arXiv:0708.2628v2 [math.GR] 16 Dec 2007
.
TWISTED CONJUGACY CLASSES IN SYMPLECTIC
GROUPS, MAPPING CLASS GROUPS AND BRAID
GROUPS
(INCLUDING AN APPENDIX WRITTEN WITH
FRANCOIS DAHMANI)
ALEXANDER FEL’SHTYN AND DACIBERG L. GONC¸ ALVES
Abstract. We prove that the symplectic group Sp(2n, Z) and the map-
ping class group Mod
S
of a compact surface S satisfy the R
∞
property.
We also show that B
n
(S), the full braid group on n-strings of a surface
S, satisfies the R
∞
property in the cases where S is either the compact
disk D, or the sphere S
2
. This means that for any automorphism φ of G,
where G is one of the above groups, the number of twisted φ-conjugacy
classes is infinite.
Contents
1. Introduction 2
2. Preliminaries 3
3. Automorphisms of the symplectic group 6
4. Automorphisms of the m apping class group 12
5. Automorphisms of the braid groups of S
2
and disk D 14
6. Appendix: Geometric group theory and R
∞
property for
mapping class group 15
References 18
Date: February 1, 2008.
2000 Mathematics Subject Classification. 20E45;37C25; 55M20.
Key words and phrases. Reidemeister number, twisted conjugacy classes, Braids group,
Mapping class group, Symplectic group.
This work was initiated d uring our visit to the University of British Columbia, Van-
couver in April 2007 and the visit of the fi rst author to the Universit´e Paul Sabatier,
Toulouse in May-June 2007 and was completed during conferences in Bendlewo and War-
saw in July-August 2007.
1
2 ALEXANDER FEL’SHTYN AND DACIBERG L. GONC¸ ALVES
1. Introduction
Let φ : G → G be an automorphism of a group G. A class of equiv-
alence relation defined by x ∼ gxφ(g
−1
) is called the Reidemeister class ,
φ-conjugacy class, or twisted conjugacy class of φ. The number R(φ) of
Reidemeister classes is called the Reidemeister number of φ. The interest
in twisted conjugacy relations has its origins, in particular, in the Nielsen-
Reidemeister fixed point theory (see, e.g. [43, 35, 12]), in S elberg theory
(see, eg. [46, 1]), and in Algebraic Geometry (see, e.g. [30]).
A current significant problem in this area is to obtain a twisted analogue
of the Burnside-Froben ius theorem [16, 12, 19, 20, 49, 18, 17], that is, to show
the coincidence of th e Reidemeister number of φ and the number of fixed
points of the induced homeomorphism of an appropriate dual object. One
step in this process is to describe the class of groups G f or which R(φ) = ∞
for any automorphism φ : G → G.
The work of discovering which groups belong to this class of groups was
begun by Fel’shtyn and Hill in [16]. It was later shown by various authors
that the following groups belong to this class: (1) non-elementary Gro-
mov hyperbolic groups [13, 39]; (2) Baumslag-Solitar groups BS(m, n) =
ha, b|ba
m
b
−1
= a
n
i except f or BS(1, 1) [14]; (3) generalized Baumslag-Solitar
groups, that is, finitely generated groups which act on a tree with all edge
and vertex stabilizers infinite cyclic [38]; (4) lamplighter groups Z
n
≀ Z if
and only if 2|n or 3|n [27]; (5) the solvable generalization Γ of BS(1, n)
given by the short exact sequence 1 → Z[
1
n
] → Γ → Z
k
→ 1 as well as any
group quasi-isometric to Γ [47], such groups are quasi-isometric to BS(1, n)
[48] ( note however th at the class of groups for which R(φ) = ∞ for any
automorphism φ is not closed under quasi-isometry); (6) saturated weakly
branch groups, including the Grigorchuk group and the Gupta-Sidki group
[21]; (7)The R. Thompson group F [4].
The paper [47] suggests a terminology for this property, which we would
like to follow . Namely, a group G has property R
∞
if all of its automor-
phisms φ satisfy R(φ) = ∞.
For the immediate consequences of the R
∞
property in topological fixed
point theory see, e.g., [48].
TWISTED CONJUGACY CLASSES 3
In the present paper we prove that the symplectic group Sp(2n, Z) and the
mapping class group M od
S
of a compact surface S have the R
∞
property.
We also show that B
n
(S), the full braid group on n-strings of a compact
surface S, satisfy the R
∞
property in the cases where S is either the compact
disk D, or the sp here S
2
. The resu lts of the present paper indicate that
the further study of Reidemeister theory for these groups should go along
the lines similar to those of the infinite case. On the other hand, th ese
results redu ces the class of groups for which the twisted Burnside-Frobenius
conjecture [16, 19, 20 , 49, 18, 17] has yet to be verified.
The paper is organized into 6 sections. In section 2 we describe a very
naive procedure to decide whether a group has the R
∞
property. Also we
recall some known relations between mapping class groups and braid groups
which will be u s ed later. In section 3 we show that the sy mplectic group has
the R
∞
property. In section 4 we give algebraic proof of the R
∞
property
for the Mappin g class group of the closed surfaces. There we use the short
exact sequence
1 → I
n
→ Mod
S
→ Sp(2n, Z) → 1,
where I
n
is the Torelli group. In section 5 we show that, with a few ex-
ceptions, the br aids groups of the disk D and the s phere S
2
have the R
∞
property.
In Appendix, written with Francois Dahmani, we use geometric methods
to show that the mapping class group of the compact su rfaces has the R
∞
property with few obvious exceptions.
Acknowledgments The first author would like to thank V. Guirard el,
J. Guaschi, M. Kapovich and J.D. McCarthy for stimulating discussions and
comments. He also thanks the Universit´e Paul Sabatier, Toulouse for its kind
hospitality and support while the part of this work has been completed.
2. Preliminaries
Consider a group extension respecting the homomorphism φ:
4 ALEXANDER FEL’SHTYN AND DACIBERG L. GONC¸ ALVES
(2.1)
0
//
H
i
//
φ
′
G
p
//
φ
G/H
φ
//
0
0
//
H
i
//
G
p
//
G/H
//
0,
where H is a normal subgroup of G. First, let us remark that the Reide-
meister classes of φ in G are m ap ped epimorphically on classes of
φ in G/H.
Indeed,
(2.2) p(eg)p(g)
φ(p(eg
−1
)) = p(eggφ(eg
−1
).
Moreover, if R(φ) < ∞, then the previous remark implies R(φ) < ∞.
We next give a criterion for a group to have the R
∞
property.
Lemma 2.1 Suppose that a group G has an infinite number of conjugacy
classes. Then any inner au tomorphism also has infinite Reidemeister num-
ber. Moreover, if {φ
j
}
j∈Out(G)
is a subset of Aut(G) which contains single
representatives for each coset Aut(G)/Inn(G) and R(φ
j
) = ∞ for all j, then
G has the R
∞
property.
Proof. The first p art of the statement was proved in [4]. The second part
follows easily using a similar argument which we now recall. Let φ be an
automorphism and θ an element of th e group. Then we have two auto-
morphisms, namely φ and the composite of φ with the inner automorphism
which is conjugation by θ, which we denote by θ ◦ φ. We claim that mul-
tiplication by θ
−1
on the right provides a bijection between the set of Rei-
demeister classes of φ and those of θ ◦ φ. Consider two elements where
the firs t is denoted by α and the second is of the form βαφ(β)
−1
for some
β ∈ G. We claim that the two elements αθ
−1
and the βαφ(β)
−1
θ
−1
are in
the same θ ◦ φ Reidemeister classes. To show this, we write class of αθ
−1
as
βαθ
−1
((θ ◦ φ(β))
−1
) = βαθ
−1
θφ(β)
−1
θ
−1
= βαφ(β)
−1
θ
−1
. Thus αθ
−1
and
βαφ(β)
−1
θ
−1
are in the same Reidemeister class of θ ◦φ. Similarly, multipli-
cation by θ on the right provides a bijection between the set of Reidemeister
classes of θ ◦ φ and those of φ. One correspondence is the inverse of the
other and the result follow s.
TWISTED CONJUGACY CLASSES 5
Lemma 2.2 Let φ ∈ Aut(G) and Q
i
be an infinite f amily of quotients of
G,
1 → K
i
→ G → Q
i
→ 1,
such that φ(K
i
) ⊂ K
i
. If the sequence of numbers R(φ
i
) is unbounded,
where φ
i
is the induced h omomorphism on the quotient Q
i
, then R(φ) = ∞.
Proof. Since p : G → Q
i
is surjective, we have R(φ) ≥ R(φ
i
) . If R(φ) is
finite then the sequence R(φ
i
) is bounded. Hence the result follows.
Despite the fact that the Lemma 2.2 is quite obvious, it is usefull for its
ability to find such quotients and to estimate Reidemeister number on the
quotient. We will apply Lemma 2.2 in a situation wh ere the quotients are
finite groups.
We will now state a result which relates braid groups and mapping class
groups. We will use this relation to study the braid group of the sphere.
Following s ection 2 of [45], let S be a 2-manifold with n distinguished points
in its interior. Let H(S, n) denotes the space of homeomorphisms of S, which
fix pointwise the n distinguished points. If n = 0, we set H(S) = H(S, 0).
We define G(S, n) = π
0
(H(S, n)) and, if n = 0, we set G(S) = G(S, 0).
If S is the sphere S
2
we have (see the bottom of the page 615 in [45]):
Theorem 2.3 If S is S
2
then we have the short exact sequence
1 → Z
2
→ B
r
(S
2
) → G(S
2
r
) → G(S
2
) → 1
where S
2
r
= S
2
− r open disks.
It is well known that G(S
2
) = Z
2
, where the nontrivial element is rep-
resented by the isotopy class of an orientation reversing homeomorphism of
the sphere. So the preimage of the element id ∈ G(S
2
) with respect to the
projection G(S
2
r
) → G(S
2
) gives the mappping class group Mod
S
2
r
(see sec-
tion 4 for the definition of Mod
S
). The above sequence implies immed iately
the following short exact sequence:
1 → Z
2
→ B
r
(S
2
) → M od
S
2
r
→ 1.
This sequence will used in section 5.