On faithful actions of groups and semigroups by orientation-preserving plane isometries
Feitful representations of two generated free groups and free semigroups by orientation-preserving plane isometries constructed.
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Інститут прикладної математики і механіки НАН України
2003
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irk-123456789-1557252019-06-18T01:28:39Z On faithful actions of groups and semigroups by orientation-preserving plane isometries Vorobets, Y. Feitful representations of two generated free groups and free semigroups by orientation-preserving plane isometries constructed. 2003 Article On faithful actions of groups and semigroups by orientation-preserving plane isometries / Y. Vorobets // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 4. — С. 118–125. — англ. 1726-3255 2000 Mathematics Subject Classification: 20E05, 20F32, 20M05, 20M30. http://dspace.nbuv.gov.ua/handle/123456789/155725 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Feitful representations of two generated free
groups and free semigroups by orientation-preserving plane isometries constructed. |
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Vorobets, Y. |
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Vorobets, Y. On faithful actions of groups and semigroups by orientation-preserving plane isometries Algebra and Discrete Mathematics |
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Vorobets, Y. |
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Vorobets, Y. |
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On faithful actions of groups and semigroups by orientation-preserving plane isometries |
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On faithful actions of groups and semigroups by orientation-preserving plane isometries |
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On faithful actions of groups and semigroups by orientation-preserving plane isometries |
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On faithful actions of groups and semigroups by orientation-preserving plane isometries |
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On faithful actions of groups and semigroups by orientation-preserving plane isometries |
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on faithful actions of groups and semigroups by orientation-preserving plane isometries |
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Інститут прикладної математики і механіки НАН України |
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2003 |
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On faithful actions of groups and semigroups by orientation-preserving plane isometries / Y. Vorobets // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 4. — С. 118–125. — англ. |
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Algebra and Discrete Mathematics |
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AT vorobetsy onfaithfulactionsofgroupsandsemigroupsbyorientationpreservingplaneisometries |
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2025-07-14T07:57:33Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2003). pp. 118 – 125
c© Journal “Algebra and Discrete Mathematics”
On faithful actions of groups and semigroups by
orientation-preserving plane isometries
Yaroslav Vorobets
Communicated by V. M. Usenko
Dedicated to R. I. Grigorchuk on the occasion of his 50th birthday
Abstract. Feitful representations of two generated free
groups and free semigroups by orientation-preserving plane isome-
tries constructed.
Let G+ denote the group of orientation-preserving isometries of Eu-
clidean plane. G+ is a locally compact Lie group, it consists of rotations
and translations. Let G be a countable group or semigroup. An action
of the (semi)group G on the plane by orientation-preserving isometries
is a homomorphism d : G → G+. Let x be a point in the plane. The
orbit of x under the action d is the sequence Od(x) = {d(g)x}g∈G in-
dexed by elements of G. Suppose G is finitely generated and g1, . . . , gk
is some fixed set of its generators. Then the action d is uniquely deter-
mined by isometries A1 = d(g1), . . . , Ak = d(gk), and we denote it by
G[A1, . . . , Ak]. In general, the action G[A1, . . . , Ak] may not exist for
some k-tuples (A1, . . . , Ak) of isometries. It does exist in the case G is
the free semigroup FSGk or the free group FGk with k generators.
The action d is called faithful if it is a monomorphism. Suppose
d(g1)x = d(g2)x for some g1, g2 ∈ G and a point x. If g1 6= g2 and the
action d is faithful, then d(g1)d(g2)
−1 is a nontrivial rotation and x is its
fixed point. Thus d is faithful implies there exists a countable subset Sd
of the plane such that for any x /∈ Sd all points of the orbit Od(x) are
distinct.
2000 Mathematics Subject Classification: 20E05, 20F32, 20M05, 20M30.
Key words and phrases: free groups, free semigroups, plane isometries, group
actions, semigroup actions.
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.Y. Vorobets 119
Theorem 1. For a generic pair (A, B) ∈ G2
+ (both in the sense of mea-
sure and of category), the action FSG2[A, B] is faithful.
Theorem 2. Suppose A is a nonzero translation and B is a rotation by
an angle ϕ. Then the action FSG2[A, B] is faithful if and only if cos ϕ
is a transcendent number.
The action FG2[A, B] can never be faithful for the following reason.
For any group G, let G′ denote the commutant of G, that is, the group
generated by commutators XY X−1Y −1, where X, Y ∈ G. By G′′ we
denote the commutant of G′. It is easy to see that the group G′
+ consists
of translations, hence the group G′′
+ is trivial. Therefore every action of
the group FG2 of the form FG2[A, B] descends to an action of the group
G2 = FG2/FG′′
2 (the free 2-step-solvable group with two generators).
Theorem 3. For a generic pair (A, B) ∈ G2
+ (both in the sense of mea-
sure and of category), the action G2[A, B] is faithful.
We proceed to the proofs of Theorems 1, 2, and 3.
A finite sequence x0, x1, . . . , xk of points of the lattice Z
2 is called a
path if x0 = (0, 0) and |xj − xj−1| = 1 for j = 1, . . . , k. Ordered pairs
(xj−1, xj), 1 ≤ j ≤ k, are called links of the path. The set of all paths is
denoted by P . A path x0, x1, . . . , xk is closed if its endpoint xk coincides
with x0. The set of all closed paths is denoted by P ′.
Let x1 and x2 be neighboring points of the lattice Z
2 and γ ∈ P .
Denote by nγ(x1, x2) the number of times when the pair (x1, x2) occurs
as a link of the path γ. Let P ′′ be the set of paths γ ∈ P such that
nγ(x1, x2) = nγ(x2, x1) for any x1, x2 ∈ Z
2, |x2 − x1| = 1. Clearly,
P ′′ ⊂ P ′.
Now let us assign a path γ(g) ∈ P to an arbitrary element g ∈ FG2.
Let a and b be generators of FG2. Introduce vectors ea = (1, 0), ea−1 =
(−1, 0), eb = (0, 1), eb−1 = (0,−1). Every element g ∈ FG2 can be
represented in the form ckck−1 . . . c1, where cj ∈ {a, b, a−1, b−1}, j =
1, 2, . . . , k. Choose γ(g) to be the path x0, x1, . . . , xk such that x0 = (0, 0)
and xj − xj−1 = ecj
, 1 ≤ j ≤ k. Obviously, each path γ ∈ P is assigned
to a unique element of the group FG2. However the path γ(g) is not
determined in a unique way by g. Still, some crucial features of γ(g)
depend only on an element g ∈ FG2. These are the endpoint of γ(g) and
differences nγ(g)(x1, x2) − nγ(g)(x2, x1) for all x1, x2 ∈ Z
2, |x1 − x2| = 1.
Given g ∈ FG2, the set of paths assigned to g contains a unique path of
the shortest length. The number of links in this shortest path is called
the length of g.
Lemma 1. Suppose g ∈ FG2. Then g ∈ FG′
2 if and only if γ(g) ∈ P ′,
and g ∈ FG′′
2 if and only if γ(g) ∈ P ′′.
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Proof. Let g, h ∈ FG2. Suppose x0, x1, . . . , xk is the path γ(g) and
y0, . . . , ym is the path γ(h). Then the sequence x0, x1, . . . , xk, xk +y1, . . . ,
xk +ym is the path γ(hg) and x0 = xk−xk, xk−1−xk, . . . , x0−xk = −xk
is the path γ(g−1). Let N1 : FG2 → Z
2 be the map taking each g ∈ FG2
to the endpoint of the path γ(g). It is easy to observe that N1 is a ho-
momorphism. Let H1 denote the kernel of N1. Then g ∈ H1 if and only
if γ(g) ∈ P ′. Clearly, H1 is a normal subgroup of FG2 and FG′
2 ⊂ H1.
Take any element g ∈ H1 of positive length. The element g is uniquely
represented as ckck−1 . . . c1, where cj ∈ {a, b, a−1, b−1}, 1 ≤ j ≤ k, and
k is the length of g. Since g ∈ H1, we have cm = c−1
1 for some m,
1 < m ≤ k. By construction, m > 2. Set h = cm−1cm−2 . . . c2. Then the
element g1 = gc−1
1 h−1c1h = ck . . . cm+1cm−1 . . . c2 is of length at most
k − 2. Moreover, g1 ∈ H1 since c−1
1 h−1c1h ∈ FG′
2. The inductive argu-
ment yields that H1 = FG′
2.
Let L denote the set of ordered pairs (x1, x2) such that x1, x2 ∈
Z
2 and |x1 − x2| = 1. For any path γ ∈ P the collection of numbers
nγ(x1, x2) − nγ(x2, x1), (x1, x2) ∈ L, can be considered as an element
of the group Z
L. Since differences nγ(g)(x1, x2) − nγ(g)(x2, x1) depend
only on g ∈ FG2, we have a well-defined map N2 : FG2 → Z
L. The
restriction of the map N2 to the subgroup H1 = FG′
2 is a homomorphism.
By H2 denote the kernel of this restriction. Clearly, g ∈ H2 if and only
if γ(g) ∈ P ′′. It is easy to observe that H2 is a normal subgroup of
FG2 and FG′′
2 ⊂ H2. We claim that H2 = FG′′
2, i.e., any element
g ∈ H2 belongs to FG′′
2. The claim is proved by induction on the length
k of the element g. In the case k = 0, there is nothing to prove. Now
let k > 0 and suppose the claim is true for all elements of length less
than k. There is a unique representation g = ckck−1 . . . c1 such that
cj ∈ {a, b, a−1, b−1}, 1 ≤ j ≤ k. Denote by γ the path x0, x1, . . . , xk such
that x0 = (0, 0) and xj − xj−1 = ecj
, 1 ≤ j ≤ k. Then γ ∈ P ′′ since
g ∈ H2. In particular, there exists an index l > 0 such that the points
x0, x1, . . . , xl−1 are distinct while xl = xm for some m < l. Set g1 =
cm−1 . . . c1ck . . . cm. Then g1 = cm−1 . . . c1g(cm−1 . . . c1)
−1 ∈ H2 and the
length of g1 is at most k. The path γ(g1) can be chosen as y0, y1, . . . , yk,
where yi = xi+m − xm for 0 ≤ i ≤ k − m and yi = xi−k+m − xm for
i > k − m. Since γ(g1) ∈ P ′′, there exists n > 0 such that yn−1 = y1
and yn = y0. By construction, the points y0, y1, . . . , yl−m−1 are distinct
and yl−m = y0, hence n > l − m. The sequences y0, y1, . . . , yl−m, and
yl−m, . . . , yn, and yn, . . . , yk are closed paths. They are assigned to some
elements h1, h2, h3 ∈ FG′
2, respectively. Clearly, g1 = h3h2h1. Since
yn−1 = y1, the element g2 = h3h1h2 is of length at most k−2. Moreover,
g2 = g1h
−1
1 h−1
2 h1h2 ∈ H2 as h−1
1 h−1
2 h1h2 ∈ FG′′
2. By the inductive
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assumption, g2 ∈ FG′′
2. Then g1 ∈ FG′′
2. Since g and g1 are conjugated,
we have g ∈ FG′′
2. The claim is proved.
Let P ′′
1 denote the set of paths γ ∈ P such that nγ(x, x + ea) =
nγ(x+ ea, x) for every x ∈ Z
2 and P ′′
2 denote the set of paths γ ∈ P such
that nγ(x, x + eb) = nγ(x + eb, x) for every x ∈ Z
2.
Lemma 2. P ′′
1 ∩ P ′ = P ′′
2 ∩ P ′ = P ′′.
Proof. Obviously, P ′′ = P ′′
1 ∩ P ′′
2 . For every path γ ∈ P ′ and every
x ∈ Z
2 we have the equality nγ(x, x+ea)+nγ(x, x−ea)+nγ(x, x+eb)+
nγ(x, x−eb) = nγ(x+ea, x)+nγ(x−ea, x)+nγ(x+eb, x)+nγ(x−eb, x).
If, moreover, γ ∈ P ′′
1 , then nγ(x, x + ea) = nγ(x + ea, x) and nγ(x, x −
ea) = nγ(x − ea, x), hence nγ(x, x + eb) = nγ(x + eb, x) if and only if
nγ(x − eb, x) = nγ(x, x − eb). By the inductive argument we obtain that
the equalities nγ(x, x+eb) = nγ(x+eb, x) and nγ(x+keb, x+(k+1)eb) =
nγ(x+(k +1)eb, x+ keb) are equivalent for any γ ∈ P ′′
1 ∩P ′, any x ∈ Z
2,
and any integer k. Since nγ(x+keb, x+(k+1)eb) = nγ(x+(k+1)eb, x+
keb) = 0 for large k, the equality nγ(x, x + eb) = nγ(x + eb, x) holds.
Thus, P ′′
1 ∩ P ′ ⊂ P ′′
2 . The relation P ′′
2 ∩ P ′ ⊂ P ′′
1 is established in the
same way. The lemma is proved.
Let A, B ∈ G+ be noncommuting (counterclockwise) rotations by an-
gles ϕ and ψ, respectively. We assume that the angles ϕ and ψ are not
multiples of 2π.
Lemma 3. Suppose the action G2[A, B] is not faithful. Then there exists
a nonzero polynomial Q in two variables with integer coefficients such that
Q(eiϕ, eiψ) = 0.
Proof. Let x0 be the fixed point of the rotation B. Let Rα denote the
rotation by an angle α around the point x0. Let T (y) denote the trans-
lation by a vector y ∈ R
2. We have B = Rψ and A = RϕT (z), where
z is a nonzero vector. Set d = FG2[A, B]. Given an element g ∈ FG2,
let (m, k) be the endpoint of the path γ(g). It is easy to observe that
d(g) = Rmϕ+kψT (y) for some y ∈ R
2. Then
d(ag) = Ad(g) = R(m+1)ϕ+kψT (y + R−mϕ−kψz),
d(a−1g) = A−1d(g) = R(m−1)ϕ+kψT (y − R−(m−1)ϕ−kψz),
d(bg) = Bd(g) = Rmϕ+(k+1)ψT (y),
d(b−1g) = B−1d(g) = Rmϕ+(k−1)ψT (y).
These relations along with the inductive argument allow us to calculate
the isometry d(g) for every g ∈ FG2. We obtain d(g) = Rm1ϕ+k1ψT (y),
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where (m1, k1) is the endpoint of the path γ(g) and
y =
∑
(m,k)∈Z2
(
nγ(g)((m, k), (m+1, k))−nγ(g)((m+1, k), (m, k))
)
R−mϕ−kψz.
Suppose the isometry d(g) is the identity. Then m1ϕ + k1ψ is a mul-
tiple of 2π and y = 0. The first condition is equivalent to the equality
ei(m1ϕ+k1ψ) = 1. Since z is a nonzero vector, the condition y = 0 is
equivalent to the equality
∑
(m,k)∈Z2
(
nγ(g)((m, k), (m+1, k))−nγ(g)((m+1, k), (m, k))
)
e−i(mϕ+kψ) = 0.
If γ(g) /∈ P ′ ∩ P ′′
1 , then the two equalities imply there exists a
nonzero polynomial Q in two variables with integer coefficients such that
Q(eiϕ, eiψ) = 0. On the other hand, if γ(g) ∈ P ′ ∩P ′′
1 , then g ∈ FG′′
2 due
to Lemmas 1 and 2.
Finally, we can guarantee that at least one of the following conditions
holds: (i) there exists a nonzero polynomial Q in two variables with inte-
ger coefficients such that Q(eiϕ, eiψ) = 0; (ii) the isometry FG2[A, B](g)
is the identity if and only if g ∈ FG′′
2. The condition (ii) means the action
G2[A, B] is faithful.
Lemma 4. There exist Fσ-sets S1, S2 ∈ R
2 such that:
(i) the section {β | (α, β) ∈ S1} is at most countable for any α ∈ R,
(ii) the section {α | (α, β) ∈ S2} is at most countable for any β ∈ R,
(iii) the action G2[A, B] is faithful whenever (ϕ, ψ) /∈ S1 ∪ S2.
Proof. Let Q be a nonzero polynomial in two variables with integer coef-
ficients. Clearly, the set Z(Q) = {(z1, z2) ∈ C
2 | Q(z1, z2) = 0} is closed.
The expression Q(z1, z2) is uniquely represented in the form
p0(z2)z
m
1 + p1(z2)z
m−1
1 + · · · + pm−1(z2)z1 + pm(z2),
where p0, p1, . . . , pm (m ≥ 0) are polynomials in one variable with inte-
ger coefficients and, moreover, p0 is a nonzero polynomial. Set P (Q) =
{(z1, z2) ∈ C
2 | p0(z2) = 0}, Z1(Q) = Z(Q) ∩ P (Q), and Z2(Q) =
Z(Q) \ Z1(Q). Since p0 is a nonzero polynomial, the set P (Q) is the
union of a finite number of parallel planes in C
2. Then the set Z1(Q)
is closed and the section {z2 | (z1, z2) ∈ Z1(Q)} is at most finite for
any z1 ∈ C. Given ε > 0, let Pε(Q) denote ε-neighborhood of the set
P (Q). Obviously, the set Z2(Q) \Pε(Q) is closed for any ε > 0, therefore
Z2(Q) is an Fσ-set. Take any z2 ∈ C. If p0(z2) 6= 0, then the section
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{z1 | (z1, z2) ∈ Z2(Q)} = {z1 | (z1, z2) ∈ Z(Q)} contains at most m
elements. If p0(z2) = 0, then the section {z1 | (z1, z2) ∈ Z2(Q)} is empty.
Set Z1 =
⋃
Q Z1(Q) and Z2 =
⋃
Q Z2(Q), where both unions are
over all nonzero polynomials in two variables with integer coefficients.
Since there are only countably many such polynomials, it follows from
the above that Z1 and Z2 are Fσ-sets. Moreover, for any z1 ∈ C the
section {z2 | (z1, z2) ∈ Z1} is at most countable, and for any z2 ∈ C the
section {z1 | (z1, z2) ∈ Z2} is at most countable.
Define a map E : R
2 → C
2 by the relation E(α, β) = (eiα, eiβ) for
any α, β ∈ R
2. Set S1 = E−1(Z1) and S2 = E−1(Z2). The map E
is continuous and the preimage E−1(z) of any point z ∈ C
2 is at most
countable. It follows that S1 and S2 are Fσ-sets satisfying conditions (i)
and (ii).
Recall that A and B are noncommuting rotations by the angles ϕ
and ψ, respectively. Suppose (ψ, ϕ) /∈ S1 ∪ S2. Then Q(eiϕ, eiψ) 6= 0 for
each nonzero polynomial Q in two variables with integer coefficients. By
Lemma 3, the action G2[A, B] is faithful. Thus condition (iii) holds.
Proof of Theorem 3
Let x0 be a point in Euclidean plane. For any α ∈ R and any y ∈ R
2,
let Rα denote the (counterclockwise) rotation by the angle α around the
point x0 and T (y) denote the translation by the vector y. Define a map
D : R×R
2 → G+ by the relation (α, y) 7→ RαT (y). The map D descends
to a map D0 : R/2πZ × R
2 → G+, which is a diffeomorphism. Let
S1, S2 ⊂ R
2 be Fσ-sets satisfying conditions (i), (ii), and (iii) of Lemma
4. We can assume without loss of generality that S1 and S2 are invariant
under translations from (2πZ)2. Set S0 = R
2 \ (S1 ∪ S2). It follows from
the conditions (i) and (ii) that S0 is a Gδ-subset of R
2 which is dense and
of full measure. Finally, let S denote the set of pairs (A, B) ∈ G2
+ such
that A = RϕT (y) and B = RψT (z), where (ϕ, ψ) ∈ S0, ϕ and ψ are not
multiples of 2π, and y 6= z. Since D0 is a diffeomorphism, it follows that
S is a dense Gδ-subset of full measure of G2
+. This means that a pair
(A, B) ∈ S is generic both in the sense of measure and of category. By
construction, A and B are nontrivial rotations that do not commute. By
Lemma 4, the action G2[A, B] is faithful.
Lemma 5. Generators of the group FG2/FG′′
2 generate a free subsemi-
group.
Proof. Let a and b be generators of the group FG2. By H denote the
semigroup generated by a and b. Suppose g1, g2 ∈ H. We have to prove
that g−1
2 g1 ∈ FG′′
2 only if g1 = g2. Let x0, x1, . . . , xk be the path γ(g1)
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and y0, y1, . . . , yn be the path γ(g2). Without loss of generality it can be
assumed that all links (xj−1, xj) and (yj−1, yj) are of the form (x, x+ ea)
or (x, x + eb). If g−1
2 g1 ∈ FG′
2, then n = k, xk = yk, and x0, x1, . . . , xk =
yk, . . . , y1, y0 is the path γ(g−1
2 g1). Obviously, nγ(g−1
2
g1)(xj−1, xj) = 1 for
any j = 1, . . . , k, while nγ(g−1
2
g1)(xj , xj−1) = 1 only if (xj−1, xj) is a link
of the path γ(g2). It follows from Lemma 1 that g−1
2 g1 ∈ FG′′
2 only if
g1 = g2.
Proof of Theorem 1
Let a and b be generators of the free group FG2. Let p : FG2 → G2 =
FG2/FG′′
2 be the natural projection. The elements p(a) and p(b) are
generators of the group G2. By Lemma 5, the semigroup generated by
p(a) and p(b) is free. It follows easily that for any A, B ∈ G+ the action
FSG2[A, B] is faithful whenever the action G2[A, B] is faithful. Thus
Theorem 1 is a corollary of Theorem 3.
Proof of Theorem 2
Let x0 be the fixed point of the rotation B. Denote by Rα the rota-
tion by an angle α around the point x0. Denote by T (y) the translation
by a vector y ∈ R
2. We have B = Rϕ and A = T (z), where z is
a nonzero vector. Let a and b be generators of the semigroup FSG2.
An arbitrary element g ∈ FSG2 can be uniquely represented in the
form bmkabmk−1a . . . bm1abm0 , where m0, m1, . . . , mk are nonnegative in-
tegers. It is easy to observe that FSG2[A, B](g) = RαgT (yg), where
αg = ϕ
∑k
j=0 mj and
yg = R−m0ϕz + R−(m0+m1)ϕz + · · · + R−(m0+m1+···+mk−1)ϕz.
Let h = blsabls−1a . . . bl1abl0 be an element of FSG2 different from g.
Suppose that FSG2[A, B](h) = FSG2[A, B](g). Then αh − αg is a mul-
tiple of 2π and yh = yg. The first condition is equivalent to the equality
ei(l0+l1+···+ls)ϕ = ei(m0+m1+···+mk)ϕ, while the second condition is equiv-
alent to the equality
e−il0ϕ + e−i(l0+l1)ϕ + · · · + e−i(l0+···+ls−1)ϕ =
e−im0ϕ + e−i(m0+m1)ϕ + · · · + e−i(m0+···+mk−1)ϕ.
Since the sequences m0, m1, . . . , mk and l0, l1, . . . , ls are different, the two
equalities imply e−iϕ is an algebraic number. Thus the action FSG2[A, B]
can be not faithful only if the number e−iϕ is algebraic.
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.Y. Vorobets 125
Now suppose e−iϕ is an algebraic number. Then there exist two dif-
ferent nondecreasing sequences m0, m1, . . . , mk and l0, l1, . . . , ls of non-
negative integers such that
e−im0ϕ + e−im1ϕ + · · · + e−imkϕ = e−il0ϕ + e−il1ϕ + · · · + e−ilsϕ.
Choose a positive integer M such that mk ≤ M and ls ≤ M . We can
observe that FSG2[A, B](g) = FSG2[A, B](h), where
g = bM−mkabmk−mk−1a . . . bm1−m0abm0 ,
h = bM−lsabls−ls−1a . . . bl1−l0abl0 .
The elements g and h of the semigroup FSG2 are different, therefore the
action FSG2[A, B] is not faithful.
It remains to observe that, given a real number α, the numbers e−iα,
sinα and cos α are either all algebraic or all transcendent.
Contact information
Y. Vorobets Pidstryhach Institute for Applied Problems
of Mechanics and Mathematics of Ukrainian
NAS, Lviv, Ukraine
E-Mail: vorobets@lviv.litech.net
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