Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups
Let φ : G → G be a group endomorphism where G is a finitely generated group of exponential growth, and denote by R(φ) the number of twisted φ-conjugacy classes. Fel’shtyn and Hill [7] conjectured that if φ is injective, then R(φ) is infinite. This conjecture is true for automorphisms of non-elem...
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2006
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| Cite this: | Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups / A. Fel’shtyn, D.L. Goncalves // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 36–48. — Бібліогр.: 22 назв. — англ. |
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| author | Fel’shtyn, A. Goncalves, D.L. |
| author_facet | Fel’shtyn, A. Goncalves, D.L. |
| citation_txt | Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups / A. Fel’shtyn, D.L. Goncalves // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 36–48. — Бібліогр.: 22 назв. — англ. |
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| description | Let φ : G → G be a group endomorphism where
G is a finitely generated group of exponential growth, and denote
by R(φ) the number of twisted φ-conjugacy classes. Fel’shtyn and
Hill [7] conjectured that if φ is injective, then R(φ) is infinite. This
conjecture is true for automorphisms of non-elementary Gromov
hyperbolic groups, see [17] and [6]. It was showed in [12] that the
conjecture does not hold in general. Nevertheless in this paper,
we show that the conjecture holds for injective homomorphisms for
the family of the Baumslag-Solitar groups B(m,n) where m 6= n
and either m or n is greater than 1, and for automorphisms for the
case m = n > 1. family of the Baumslag-Solitar groups B(m,n)
where m 6= n.
|
| first_indexed | 2025-11-25T20:56:21Z |
| format | Article |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2006). pp. 36 – 48
c© Journal “Algebra and Discrete Mathematics”
Twisted conjugacy classes of Automorphisms of
Baumslag-Solitar groups
Alexander Fel’shtyn, Daciberg L. Gonçalves
Communicated by R. I. Grigorchuk
Abstract. Let φ : G → G be a group endomorphism where
G is a finitely generated group of exponential growth, and denote
by R(φ) the number of twisted φ-conjugacy classes. Fel’shtyn and
Hill [7] conjectured that if φ is injective, then R(φ) is infinite. This
conjecture is true for automorphisms of non-elementary Gromov
hyperbolic groups, see [17] and [6]. It was showed in [12] that the
conjecture does not hold in general. Nevertheless in this paper,
we show that the conjecture holds for injective homomorphisms for
the family of the Baumslag-Solitar groups B(m,n) where m 6= n
and either m or n is greater than 1, and for automorphisms for the
case m = n > 1. family of the Baumslag-Solitar groups B(m,n)
where m 6= n.
1. Introduction
The fixed point classes of a surface homeomorphism were introduced by
J. Nielsen in [20]. Subsequently, K. Reidemeister [21] laid for any map of
compact polyhedron the algebraic foundation for what is now known as
Nielsen fixed point theory. As result of Reidemeister’s work, the twisted
conjugacy classes of a group homomorphism was introduced. It turns out
This work was initiated during the visit of the second author to Siegen University
from September 13 to September 20, 2003. The visit was partially supported by a grant
of the “Projeto temático Topologia Algébrica e Geométrica-FAPESP". The second
author would like to thank Professor U. Koschorke for making this visit possible and
for the hospitality.
2000 Mathematics Subject Classification: 20E45, 37C25, 55M20.
Key words and phrases: Reidemeister number, twisted conjugacy classes,
Baumslag-Solitar groups.
A. Fel’shtyn, D. L. Gonçalves 37
that the fixed point classes of a map can easily be identified with the con-
jugacy classes of liftings of this map to the universal covering of compact
polyhedron, and these last ones with the twisted conjugacy classes of the
homomorphism induced in the fundamental group of the polyhedron. Let
G be a finitely generated group and φ : G → G an endomorphism. Two
elements α, α′ ∈ G are said to be φ−conjugate if there exists γ ∈ G with
α′ = γαφ(γ)−1. The number of twisted φ-conjugacy classes is called the
Reidemeister number of an endomorphism φ, denoted by R(φ). If φ
is the identity map then the φ-conjugacy classes are the usual conjugacy
classes in the group G. Let X to be a connected, compact polyhedron and
f : X → X to be a continuous map. Reidemeister number R(f), which
is simply the cardinality of the set of twisted φ-conjugacy classes where
φ = f# is the induced homomorphism on the fundamental group, is rele-
vant for the study of fixed points of f in the presence of the fundamental
group. In fact the finiteness of Reidemeister number plays an important
role. See for example [22], [13], [7], [9] and the introduction of [12]. It is
proved in [8] for a wide class of groups including polycyclic and finitely
generated polynomial growth groups that the Reidemeister number of an
automorphism φ is equal to the number of finite-dimensional fixed points
of φ̂ on the unitary dual, if the Reidemeister number is finite. This theo-
rem is a natural generalization to infinite groups of the classical Burnside
theorem which says that the number of classes of irreducible represen-
tations of a finite group is equal to the number of conjugacy classes of
elements of this group.
From another side for any automorphism φ : G → G of a non-
elementary Gromov hyperbolic group G the number of twisted φ-
conjugacy classes is infinite see [17] and [6]. Furthermore, using co-Hofian
property, it was showed in [6] that if in addition G is torsion-free and
freely indecomposable then for every injective endomorphism φ, R(φ) is
infinite. This result gives supportive evidence to a conjecture of [7] which
states that R(φ) = ∞ if G is a finitely generated torsion-free group with
exponential growth.
This conjecture was showed to be false in general. In [12] was con-
structed automorphisms φ : G→ G on certain finitely generated torsion-
free exponential growth groups G that are not Gromov hyperbolic with
R(φ) <∞.
In the present paper we study this problem for a family of finitely
generated groups which have exponential growth but are not Gromov
hyperbolic. They are the Baumslag-Solitar groups. We recall its defini-
tion. They are indexed by pairs of natural numbers and have the following
38 Twisted classes of Baumslag-Solitar groups
presentation:
B(m,n) = 〈a, b : a−1bma = bn〉,m, n > 0.
It is easy to see that B(m,n) is isomorphic to B(n,m). This family has
different features from the one given in [12], which is a family of groups
which are metabelian having as kernel the group Z
n. So they contain a
subgroup isomorphic to Z+Z. In the case of Baumslag-Solitar groups this
happens if and only if m = n. For m = n = 1 the group B(1, 1) = Z + Z
does not have exponential growth. So without loss of generality we will
consider m ≤ n, and for m = n n > 1. For more about these groups see
[1, 4].
Some results in this work could be obtained using the description of
the Automorphism group of a Baumslag-Solitar group (for those, see [10]
and [19]).
Our main result is:
Theorem For any injective homomorphism of B(m,n) where m 6= n we
have that the Reidemeister number is infinite if either m > 1 or n > 1.
Also the same holds for automorphisms of B(m,m) if m > 1.
This result corresponds to the three results Theorem 3.4, Theorem
4.4 and Theorem 5.4 for the various values of m and n.
We have been informed by G. Levitt that our results can be gen-
eralized when looked at in the context of generalized Baumslag-Solitar
groups (fundamental groups of graphs of groups with all edge and vertex
groups infinite cyclic) where the basic techniques used are mostly due to
Max Forester.
The paper is organized into four sections. In section 2, we develop
some preliminaries about the Reidemeister classes of a pair of homomor-
phism between short exact sequences. In section 3 we study the case
B(1, n) for n > 1. The main result is Theorem 3.4. In section 4, we
consider the cases B(m,n) for 1 < m < n. The main result is Theorem
4.4. In section 5, we consider the cases B(n, n) for n > 1. The main
result is Theorem 5.3.
Acknowledgments. The authors would like to thank G. Levitt
for his helpful comments improving earlier version of this manuscript.
The first author would like to thank B. Bowditch, T. Januszkiewicz, M.
Kapovich and E. Troitsky for stimulating discussions and comments. The
second author would like to express his thanks to Dessislava Kochloukova
for very helpfull discussions.
The first author would like to thank the Max-Planck-Institute für Math-
ematik, Bonn for kind hospitality and support.
A. Fel’shtyn, D. L. Gonçalves 39
2. Preliminaries
In this section we recall some facts about Reidemeister classes of a pair of
homomorphism of a short exact sequences. The set of the Reidemeister
classes will be denoted by R[ , ] and the number of such classes by R( , ).
This will be used for the case where the two sequences are the same and
one of the homomorphism is the identity. The main reference for this
section is [11] and more details can be found there.
Let us consider a diagram of two short exact sequences of groups and
maps between these two sequences:
1 → H1
i1→ G1
p1
→ Q1 → 1
f ′ ↓↓ g′ f ↓↓ g f ↓↓ g
1 → H2
i2→ G2
p2
→ Q2 → 1
(2.1)
where f ′ = f |H1
, g′ = g|H1
.
We recall that the Reidemeister classes R[f1, f2] relative to homomor-
phisms f1, f2 : K → π are the equivalence classes of elements of π given
by the following relation: α ∼ f2(τ)αf1(τ)
−1 for α ∈ π and τ ∈ K.
Also the diagram (2.1) above will provide maps between sets
R[f ′, g′]
î2→ R[f, g]
p̂2
→ R[f, g]
where the last map is clearly surjective. As obvious consequence of this
fact will be used to solve some of the cases that we are going to discuss,
and will appear below as Corollary 2.2. For the remain cases we need fur-
ther information about the above sequence and we will use the Corollary
2.4.
Proposition 1.2 in [11] says
Proposition 2.1 Given the diagram (2.1) we have a short sequence of
sets
R[f ′, g′]
î2→ R[f, g]
p̂2
→ R[f, g]
where p̂2 is surjective and p̂−1
2 [1] = im (̂i2) where 1 is the identity element
of Q2.
An immediate consequence of this result is
Corollary 2.2 If R(f, g) is infinite then R(f, g) is also infinite.
Proof. Since p̂2 is surjective the result follows.
40 Twisted classes of Baumslag-Solitar groups
In order to study the injectivity of the map î2, for each element α ∈ Q2
let H2(α) = p−1
2 (α), Cα = {τ ∈ Q1|g(τ)αf(τ−1) = α} and let Rα[f ′, g′]
be the set of equivalence classes of elements of H2(α) by the equivalence
relation β ∼ g(τ)βf(τ−1) where β ∈ H2(α) and τ ∈ p−1
1 (Cα). Finally let
R[fα, gα] be the set of equivalence classes of elements of H2(α) given by
the relation β ∼ g(τ)βf(τ−1) where β ∈ H2(α) and τ ∈ G1.
Proposition 1.2 in [11] says
Proposition 2.3 Two classes of R(fα, gα) represent the same class of
R(f, g) if and only if they belong to the same orbit by the action of
Cα. Further the isotropy subgroup of this action at an element [β] is
G[β] = p1(Cβ) ⊂ Cα where β ∈ [β].
An immediate consequence of this result is
Corollary 2.4 If Cα is finite and R(fα, gα) is infinite, for some α, then
R(f, g) is also infinite. In particular this is the case if Q2 is finite.
Proof. The orbits of the action of Cα on R[fα, gα] are finite. So we have
an infinite number of orbits. The last part is clear from the first part and
the result follows.
3. The cases B(1, n)
As pointed out in the introduction, let n > 1 and B(1, n) = 〈a, b :
a−1ba = bn, n > 1〉.
Recall from [4] that the Baumslag-Solitar groups B(1, n) are finitely-
generated solvable groups which are not virtually nilpotent.These groups
have exponential growth[16] and they are not Gromov hyperbolic . Fur-
thermore, those groups are metabelian and torsion free.
Consider the homomorphisms | |a : B(1, n) −→ Z which associates
for each word w ∈ B(1, n) the sum of the exponents of a in the word. It
is easy to see that this is a well defined map into Z which is surjective.
Proposition 3.1 We have a short exact sequence
0 −→ K −→ B(1, n)
| |a
−→ Z −→ 1,
where K, the kernel of the map | |a, is the set of the elements which have
the sum of the powers of a equal to zero. Furthermore, B(1, n) = K ⋊Z
- (semidirect product).
Proof. The first part is clear. The second part follows because Z is free,
so the sequence splits.
Proposition 3.2 The kernel K coincide with N〈b〉 which is the normal-
izer of 〈b〉 in B(1, n).
A. Fel’shtyn, D. L. Gonçalves 41
Proof. We have N〈b〉 ⊂ K. But the quotient B/N〈b〉 has the following
presentation: ā−1b̄ā = b̄n, b̄ = 1. Therefore this group is isomorphic to Z
and the natural projection coincides with the map | |a under the obvious
identification of Z with B/N〈b〉. Consider the commutative diagram
0→N〈b〉→ B(1, n)→ B/N〈b〉→ 1
↓ ↓ ↓
0→K→ B(1, n)→ Z→ 1
where the last vertical map is an isomorphism. From the five Lemma the
result follows.
We recall that the groups B(1, n) are metabelian. Further B(1, n)
is isomorphic to Z[1/n] ⋊θ Z where the action of Z on Z[1/n] is given
by θ(1)(x) = x/n. To see this, first observe that the map defined by
φ(a) = (0, 1) and φ(b) = (1, 0) extends to a unique homomorphism φ :
B → Z[1/n] ⋊ Z which is clearly surjective. It suffices to show that this
homomorphism is injective. Given a word w = ar1bs1 ...artbst such that
r1 + ... + rt = 0, so a word on K, using the relation of the group this
word is equivalent to bs1/nr1 bs2/nr1+r2 ....bst−1/nr1+...+rt−1
.bst . If we apply
φ to this element and assume that it is in the kernel of φ, we obtain that
the sum of the powers s1/n
r1 + s2/n
r1+r2 + ....+ st−1/n
r1+...+rt−1 + st is
zero. But this means that w is the trivial element, hence φ restricted to
K is injective. Therefore the result follows.
Proposition 3.3 Any homomorphism φ : B(1, n) → B(1, n) is a homo-
morphism of the short exact sequence given in Proposition 3.2.
Proof. Let φ̄ be the homomorphism induced by φ on the abelianized
of B(1, n). The abelianized of B(1, n), denoted by B(1, n)ab, is iso-
morphic to Zn−1 + Z. The torsion elements of B(1, n)ab form a sub-
group isomorphic to Zn−1 which is invariant under any homomorphism.
The preimage of this subgroup under the projection to the abelianized
B(1, n) → B(1,m)ab is exactly the subgroup N(b), i.e. the elements rep-
resented by words where the sum of the powers of a is zero. So it follows
that N(b) is mapped into N(b) and the result follows.
Theorem 3.4 For any injective homomorphism of B(1, n) we have that
the Reidemeister number is infinite.
Proof. Let φ be a homomorphism. By Propositon 3.3 it is a homomor-
phism of short exact sequence. The induced map on the quotient is a
nontrivial endomomorphism of Z. If the induced homomorphism φ̄ is
the identity, by Corollary 2.2 the number of Reidemeister classes is in-
finite. Since φ is also injective, we can assume that φ̄ is multiplication
42 Twisted classes of Baumslag-Solitar groups
by k 6= 0, 1. Now we claim that there is no injective homomorphism
of B(1, n) such that the induced on the quotient is multiplication by k
with k 6= 0, 1. When we apply the homomorphism φ to the relation
a−1ba = bn, using the description of B(1, n) → Z[1/n] ⋊ Z we get:
a−kφ(b)ak = (nkφ(b), 0) = (nφ(b), 0), which implies that either n1−k = 1
or φ(b) = 0. So the result follows.
4. The case B(m,n), 1 < m < n
The groups in this section are more complicated than the ones in the
previous section. Nevertheless in order to get the results we will use a
similar procedure as the one in section 3. As pointed out in the introduc-
tion let 1 < m < n and B(m,n) = 〈a, b : a−1bma = bn〉.
Recall that such groups are non-virtually solvable.
Consider the homomorphisms | |a : B(m,n) −→ Z which associates
to each word w ∈ B(m,n) the sum of the powers of a in the word. It is
easy to see that this is a well defined map into Z which is surjective.
Proposition 4.1 We have a short exact sequence
0 −→ K −→ B(m,n) −→ Z −→ 1,
where K, the kernel of the map | |a, is the set of the elements which have
the sum of the powers of a equals to zero. Furthermore, B(m,n) = K⋊Z
- semidirect product where the action is given with respect to some fixed
section.
Proof. The first part is clear. The second part follows because Z is free so
the sequence splits. Since the kernel K is not abelian the action is defined
with respect to a specific section (see [2]) and the result follows.
Proposition 4.2 The kernel K coincides with N〈b〉 which is the normal-
izer of 〈b〉 in B(m,n).
Proof. Similar to Propositon 3.2.
Propositon 4.3 Any homomorphism φ : B(m,n) → B(m,n) is a homo-
morphism of the short exact sequence given in Proposition 4.1.
Proof. Let φ̄ be the homomorphism induced by φ on the abelianized
of B(1, n). The abelianized of B(m,n), denoted by B(m,n)ab, is iso-
morphic to Zn−m + Z. The torsion elements of B(m,n)ab form a sub-
group isomorphic to Zn−m which is invariant under any homomorphism.
The preimage of this subgroup under the projection to the abelianized
A. Fel’shtyn, D. L. Gonçalves 43
B(1, n) → B(1,m)ab is exactly the subgroup N(b), i.e. the elements rep-
resented by words where the sum of the powers of a is zero. So it follows
that N(b) is mapped into N(b) and the result follows.
In order to have a homomorphism φ of B(m,n) which has finite Rei-
demeister number, the induced map on the quotient Z must be different
from the identity by the same argument used in the proof of Theorem
3.4. So we will assume this from now on that φ is not the identity.
Now we will give a presentation of the group K. The group K is
generated by the elements gi = a−ibai i ∈ Z and they satisfy the following
relations: {1 = a−j(a−1bmab−n)aj = gm
j+1g
−n
j } for all integers j. This
presentation is a consequence of the Bass-Serre theory, see [5] Theorem
27 page 211.
Now we will prove the main result of this section. Denote by Kab the
abelianized of K.
Theorem 4.4 For any injective homomorphism of B(m,n) we have that
the Reidemeister number is infinite.
Proof. Let us consider the short exact sequence obtained from the short
exact sequence given in Proposition 4.1 by taking the quotient with the
commutators subgroup of K, i.e.
0 −→ Kab −→ B(m,n)/[K,K] −→ Z −→ 1.
So we obtain a short exact sequence where the kernelKab is abelian. From
the presentation of K we obtain a presentation of Kab given as follows:
It is generated by the elements gi i ∈ Z and they satisfy the following
relations: {1 = gm
j+1g
−n
j , gigj = gjgi} for all integers i, j. This is the
same as the quotient of the free abelian group generated by the elements
gi, i ∈ Z(so the direct sum of Z ′s indexed by Z), module the subgroup
generated by the relations {1 = gm
j+1g
−n
j }. So an element can be regarded
as an equivalence classe of a sequence of integers indexed by Z, where
the elements of the sequence are zero but for finite number. By abuse of
notation we denote the induced homomorphism on B(m,n)/[K,K] also
by φ. As in Theorem 3.5, we apply the injective homomorphism φ to
the relation a−1bma = bn regarded as elements of B(m,n)/[K,K] and we
obtain φ(a−1(b)ma) = φ(bn) = φ(b)n. Since the original homomorphism
is injective it follows that the induced homomorphisms on the quotient
and on K are also injective. But this implies that the homomorphism on
Kab is also injective and it follows that φ is injective on B/[K,K]. As in
Theorem 3.5, we can assume that φ(a) = akθ for θ ∈ Kab and k 6= 0, 1,
otherwise either the map would not be injective or it follows immediatly
that the Reidemeister number is infinite, respectively. Since the kernel
44 Twisted classes of Baumslag-Solitar groups
of the extension is abelian the equation becomes θ−1a−kφ((b)m)akθ =
a−kφ((b)m)ak = φ(bn) = nφ(b). The element φ(b) is the equivalence class
of a sequence of integers indexed in Z (this comes from the presentation of
Kab) where only a finite number of elements of the sequence is different
from zero. So let φ(b) = (ni1 , ..., nir) where i1 < i2 < ... < ir and
nis 6= 0 for all s = 1, ..., r, and let t = ir − i1. In the group Kab we
have identifications and ξn
l = ξm
l−1 (this follows from the presentation of
the group). So if we take the power bmnt+1
, then φ(bmnt+1
) = φ(bn
t
)mn.
From the relations it follows that φ(bn
t
) = φ(b)nt
= (ntni1 , ..., n
tnir) = b0
for some b0 ∈ Zi1 . Now consider the equation
a−kφ((bn
t
)mk
)ak = φ(bn
t+k
) = (φ(bn
t
))nk
= (b0)
nk
.
But
φ(a−1(bn
t
)mk
)a) = φ((bn
t+1
)mk−1
) =
= (φ(bnt))nmk−1
= (b0)
nmk−1
.
So we have either nk = nmk−1, which implies m = n, or b0 is the trivial
element. Since neither cases can happen, the result follows.
5. The case B(n, n), n > 1
As it was point out before if n = 1 the group is Z + Z. Then we have
even an automorphism that has a finite number of Reidemeister classes.
For n > 1 we describe the groups B(n, n) as certain extensions in order
to study the automorphims. At the end we show that any automorphism
have infinite Reidemeister number.
These groups, in contrast with the other Baumslag-Solitar groups
already considered, have subgroups isomorphic to Z + Z. Let us point
out that for n = 2 this is not the fundamental group of the Klein bottle.
There is a surjection from B(2, 2) into the fundamental group of the Klein
bottle.
We start by describing these groups. Let | |b : B(n, n) → Z be the ho-
momorphism which associates to a word the sum of the powers of b which
appears in the word. This is a well defined surjective homomorphism and
we have:
Proposition 5.1 There is a splitting short exact sequence:
0 → F → B(n, n) → Z → 1,
where F is the free group in n generators x1, ..., xn and the last map is
| |b. Further, the action of the generator 1 ∈ Z is the automorphism of
F which sends xj to xj+1 for j < n, and xn to x1.
A. Fel’shtyn, D. L. Gonçalves 45
Proof. Let F⋊Z be the semi-direct product of F by Z where F is the free
group on the set x1, ..., xn and the action is given by the automorphism
of F which maps xj to xj+1 for j < n and xn to x1. We will show that
B(n, n) is isomorphic to F ⋊ Z. For, consider the map ψ : {a, b} →
F ⋊ Z, which sends a to x1 and b to 1 ∈ Z. This map extends to a
homomorphism B(n, n) → F ⋊ Z, which we also denote by ψ, since the
relation which defines the group B(n, n) is preserved by the map. Also ψ
is a homomorphism of short exact sequences. The map restricted to the
kernel of | |b is surjective to the free group F . Also the kernel admits a
set of generators with cardinality n. So the map restrict to the kernel is
an isomorphism and the result follows.
Proposition 5.2 The center of B(n, n) is the subgroup generated by bn
and any injective homomorphism φ : B(n, n) → B(n, n) leaves the center
invariant.
Proof. For the first part, from Proposition 5.1 we know that B(n, n) is of
the form F ⋊ Z. Let (w, br) ∈ F ⋊ Z be in the center and (v, 1) ∈ F ⋊ Z
where v is an arbitrary element of F . We have (w, br)(v, 1) = (w.br(v), br)
and (v, 1)(w, br) = (v.w, br). We can assume that w is a word written
in the reduced form which starts with xni
i for some 1 ≤ i ≤ n. Let r0
be the integer 0 ≤ r0 ≤ n − 1 congruent to n. Now we consider several
cases. I- Let r0 = 0. Then take v = xi+1 if i < n or v = x1 if i = n.
We claim that w.br(v) 6= v.w, so the elements do not commute. To see
that they do not commute observe first that v.w is in the reduced form.
If w.br(v) is not reduced they can not be equal. If it is reduced, also they
can not be equal since they start with different letters. The argument
above does not work if w = 1, but this is the case where the element is
in the center. II- Let r0 6= 0 and w 6= 1. Then take v = xni
i . Again v.w is
in the reduced form which stars with x2ni
i . If w.br(v) is not reduced they
can not be equal. If it is reduced, also they can not be equal since they
start with different power of xi, even if the word contains only one letter
since br(v) is not a power of xi(r is not congruent to 0 mod n). III- Let
r0 6= 0 and w = 1. Then r = kn+ r0 and from the relation of the group
follows a−1bra = a−1bkn+r0ra = bkna−1br0a. But a−1br0a = br0 implies
br0ab−r0 = a which in terms of the notation of the Propositon 5.1 means
x1 = xr0
which is a contradition. So the result follows.
For the second part we have to show that if an element commutes
with im(φ) then it is of the form bkn for some k 6= 0. Since φ(bn) has
the property which commutes with im(φ), the result will follows from the
claim above. First observe that the image of φ is a non-abelian subgroup
and φ(bn) is of the form (w, bk1n) where k1 is possible zero. It is not
46 Twisted classes of Baumslag-Solitar groups
hard to see that for an element of the form (w, bk1n) where w 6= 1 its
centrelizer is Z + Z, namely genereted by bn and the highest root of w
in F . So there is an element in the image which doesn’t commute with
(w, bk1n) unless w = 1. But in this case k1 has to be different from zero
since the homomorphism is injective and the result follows.
Now we consider the group which is the quotient of F⋊Z by the center,
where the center is the subgroup < bn >. This quotient is isomorphic to
F ⋊ Zn where we denote the image of the generator b in Z by b̄ in Zn.
Propositon 5.3 Any homomorphism of the group F ⋊Zn such that the
restriction to F is an automorphism has infinite Reidemeister number.
Proof. We know that F is the free group in the letters x1, ..., xn and let
θ : F ⋊Z → Zn be the homomorphism defined by θ(xi) = 1 and θ(b̄) = 0.
The kernel of this homomorphism defines a subgroup of F ⋊ Z of index
n which is isomorphic to F ′
⋊ Zn where F ′ is the kernel of the homomor-
phism θ restricted to F . Now we claim that F ′ is invariant with respect to
any homorphism, i.e. F ′ is characteristic. Let (w, 1̄) be an arbitrary el-
ement of the subgroup F ′ with w 6= 1. First observe that θ(φ(xi)) =
θ(φ(x1)) for all i. This folows by induction. Since xi+1 = b̄.xi.b̄
−1
follows θ(φ(xi+1)) = θ(φ(b̄)).θ(φ(xi)).θ(φ(b̄−1)) = θ(φ(xi)). Therefore
θ(φ(w, 1̄)) = θ((w, 1̄))θ(φ(x1)) and follows that the subgroup is invari-
ant. Based on this the given homomorphism φ provides a map of the
short exact sequence
0 → F ′ → F ⋊ Zn → Zn + Zn → 0
where the restriction to the kernel is an automorphism of a free group of
finite rank. So by the Corollary 2.4 the result follows.
Now we proof the main result.
Theorem 5.4 Any automorphism φ of B(m,n) has infinite Reidemeister
number.
Proof. Given an automorphism from Propositon 5.2 it induces a surjec-
tive homomorphism on F⋊Zn, which we denote by φ̄. Let (w, 1̄) ∈ F⋊Zn
be an element of the kernel. So the image of the element (w, 1) by φ be-
longs nZ. Any multiple of (w, 1̄) also belongs to the kernel. Since φ is
injective, the restriction to < bn > is injective so the image is a non-trivial
subgroup. So some non-trivial multiple of φ(w, 1) belongs to the image
of φ′, where φ′ is the restriction of φ to the subgroup nZ. This contradict
the fact that φ is injective unless the multiple of (w, 1) is trivial. Since
F is torsion free follows that (w, 1) must be zero and we conclude that
A. Fel’shtyn, D. L. Gonçalves 47
the restriction of φ̄ to F is injective. Now from Proposition 5.3 the result
follows.
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Contact information
A. Fel’shtyn Instytut Matematyki, Uniwersytet Szczecin-
ski, ul. Wielkopolska 15, 70-451 Szczecin,
Poland and Boise State University, 1910
University Drive, Boise, Idaho, 83725-155,
USA
E-Mail: felshtyn@diamond.boisestate.edu,
felshtyn@mpim-bonn.mpg.de
D. L. Gonçalves Dept. de Matemática - IME - USP, Caixa
Postal 66.281 - CEP 05311-970, São Paulo -
SP, Brasil
E-Mail: dlgoncal@ime.usp.br
Received by the editors: 30.01.2006
and in final form 24.11.2006.
|
| id | nasplib_isofts_kiev_ua-123456789-157372 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-25T20:56:21Z |
| publishDate | 2006 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Fel’shtyn, A. Goncalves, D.L. 2019-06-20T03:07:55Z 2019-06-20T03:07:55Z 2006 Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups / A. Fel’shtyn, D.L. Goncalves // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 3. — С. 36–48. — Бібліогр.: 22 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20E45, 37C25, 55M20. https://nasplib.isofts.kiev.ua/handle/123456789/157372 Let φ : G → G be a group endomorphism where G is a finitely generated group of exponential growth, and denote by R(φ) the number of twisted φ-conjugacy classes. Fel’shtyn and Hill [7] conjectured that if φ is injective, then R(φ) is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups, see [17] and [6]. It was showed in [12] that the conjecture does not hold in general. Nevertheless in this paper, we show that the conjecture holds for injective homomorphisms for the family of the Baumslag-Solitar groups B(m,n) where m 6= n and either m or n is greater than 1, and for automorphisms for the case m = n > 1. family of the Baumslag-Solitar groups B(m,n) where m 6= n. This work was initiated during the visit of the second author to Siegen University from September 13 to September 20, 2003. The visit was partially supported by a grant of the “Projeto tem´atico Topologia Alg´ebrica e Geom´etrica-FAPESP". The second author would like to thank Professor U. Koschorke for making this visit possible and for the hospitality. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups Article published earlier |
| spellingShingle | Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups Fel’shtyn, A. Goncalves, D.L. |
| title | Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups |
| title_full | Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups |
| title_fullStr | Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups |
| title_full_unstemmed | Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups |
| title_short | Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups |
| title_sort | twisted conjugacy classes of automorphisms of baumslag-solitar groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/157372 |
| work_keys_str_mv | AT felshtyna twistedconjugacyclassesofautomorphismsofbaumslagsolitargroups AT goncalvesdl twistedconjugacyclassesofautomorphismsofbaumslagsolitargroups |