Infinitely iterated wreath products of metric spaces
The construction of the finitary wreath product of metric spaces and its completion, the infinitely iterated wreath product of metric spaces are introduced. They full isometry groups are described. Some properties and examples of these constructions are considered.
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Oliynyk, B. 2019-06-09T13:44:27Z 2019-06-09T13:44:27Z 2013 Infinitely iterated wreath products of metric spaces / B. Oliynyk // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 48–62. — Бібліогр.: 14 назв. — англ. 1726-3255 2010 MSC:54E40, 54B10, 54H15, 20E22. https://nasplib.isofts.kiev.ua/handle/123456789/152263 The construction of the finitary wreath product of metric spaces and its completion, the infinitely iterated wreath product of metric spaces are introduced. They full isometry groups are described. Some properties and examples of these constructions are considered. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Infinitely iterated wreath products of metric spaces Article published earlier |
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The construction of the finitary wreath product of metric spaces and its completion, the infinitely iterated wreath product of metric spaces are introduced. They full isometry groups are described. Some properties and examples of these constructions are considered.
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Infinitely iterated wreath products of metric spaces / B. Oliynyk // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 48–62. — Бібліогр.: 14 назв. — англ. |
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2025-11-25T20:56:19Z |
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1850543574378086400 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 15 (2013). Number 1. pp. 48 – 62
c© Journal “Algebra and Discrete Mathematics”
Infinitely iterated wreath products
of metric spaces
Bogdana Oliynyk
Communicated by V. I. Sushchansky
Abstract. The construction of the finitary wreath product
of metric spaces and its completion, the infinitely iterated wreath
product of metric spaces are introduced. They full isometry groups
are described. Some properties and examples of these constructions
are considered.
Introduction
Let s : R+ → R+ be a strictly increasing continuous function with
s(0) = 0, called a scale. A space s(X) which arises from a metric space
(X, dX) by replacing the metric dX by s(dX) is called a metric transform
of (X, dX). This notion for metric spaces was introduced by Blumenthal
in [1]. Metric transforms was studied in many papers, in particular, metric
transforms of Euclidean spaces into subsets of Hilbert space have been
investigated by Schoenberg and von Neumann ([2], [3]).
In general case a metric transform s(X) of a space (X, dX) may not
be a metric space. But if s(t) is differentiable scale and the derivative
s′ is non-increasing then s(dX) is a metric. Metric spaces (X, dX) and
(Y, dY ) are called isomorphic ([4]) if there exist a bijections g : X → Y
and a scale s, such that for arbitrary u, v ∈ X
dX(u, v) = s(dY (g(u), g(v))),
2010 MSC: 54E40, 54B10, 54H15, 20E22.
Key words and phrases: Metric space, Wreath product, Isometry group.
B. Oliynyk 49
i.e. the space X and the metric transform s(Y ) are isometric. In this case
the space (X, dX) is denoted by s(Y ). If the space (X, dX) is isomorphic
to some subspace of the space (Y, dY ), then we say that (X, dX) can
be isomorphically embedded in the space (Y, dY ). Isomorphic spaces are
topologically equivalent. Note, that isometry groups of isomorphic metric
spaces are isomorphic.
In this paper we introduce two constructions of uniformly discrete
metric spaces with finite diameters using the notion of isomorphism of
metric spaces. The first one is the finitary wreath product of metric spaces.
The second one is its completion and can be introduced independently
from the first one. We call it the infinitely iterated wreath product of
metric spaces. This construction can be regarded as a generalization of
the boundary ∂T of the infinite spherically homogeneous rooted tree T
(see [5]). We also describe the isometry group, some properties and some
examples of finitary and infinitely iterated wreath products of metric
spaces.
It is a well-known problem for a given permutation group (G, X)
to find or just to prove the existence of a discrete structure X (e.g.,
graph, metric space, ordered set, etc.) such that its automorphism group
Aut(X) is isomorphic to (G, X) as a transformation group, see [6]. Then
a natural question arises. Assume that some transformation groups are
realized as automorphism groups of certain structures. What can be
said about realization of different constructions (e.g., direct or wreath
products) of these groups? We partially answer this question. For arbitrary
transformation groups (G1, X1), (G2, X2), . . ., which are isometry groups
of uniformly discrete metric spaces of finite diameters, we construct a
metric space of finite diameter such that the wreath product of the given
transformation groups is isomorphic to the isometry group of this space
as a transformation group.
Recall that a metric space (U, dU ) is called isomorphically universal
for a collection M of metric spaces if any metric space (X, dX) from M
is isomorphically embeddable in (U, dU ). The space l2 is isomorphically
universal for finite metric spaces [4]. We define a continuum family of
countable metric spaces isomorphically universal for finite metric spaces.
The isometry groups of these spaces contain isomorphic copies of each
countable residually finite group.
We show that for any finite group G there exists a self-similar metric
space X such that the infinitely iterated wreath power ≀∞i=1G of this group
is isomorphic to the isometry group of X.
50 Infinitely iterated wreath products of metric spaces
1. Preliminaries
Several constructions discussed below are based on the notion of the
wreath product of metric spaces ([7]). To define it we need a few definitions.
Recall that a metric space (X, dX) is said to be uniformly discrete
if there exists a real number r > 0 such that for any different points
x, y ∈ X the inequality dX(x, y) > r holds.
Let (X, dX) be a uniformly discrete metric space and (Y, dY ) be a
metric space with finite diameter. Assume that for a positive number r
and arbitrary points x1, x2 ∈ X, x1 6= x2, the inequality dX(x1, x2) ≥ r
holds.
Let s(x) be a scale satisfying the inequality diam(s(Y )) < r. Such a
function exists, for the diameter of the space Y is finite.
Define a metric ρs on the cartesian product X × Y by the rule:
ρs((x1, y1), (x2, y2)) =
{
dX(x1, x2), if x1 6= x2
s(dY (y1, y2)), if x1 = x2
.
This metric space is called the wreath product of metric spaces (X, dX)
and (Y, dY ) and denoted by XwrY . The metric space provided by this
construction is unique up to isomorphism, that is it does not depend on
the choice of the scale s(t).
The following lemma is easily verified.
Lemma 1. 1) The wreath product of metric spaces (X, dX) and (Y, dY )
contains isomorphic copies of both spaces (X, dX) and (Y, dY ).
2) Let (X, dX), (Y, dY ), (W, dW ) be metric spaces such that (X, dX),
(Y, dY ) are uniformly discrete and (Y, dY ), (W, dW ) have finite di-
ameters. Then for any admissible scales s1, s2, s3, s4 spaces
(Xwrs1
Y )wrs2
W and Xwrs3
(Y wrs4
W )
are isomorphic, i.e. the operation of wreath product of metric spaces
is associative.
2. Construction
Let (X1, d1), (X2, d2), . . . be an infinite sequence of uniformly discrete
metric spaces of finite diameters. Assume that r1, r2, . . . is an infinite
B. Oliynyk 51
sequence of positive numbers such that for arbitrary points a, b ∈ Xi,
a 6= b, the inequalities
di(a, b) ≥ ri, i ≥ 1 (1)
hold.
Fix an infinite sequence of scales
α = (s2(x), s3(x), s4(x), . . .)
such that
diam(s2(X2)) < r1, diam(si(Xi)) < si−1(ri−1), i ≥ 3. (2)
By Lemma 1 we can consider the n-iterated wreath product of metric
spaces using corresponding scales from sequence α. Denote the n-iterated
wreath product of metric spaces X1, . . ., Xn by
wrn
i=1(α)Xi.
Fix a sequence of points x0
i ∈ Xi, i ≥ 1. One can define an isometric
embedding
ηn : wrn
i=1(α)Xi → wrn+1
i=1 (α)Xi
given by the rule
ηn(x1, . . . , xn) = (x1, . . . , xn, x0
n+1).
Then we have a directed system
〈wrn
i=1(α)Xi, ηn〉, n ∈ N.
Denote the limit space of this system by −→wr∞
i=1(α)Xi and by wr∞
i=1(α)Xi
its completion. We call the metric spaces −→wr∞
i=1(α)Xi the finitary wreath
product of metric spaces (X1, d1), (X2, d2), . . . with respect to the sequence
of scales α and the space wr∞
i=1(α)Xi the infinitely iterated wreath product
of the metric spaces (X1, d1), (X2, d2), . . . with respect to the sequence
of scales α.
Denote the set
∏∞
i=1 Xi by X. Define a subset X̃ of X as a set of all
sequences (x1, x2, . . .) such that for some i ∈ N the equalities
xj = x0
j , j ≥ i,
holds. Then the finitary wreath product and the infinitely iterated wreath
product of metric spaces (X1, d1), (X2, d2), . . . with the sequence of scales
52 Infinitely iterated wreath products of metric spaces
α can be described as metric spaces defined on the sets X̃ and X corre-
spondingly, where the metric is defined by the rule:
ρα((a1, a2, a3, . . .), (b1, b2, b3, . . .)) =
=
d1(a1, b1), if a1 6= b1;
s2(d2(a2, b2)), if a1 = b1 and a2 6= b2;
s3(d3(a3, b3)), if a1 = b1, a2 = b2, a3 6= b3;
. . . . . . . . . .
(3)
The infinitely iterated wreath product of metric spaces (X1, d1), (X2, d2), . . .
is homeomorphic to the projective limit of finitely iterated wreath prod-
ucts of metric spaces wrn
i=1Xi, n ≥ 1, with natural projections, where
wr1
i=1Xi = X1.
Proposition 1. Let (X1, d1), (X2, d2), . . ., and (Y1, b1), (Y2, b2), . . . be
sequences of uniformly discrete metric spaces of finite diameters, α1 and
α2 be sequences of scales such that the inequalities (2) hold for both
of them. If for each i, i ≥ 1, spaces Xi and Yi are isomorphic then
spaces wr∞
i=1(α1)Xi and wr∞
i=1(α2)Yi (−→wr∞
i=1(α1)Xi and −→wr∞
i=1(α2)Yi) are
isomorphic as well.
Proof. Let h1(x), h2(x), . . . be a sequence of scales such that for each i,
i ≥ 1, Yi = hi(Xi). Assume that
α1 = (s2(x), s3(x), . . .), α2 = (g2(x), g3(x), . . .).
Define an infinite sequence of numbers q2, q3, . . . such that
qn = sup
u,v∈Yn
{gn(bn(u, v)}, n ≥ 2.
As α2 satisfies inequalities (2), the following inequalities hold
q2 < r1, qn < gn−1(rn−1), n ≥ 3.
Define a new function S̃(x) on the R+ by the rule:
B. Oliynyk 53
S̃(x) =
x, if x > r1;
r1 +
s2(h2(g−1
2
(q2)))−r1
q2−r1
(x − r1), if q2 ≤ x ≤ r1;
s2(h2(g−1
2 (x))), if g2(r2) < x < q2;
. . . . . . . . .
sn(hn(g−1
n (qn))) + βn(x − qn), if qn ≤ x ≤ gn−1(rn−1)
sn(hn(g−1
n (x))), if gn(rn) < x < qn;
. . . . . . . . . ,
where
βn =
sn−1(hn−1(g−1
n−1(rn−1))) − sn(hn(g−1
n (qn)))
gn−1(rn−1) − qn
, for all n ≥ 3,
i.e. if q2 ≤ x ≤ r1 or qn ≤ x ≤ gn−1(rn−1), then the graph of this
function is the line segment joining points (q2, s2(g−1
2 (q2)) and (r1, r1), or
(qn, sn(g−1
n (qn)) and (gn−1(rn−1), sn−1(g−1
n−1(rn−1))), n ≥ 3, respectively.
It is clear that S̃(x) is a scale and S̃(ρα1
) = ρα2
. Hence, spaces
wr∞
i=1(α1)Xi and wr∞
i=1(α2)Yi (−→wr∞
i=1(α1)Xi and −→wr∞
i=1(α2)Yi) are iso-
morphic.
Therefore, to consider metric spaces −→wr∞
i=1(α)Xi or wr∞
i=1(α)Xi up to
isomorphism we can assume that the corresponding sequence of scales
α is fixed. In this case we denote the infinitely iterated wreath product
and finitary wreath product of metric spaces X1, X2, . . . by wr∞
i=1Xi and
−→wr∞
i=1Xi correspondingly.
As (Xi, di), i ≥ 1, are uniformly discrete metric spaces, the metric di
induces the discrete topology on Xi, for all i ≥ 1. The metric ρα induces
the topology σ on
∏∞
i=1 Xi. The topology σ coincides with the Tykhonov’s
product topology on the product
∏∞
i=1 Xi of discrete spaces. If |Xi| < ∞,
i ≥ 1, then the space wr∞
i=1Xi and the Cantor space are homeomorphic.
In this case the space wr∞
i=1Xi is a compact totally disconnected metric
space, while the space −→wr∞
i=1Xi is a countable everywhere dense subspace
in wr∞
i=1Xi.
3. Characterization
Let T be an infinite spherically homogeneous rooted tree with the root
v0. Recall the definition of the space ∂T of paths in T , i.e. the boundary
of T (see, e.g., [5]). For every nonnegative integer l the l-th level is the set
54 Infinitely iterated wreath products of metric spaces
Vl of all vertices v ∈ V (T ) such that the length of the unique simple path
connecting v and v0 in T equals l. The tree T is uniquely defined by its
spherical index, i.e. by an finite sequence of cardinal numbers [k1; k2; . . . , ],
where ki is the number of edges joining a vertex of the i − 1-th level with
vertices of the i-th level. A rooted path is an infinite sequence of vertices
(v0, v1, . . . , vn, . . .) such that the vertices vi, vi+1 are connected by an edge
for every i, i ≥ 0. The metric space ∂T is the set of all infinite rooted
paths of T with the ultrametric ρ, defined by the rule:
ρ(γ1, γ2) = 1/(m + 1),
where m is the length of the common beginning of rooted paths γ1 and
γ2.
Recall, that a metric space is called discrete if all non-zero distances
in this space equal 1.
Proposition 2. Let T be an infinite spherically homogenous rooted tree
with spherical index [k1; k2; . . . , ]. Assume that (X1, d1), (X2, d2), . . . are
discrete metric spaces, such that |Xi| = ki, i ≥ 1. Then there exists a
sequences of scales α such that spaces ∂T and wr∞
i=1(α)Xi are isometric.
To prove this statement it is sufficient to pick α = (1
2x, 1
3x, 1
4x, . . .).
Let now (X1, d1), (X2, d2), . . . be uniformly discrete metric spaces of
finite diameters. Consider an infinite spherically homogenous rooted tree
T with spherical index [|X1|; |X2|; . . . , ]. Fix an infinite sequence of scales
α = (s2(x), s3(x), s4(x), . . .)
such that inequalities (2) hold. Let s1(x) = x, x ∈ R+. We can introduce
a natural metric on the set ∂T of all rooted path of tree T . For arbitrary
paths γ1 = (v0, u1, u2, . . .), γ2 = (v0, v1, v2, . . .) we put
σ(γ1, γ2) =
{
sn+1(dn+1(vn, un)), if γ1 6= γ2;
0, if γ1 = γ2,
where n is the length of the common beginning of rooted paths γ1 and γ2.
Proposition 3. The infinitely iterated wreath product of metric spaces
(X1, d1), (X2, d2), . . . with the sequence of scales α is isometric to the
space (∂T, σ).
B. Oliynyk 55
Note that for arbitrary i, i ≥ 1 the space (Xi, di) is isomorphically
embeddable in the space (∂T, σ). Indeed, fix a point aj from the space
Xj , j ≥ 1, j 6= i. Then the subspace of paths
(v0, a1, . . . , ai−1, xi, ai+1, . . .), xi ∈ Xi
is isomorphic to the space (Xi, di). Such a subspace of (∂T, σ) will be
called naturally isomorphic to the metric space (Xi, di).
Lemma 2. Let u = (u1, u2, . . .), v = (v1, v2, . . .), w = (w1, w2, . . .) be
different points of the space wr∞
i=1Xi. The points u, v, w are vertices
of a scalene triangle iff there exists k such that u1 = v1 = w1, . . .,
uk−1 = vk−1 = wk−1, uk 6= vk, wk 6= vk, uk 6= wk and the triangle uk, vk,
wk is scalene in the space Xk.
Proof. Let u, v, w ∈ wr∞
i=1Xi be vertices of a scalene triangle. Assume that
there exist k and l, k 6= l, such that u1 = v1 = w1, . . ., uk−1 = vk−1 = wk−1,
uk 6= vk, wk = vk, . . ., wl−1 = vl−1, wl 6= vl. Using (3) we obtain
ρα(u, v) = ρα((u1, u2, . . .), (v1, v2, . . . , )) = sk(dk(uk, vk)),
ρα(u, w) = ρα((u1, u2, . . .), (w1, w2, . . . , )) = sk(dk(uk, wk)).
Therefore, ρα(u, v) = ρα(u, w).
The converse statement directly follows from the definition of the
space wr∞
i=1Xi.
Proposition 4. (A) The space wr∞
i=1Xi is totally disconnected.
(B) The space wr∞
i=1Xi is compact iff for each i ≥ 1 the space Xi is
finite.
(C) The space wr∞
i=1Xi is separable iff for each i ≥ 1 the space Xi is
countable or finite.
(D) The space wr∞
i=1Xi is ultrametric iff for each i ≥ 1 the space Xi is
ultrametric.
Proof. (A) The space wr∞
i=1Xi is a product of totally disconnected spaces.
Hence wr∞
i=1Xi is totally disconnected.
(B) It follows from the Tykhonov’s compactness theorem that the
product
∏∞
i=1 Xi is compact iff for all i ≥ 1 the space Xi is compact. As
(X1, d1), (X2, d2), . . ., is an infinite sequence of uniformly discrete metric
spaces of finite diameters, Xi is compact iff Xi is finite.
56 Infinitely iterated wreath products of metric spaces
(C) For each j ≥ 1 fix a point aj from Xj . Consider the subspace of all
sequences (x1, x2, . . .), xi ∈ Xi, such that for some number m equalities
xi = ai hold, i ≥ m. Then this subspace is a countable everywhere dense
subset of wr∞
i=1Xi.
Conversely, assume that the space Xj is not countable. Then it follows
from inequalities (1) that Xj is not separable. Therefore wr∞
i=1Xi is not
separable.
(D) The proof directly follows from Lemma 2.
4. The isometry group
For the next theorem we need a few definitions. Let (G1, X1),
(G2, X2), . . . be an infinite sequence of transformation groups. Follow-
ing [8] the transformation group (G,
∏∞
i=1 Xi) = ≀∞i=1(Gi, Xi) is called
infinitely iterated wreath product of groups (G1, X1), (G2, X2), . . . if for
all elements u ∈ G the following conditions hold:
1) if (x1, . . . , xn, . . .)u = (y1, . . . , yn, . . .), then for all i ≥ 1 the value
of yi depends only on x1, . . ., xi;
2) for fixed x1, . . . , xi−1 the mapping gi(x1, . . . , xi−1) defined by the
equality
gi(x1, . . . , xi−1)(xi) = yi, xi ∈ Xi
is a transformation of the set Xi that belongs to Gi.
It follows from this definition that each element u ∈ G can be written
as an infinite sequence, called tableaux:
u = [g1, g2(x1), g3(x1, x2), . . .],
where g1 ∈ G1, gi(x1, . . . , xi−1) ∈ G
X1×...×Xi−1
i , i ≥ 2. Each element
u ∈ G acts on (m1, m2, m3 . . .) ∈
∏∞
i=1 Xi by the rule
(m1, m2, m3 . . .)u = (mg1
1 , m
g2(m1)
2 , m
g3(m1,m2)
3 , . . .).
Theorem 1. The isometry group of the infinitely iterated wreath product
of metric spaces (Xn, dn), n ≥ 1, is isomorphic as a transformation group
to the infinitely iterated wreath product of isometry groups of these spaces
(Isom(wr∞
i=1Xi),
∞∏
i=1
Xi) ≃ ≀∞i=1(IsomXi, Xi).
B. Oliynyk 57
Proof. Consider arbitrary transformation
u = [g1, g2(x1), . . . , gn(x1, . . . , xn−1), . . .] ∈ ≀∞i=1(Gi, Xi).
We shall show that u is an isometry of the space wr∞
i=1Xi. By the definition
of the wreath product of permutation groups the element u acts on
∏∞
i=1 Xi.
Therefore, it is sufficient to show that u preserves the metric ρα. Indeed,
from (3) we have
ρα((a1, a2, a3, . . .)u, (b1, b2, b3, . . .)u) =
= ρα((ag1
1 , a
g2(a1)
2 , a
g3(a1,a2)
3 , . . .), (bg1
1 , b
g2(b1)
2 , b
g3(b1,b2)
3 , . . .)) =
=
d1(ag1
1 , bg1
1 ), if ag1
1 6= bg1
1 ;
s2(d2(a
g2(a1)
2 , b
g2(b1)
2 )), if ag1
1 = bg1
1 and a
g2(a1)
2 6= b
g2(b1)
2 ;
s3(d3(a
g3(a1,a2)
3 , b
g3(b1,b2)
3 )), if ag1
1 = bg1
1 , a
g2(a1)
2 = b
g2(b1)
2 ,
a
g3(a1,a2)
3 6= b
g3(b1,b2)
3 ;
. . . . . . . . . .
(4)
As g1 ∈ IsomX1, ag1
1 = bg1
1 iff a1 = b1. Hence, ag1
1 = bg1
1 iff g2(a1) = g2(b1).
As g2(a1) ∈ IsomX2, from equalities a1 = b1 and a
g2(a1)
2 = b
g2(b1)
2 it follows
that a2 = b2 and so on. Then similarly, using (4), we get
ρα((a1, a2, a3, . . .)u, (b1, b2, b3, . . .)u) =
d1(a1, b1), if a1 6= b1;
s2(d2(a2, b2)), if a1 = b1 and a2 6= b2;
s3(d3(a3, b3)), if a1 = b1, a2 = b2, a3 6= b3;
. . . . . . . . .
=
= ρα((a1, a2, a3, . . .), (b1, b2, b3, . . .)).
Therefore, u is an isometry of wr∞
i=1Xi.
Let now ϕ be an isometry of wr∞
i=1Xi. Consider points (a1, a2, a3, . . .),
(b1, b2, b3, . . .) of wr∞
i=1Xi such that
ϕ((a1, a2, a3, . . .)) = (y1, y2, y3, . . .), ϕ((b1, b2, b3, . . .)) = (z1, z2, z3, . . .),
for some (y1, y2, y3, . . .), (z1, z2, z3, . . .) ∈ wr∞
i=1Xi. We have
ρα((a1, a2, . . . , an, . . .), (b1, b2, . . . , bn, . . .)) = sj(dj(aj , bj)), (5)
58 Infinitely iterated wreath products of metric spaces
where j is those number for which a1 = b1, . . ., aj−1 = bj−1, aj 6= bj .
Similarly
ρα(ϕ(a1, a2, . . . , an, . . .), ϕ(b1, b2, . . . , bn, . . .)) =
= ρα((y1, y2, . . . , yn, . . .), (z1, z2, . . . , zn, . . .)) = sl(dl(yl, zl)), (6)
where l is the number for which y1 = z1, . . ., yl−1 = zl−1, yl 6= zl. Using
(3), (5), (6), we get l = j.
Hence, for all i ≥ 1 the values of yi depend only of the values of a1, . . .,
ai. Therefore, there exists a tableaux [g1, g2(x1), g3(x1, , x2), . . .] such that
g1 ∈ IsomX1, gi(x1, . . . , xi−1) ∈ (IsomXi)
X1×...×Xi−1 , i > 1. Moreover,
the n-coordinate tableaux [g1, g2(x1), . . . , gn(x1, . . . , xn−1)] acts on X as
ϕ does. This completes the proof.
The next corollaries follow immediately from Theorem 1.
Corollary 1. Let (G1, X1), (G2, X2), . . . be an infinite sequence of trans-
formation groups. If each of the groups (Gi, Xi), i ≥ 1 is the isometry
group of some uniformly discrete metric space with finite diameter then
the wreath product ≀∞i=1(Gi, Xi) is isomorphic as a transformation group
to the isometry group of a metric space of finite diameter.
Corollary 2. Let G1, G2, . . . be an infinite sequence of finite groups. Then
the wreath product ≀∞i=1Gi of these groups is isomorphic to the isometry
group of a totally disconnected compact metric space of finite diameter.
Proof. Each finite group is isomorphic to the isometry group of some finite
metric space. This fact follows, for example, from Frucht’s theorem [9].
Let (X1, d1), (X2, d2), . . ., be an infinite sequence of finite metric spaces
such that Gi ≃ IsomXi. Then it follows from Theorem 1 that
≀∞i=1IsomXi ≃ Isom(wr∞
i=1Xi).
Therefore ≀∞i=1Gi ≃ Isom(wr∞
i=1Xi). Moreover, it follows from Proposition
4 that the space wr∞
i=1Xi is totally disconnected compact and has finite
diameter.
Corollary 3. If for each i ≥ 1 the space Xi is homogeneous, then the
space wr∞
i=1Xi is homogeneous too.
B. Oliynyk 59
5. Examples
5.1. Wreath products of Hamming spaces
Denote by Hm the m-dimensional cube equipped with the Hamming
distance, i.e. the set of all binary m-tuples (a1, . . . , am), ai ∈ {0, 1},
1 ≤ i ≤ m, with the Hamming metric dHm
:
dHm
(x̄, ȳ) =
m∑
i=1
|xi − yi|,
where x̄, ȳ ∈ {0, 1}m. Let Θ be the set of all infinite increasing sequences
of natural numbers. For any m̄ ∈ Θ, m̄ = (m1, m2, . . .), we can fix an
infinite sequence of scales
α(m̄) = (s2(x), s3(x), s4(x), . . .),
such that
sk(t) =
t
∏k
i=2(mi + 1)
, k ≥ 2.
As diamHmi
= mi, inequalities (2) hold for the sequence of Hamming
spaces
Hm1
, Hm2
, . . . .
Hence, we can consider the space −→wr∞
i=1(α)Hmi
. Denote this space by
UH(m̄).
Since for every i, i ≥ 1, the space Hmi
is finite, the space UH(m̄) is
countable. From (3) it follows that diam(UH(m̄)) = m1.
Proposition 5. Let m̄, k̄ ∈ Θ, m̄ 6= k̄. Then spaces UH(m̄) and UH(k̄)
are not isomorphic.
Proof. Let m̄ = (m1, m2, . . .), k̄ = (k1, k2, . . .). Assume that m1 = k1, . . .,
mi−1 = ki−1, but mi 6= ki. Then the (m1 + . . .+mi−1 +1)th largest values
of metrics in spaces UH(m̄) and UH(k̄) are achieved on different numbers
of points. Hence, the spaces UH(m̄) and UH(k̄) are not isomorphic.
Theorem 2. Let m̄ ∈ Θ. Then UH(m̄) is a homogeneous countable
metric space, which contains an isomorphic copy of arbitrary finite metric
space. Any countable residually finite group G is isomorphic to some
subgroup of the isometry group of the space UH(m̄).
60 Infinitely iterated wreath products of metric spaces
Proof. Let m̄ = (m1, m2, . . .). Each space Hmi
is homogeneous. Hence,
from Corollary 3 we obtain that the space UH(m̄) is homogeneous.
Let X be an n-point metric space. It follows from [10] that there
exists an isomorphic embedding of X into the Hamming space Hm, where
m = 1
2((n
2 )2 − (n
2 ) − 2)(n2 − 2n + 7). Note, that for finite metric spaces the
definitions of their isomorphism used in this paper and in [10] coincide.
Since m̄ is an infinite increasing sequences of natural numbers there exists
a number mj such that the space X is isomorphic to some subspace Hmj
,
and hence it is isomorphic to some subspace of UH(m̄).
The isometry group IsomHm of the space Hm is isomorphic to the
wreath product Sm ≀ S2 of symmetric groups Sm and S2 (see, e.g. [11]).
Hence, from Theorem 1 we have IsomUH(m̄) ≃ ≀∞i=1(Smi
≀ S2). Therefore,
≀∞i=1(Smi
) < IsomUH(m̄). Since m̄ is an infinite increasing sequences of
natural numbers arbitrary countable residually finite group G is isomor-
phic to some subgroup of ≀∞i=1(Smi
) (see [12]).
5.2. Self-similarity
For definitions and basic properties of self-similar sets see, for in-
stance, [13].
Let (W, dW ) be a finite metric space, W = {w1, . . . , wn}, such that
d = min
x,y∈W,x 6=y
dW (x, y) > 1.
Assume that D is positive real number such that D = diam(W ).
Consider the sequence of spaces W, W, . . .. Let X =
∏∞
i=1 W . Fix the
sequence of scales
α = (
d
D + 1
t,
d
(D + 1)2
t,
d
(D + 1)3
t, . . .).
Then inequalities (2) hold and we can consider the infinitely iterated
wreath product wr∞
i=1(α)W .
Since W is finite, the space wr∞
i=1(α)W is compact. If space W is
discrete then wr∞
i=1(α)W and the space ∂T of rooted paths in the n-
regular rooted tree T are isometric. Hence, in this case we immediately
obtain that the space wr∞
i=1(α)W is self-similar (see, for instance, [5]).
But if W is not a discrete metric space, then we can similarly show that
wr∞
i=1(α)W is self-similar as well. Indeed, for all i, 1 ≤ i ≤ n, we define a
map fi : X → X by the rule:
fi(u1, u2, u3, . . .) = (wi, u1, u2, u3, . . .).
B. Oliynyk 61
Then fi is a contraction with respect to the metric ρα. Since X =⋃n
i=1 fi(X) the space wr∞
i=1(α)W is self-similar with respect to f1, . . . , fn
(see [13, Theorem 1.1.4]).
From Theorem 1 it follows that the isometry group Isomwr∞
i=1(α)W
of the space wr∞
i=1(α)W is isomorphic to the infinite wreath product
≀∞i=1IsomW . Since
Isom(wr∞
i=1(α)W ) = IsomW ≀ Isom(wr∞
i=1(α)W )
the group Isom(wr∞
i=1(α)W ) acts on X self-similarly (see [14, Definition
1.5.4] for details).
Note that all spaces wr∞
i=1(α)W are homeomorphic to the Cantor
space. If (W, dW ) is the space with trivial isometry group, then from
Theorem 1 it follows that wr∞
i=1(α)W is a self-similar set with trivial
isometry group.
Proposition 6. Let G be a finite group. Then there exists a self-similar
metric space X such that ≀∞i=1G ≃ Isom(X).
The proof of this proposition follows from Corollary 2.
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Contact information
B. Oliynyk Department of Mechanics and Mathematics,
Kyiv Taras Shevchenko University, Kyiv, 01033,
Ukraine
E-Mail: bogdana.oliynyk@gmain.com
Received by the editors: 20.07.2012
and in final form 16.10.2012.
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