Classification of the local isometry groups of rooted tree boundaries
Classification of the groups of local isometries of the rooted tree boundaries is established for transitive groups.
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Інститут прикладної математики і механіки НАН України
2007
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | Classification of the local isometry groups of rooted tree boundaries / Y. Lavrenyuk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 2. — С. 104–110. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1573612019-06-21T01:30:25Z Classification of the local isometry groups of rooted tree boundaries Lavrenyuk, Y. Classification of the groups of local isometries of the rooted tree boundaries is established for transitive groups. 2007 Article Classification of the local isometry groups of rooted tree boundaries / Y. Lavrenyuk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 2. — С. 104–110. — Бібліогр.: 6 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 05C05, 20E08, 20F28. http://dspace.nbuv.gov.ua/handle/123456789/157361 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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English |
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Classification of the groups of local isometries of
the rooted tree boundaries is established for transitive groups. |
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Article |
| author |
Lavrenyuk, Y. |
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Lavrenyuk, Y. Classification of the local isometry groups of rooted tree boundaries Algebra and Discrete Mathematics |
| author_facet |
Lavrenyuk, Y. |
| author_sort |
Lavrenyuk, Y. |
| title |
Classification of the local isometry groups of rooted tree boundaries |
| title_short |
Classification of the local isometry groups of rooted tree boundaries |
| title_full |
Classification of the local isometry groups of rooted tree boundaries |
| title_fullStr |
Classification of the local isometry groups of rooted tree boundaries |
| title_full_unstemmed |
Classification of the local isometry groups of rooted tree boundaries |
| title_sort |
classification of the local isometry groups of rooted tree boundaries |
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Інститут прикладної математики і механіки НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/157361 |
| citation_txt |
Classification of the local isometry groups of rooted tree boundaries / Y. Lavrenyuk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 2. — С. 104–110. — Бібліогр.: 6 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT lavrenyuky classificationofthelocalisometrygroupsofrootedtreeboundaries |
| first_indexed |
2025-07-14T09:48:13Z |
| last_indexed |
2025-07-14T09:48:13Z |
| _version_ |
1837615278040547328 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2007). pp. 104 – 110
c© Journal “Algebra and Discrete Mathematics”
Classification of the local isometry groups of
rooted tree boundaries
Yaroslav Lavrenyuk
Dedicated to V.I. Sushchansky on the occasion of his 60th birthday
Abstract. Classification of the groups of local isometries of
the rooted tree boundaries is established for transitive groups.
1. The notion of local isometry is defined naturally on a metric space. A
bijection is a local isometry if it acts as an isometry in a neighborhood of
each point. More precisely, let (X1, d1) and (X2, d2) be metric spaces. A
bijection α : X1 → X2 is called a local isometry if for every x ∈ X1 there
exists a neighborhood Ux of x such that for every x1, x2 ∈ Ux we have
d2(x
α
1 , xα
2 ) = d1(x1, x2).
It is clear that the set of local isometries of a metric space (X, d) is a
group.
The group of the local isometries of the boundary of a rooted tree
is investigated in [LS], [Lav], [MI]. In particular, in [Lav] it was proved
that if the group of the local isometries of the boundary of a locally finite
rooted tree is transitive, then it is complete, i.e., all its automorphisms
are inner and its center is trivial.
It is known that the full isometry groups of the boundary of a locally
finite spherically homogeneous rooted tree are isomorphic if and only if
the trees are isometric [Nek, Proposition 2.10.7]. The main result of this
work is the analogous assertion for the local isometries.
2. We start with necessary definitions on rooted trees and groups acting
on such trees. Let T be a locally finite rooted tree with the root vertex
2000 Mathematics Subject Classification: 05C05, 20E08, 20F28.
Key words and phrases: rooted trees, ultrametrics, local isometry, classifica-
tion.
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.Y. Lavrenyuk 105
v0. For every two vertices u, v of the tree T (u, v ∈ V (T )) we define the
distance between u and v, written d(u, v), to be the length of the shortest
path connecting them. For an integer n ≥ 0 we define the nth level (the
sphere of the radius n) to be the set
Vn(T ) = {v ∈ V (T ) : d(v0, v) = n} .
If the degree of a vertex v ∈ Vn(T ) depends only on n, then the
tree T is called spherically homogeneous. Spherical index of a spherically
homogeneous tree T is the sequence Ω = (a0, a1, . . .), where a0 is degree
of the root and an + 1 is degree of any vertex of n-th level.
Let T be a spherically homogeneous rooted tree with the root v0 and
spherical index Ω. All such trees are isomorphic to the tree TΩ with the
set of vertices equal to the set of all finite sequences (i0, i1, . . . , in−1),
where ik ∈ {1, 2, . . . , ak} and n ≥ 0 is an integer. We also include the
empty sequence (corresponding to n = 0). Two vertices are adjacent if
and only if they are of the form (i0, . . . , in−1), (i0, . . . , in−1, in).
An end of a rooted tree is an infinite path starting in the rooted vertex
and having no repetitions. We will denote by ∂T (boundary) the set of
all ends of T .
Let λ̄ = {λn}
∞
n=0 be a strictly decreasing sequence of positive numbers
tending to zero. We can introduce a natural ultrametric on ∂T by putting
ρ(x1, x2) = λn, where n is the length of the maximal common part of
the paths x1 and x2. The topology induced by the metrics ρ (or for
convenience λ̄) is compact, totally disconnected and has a base of open
sets (balls or cylinder sets) of the form Uv = {x ∈ ∂T | v ∈ x}, v ∈ V (T ).
This compact ultrametric space will be denoted by (∂T, λ̄) or simply by
∂T .
3. In this work we study the classification problem of the full local
isometry groups of the boundaries of rooted trees. We prove the following
theorem.
Theorem 1. Let T1 and T2 be locally finite rooted trees such that the
groups of the local isometries act on their boundaries transitively. The
full local isometries groups of (∂T1, µ̄1) and (∂T2, µ̄2) are isomorphic if
and only if (∂T1, λ̄1) and (∂T2, λ̄2) are locally isometric for some metrics
λ̄1 and λ̄2. In other words there are positive integers i and j, such that
for every natural s the equality
|Vi+s(T1)| = |Vj+s(T2)|
holds.
In the work [Lav] a sufficient condition for isomorphism for the local
isometry groups was established
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.106 Classification of isometry groups
Proposition 2 ([Lav]). If there exist ultrametrics µ̄1 and µ̄2 such that the
spaces (∂T1, µ̄1) and (∂T2, µ̄2) are locally isometric then their full local
isometry groups are isomorphic.
In that work it was also established that the boundary of a rooted
tree on which the local isometries act transitively is locally isometric to
the boundary of some spherically homogeneous tree.
Proposition 3 ([Lav]). If the group LIsom(∂T, λ̄) acts transitively on ∂T
then there is a spherically homogeneous tree TΩ and a sequence λ̄Ω such
that (∂T, λ̄) and (∂TΩ, λ̄Ω) are locally isometric.
Due to Propositions 2 and 3, in order to prove Theorem 1, we only
need to show that if the groups of local isometries of the boundaries ∂T1
and ∂T2 of a spherically homogeneous tree are isomorphic then (∂T1, λ̄1)
and (∂T2, λ̄2) are locally isometric for some metrics λ̄1 and λ̄2.
4. Let us fix some notations. Let Ti (i = 1, 2) be spherically homogeneous
trees with spherical indices containing no ones. We will denote ∂Ti by Xi.
We assume that the full local isometry groups of X1 and X2 are isomor-
phic and φ : LIsom X1 → LIsom X2 is an isomorphism. By Rubin’s the-
orem [Rub, Corollary 3.13c] there exists a homeomorphism h : X1 → X2
such that for every g ∈ LIsom X1 the equality φ(g) = hgh−1 holds. The
action of LIsom Xi on Xi is ergodic with respect to the Bernoulli measure.
Actually the only probabilistic measure on Xi with respect to which the
action of LIsom Xi is ergodic is the Bernoulli measure (see [GNS]). Thus,
we can fix the unique probabilistic LIsom Xi-invariant measures mi on
sigma-algebras of Borel subsets of Xi.
Lemma 4. For the homeomorphism h and the ball U1
v (v ∈ V (T1)) there
are positive integers l and k such that
h(U1
v ) =
k
⋃
i=1
U2
vi
,
where {v1, . . . , vk} ⊂ Vl(T2).
Proof. Since U1
v is compact and balls in X2 do not intersect or one con-
tains the other, we have what is required.
Lemma 5. For every ball U1
v (v ∈ Vk(T1)) (k ∈ N) the equality
m2(h(U1
v )) = m1(U
1
v ),
holds.
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.Y. Lavrenyuk 107
Proof. The measures m1, m2 are probabilistic, i.e. m1(X1) = m2(X2) =
1. Since m1 is an LIsom X1-invariant measure,
m1(X1) = m1
⊔
w∈Vk(T1)
U1
w
= |Vk(T1)|m1(U
1
v ).
Since every local isometry preserves the measure, and φ(g) = hgh−1 for
all g ∈ LIsom X1 we have m2(h(U1
u)) = m2(h(U1
w)) for any u, w ∈ Vk(T1).
Therefore
m2(X2) = m2(h(X1)) = m2
h
⊔
w∈Vk(T1)
U1
w
=
= m2
⊔
w∈Vk(T1)
h(U1
w)
= |Vk(T1)|m2(h(U1
v )).
Thus, m2(h(U1
v )) = m1(U
1
v ).
Lemma 6. Let w ∈ Vm(T2) and v ∈ Vl(T1) be such that h−1(U2
w) = U1
v .
Then |Vm(T2)| = |Vl(T1)|.
Proof. By Lemma 5 we have m2(U
2
w) = m1(U
1
v ). Therefore
|Vm(T2)|m2(U
2
w) = m1(U
1
v )|Vl(T1)| = 1.
Thus we have |Vm(T2)| = |Vl(T1)|.
Let (X, d) be a metric space. A bijection α : X1 → X2 is called a
uniform local isometry if there exists δ > 0 such that
d2(x
α
1 , xα
2 ) = d1(x1, x2)
for all x1, x2 ∈ X1 such that d1(x1, x2) < δ. It is easy to see that for a
compact spaces every local isometry is a uniform local isometry.
Lemma 7. There exist ultrametrics µ̄1 and µ̄2 such that the spaces (∂T1, µ̄1)
and (∂T2, µ̄2) are locally isometric if and only if there exist i and j, such
that for every positive integer s the equality
|Vi+s(T1)| = |Vj+s(T2)|
holds.
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.108 Classification of isometry groups
Proof. Obviously, if the equality holds then the spaces (∂T1, µ̄1) and
(∂T2, µ̄2) are locally isometric for some ultrametrics µ̄1 and µ̄2. The
remaining implication follows from uniformity of local isometry.
Lemma 8. The spaces X1 and X2 are locally isometric for some ultra-
metrics.
Proof. Let {vi, i ∈ N} ⊂ V (T2) be infinite sequence of vertices such that
vi � vj for all i 6= j. Let h−1(U2
vi
) =
⊔ki
j=1 U1
uij
, where {ui1, . . . , uiki
} ⊂
Vli(T1).
We also consider infinite sequence of vertices {wi, i ∈ N} ⊂ V (T2)
such that wi ≺ vi and m2(U
2
wi
) < m1(U
1
uij
) for all i and j. We can assume
that m2(U
2
wi
) > m2(U
2
wi+1
), that is wi closer to the root than wi+1. We
suppose that h−1(U2
wi
) is not a ball for all i. We will show that there
exists a sequence {gi, i ∈ N} ⊂ Isom X1 such that
1. gi acts trivially on X1 \ h−1(U2
vi
).
2. φ(gi)(U
2
wi
) = hgih
−1(U2
wi
) is not a ball.
There exist only two possibilities for h−1(U2
wi
):
1. There exist j1 and j2 (1 ≤ j1, j2 ≤ ki) such that h−1(U2
wi
)∩U1
uijs
6= ∅
(s = 1, 2).
2. There exists j (1 ≤ j ≤ ki) such that h−1(U2
wi
) $ U1
uij
.
In the first case we choose gi as follows: gi acts trivially on X1 \h−1(U2
vi
)
and on U1
uij1
and move some points of U1
uij2
in such a way that gi(U
1
uij2
) =
U1
uij2
and gi(h
−1(U2
wi
)∩U1
uij2
) 6= h−1(U2
wi
)∩U1
uij2
. Since h−1(U2
wi
)∩U1
uij2
is strictly contained in U1
uij2
, we have that there exists gi with the required
properties.
In the second case we choose gi as follows: gi acts trivially on X1 \
h−1(U2
vi
) and stabilizes U1
uij
and gi(h
−1(U2
wi
)) 6= h−1(U2
wi
). Since h−1(U2
wi
)
is strictly contained in U1
uij
, we have that there exists gi with required
properties.
Since h−1(U2
vi
) ∩ h−1(U2
vj
) = ∅ for i 6= j we have that the infinite
product g1g2g3 . . . defines an isometry g of the metric space X1. Thus
φ(g) is a local isometry of X2. Therefore images of the balls with “small”
radii are balls for the mapping φ(g). That is, there is a positive integer
r, such that for every t > r and every vertex v ∈ Vt(T2) there exists
w ∈ V (T2) such that
φ(g)(U2
v ) = U2
w. (0.1)
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.Y. Lavrenyuk 109
By the construction, φ(gi)(U
2
wi
) is not a ball for any positive integer
i. Hence φ(g)(U2
wr+1
) is not a ball either. Since wr+1 is in the level
with number greater than r, we get a contradiction with (0.1). Thus
our assumption is not true. So for any n > r the image h−1(U2
wn
) is a
ball. Let m be such that wn is in Vm(T2). By Lemma 6 we have that
|Vm(T2)| = |Vl(T1)| for some positive integer l.
Suppose that there exists an infinite increasing sequence of positive
integers {li, i ∈ N} such that for every i and m the inequality |Vli(T2)| 6=
|Vm(T1)| holds. We can consider a sequence {uj , j ∈ N} ⊂ V (T2) such
that for every positive integer j the conditions uj ≺ wj and uj ∈ Vk(T2)
hold for k ∈ {li, i ∈ N}. Since m2(U
1
ui
) < m2(U
2
wi
) < m1(U
1
ui1
) for
all i, by the supposition and by Lemma 5, we have that h(U1
ui
) is not
a ball. Therefore we have a contradiction with what we proved above.
Thus for every m greater than some n the equality |Vm(T2)| = |Vl(T1)|
holds for some positive integer l. Since h is a homeomorphism and φ is
an isomorphism the converse statement is true: for every l greater than
some k the equality |Vl(T1)| = |Vm(T2)| holds for some positive integer m.
Therefore there exist i0 and j0 such that for every natural s the following
equality holds:
|Vi0+s(T1)| = |Vj0+s(T2)|.
So the spaces X1 and X2 are locally isometric for some metrics.
Now the proof of Theorem 1 follows from Lemmas 7, 8 and Proposi-
tions 2 and 3.
Corollary 9. If the ultrametrics on X1 and X2 are such that these spaces
are locally isometric, then the homeomorphism h is a local isometry.
Proof. Let f : X1 → X2 be a local isometry. Then hf−1 induces an
automorphism of G1. Since G1 is a complete group we have that hf−1 is
a local isometry. So h is also a local isometry.
References
[GNS] R. Grigorchuk, V. Nekrashevich, V. Sushchansky, Automata, dynamical sys-
tems, and groups. (Russian) Tr. Mat. Inst. Steklova 231 (2000), Din. Sist., Avtom.
i Beskon. Gruppy, 134–214; translation in Proc. Steklov Inst. Math. 2000, no. 4
(231), 128–203
[Lav] Lavrenyuk Ya. On automorphisms of local isometry groups of compact ultra-
metric spaces. International Journal of Algebra and Computation, 15(5-6) 2005,
1013–1024.
[LS] Ya.V.Lavrenyuk, V.I.Sushchansky Automorphisms of homogeneous symmetric
groups and hierarchomorphisms of rooted trees, Algebra and Discrete Mathemat-
ics, N. 4 (2003), 33–49.
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.110 Classification of isometry groups
[MI] B.Majcher-Iwanow Equivalence relations induced by some locally compact
groups of homeomorphisms of 2
N. Colloq. Math. 103 (2005), no. 2, 287–301.
[Nek] V.Nekrashevych Self-similar groups. AMS: Mathematical Surveys and Mono-
graphs, Vol.117, 231 pp., 2005.
[Rub] Rubin M. On the reconstruction of topological spaces from their groups of home-
omorphisms. Transactions of the AMS, Vol. 312, No 2. April 1989, 487–538.
Contact information
Y. Lavrenyuk Kyiv Taras Shevchenko University, Volody-
myrska st. 64, 01033, Kyiv, Ukraine
E-Mail: ylavrenyuk@univ.kiev.ua
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