UNIQUENESS OF ROOTS UP TO CONJUGACY FOR SOME ...

4 EON-KYUNG LEE AND SANG-JIN LEE
α 1 = σ 2 1 β 1 = σ 2 2 α 2 = σ 2 1σ 4 2 β 2 = σ 2 2σ 4 1
Figure 3. α i , β i ∈ B n,1 are conjugate in B n but not in B n,1 .
Example 1.7. Consider the 1-pure 3-braids which are depicted in Figure 3:
{
{
α 1 = σ1,
2 α 2 = σ 2
β 1 = σ2,
2 and
1σ2,
4
β 2 = σ2σ 2 1.
4
Because ∆α i ∆ −1 = β i for i = 1, 2, where ∆ = σ 1 σ 2 σ 1 , the braid α i is conjugate to β i in B 3 .
However, α i is not conjugate to β i in B 3,1 for i = 1, 2 because lk(α 1 ) = lk(α 2 ) = 1, lk(β 1 ) = 0 and
lk(β 2 ) = 2. Note that α 1 and β 1 are reducible, and that α 2 and β 2 are pseudo-Anosov.
2. Preliminaries
Here, we review basic definitions and results on braids. See [Art25, Bir74, Thu88, FLP79, LL08].
Let D 2 = {z ∈ C : |z| ≤ n + 1}, and let D n be the n-punctured disk D 2 \ {1, 2, . . . , n}. The Artin
braid group B n is the group of automorphisms of D n that fix the boundary pointwise, modulo
isotopy relative to the boundary. Geometrically, an n-braid can be interpreted as an isotopy class
of the collections of pairwise disjoint n strands l = l 1 ∪· · ·∪l n ⊂ D 2 ×[0, 1] such that l ∩(D 2 ×{t})
consists of n points for each t ∈ [0, 1], and, in particular, it is {(1, t), . . . , (n, t)} for t ∈ {0, 1}. The
admissible isotopies lie in the interior of D 2 × [0, 1]. The center of the n-braid group B n is infinite
cyclic generated by ∆ 2 , where ∆ = σ 1 (σ 2 σ 1 ) · · · (σ n−1 · · · σ 1 ).
The well-known Nielsen-Thurston classification of mapping classes of punctured surfaces into
periodic, reducible and pseudo-Anosov ones [Thu88, FLP79] yields an analogous classification of
braids: an n-braid α is periodic if some power of α is central; α is reducible if there exists an
essential curve system in D n which is invariant up to isotopy under the action of α; α is pseudo-
Anosov if no non-trivial power of α is reducible.
Lemma 2.1. Let α, β ∈ B n be such that α k = β k for a nonzero integer k. Then
(i) α and β are of the same Nielsen-Thurston type;
(ii) if α is pseudo-Anosov, then α = β.
Proof. (i) is well known. (ii) was proved by González-Meneses [Gon03].
□
2.1. Periodic braids. Let δ = σ n−1 · · · σ 1 and ɛ = δσ 1 , then δ n = ∆ 2 = ɛ n−1 . (If we need to
specify the number of strands, we will write δ = δ (n) , ɛ = ɛ (n) and ∆ = ∆ (n) .) Note that δ and ɛ
are represented by rigid rotations of the n-punctured disk as in Figure 4 when the punctures are
at the center of the disk or on a round circle centered at the origin. By Brouwer, Kerékjártó and
Eilenberg, it is known that an n-braid α is periodic if and only if it is conjugate to a power of
either δ or ɛ [Bro19, Ker19, Eil34, BDM02].
Lemma 2.2. An n-braid α is periodic if and only if α is conjugate to either δ m or ɛ m for some
integer m. Further, if α is periodic and non-central, then exactly one of the following holds.
(i) α is conjugate to δ m for some m ≢ 0 (mod n). In this case, α has no pure strand.
(ii) α is conjugate to ɛ m for some m ≢ 0 (mod n − 1). In this case, α has only one pure
strand.