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On Representations of Braids as Automorphisms of Free Groups and Corresponding Linear Representations

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  • Sobolev Institute of Mathematics, Russian Academy of Scinces, Russia, Novosibirsk

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ON REPRESENTATIONS OF BRAIDS AS AUTOMORPHISMS
OF FREE GROUPS AND CORRESPONDING LINEAR
REPRESENTATIONS
VALERIY G. BARDAKOV AND PAOLO BELLINGERI
Abstract. In this survey we explore relationships between several different
representations of braid groups as automorphisms of free groups as well as
induced linear representations.
1. Introduction
In his first seminal paper on braid groups [1], Artin proposed an interpretation
of the braid group Bnas a group of automorphisms of the free group Fn. This
representation has several important properties: for instance it gives an immediate
solution for the word problem, and, using Fox derivatives, one can construct Burau
representation. Actually, the relevance of Artin representation in the study of
braids, mapping class groups and knots is impressive and it motivated to look
forward for generalizations or other “geometric” representations (see for instance
[2, 5, 13–15]).
There are several other faithful representations of braids in terms of automor-
phisms of free groups: in the following we will recall in particular Perron-Vannier
representation [13], Wada representations [15] and we will propose a new repre-
sentation, that we will call Fenn-Rolfsen-Zhu representation because inspired from
[7].
We refer to [2] for a complete survey on braids seen as automorphisms of free
groups and for algebraical proofs of well known results arising from this approach:
the main aim of this note is to construct some reductions and extensions of above
mentioned representations and to provide several algebraical relations between
them. In particular we will show that they are all faithful, that extended Artin
representation is conjugated to Fenn-Rolfsen-Zhu representation (Theorem 2.3),
that reduced Fenn-Rolfsen-Zhu representation is conjugated to Artin representa-
tion (Proposition 2.5) and that extended Perron-Vannier representation is actually
a Wada representation (Theorem 4.1). At the end of Section 2 we will construct a
family of representations of Bncontaining reduced Artin representation, extended
Artin representation and Fenn-Rolfsen-Zhu representation. In Section 5 we will
2000 Mathematics Subject Classification. Primary: 20F36, 20F05, 20F10.
The research of the first author was partially supported by by Laboratory of Quantum Topology
of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020), RFBR-
14-01-91052, RFBR-13-01-00513 and Indo-Russian RFBR-13-01-92697.
The research of the second author was partially supported by French grant ANR-11-JS01-002-
01.
The authors thank Warren Dicks and Eddy Godelle for an interesting interpretation of Perron-
Vannier representation in terms of transvection automorphisms (Remark 3.4).
1
2 BARDAKOV AND BELLINGERI
provide linear representation of Bninduced by Perron-Vannier and Fenn-Rolfsen-
Zhu representations (see in particular Proposition 5.4): as a corollary we will show
that extended Perron Vannier representation is not equivalent to extended Artin
representation (Proposition 5.3).
2. Representations of Artin and Fenn-Rolfsen-Zhu
Let Fmbe the free group of rank nwith the set of free generators {x1, x2,...,
xm}. Assume also that Aut(Fm) is the automorphism group of Fm.
In the following we will show extensions and reductions of several representations
of Bninto Aut(Fm) (for some m) and we will establish relations between them; in
particular we will remark when they are conjugated.
Definition 2.1. Let n, m > 1. Two representations ρ, ρ0:BnAut(Fm)are
conjugated if there exists an automorphism χ:FmFmsuch that χρ(β)χ1=
ρ0(β)for all βBn.
We will consider also a weaker notion of equivalence for representations (see
Definition 1.4 of [5]).
Definition 2.2. Let n, m > 1. Two representations ρ, ρ0:BnAut(Fm)are
equivalent if there exist automorphisms χ:FmFmand µ:BnBnsuch that
χρ(β)χ1=ρ0(µ(β)) for all βBn.
When n=mthe first famous example of representation is the Artin representation
of Bn. This representation
ρA:BnAut(Fn),
due to Artin himself, is defined associating to any generator σi, for i= 1,2, . . . , n1,
of Bnthe following automorphism of Fn:
ρA(σi) :
xi7−xixi+1 x1
i,
xi+1 7−xi,
xl7−xl, l 6=i, i + 1.
Let us recall also that there is a geometrical interpretation of ρA: the braid group
Bnis isomorphic to the mapping class group of the n-punctured disk, that we denote
by Dn, and Artin representation therefore corresponds to the induced action of Bn
on π1(Dn) = Fn.
We will consider an extension of Artin representation: eρA:BnAut(Fn+1),
where Fn+1 =hx0, x1, . . . , xni, defining
eρA(σi) :
xi7−xixi+1 x1
i,
xi+1 7−xi,
xl7−xl, l 6=i, i + 1.
for all generators x0, x1, x2, . . . , xn.
In [7], Fenn, Rolfsen and Zhu constructed an action of Bnon particular arcs of
the n-punctured disk Dn. This construction can be described as follows.
Let P=p1, . . . , pnbe the set of punctures of Dnand let Abe an oriented arc
with endpoints in P. To Awe can associate a word in the symbols I0, I1, . . . , In,
I1
0, I1
1, . . . , I1
n. Let suppose that the punctures p1, . . . , pnon Dnare on the real
line and let s0, . . . , sn+1 be the segments on Figure 1. Assume that Ais transverse
to the real line: starting from the initial point of A, say pk, and write Imwhen
ON REPRESENTATIONS OF BRAIDS AS AUTOMORPHISMS OF FREE GROUPS 3
Acrosses the segment smwith increasing imaginary part and write I1
motherwise.
Since Bnacts on (isotopy classes of) arcs, Bnacts on the word w(A) associated to
A.1
s
n
p p p
s
s0
1 2
1
n
Figure 1. The group Bnacts on the segments s0, . . . , sn+1.
In particular the generator σiacts on any letter of w(A) (suppose that Ahas
different endpoints than iand i+ 1) as follows:
Ii7−Ii1I1
iIi+1,
I1
i7−I1
i+1 IiI1
i1,
I±1
l7−I±1
l, l 6=i.
This construction inspired us the following representation:
ρF:BnAut(Fn+1), Fn+1 =hy0, y1, . . . , yni,
which maps the generators of Bnto the following automorphisms
ρF(σi) : yi7−yi1y1
iyi+1,
yl7−yl, l 6=i.
Remark that previous representation is evidently equivalent (see also the end of
the section) to the reduced Artin representation (see [4, p. 121]) ρRA :Bn
Aut(Fn+1):
ρRA(σi) : yi7−yi+1y1
iyi1,
yj7−yj, j 6=i, i= 1,2, . . . , n 1.
One can easily check that ρRA(σi) = ρF(σ1
i). The relation between represen-
tations ρAand ρFcan be algebraically described as follows.
Theorem 2.3. The representation ρF:BnAut(Fn+1)is conjugated to the
representation eρA:BnAut(Fn+1). In particular, ρFis faithful.
Proof. We define the elements y0=x0, y1=x1
1y0=x1
1x0, y2=x1
2y1=
x1
2x1
1x0, . . . , yn=x1
nyn1=x1
nx1
n1. . . x1
1x0in Fn+1 =hx0, x1, . . . , xni. It
is evident that these elements is a basis of Fn+1 and the old basis can be express
from new by the rules
x0=y0, x1=y0y1
1, x2=y1y1
2, . . . , xn=yn1y1
n.
1We did not show that w(A) is invariant up to isotopy. In [7] it is explained how to associate
an unique word w(A) to a given arc A.
4 BARDAKOV AND BELLINGERI
Let us find the Artin representation in the new basis {y0, y1, . . . , yn}.We have 2
y
eρA(σi)
k=yk, k < i,
y
eρA(σi)
i= (x1
iyi1)
eρA(σi)=xix1
i+1x1
iyi1=yi1y1
iyi+1,
y
eρA(σi)
i+1 = (x1
i+1yi)
eρA(σi)=x1
iyi1y1
iyi+1 =yi+1,
y
eρA(σi)
l= (x1
lyl1)
eρA(σi)=yl, l > i + 1.
Hence the Artin representation in the bases {y0, y1, . . . , yn}coincides with the rep-
resentation of Fenn-Rolfsen-Zhu.
We can reformulate previous theorem saying that the representations eρAand ρF
are conjugated (see Definition 2.1).
Let ϕAF Aut(Fn+1) be the automorphism such that
ϕ1
AF eρAϕAF =ρF,
or, in other words, such that for any generator xiof Fn+1 and for any σjBnwe
get
xϕ1
AF
eρA(σj)ϕAF
i=xρF(σj)
i, i = 0,1, . . . , n, j = 1,2, . . . , n 1.
We can also determine ϕAF :
ϕAF :
x07−x0,
x17−x0x1
1,
x27−x1x1
2,
.
.
.
xn7−xn1x1
n.
Then
ϕ1
AF :
x07−x0,
x17−x1
1x0,
x27−x1
2x1
1x0,
.
.
.
xn7−x1
nx1
n1. . . x1
1x0,
and we can check the formulas
xϕ1
AF
eρA(σj)ϕAF
i=xρF(σj)
i, i = 0,1, . . . , n, j = 1,2, . . . , n 1.
We know that every automorphism in ρA(Bn) fixes the product x1x2. . . xn.
Hence every automorphism in eρA(Bn) fixes the product xk
0x1x2. . . xnfor arbitrary
integer k. For the automorphisms in ρF(Bn) we have a similar result.
Corollary 2.4. Any automorphism in ρF(Bn)fixes elements win the subgroup
hx0, xni.
Proof. It is enough to show that elements w= (xk
0x1x2. . . xn)ϕAF =xk+1
0x1
n,
where kZ, are fixed by ρF(Bn). We know that any element xk
0x1x2. . . xnis
fixed by every automorphism in eρA(Bn). Hence, if we define wby the formula
w= (xk
0x1x2. . . xn)ϕAF then
wϕ1
AF
eρA(Bn)ϕAF = (xk
0x1x2. . . xn)
eρA(Bn)ϕAF = (xk
0x1x2. . . xn)ϕAF =w.
2In the following, given ρ:BnAut(Fm), we will note by xρ(y)the action on xFmby
ρ(y).
ON REPRESENTATIONS OF BRAIDS AS AUTOMORPHISMS OF FREE GROUPS 5
Since,
ϕ1
AF eρA(Bn)ϕAF =ρF(Bn),
then
wρF(Bn)=w.
We can define a representation ρRF :BnAut(Fn) that is the composition
of ρFand the homomorphism which forgets y0, i.e.
ρRF (σ1) : y17−y1
1y2,
yj7−yj, j 6= 1,
ρRF (σi) : yi7−yi1y1
iyi+1,
yj7−yj, j 6= 1,1< i n1.
We can provide a result similar to Theorem 2.3, relating Artin representation
ρAto representation ρRF .
Proposition 2.5. The representation ρRF is conjugated to the Artin representation
ρA. In particular, the representation ρRF is faithful.
Proof. Take the new generators of Fn=hx1, x2, . . . , xni:
y1=x1
1, y2=x1
2y1=x1
2x1
1, . . . , yn=x1
nyn1=x1
n. . . x1
1.
Express the old generators
x1=y1
1, x2=y1y1
2, . . . , xn=yn1y1
n.
Then the representation ρAin the new generators has the form
ρA(σ1) : y17−y1
1y2,
yj7−yj, j 6= 1,
ρA(σi) : yi7−yi1y1
iyi+1,
yj7−yj, j 6= 1,1< i n1.
Therefore we recover representation ρRF .
Generalizing previous results, we can construct a larger family of faithful repre-
sentations that contains ρFand ρRA. For this purpose define the family of auto-
morphisms
ϕε,µ,k :
x07−xε0
0,
x17−(xk1
0xε1
1)µ1,
x27−(xk2
1xε2
2)µ2,
.
.
.
xn7−(xkn
n1xεn
n)µn,
where
ε= (ε0, ε1, . . . , εn), εi∈ {±1}, µ = (µ1, µ2, . . . , µn), µj∈ {±1},
k= (k1, k2, . . . , kn)Zn.
6 BARDAKOV AND BELLINGERI
Then the inverse automorphism to ϕε,µ,k is equal to
ϕ1
ε,µ,k :
x07−xε0
0,
x17−(xε0k1
0xµ1
1)ε1,
x27−((xε0k1
0xµ1
1)ε1k2xµ2
2)ε2,
.
.
.
xn7−((. . . ((xε0k1
0xµ1
1)ε1k2xµ2
2)ε2. . . xµn1
n1)εn1knxµn
n)εn,
and we can define a representation of Bnby the rule
ϕ1
ε,µ,k eρA(Bn)ϕε,µ,k.
In particular, if we take
ε= (1,1,1,...,1) Zn+1, µ = (1,1,...,1) Zn, k = (1,1,...,1) Zn,
then we can define
ϕ1
ε,µ,k eρA(Bn)ϕε,µ,k
which is exactly the representation ρF(Bn); if we take
ε= (1,1,...,1) Zn+1, µ = (1,1,...,1) Zn, k = (1,1,...,1) Zn,
then
ϕ1
ε,µ,k eρA(Bn)ϕε,µ,k
is the reduced Artin representation ρRA.
3. Perron-Vannier representation
Another interesting faithful representation of the braids as automorphisms of
free groups is the Perron-Vannier representation [13].
This representation becomes from the mapping which sends Bninto the mapping
class group of the surface Σ shown in Figure 2, where any generator σiof Bnis
sent into the Dehn twist τialong the curve ci. Perron-Vannier representation is
therefore given by the induced action of τi(i= 1, . . . , n 1) on π1(Σ) = Fn1(see
[6] for a detailed description of this action and for the geometrical interpretation of
Σ as a branched 2-fold cover of C).
ci
1
cn
1
c
Figure 2. The generator σiof Bnis sent into the Dehn twist τi
along the curve ci.
ON REPRESENTATIONS OF BRAIDS AS AUTOMORPHISMS OF FREE GROUPS 7
Perron-Vannier representation ρP:BnAut(Fn1) is algebraically defined as
follows:
ρP(σ1) : x1x1,
xjx1
1xj, j 6= 1,
and for 2 in1,
ρP(σi) :
xi17−xi,
xi7−xix1
i1xi,
xj7−xj, j 6=i1, i.
The faithfulness of ρPwas proven in [13] with topological arguments (see [6] for
an algebraical proof). Starting from Perron-Vannier representation in [6] was con-
structed another faithful representation ρCP :BnAut(Fn1) given algebraically
by:
ρCP (σi) :
yi17−yi1yi,
yi+1 7−y1
iyi+1
yj7−yj, j 6=i1, i + 1,
where Fn1=hy1, . . . , yn1i.
In particular, according to the previous definition, we have that:
ρCP (σ1) :
y17−y1,
y27−y1
1y2
yj7−yj, j > 2.
and
ρCP (σn1) :
yj7−yj, j < n 2,
yn27−yn2yn1,
yn17−yn1
We are interested to construct two extensions of Perron-Vannier representations
in Aut(Fn) and Aut(Fn+1).
Proposition 3.1. The following representation ρ(1)
CP :BnAut(Fn)is faithful:
ρ(1)
CP (σ1) :
y17−y1,
y27−y1
1y2
yj7−yj, j > 2.
and for i > 1:
ρ(1)
CP (σi) :
yi17−yi1yi,
yi+1 7−y1
iyi+1
yj7−yj, j 6=i1, i + 1.
where Fn=hy1, . . . , yn1, yni.
Proof. The map FnFn1which ”forgets” yninduces a projection of q:ρ(1)
CP (Bn)
ρCP (Bn) such that qρ(1)
CP =ρC P . Since ρC P :BnAut(Fn1) is faithful, also
ρ(1)
CP is injective.
8 BARDAKOV AND BELLINGERI
Proposition 3.2. The following representation ρ(2)
CP :BnAut(Fn+1)is faithful:
ρ(2)
CP (σi) :
yi17−yi1yi,
yi+1 7−y1
iyi+1
yj7−yj, j 6=i1, i + 1.
where Fn+1 =hy0, y1, . . . , yn1, yni.
Proof. We consider the map Fn+1 Fn1which ”forgets” y0and ynand we
proceed as in previous proposition
A straightforward consequence of the definition of ρ(2)
CP is the following result.
Corollary 3.3. The images of braids via ρ(2)
CP preserve the product y0· · · yn.
In Proposition 5.3 we will show that ρFand extended Perron-Vannier represen-
tation ρ(2)
CP are not equivalent.
Question [12, Question 17.14]. Let n4. Find the minimal number mfor
which there is a faithful representation BnAut(Fm). In particular, is it true
that there is no a faithful representation B4Aut(F2)?
The answer on the second part of this question is negative. For B4there is not
a faithful representation in Aut(F2) and then the minimal number for B4is m= 3.
Indeed, B4contains Z3, which is generated by σ1, σ3and (σ1σ2σ3)4. On the other
side the group Aut(F2) is an extension of Out(F2) = GL2(Z) by F2. Since GL2(Z)
and F2do not contain Z2, Aut(F2) does not contain Z3.
A (non faithful) representation B4Aut(F2) was given in [11]: this represen-
tation was extended in [10] to a (non faithful) representation B2gAut(F2g2)
arising from the action of B2gon a particular ramified double covering.
Remark 3.4. Inspired by the work in [11], Godelle proposed a representation ρG:
BnAut(Fn1), based on the notion of transvection automorphisms [8]. The
representation ρGis defined as follows:
ρG(σi) :
xi17−xixi1,
xi+1 7−xixi+1 ,
xj7−xj, j 6=i±1.
for i1 (mod 4);
ρG(σl) :
xl17−x1
lxl1,
xl+1 7−x1
lxl+1,
xj7−xj, j 6=l±1.
for l2 (mod 4);
ρG(σp) :
xp17−xp1xp,
xp+1 7−xp+1 xp,
xj7−xj, j 6=p±1.
for p3 (mod 4);
ρG(σq) :
xq17−xq1x1
q,
xq+1 7−xq+1 x1
q,
xj7−xj, j 6=q±1.
ON REPRESENTATIONS OF BRAIDS AS AUTOMORPHISMS OF FREE GROUPS 9
for q0 (mod 4),
and where Fn1=hx1, . . . , xn1i. If we change the basis of Fn1replacing xiby
yi=x1
ifor i1 (mod 4),xlby yl=xlfor l2 (mod 4),xpby yp=xpfor
p3 (mod 4) and xqby yq=x1
qfor q0 (mod 4) we obtain the Perron-Vannier
representation ρCP .
4. Local type representations
In [15] Wada introduced a family of representations of Bnin Aut(Fn) of the
following special form: any generator σiof Bnacts trivially on generators of Fn
except a pair of generators:
xσi
i=u(xi, xi+1),
xσi
i+1 =v(xi, xi+1),
xσi
j=xjj6=i, i + 1 ,
where uand vare now words in the generators xi, xi+1, with hxi, xi+1i ' F2.
Wada named them as shift type representations, but they are usually known as
representation of local type.
Wada found seven families of representations of local type (we denote by ψjthe
corresponding representation):
Type 1, ψ1:u(xi, xi+1) = xiand v(xi, xi+1 ) = xi+1;
Type 2, ψ2:u(xi, xi+1) = xi+1 and v(xi, xi+1 ) = x1
i;
Type 3, ψ3:u(xi, xi+1) = x1
i+1 and v(xi, xi+1) = x1
i;
Type 4, ψ4,h:u(xi, xi+1 ) = xh
ixi+1xh
iand v(xi, xi+1) = xi;
Type 5, ψ5:u(xi, xi+1) = xix1
i+1xiand v(xi, xi+1 ) = xi;
Type 6, ψ6:u(xi, xi+1) = xixi+1 xiand v(xi, xi+1) = x1
i;
Type 7, ψ7:u(xi, xi+1) = x2
ixi+1 and v(xi, xi+1) = x1
i+1x1
ixi+1.
Types 1–3 are obviously not faithful, while Types 4–7 are faithful ([14], see
also Remark 9.8 in [2] and June 19/2011 addenda in [3] for a useful survey on
proofs of faithfulness) and can be used to define link invariants [5,15]. The Artin
representation is a particular case of representation of local type (ρA=ψ4,1).
Wada conjectured that above families where the only local type representations,
up to two symmetries, the involution of the free group Fnsending any generator
xiinto its inverse and the involution of the braid group Bnsending any generator
σjinto its inverse: this conjecture was recently proved by Ito [9].
Actually the family of local type representations proposed by Wada is redundant:
in [14] was remarked that Type 5 and Type 6 are conjugated and in [5] (Proposition
A.1) was proved that type 5 and type 7 were equivalent, more precisely that it
exists an automorphism χ:FnFnsuch that χψ7(σi)χ1=ψ5(µ(σi)),
i= 1,2, . . . , n 1, where µ:BnBnis the involution sending σiinto σ1
i.
Similarly to this result we can prove a relation between Perron-Vannier repre-
sentations and representations of local type.
Theorem 4.1. The extended Perron-Vannier representation ρ(1)
CP defined in Propo-
sition 3.1 is equivalent to Wada representation ψ5.
Proof. Take in the group Fn=hx1, x2, . . . , xninew basis
y1=x1x1
2, y2=x2x1
3, . . . , yn1=xn1x1
n, yn=xn.
10 BARDAKOV AND BELLINGERI
We have the following action in this basis
ψ5(σ1) :
y17−y1,
y27−y1y2,
yj7−yj, j 3,
and
ψ5(σi) :
yi17−yi1y1
i,
yi+1 7−yiyi+1 ,
yj7−yj, j 6=i1, i + 1,
i= 2,3, . . . , n 1.
Define the new representation ψ
5:BnAut(Fn) by the rule
σi7−(ψ5(σi))1, i = 1,2, . . . , n 1,
then we get
ψ
5(σ1) :
y17−y1,
y27−y1
1y2,
yj7−yj, j 3,
and
ψ
5(σi) :
yi17−yi1yi,
yi+1 7−y1
iyi+1,
yj7−yj, j 6=i1, i + 1,
i= 2,3, . . . , n 1.
which is exactly Perron-Vannier representation ρ(1)
CP .
5. Linear representations, which are induced by the representations
of Fenn-Rolfsen-Zhu and Perron-Vannier
Consider the composition of homomorphisms
Bn
ρF
Aut(Fn+1 )π
GLn+1 (Z)
and denote this composition by ρF=ρFπ, i.e.
ρF:BnGLn+1 (Z).
Also, denote
ri=ρF(σi), i = 1,2, . . . , n 1.
It is easy to check that
ri=
Ii10 0
110
001 0 0
011
0 0 Ini1
, i = 1,2, . . . , n 1,
where Ikis the unit matrix of the order k.
Proposition 5.1. The image of Bnunder the homomorphism ρF:BnGLn+1 (Z)
is isomorphic to the symmetric group Sn.
Proof. It is evident that r2
i=In+1 for all i= 1,2, . . . , n 1.Also, since ρFis a
homomorphism then elements riare satisfy the braid relations. Hence
ρF(Bn) = hρF(σ1), ρF(σ2), . . . , ρF(σn1)i=hr1, r2, . . . , rn1i=Sn.
ON REPRESENTATIONS OF BRAIDS AS AUTOMORPHISMS OF FREE GROUPS 11
Recall that the extended Perron-Vannier representation ρ(2)
CP :BnAut(Fn+1 ),
was defined as follows:
ρ(2)
CP (σi) :
yi17−yi1yi,
yi+1 7−y1
iyi+1
yj7−yj, j 6=i1, i + 1.
In this section we will prove that ρ(2)
CP is not equivalent to the Artin representation
and hence it is not equivalent to the representation of Fenn-Rolfsen-Zhu. To do this
define the homomorphism
ρP:Bn
ρ(2)
CP
Aut(Fn+1 )π
GLn+1 (Z)
and find the matrix
si=ρP(σi), i = 1,2, . . . , n 1.
We see that
si=
Ii10 0
1 0 0
01 1 10
0 0 1
0 0 Ini1
.
Hence
ρP(Bn) = hs1, s2, . . . , sn1i.
The following lemma is trivial:
Lemma 5.2. For any integer kholds
sk
i=
Ii10 0
1 0 0
0k1k0
0 0 1
0 0 Ini1
, i = 1,2, . . . , n 1.
In particular, sihas infinite order.
Using this lemma we can prove
Proposition 5.3. The representations ρ(2)
CP and eρAare not equivalent.
Proof. Assume, that they are equivalent. Hence, there is an automorphism ψ
Aut(Fn+1) such that
ψ1eρAψ=ρ(2)
CP .
In particular,
ψ1eρA(σi)ψ=ρ(2)
CP (σi), i = 1,2, . . . , n 1.
This is an equality in Aut(Fn+1) and hence under the action of the homomorphism
π: Aut(Fn+1)GLn+1(Z)
it goes to an equality
π(ψ1)π(eρA(σi))π(ψ) = π(ρ(2)
CP (σi))
in GLn+1(Z).Since π(eρA(σi)) = ri, π(ρ(2)
CP (σi)) = sithen we have the equality
π(ψ1)riπ(ψ) = si
12 BARDAKOV AND BELLINGERI
for some matrix π(ψ)GLn+1(Z).But we know that r2
i=Inand sihas infinite
order. Hence this equality doesn’t hold.
We end this section with a Burau-like representation associated to ρF: more
precisely we will associate to
ρF:BnAut(Fn+1 )
a linear representation
ρF:BnGLn+1 (Z[t±
0, t±
1, . . . , t±
n]).
We will use the Magnus representation [4].
Define a ring homomorphism
τ:ZFn+1 Z[t±
0, t±
1, . . . , t±
n],
by the rule τ(yi) = ti, i = 0,1, . . . , n, and extending by linearity.
To use the Magnus representation the following equations must be true
τ(yρF(σj)
i) = τ(yi), i = 0,1, . . . , n, j = 1,2, . . . , n 1.
If i6=jthen
yρF(σj)
i=yi
and our equation is true. If i=jthen we have
τ(yρF(σj)
j) = τ(yj1y1
jyj+1) = tj1t1
jtj+1.
On the other hand τ(yj) = tj.Hence we have the system of equations
tj1t1
jtj+1 =tj, j = 1,2, . . . , n 1.
From this system we find
tj=n
qtnj
0tj
n, j = 1,2, . . . , n 1.
The linear representation ρFmaps any element from Bnto an automorphism of
free n+ 1-dimension Z[t±
0, t±
1, . . . , t±
n]-module with basis {v0, v1, . . . , vn}. A braid
βBnmaps to automorphism
ρF(β) : vi7−
n
X
j=0
τ ∂yρF(β)
i
∂yj!vj, i = 0,1, . . . , n.
It is evident that is enough define the automorphisms ρF(σk), k = 1,2, . . . , n 1.
We will write yσk
iinstead of yρF(σk)
i. Calculating the Fox derivatives.
We see that if i6=kthen yσk
i=yiand
∂yσk
i
∂yj
=0, i 6=j,
1, i =j.
If k=ithen we have
yσi
i=yi1y1
iyi+1
and ∂yσi
i
∂yj
= 0, j 6=i1, i, i + 1;
∂yσi
i
∂yi1
= 1,∂yσi
i
∂yi
=yi1y1
i,∂yσi
i
∂yi+1
=yi1y1
i.
ON REPRESENTATIONS OF BRAIDS AS AUTOMORPHISMS OF FREE GROUPS 13
Applying the map τ, we will have
ρF(σi) : vi7−vi1ti1t1
ivi+ti1t1
ivi+1,
vk7−vk, k 6=i,
for all i= 1,2, . . . , n 1.
Since tj=n
qtnj
0tj
nthen
ti1t1
i=n
qtn(i1)
0ti1
n/n
qtni
0ti
n=n
pt0/tn.
Hence, if we define t=n
pt0/tnthen we have:
Proposition 5.4. There exists a linear representation
ρF:BnGLn+1 (Z[t±1]),
which is defined as follows:
ρF(σi) : vi7−vi1tvi+tvi+1 ,
vk7−vk, k 6=i,
for all i= 1,2, . . . , n 1.
References
1. E. Artin, Theorie der Z¨opfe Abh. Math. Sem. Univ. Hamburg 4(1925), no. 1, 47–72.
2. L. Bacardit and W. Dicks, Actions of the braid group, and new algebraic proofs of results of
Dehornoy and Larue. Groups – Complexity – Cryptology 1(2009), 77–129.
3. http://mat.uab.cat/dicks/bacardit.html.
4. J. S. Birman, Braids, links and mapping class group, Princeton–Tokyo: Univ. press, 1974.
5. J. Crisp and L. Paris, Representations of the braid group by automorphisms of groups, in-
variants of links, and Garside groups. Pacific J. Math. 221 (2005), no. 1, 1–27.
6. J. Crisp and L. Paris, Artin groups of type B and D, Adv. Geom. 5(2005), 607–636.
7. R. Fenn, D. Rolfsen and J. Zhu, Centralisers in the braid group and singular braid monoid,
Enseign. Math. 42 (1996), 75–96.
8. E. Godelle, Repr´esentation par des transvections des groupes d’Artin-Tits. Groups Geom.
Dyn. 1(2007), no. 2, 111–133.
9. T. Ito, The classification of Wada-type representations of braid groups. J. Pure Appl. Algebra
217 (2013), no. 9, 1754–1763.
10. C. Kassel, On an action of the braid group B2g+2 on the free group F2g.Internat. J. Algebra
Comput. 23 (2013), no. 4, 819–831.
11. C. Kassel, C. Reutenauer, Sturmian morphisms, the braid group B4, Christoffel words and
bases of F2,. Ann. Mat. Pura Appl. 186 (2007), no. 2, 317–339.
12. The Kourovka Notebook, Unsolved Problems in Group Theory, 18th ed., Sobolev Institute of
Mathematics, Novosibirsk, 2014.
13. B. Perron and J. P. Vannier, Groupes de monodromie g´eom´etrique des singularit´es simples,
Math. Ann. 306 (1996), 231-245.
14. V. Shpilrain, Representing braids by automorphisms. Internat. J. Algebra Comput. 11 (2001),
no. 6, 773–777.
15. M. Wada, Group invariants of links. Topology 31 (1992), no. 2, 399–406.
Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk 630090,
Russia and Laboratory of Quantum Topology, Chelyabinsk State University, Brat’ev
Kashirinykh street 129, Chelyabinsk 454001, Russia.
E-mail address:bardakov@math.nsc.ru
Laboratoire de Math´
ematiques Nicolas Oresme, CNRS UMR 6139, Universit´
e de Caen
BP 5186, F-14032 Caen, France.
E-mail address:paolo.bellingeri@unicaen.fr
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