}}$ for all $h\in H_{0}.$ For $k\in\omega$, we define
continuous maps $\phi^{2k}:H_{0}\rightarrow{\mathbb{R}}$ and
$\phi^{2k+1}:H_{0}\rightarrow{\mathbb{T}}$ by

$\phi^{j}(h)=\left\\{\begin{array}[]{ll}h(2k),&j=2k,\cr
h(2k+1),&j=2k+1.\end{array}\right.$

Now fix $g_{0},g_{1}\in f({\mathbb{R}}^{m})$ and find
$a_{0},a_{1}\in{\mathbb{R}}^{m}$ such that $f(a_{0})=g_{0}$ and
$f(a_{1})=g_{1}$. For $t\in[0,1]$ and $l\in\\{1,2\\}$, define
$a^{l}(t)\in{\mathbb{R}}^{m}$ as

$a^{l}(t)=\left\\{\begin{array}[]{ll}a_{0}+t(a_{1}-a_{0}),&l=1,\cr
t(a_{0}+a_{1}),&l=2.\end{array}\right.$

Then it is easy to construct a continuous functions
$F^{l}_{j}:[0,1]\rightarrow{\mathbb{R}}$ for $l\in\\{1,2\\}$ and $k\in\omega$
such that

$F^{l}_{2k}(t)=\phi^{2k}(S(f(a^{l}(t)))),\quad\exp({\rm
i}F^{l}_{2k+1}(t))=\phi^{2k+1}(S(f(a^{l}(t)))).$

Note that $S(f(a^{2}(0)))=1_{H}$. We can assume that $F^{2}_{j}(0)=0$ for each
$j$.

For any $j\in\omega$, the proof of [5, Lemma 6.17] gives

$F^{1}_{j}(1/2)=F^{1}_{j}(0)+(F^{1}_{j}(1)-F^{1}_{j}(0))/2=(F^{1}_{j}(0)+F^{1}_{j}(1))/2,$
$F^{2}_{j}(1/2)=F^{2}_{j}(1)/2.$

It follows that

$S(f(a^{1}(1/2)))^{2}=S(f(a_{0}))S(f(a_{1}))=S(g_{0})S(g_{1}),$
$S(f(a^{2}(1/2)))^{2}=S(f(a_{0}+a_{1}))=S(g_{0}g_{1}).$

Since $a^{1}(1/2)=a^{2}(1/2)$, we have $S(g_{0})S(g_{1})=S(g_{0}g_{1})$.

So, the map $S:f({\mathbb{R}}^{m})\rightarrow H_{0}$ is a continuous
homomorphism. ∎

Let us recall the structure of Hausdorff locally compact abelian groups. Let
$G$ be a Hausdorff locally compact abelian group, then $G$ is topologically
isomorphic to the group ${\mathbb{R}}^{n}\times H$, where $H$ is a locally
compact abelian group containing a compact open subgroup (cf. [10, Theorem
24.30]). Moreover, if $G$ is also connected, then it is a direct product of a
compact connected abelian group $K$ and the group ${\mathbb{R}}^{n}$ (cf. [10,
Theorem 9.14]). Any locally compact connected metrizable abelian group can be
embedded as a closed subgroup of
${\mathbb{R}}^{n}\times{\mathbb{T}}^{\omega}$. In particular, all compact
metrizable abelian groups can be embedded in ${\mathbb{T}}^{\omega}$ (see Page
119 of [1]). $G$ is said to be solenoidal if there is a continuous
homomorphism $f:$ ${\mathbb{R}}\rightarrow G$ such that $f({\mathbb{R}})$ is
dense in $G$ (see [10, (9.2)]). It is well known that a compact metrizable
abelian group is solenoidal iff it is connected (see Page 13 and Proposition
5.16 of [1]). Thus for each locally compact connected metrizable abelian group
$G$, there is a continuous homomorphism $f:$ ${\mathbb{R}}^{m}\rightarrow G$
which satisfies $\overline{f({\mathbb{R}}^{m})}=G$. For more details on
locally compact abelian groups, we refer to [1, 10].

By applying Lemma 2.6 for locally compact abelian Polish groups, we get the
following results.

######  Theorem 2.7.

Let $G$ and $H$ be two locally compact abelian Polish groups. If
$E(G)\leq_{B}E(H)$, then there is a continuous homomorphism
$S:G_{0}\rightarrow H_{0}$ such that $\ker(S)$ is non-archimedean.

###### Proof.

If $E(G)\leq_{B}E(H)$, then $E_{*}(G)\leq_{B}E_{*}(H)$. Without loss of
generality we may assume that $G_{0}$ is nontrivial. First note that $H_{0}$
can be embedded into ${\mathbb{R}}^{n}\times{\mathbb{T}}^{\omega}$. Thus we
may assume without loss of generality that
$H_{0}\subseteq{\mathbb{R}}^{\omega}\times{\mathbb{T}}^{\omega}$. Let $f$ be a
continuous homomorphism from ${\mathbb{R}}^{m}$ to $G_{0}$ with
$\overline{f({\mathbb{R}}^{m})}=G_{0}$. Then by Lemma 2.6 there exists a
continuous map $S:G_{0}\rightarrow H_{0}$ such that the map $S$ restricted on
$f({\mathbb{R}}^{m})$ is a homomorphism to $H_{0}$. Since
$f({\mathbb{R}}^{m})$ is dense in $G_{0}$, we see that $S$ is a homomorphism
from $G_{0}$ to $H_{0}$.

Then we only need to check that $\ker(S)$ is non-archimedean. Assume toward a
contradiction that $\ker(S)$ is not non-archimedean.

Note that $\ker(S)$ is an abelian Polish group. Fix a compatible two sided
invariant metric on $\ker(S)$. Let $V_{k}\subseteq\ker(S),\,k\in\omega$ be an
open symmetric neighborhood base of $1_{\ker(S)}=1_{G}$ with
$\lim_{k}{\mathrm{diam}\,}(V_{k})=0$. Then there exists a $k_{0}\in\omega$
such that $V_{k_{0}}$ does not contain any open subgroup of $\ker(S)$. Since
$V_{k}$ is symmetric, $\bigcup_{m}V_{k}^{m}$ is an open subgroup of $\ker(S)$,
so $\bigcup_{m}V_{k}^{m}\nsubseteq V_{k_{0}}$ for each $k$. Thus we can find
an $m_{k}\in\omega$ and $g_{k,0},\dots,g_{k,m_{k}-1}\in V_{k}$ such that
$g_{k,0}g_{k,1}\cdots g_{k,m_{k}-1}\notin V_{k_{0}}$.

Denote $M_{-1}=0$ and $M_{k}=m_{0}+m_{1}+\dots+m_{k}$ for $k\in\omega$. Now
for $n\in\omega$, define

$x(n)=\left\\{\begin{array}[]{ll}g_{k,j},&n=M_{k-1}+j,0\leq j<m_{k},\cr
1_{G},&\mbox{otherwise.}\end{array}\right.$

Therefore $xE_{*}(G)1_{G^{\omega}}$ fails. Note that we have
$\lim_{n}x(n)=1_{G}$ and $S(x(n))=1_{H}$ for each $n$. So it is trivial that
$S(x)E_{*}(H_{0})S(1_{G^{\omega}})$, where $S(x)(n)=S(x(n))$, contradicting
Lemma 2.4. ∎

In paricular, if $G$ is compact connected, then the converse of Theorem 2.7 is
also true.

######  Theorem 2.8 (Rigid Theorem).

Let $G$ be a compact connected abelian Polish group and $H$ a locally compact
abelian Polish group. Then $E(G)\leq_{B}E(H)$ iff there is a continuous
homomorphism $S:G\rightarrow H$ such that $\ker(S)$ is non-archimedean.

###### Proof.

Let $S$ be a continuous homomorphism from $G$ to $H$ such that $\ker(S)$ is
non-archimedean. Since $G$ is compact, $S(G)$ is a compact, thus closed
subgroup of $H$. So we have $E(S(G))\leq_{B}E(H)$.

Note that $S(G)$ is also a compact connected abelian Polish group. Let $f$ be
a continuous homomorphism $f:{\mathbb{R}}\rightarrow S(G)$ such that
$\overline{f({\mathbb{R}})}=S(G)$. Then $\ker(f)$ is a proper closed subgroup
of ${\mathbb{R}}$. Hence $\ker(f)$ is a discrete group. This gives that the
interval $[0,1]$ embeds in $S(G)$. Then by Lemma 2.3, we get that
$E(G)\leq_{B}E(S(G))\leq_{B}E(H)$.

On the other hand, if $E(G)\leq_{B}E(H)$, by Theorem 2.7, there is a
continuous homomorphism $S:G_{0}\to H_{0}$ such that $\ker(S)$ is non-
archimedean. Since $G$ is connected, we have $G=G_{0}$. ∎

######  Corollary 2.9.

Let $G$ be a compact connected abelian Polish group and $H$ a locally compact
abelian Polish group. Suppose $H_{0}\cong{\mathbb{R}}^{n}\times K$, where $K$
is a compact connected abelian group. Then $E(G)\leq_{B}E(H)$ iff
$E(G)\leq_{B}E(K)$.

###### Proof.

$(\Leftarrow)$ part is trivial, since $E(K)\leq_{B}E(H_{0})\leq_{B}E(H)$.

$(\Rightarrow)$. If $E(G)\leq_{B}E(H)$, then there exists a continuous
homomorphism $S:G\to H$ such that $\ker(S)$ is non-archimedean. So $S(G)$ is a
connected compact subgroup of $H$, thus $S(G)\subseteq H_{0}$. Without loss of
generality, we assume that $H_{0}={\mathbb{R}}^{n}\times K$. Let
$\pi:H_{0}\to{\mathbb{R}}^{n}$ and $\pi_{K}:H_{0}\to K$ be canonical
projections. Then $\pi(S(G))$ is a compact subgroup of ${\mathbb{R}}^{n}$, so
$\pi(S(G))=\\{0\\}$. It follows that $\ker(\pi_{K}\circ S)=\ker(S)$. Applying
Theorem 2.8 on $\pi_{K}\circ S:G\to K$, we have $E(G)\leq_{B}E(K)$. ∎

Recall that a topological group $G$ is totally disconnected if
$G_{0}=\\{1_{G}\\}$. For any locally compact abelian Polish group, it is
totally disconnected iff it is non-archimedean (cf. [11, Theorem 1.34]).

For every normal space $X$, denoted by $\dim(X)$ the covering dimension of
$X$, where $\dim(X)$ is an integer $\geq-1$ or the “infinite number $\infty$”.
We omit the definition of covering dimension since it is very complexity (see
page 54 of [6]). We recall the following useful facts: $\dim(G)=n<\infty$ iff
$G$ has a totally disconnected closed subgroup $\Delta$ such that
$G/\Delta\cong{\mathbb{T}}^{n}$ iff there is a compact totally disconnected
subgroup $N$ of $G$ and a continuous surjective homomorphism
$\varphi:N\times{\mathbb{R}}^{n}\rightarrow G$ which has a discrete kernel
(see Theorem 8.22 and Corollary 8.26 of [11]). In this case, we say that $G$
is finite dimensional (cf. [11, Definitions 8.23]). Clearly, $\dim(G)=0$ iff
$G$ is totally disconnected. For more details on compact abelian groups, see
[11].

Now we recall two equivalence relations $E_{0}^{\omega}$ and $E(M;0)$ (see [4,
Definition 3.2]). The equivalence relation $E_{0}^{\omega}$ on
$2^{\omega\times\omega}$ defined by

$xE_{0}^{\omega}y\iff\forall k\exists m\forall n\geq m(x(n,k)=y(n,k)).$

Fix a metric space $M$. The equivalence relation $E(M;0)$ on $M^{\omega}$
defined by

$xE(M;0)y\iff\lim\limits_{n}d(x(n),y(n))=0.$

From the above discussions, we can establish the following theorem.

######  Theorem 2.10.

Let $G,H$ be locally compact abelian Polish groups.

  1. (1)

If $G$ is non-archimedean, then $E(G)\leq_{B}E_{0}^{\omega}$.

  2. (2)

If $G$ is not non-archimedean, then $E({\mathbb{R}})\leq_{B}E(G)$.

  3. (3)

If $G$ is not non-archimedean and $G_{0}$ is open, then
$E(G)\sim_{B}E(G_{0})$.

  4. (4)

If $n$ is a positive integer, then $E({\mathbb{T}}^{n})\leq_{B}E(G)$ iff
${\mathbb{T}}^{n}$ embeds in $G$.

  5. (5)

If $n$ is a positive integer and $G$ is compact, then $G$ is $n$-dimensional
iff $E({\mathbb{R}}^{n})<_{B}E(G)\leq_{B}E({\mathbb{T}}^{n})$.

###### Proof.

(1) It follows from [5, Theorem 3.5.(3)].

(2) Note that $G$ is not totally disconnected (cf. [11, Theorem 1.34]), so
$G_{0}$ contains at least two points. We have
$G_{0}\cong{\mathbb{R}}^{n}\times K$, where $K$ is a compact connected abelian
group. If $n>0$, it is trivial that $E({\mathbb{R}})\leq_{B}E(G)$. By
Propostion 2.2, $E(K)\leq_{B}E(G_{0})\leq_{B}E(G)$. Thus we may assume that
$G$ is compact connected and $G\subseteq{\mathbb{T}}^{\omega}$. Note that
there is a continuous homomorphism $f:$ ${\mathbb{R}}\rightarrow G$ such that
$\overline{f({\mathbb{R}})}=G$. For $g\in G\subseteq{\mathbb{T}}^{\omega}$ and
$p\in\omega$, let $\phi_{p}(g)=g(p)$. Since $G$ contains at least two points,
we can find ${p_{0}}\in\omega$ such that
$\phi_{p_{0}}(f({\mathbb{R}}))\neq\\{1_{\mathbb{T}}\\}$, so
$\phi_{p_{0}}(f({\mathbb{R}}))={\mathbb{T}}$. By [11, Corollary 8.24], the
interval $[0,1]$ embeds in $G$. Then by Theorem 2.3, we have
$E({\mathbb{R}})\leq_{B}E(G)$.

(3) By [10, §24.45], we have $G\cong G_{0}\times G/G_{0}$. Since $G_{0}$ is
open, $G/G_{0}$ is countable and discrete. By [5, Corollary 3.6], this implies
that $E(G_{0}\times G/G_{0})\sim_{B}E(G_{0})$ and thus $E(G)\sim_{B}E(G_{0})$.

(4) The “if” part follows Proposition 2.2. Assume that
$E({\mathbb{T}}^{n})\leq_{B}E(G)$. By Theorem 2.7 and [7, Corollary 2.3.4],
there is a closed subgroup $\Delta$ of ${\mathbb{T}}^{n}$ such that the group
${\mathbb{T}}^{n}/\Delta$ can be embedded in $G$, where $\Delta$ is non-
archimedean. It is obvious that ${\mathbb{T}}^{n}/\Delta$ is a locally
connected, connected and compact abelian Polish group. By [1, Proposition
8.17], ${\mathbb{T}}^{n}/\Delta\cong{\mathbb{T}}^{n}$.

(5) If $n=\dim(G)$, then we have $(N\times{\mathbb{R}}^{n})/\Delta_{1}\cong G$
and $G/\Delta_{2}\cong{\mathbb{T}}^{n}$, where $N,\Delta_{1}$, and
$\Delta_{2}$ are totally disconnected, and hence are non-archimedean. Then
Proposition 2.2 and Lemma 2.3 imply that

$E({\mathbb{R}}^{n})\leq_{B}E(N\times{\mathbb{R}}^{n})\leq_{B}E(G)\leq_{B}E({\mathbb{T}}^{n}).$

So we only need to show that $E(G)\nleq_{B}E({\mathbb{R}}^{n})$. To see this,
assume toward a contradiction that $E(G)\leq_{B}E({\mathbb{R}}^{n})$. By
Theorem 2.7, there exists a continuous homomorphism
$S:G_{0}\rightarrow{\mathbb{R}}^{n}$ such that $\ker(S)$ is non-archimedean.
Note that ${\mathbb{R}}^{n}$ has no nontrivial compact connected subgroup. So
this implies that $S(G_{0})=\\{0\\}$, contradicting that $\ker(S)$ is non-
archimedean.

On the other hand, suppose
$E({\mathbb{R}}^{n})<_{B}E(G)\leq_{B}E({\mathbb{T}}^{n})$. Let $m=\dim(G)$. By
(1) we have $m>0$. Assume for contradiction that $m=\infty$, then there exists
a continuous homomorphism $S:G_{0}\rightarrow{\mathbb{T}}^{n}$ such that
$\ker(S)$ is non-archimedean. Then we have $\dim(G_{0}/\ker(S))=\infty$, and
hence $[0,1]^{\omega}$ embeds into $G_{0}/\ker(S)$ (cf. [11, Corollary 8.24]).
By [7, Corollary 2.3.4], $S$ induce an embedding from $G_{0}/\ker(S)$ to
${\mathbb{T}}^{n}$. So $[0,1]^{\omega}$ embeds into ${\mathbb{T}}^{n}$,
contradicting that $n$ is finite. Therefore, we have $0<m<\infty$, and hence
$E({\mathbb{R}}^{m})<_{B}E(G)\leq_{B}E({\mathbb{T}}^{m})$. Then [5, Theorem
6.19] gives $m=n$. ∎

######  Remark 2.11.

Let $G$ and $H$ be two locally compact abelian Polish groups. Suppose that
$G_{0}$ is an open subgroup of $G$, and that $G_{0}$ is compact or
$G_{0}\cong{\mathbb{R}}$. Then theorems 2.7, 2.8, and 2.10.(2)–(3) imply that
$E(G)\leq_{B}E(H)$ iff there is a continuous homomorphism $S:G_{0}\rightarrow
H_{0}$ such that $\ker(S)$ is non-archimedean. This generalize Rigid Theorem,
i.e., Theorem 2.8. We don’t know whether this can be generalized to all
locally compact abelian Polish groups.

######  Question 2.12.

Does the converse of Theorem 2.7 hold for all locally compact abelian Polish
groups?

######  Question 2.13.

Let $G$ be a locally compact abelian Polish group. If $G$ is not non-
archimedean, does $E(G)\sim_{B}E(G_{0})$?

##  3\. $P$-adic solenoids

Let $P=(P(0),P(1),\dots)$ be a sequence of integers greater than $1$. Recall
that the $P$-adic solenoid $S_{P}$ is defined by

$S_{P}=\\{g\in{\mathbb{T}}^{\omega}:\forall l\,(g(l)=g(l+1)^{P(l)})\\}.$

In particular, if for each $i$, $P(i)$ is a prime number, then the $P$-adic
solenoid is denoted by $\Sigma_{P}$. Let $\mathcal{P}$ denote the set of all
primes. The group $S_{P}$ is topologically isomorphic to $\Sigma_{P^{\prime}}$
for some $P^{\prime}\in\mathcal{P}^{\omega}$. For example, we have
$S_{(4,6,8,9,\dots,9,\dots)}\cong\Sigma_{(2,2,2,3,2,2,2,3,3,\dots,3,3,\dots)}$.

It is well known that, the group $\Sigma_{P}$ is a compact connected abelian
group which is neither locally connected (cf. [8]), nor arcwise connected (see
[1, Theorem 8.27]). Every nontrivial proper closed subgroup $H$ of a $P$-adic
solenoid is totally disconnected (cf. [12, Proposition 2.7]), and thus $H$ is
non-archimedean. Clearly, $\Sigma_{P}$ is an $1$-dimensional and metrizable
group. Let $P$ and $Q$ be in $\mathcal{P}^{\omega}$. We write $Q\preceq P$
provided there is an injection $f:\omega\to\omega$ such that $Q(n)=P(f(n))$
for large enough $n$ (for more details, see [8, 9, 16]).

######  Lemma 3.1 (folklore).

Let $P$ and $Q$ be in $\mathcal{P}^{\omega}$. Then the following are
equivalent:

  1. (1)

There is a nonzero continuous homomorphism
$f:\Sigma_{P}\rightarrow\Sigma_{Q}$.

  2. (2)

There is a surjective continuous homomorphism
$g:\Sigma_{P}\rightarrow\Sigma_{Q}$.

  3. (3)

There is a surjective continuous map $h:\Sigma_{P}\rightarrow\Sigma_{Q}$.

  4. (4)

$Q\preceq P$.

###### Proof.

$(2)\Rightarrow(1)$ and $(2)\Rightarrow(3)$ are obvious. $(1)\Rightarrow(2)$
follows immediately from the fact that each nontrivial proper closed subgroup
of a $P$-adic solenoid is totally disconnected. The equivalence of $(3)$ and
$(4)$ follows from [16, Theorem 4.4].

It remains to show $(3)\Rightarrow(1)$. Let $h$ be a surjective continuous map
from $\Sigma_{P}$ to $\Sigma_{Q}$. Without loss of generality assume that
$h(1_{\Sigma_{P}})=1_{\Sigma_{Q}}$. Then there exists a continuous
homomorphism $f:\Sigma_{P}\rightarrow\Sigma_{Q}$ such that $h$ is homotopic to
$f$ (cf. [17, Corollary 2]). Since $\Sigma_{Q}$ is not arcwise connected,
$\ker(f)\neq\Sigma_{P}$. ∎

######  Theorem 3.2.

Let $P$ and $Q$ be in $\mathcal{P}^{\omega}$. Then
$E(\Sigma_{P})\leq_{B}E(\Sigma_{Q})$ iff $Q\preceq P$ iff there is a nonzero
continuous homomorphism $f:\Sigma_{P}\rightarrow\Sigma_{Q}$.

###### Proof.

Note that every nontrivial proper closed subgroup of $\Sigma_{P}$ is non-
archimedean. Then this follows from Theorem 2.8 and Lemma 3.1. ∎

Let ${\rm Fin}$ denote the set of all finite subsets of $\omega$. For
$A,B\subseteq\omega$, we use $A\subseteq^{*}B$ to denote $A\setminus B\in{\rm
Fin}$.

We prove that, for $n\in{\mathbb{N}}^{+}$, the partially ordered set
$P(\omega)/{\rm Fin}$ can be embedded into Borel equivalence relations between
$E({\mathbb{R}}^{n})$ and $E({\mathbb{T}}^{n})$.

######  Lemma 3.3.

Let $P$ be in $\mathcal{P}^{\omega}$. Then
$E({\mathbb{R}})<_{B}E(\Sigma_{P})<_{B}E({\mathbb{T}})$.

###### Proof.

By Theorem 2.10.(5), we have that
$E({\mathbb{R}})<_{B}E(\Sigma_{P})\leq_{B}E({\mathbb{T}})$.

Assume toward a contradiction that $E({\mathbb{T}})\leq_{B}E(\Sigma_{P})$.
From Theorem 2.10.(4), ${\mathbb{T}}$ embeds in $\Sigma_{P}$. This is
impossible, since $\Sigma_{P}$ is not arcwise connected and every proper
closed subgroup of $\Sigma_{P}$ is non-archimedean. ∎

Denote
$\Omega=\\{{\mathbb{R}},{\mathbb{T}},\Sigma_{P}:P\in\mathcal{P}^{\omega}\\}$.

######  Lemma 3.4.

Let $m,n\in{\mathbb{N}}^{+}$ and let
$G_{1},G_{2},\dots,G_{m},H_{1},H_{2}\dots,H_{n}\in\Omega$. Then the following
are equivalent:

  1. (1)

$E(G_{1}\times G_{2}\times\dots\times G_{m})\leq_{B}E(H_{1}\times
H_{2}\times\dots\times H_{n})$.

  2. (2)

There is a injective map
$\theta^{*}:\\{1,2,\dots,m\\}\rightarrow\\{1,2,\dots,n\\}$ such that
$E(G_{i})\leq_{B}E(H_{\theta^{*}(i)})$ for $1\leq i\leq m$.

In particular, $E(G_{1}^{m})\leq_{B}E(H_{1}^{n})$ iff $m\leq n$ and
$E(G_{1})\leq_{B}E(H_{1})$.

###### Proof.

$(2)\Rightarrow(1)$ is obvious. We only prove $(1)\Rightarrow(2)$.

Denote $G=G_{1}\times G_{2}\times\dots\times G_{m}$ and $H=H_{1}\times
H_{2}\times\dots\times H_{n}$. For $1\leq i\leq m$ and $g\in G_{i}$, we define
$e^{i}(g)\in G$ as, for $1\leq k\leq m$,

$e^{i}(g)(k)=\left\\{\begin{array}[]{ll}g,&i=k,\cr
1_{G_{k}},&\mbox{otherwise}.\end{array}\right.$

Suppose $E(G)\leq_{B}E(H)$. Since $G$ and $H$ are both connected, by Theorem
2.7, there is a continuous homomorphism $S:G\rightarrow H$ such that $\ker(S)$
is non-archimedean. For each $1\leq j\leq n$, let $\pi_{j}$ be the canonical
projection from $H$ onto $H_{j}$. Without loss of generality we assume that
there is $i_{0}\leq m$ such that, $G_{i}$ is compact for $1\leq i\leq i_{0}$,
and $G_{i}={\mathbb{R}}$ for $i_{0}<i\leq m$.

For any $1\leq i\leq i_{0}$, since $\ker(S)$ is non-archimedean, there exists
$j$ satisfying that $\pi_{j}(S(e^{i}(G_{i})))\not=\\{1_{H_{j}}\\}$. Note that
$H_{j}$ has no nontrivial proper connected compact subgroup. It follows that
$\pi_{j}(S(e^{i}(G_{i})))=H_{j}$. Now we construct a bipartite graph $G[X,Y]$
as follows. Let $X=\\{G_{1},G_{2},\dots,G_{i_{0}}\\}$,

$Y=\\{H_{j}:\exists i\,(1\leq i\leq i_{0}\mbox{ and
}\pi_{j}(S(e^{i}(G_{i})))=H_{j})\\}.$

For $G_{i}\in X$ and $H_{j}\in Y$, we put an edge between $G_{i}$ and $H_{j}$
if $\pi_{j}(S(e^{i}(G_{i})))=H_{j}$. Given $K\subseteq X$, we denote the set
of all neighbors of the vertices in $K$ by $N(K)$.

Next we show that $\left|N(K)\right|\geq\left|K\right|$ for all $K\subseteq
X$. Given $K\subseteq X$, denote

$G^{K}=\\{x\in G:x(i)=1_{G_{i}}\mbox{ for all }G_{i}\notin K\\},$
$H^{N(K)}=\\{z\in H:z(j)=1_{H_{j}}\mbox{ for all }H_{j}\notin N(K)\\}.$

Then the restriction of $S$ on $G^{K}$ is a continuous homomorphism to
$H^{N(K)}$. By Theorem 2.8, $E(G^{K})\leq_{B}E(H^{N(K)})$. Again by Theorem
2.10.(5), this implies
$E({\mathbb{R}}^{\left|K\right|})\leq_{B}E({\mathbb{T}}^{\left|N(K)\right|})$.
Then [5, Theorem 6.19] gives $\left|N(K)\right|\geq\left|K\right|$.

By Hall’s theorem (cf. [3, Theorem 16.4]), there is a injective map
$\theta^{*}:\\{1,2,\dots,i_{0}\\}\rightarrow\\{1,2,\dots,n\\}$ such that
$\pi_{\theta^{*}(i)}(S(e^{i}(G_{i})))=H_{\theta^{*}(i)}$. Since every proper
closed subgroup of $G_{i}$ is non-archimedean, from Theorem 2.8, we have
$E(G_{i})\leq_{B}E(H_{\theta^{*}(i)})$.

In the end, since $\dim(G)=m$ and $\dim(H)=n$, by Theorem 2.10.(5), we have
$E({\mathbb{R}}^{m})\leq_{B}E({\mathbb{T}}^{n})$. So $m\leq n$. Since
$E({\mathbb{R}})\leq_{B}E(H_{j})$ for each $j$, we can trivially extend
$\theta^{*}$ to an injection from $\\{1,2,\dots,m\\}$ to $\\{1,2,\dots,n\\}$
such that $E(G_{i})\leq_{B}E(H_{\theta^{*}(i)})$ for each $i$. ∎

Let $\\{\gamma_{i}\\}_{i\in\omega}$ be a nonrepeating enumeration of the set
$\mathcal{P}$. For $P\in\mathcal{P}^{\omega}$ and $\gamma_{i}\in\mathcal{P}$,
we define $t^{P}(\gamma_{i})\in\omega\cup\\{\omega\\}$ as

$t^{P}(\gamma_{i})=\left\\{\begin{array}[]{ll}\omega,&\exists^{\infty}j\in\omega\,(P(j)=\gamma_{i}),\cr\left|\\{j:P(j)=\gamma_{i}\\}\right|,&\mbox{otherwise.}\end{array}\right.$

Given $P,Q\in\mathcal{P}^{\omega}$, denote

$D(P,Q)=\\{\gamma_{i}\in\mathcal{P}:t^{P}(\gamma_{i})<t^{Q}(\gamma_{i})\\}.$

######  Lemma 3.5.

Let $P,Q\in\mathcal{P}^{\omega}$ with $E(\Sigma_{Q})\leq_{B}E(\Sigma_{P})$.
Suppose that $D(P,Q)$ is infinite. Then for $A\subseteq\omega$, there is a
group $\Sigma_{P_{A}}$ such that
$E(\Sigma_{Q})<_{B}E(\Sigma_{P_{A}})<_{B}E(\Sigma_{P})$ and for
$A,B\subseteq\omega$, we have

$A\subseteq^{*}B\Longleftrightarrow
E(\Sigma_{P_{A}})\leq_{B}E(\Sigma_{P_{B}}).$

###### Proof.

Enumerate $D(P,Q)$ as $d_{0}<d_{1}<d_{2}<\cdots$. Let
$P_{0}^{*}\in\mathcal{P}^{\omega}$ such that $P_{0}^{*}(i)=d_{3i}$ for all
$i\in\omega$.

For $L,M\in\mathcal{P}^{\omega}$, we define an element $L\oplus
M\in\mathcal{P}^{\omega}$ as

$(L\oplus M)(n)=\left\\{\begin{array}[]{ll}L(k),&n=2k,\cr
M(k),&n=2k+1.\end{array}\right.$

Given a set $A\subseteq\omega$, define $P_{A}\in\mathcal{P}^{\omega}$ as
follows. If $\omega\setminus A$ is finite, put $P_{A}=P_{0}^{*}\oplus P$;
otherwise enumerate $\omega\setminus A$ as $c_{0}<c_{1}<c_{2}<\cdots$. Define
$P_{A}^{*}\in\mathcal{P}^{\omega}$ as $P_{A}^{*}(j)=d_{1+3c_{j}}$ for
$j\in\omega.$ Then put $P_{A}=P_{A}^{*}\oplus(P_{0}^{*}\oplus P).$ By Theorem
3.2, it is easy see that
$E(\Sigma_{Q})<_{B}E(\Sigma_{P_{A}})<_{B}E(\Sigma_{P})$ for all
$A\subseteq\omega.$

For $A,B\subseteq\omega$, it is trivial to check that $A\subseteq^{*}B$ iff
$(\omega\setminus B)\setminus(\omega\setminus A)=(A\setminus B)$ is finite iff
$P_{B}\preceq P_{A}$. Again by Theorem 3.2, we have $A\subseteq^{*}B$ iff
$E(\Sigma_{P_{A}})\leq_{B}E(\Sigma_{P_{B}})$. ∎

######  Theorem 3.6.

Let $n\in\mathbb{N}^{+}$. Then for $A\subseteq\omega$, there is an
$n$-dimensional compact connected abelian Polish group $G_{A}$ such that
$E({\mathbb{R}}^{n})<_{B}E(G_{A})<_{B}E({\mathbb{T}}^{n})$ and for
$A,B\subseteq\omega$, we have

$A\subseteq^{*}B\Longleftrightarrow E(G_{A})\leq_{B}E(G_{B}).$

###### Proof.

It follows from Theorem 2.10.(5), lemmas 3.3, 3.4, and 3.5. ∎

##  4\. Dual groups

Let $G$ and $H$ be two abelian topological groups. Denote the class of all
continuous homomorphisms of $G$ to $H$ by ${\rm Hom}(G,H)$, which is an
abelian group under pointwise addition. We always equip Hom$(G,H)$ with
compact-open topology. The abelian topological group Hom$(G,{\mathbb{T}})$ is
called the dual group of $G$, denoted by $\widehat{G}$ (cf. [11, Definition
7.4]).

Let $(A,+)$ be an abelian group whose identity element denoted by $0_{A}$. We
say that $(A,+)$ is a torsion group if each element of $A$ is finite order. We
say that $(A,+)$ is torsion-free if $n\cdot g\neq 0_{A}$ for all $g\in A$ with
$g\neq 0_{A}$ and $n\in{\mathbb{N}}^{+}$. A subset $X$ of $A$ is free if any
equation $\sum_{x\in X}n_{x}\cdot x=0_{A}$ implies $n_{x}=0$ for all $x\in X$.
The torsion-free rank of $A$, written rank$(A)$, is the cardinal number
(uniquely determined) of any maximal free subset of $A$.

Each Hausdorff locally compact abelian group $G$ is reflexive, thus it is
topologically isomorphic to the double dual group $\widehat{\widehat{G}}$ (cf.
[11, Theorem 7.63]). A Hausdorff locally compact abelian group is compact and
metrizable iff its dual group is a countable discrete group (see Proposition
7.5.(i) and Theorem 8.45 of [11] ). Let $G$ be a Hausdorff compact abelian
group, then $G$ is connected iff $\widehat{G}$ is torsion-free; and $G$ is
totally disconnected iff $\widehat{G}$ is torsion (cf. [11, Corollary 8.5]).
For any finite dimensional compact abelian Polish group $G$, the covering
dimension of $A$ is equal to rank$(\widehat{G})$ (see Lemma 8.13 and Corollary
8.26 of [11]).

If $H$ is a subset of an abelian topological group $G$, then the subgroup

$H^{\perp}=\\{\gamma\in\widehat{G}:\forall x\in
H\,(\gamma(x)=1_{\mathbb{T}})\\}$

is called the _annihilator_ of $H$ in $\widehat{G}$ (cf. [11, Definition
7.12]).

Now we focus on compact connected abelian Polish groups.

######  Theorem 4.1 (Dual Rigid Theorem).

Let $G$ be a compact connected abelian Polish group and $H$ a locally compact
abelian Polish group. Then $E(G)\leq_{B}E(H)$ iff there is a continuous
homomorphism $S^{*}:\widehat{H}\to\widehat{G}$ such that $\widehat{G}/{\rm
im}(S^{*})$ is a torsion group.

###### Proof.

$(\Rightarrow)$. We assume that $E(G)\leq_{B}E(H)$. By Theorem 2.7, there is a
continuous homomorphism $S:G\rightarrow H$ such that $\ker(S)$ is non-
archimedean. This implies that there is a homomorphism $S^{*}$ from
$\widehat{H}$ to $\widehat{G}$ such that $\ker(S)\cong{{\rm
im}(S^{*})}^{\perp}$ (cf. [1, P.22 and P.23(a)]). By [11, Lemma 7.13(ii)], we
have that $\ker(S)\cong(\widehat{G}/{\rm im}(S^{*}))^{\widehat{}}$, and hence
$\widehat{\ker(S)}\cong\widehat{G}/{\rm im}(S^{*})$. Since $\ker(S)$ is non-
archimedean, thus is totally disconnected, so $\widehat{G}/{\rm im}(S^{*})$ is
a torsion group.

$(\Leftarrow)$. Since $G\cong{\widehat{\widehat{G}}}$ and
$H\cong\widehat{\widehat{H}}$, we can define $S:G\to H$ via
$(S^{*})^{*}:{\widehat{\widehat{G}}}\to{\widehat{\widehat{H}}}$ (cf. [10,
(24.41)]). Then the similar arguments as the preceding paragraph give the
desired result. ∎

######  Corollary 4.2.

Let $G$ be a compact connected abelian Polish group and $H$ a locally compact
abelian Polish group. If $E(G)\leq_{B}E(H)$, then there is a nonzero
continuous homomorphism $S^{*}:\widehat{H}\to\widehat{G}$.

###### Proof.

It follows from Theorem 4.1 and that $\widehat{G}$ is non-torsion. ∎

######  Example 4.3.

$\widehat{\mathbb{T}}\cong{\mathbb{Z}}$ (cf. [10, Examples 23.27(a)]). Fix a
$P\in\mathcal{P}^{\omega}$, then
$\widehat{\Sigma_{P}}\cong\left\\{\frac{m}{P(0)P(1)\dots
P(n)}:m\in{\mathbb{Z}},n\in{\mathbb{N}}\right\\}$ (see [10, 25.3]). In view of
Corollary 4.2, we get $E({\mathbb{T}})\nleq_{B}E(\Sigma_{P})$ again.

Recall that $\widehat{{\mathbb{Q}}}\cong S_{(2,3,4,5,6,\dots)}$ (see [10,
25.4]). We have the following.

######  Corollary 4.4.

Let $G$ be a $n$-dimensional compact abelian Polish group with
$n\in{\mathbb{N}}^{+}$. Then $E((\widehat{{\mathbb{Q}}})^{n})\leq_{B}E(G)$.

###### Proof.

By [11, Theorem 8.22.(4)], $G_{0}\cong(\widehat{{\mathbb{Q}}})^{n}/\Delta$,
where $\Delta$ is a compact totally disconnected subgroup of
$(\widehat{{\mathbb{Q}}})^{n}$. Again by Theorem 2.8, this means that
$E((\widehat{{\mathbb{Q}}})^{n})\leq_{B}E(G_{0})$, and thus
$E((\widehat{{\mathbb{Q}}})^{n})\leq_{B}E(G)$. ∎

From the arguments above, if $\Gamma$ is a countable discrete torsion-free
abelian group, then $\widehat{\Gamma}$ is a compact connected abelian Polish
group.

######  Remark 4.5.

Let $G$ be a compact connected Polish group with
$E({\mathbb{R}}^{n})\leq_{B}E(G)\leq_{B}E({\mathbb{T}}^{n})$ for some $n>0$.
By Theorem 2.10.(5), $\dim(G)=n$, so ${\rm rank}(\widehat{G})=n$. Thus
$\widehat{G}$ is isomorphic to a subgroup of $\mathbb{Q}^{n}$ (cf. [7,
Exercise 13.4.3]). In particular, if $n=1$, we have either
$G\cong{\mathbb{T}}$ or there exists a $P\in\mathcal{P}^{\omega}$ such that
$G\cong\Sigma_{P}$. The following example show that, if $n>1$, the structure
of $G$ can be more complicated.

######  Example 4.6.

There is a $2$-dimensional compact connected Polish group $G$ such that
$E(G)\nleq_{B}E(\Sigma_{P_{0}}\times\Sigma_{P_{1}}\times\dots\times\Sigma_{P_{n}})$
for $n\in{\mathbb{N}}$ and each $P_{i}\in\mathcal{P}^{\omega}$. Moreover, if
$\left|\\{i\in\omega:P(i)=2\\}\right|<\infty$, then
$E(\Sigma_{P})\nleq_{B}E(G)$.

###### Proof.

Pontryagin has constructed a countable torsion-free abelian group
$\Gamma\subseteq{\mathbb{Q}}^{2}$ whose rank is two (cf. [15, Example 2]).
Then $\widehat{\Gamma}$ is a $2$-dimensional compact connected abelian Polish
group. The group $\Gamma$ defined by its generators
$\eta,\xi_{i},(i=0,1,2\dots)$ and relations,

$None$ $2^{k_{i+1}}\xi_{i+1}=\xi_{i}+\eta,$

where $i\in\omega$ and $k_{i}\in{\mathbb{N}}^{+}$ such that
$\sup\\{k_{i}:i\in\omega\\}=\infty$.

Put $G=\widehat{\Gamma}$. We claim that
$E(G)\nleq_{B}E(\Sigma_{P_{0}}\times\Sigma_{P_{1}}\times\dots\times\Sigma_{P_{n}})$.
Otherwise, by Corollary 4.2 and [10, Theorem 23.18], there exists $i\leq n$
such that there is a nonzero continuous homomorphism $f$ from
$\widehat{\Sigma_{P_{i}}}$ to $\widehat{G}$. Note that for any $a\in$
$\widehat{\Sigma_{P_{i}}}$, there are infinitely many positive integers $n$
such that the equation $nx=a$ has a solution. But any element in $\Gamma$ does
not admit such property. This implies that
$f(\widehat{\Sigma_{P_{i}}})=\\{1_{\Gamma}\\}$ contradicting that $f$ is a
nonzero homomorphism.

Now assume that $E(\Sigma_{P})\leq_{B}E(G)$ for some
$P\in\mathcal{P}^{\omega}$. We show that $\\{i\in\omega:P(i)=2\\}$ is
infinite. By Corollary 4.2, there is a nonzero homomorphism $f$ from
$\widehat{G}$ to $\widehat{\Sigma_{P}}$. Without loss of generality we may
assume $\widehat{G}=\Gamma$ and
$\widehat{\Sigma_{P}}=\left\\{\frac{m}{P(0)P(1)\dots
P(n)}:m\in{\mathbb{Z}},n\in{\mathbb{N}}\right\\}\subseteq{\mathbb{Q}}$. From
$(*)$, a straightforward calculation shows that

$2^{k_{1}+k_{2}+\dots+k_{i}}\xi_{i}=\xi_{0}+\eta(1+2^{k_{1}}+2^{k_{1}+k_{2}}+\dots+2^{k_{1}+k_{2}+\dots+k_{i-1}}).$

So we have

$2^{k_{1}+k_{2}+\dots+k_{i}}f(\xi_{i})=f(\xi_{0})+f(\eta)(1+2^{k_{1}}+2^{k_{1}+k_{2}}+\dots+2^{k_{1}+k_{2}+\dots+k_{i-1}}).$

Note that $\lim_{i}2^{-(k_{1}+k_{2}+\dots+k_{i})}f(\xi_{0})=0$ and

$\frac{1+2^{k_{1}}+2^{k_{1}+k_{2}}+\dots+2^{k_{1}+k_{2}+\dots+k_{i-1}}}{2^{k_{1}+k_{2}+\dots+k_{i}}}f(\eta)\leq\frac{f(\eta)}{2^{k_{i}-1}}\to
0\quad(i\to\infty).$

This implies that $\lim_{i}f(\xi_{i})=0$.

Let $f(\xi_{0})=a/b$ and $f(\eta)=c/d$ for some integers $a,b,c,d$ with
$c,d>0$. Note that $2^{k_{i+1}}f(\xi_{i+1})=f(\xi_{i})+f(\eta)$. Since $f$ is
a nonzero homomorphism, there can be at most one $f(\xi_{i})=0$. For large
enough $i$, we have $f(\xi_{i})\neq 0$. So there exist integers
$m_{i},m^{\prime}_{i},c^{\prime},d^{\prime},l_{i}$ with
$m_{i},m^{\prime}_{i}\neq 0$ and $c^{\prime},d^{\prime}>0$ such that

$f(\xi_{i})=\frac{m_{i}}{2^{k_{1}+k_{2}+\dots+k_{i}}cd}=\frac{m^{\prime}_{i}}{2^{l_{i}}c^{\prime}d^{\prime}},$

where $m^{\prime}_{i}$ and $2^{l_{i}}c^{\prime}d^{\prime}$ are coprime and
$c^{\prime}|c$, $d^{\prime}|d$. It follows that

$|f(\xi_{i})|\geq\frac{1}{2^{l_{i}}c^{\prime}d^{\prime}}\geq\frac{1}{2^{l_{i}}cd}\to
0\quad(i\to\infty).$

So $l_{i}\to\infty$ as $i\to\infty$, and hence $\\{i\in\omega:P(i)=2\\}$ is
infinite. ∎

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