Erratum to The level four braid group (J. reine angew. Math. 735 (2018), 249–264)

This note is an erratum for the published version of our paper [2]. The arXiv has been updated with the corrections described here. As in our paper, let ρ be the symplectic representation of B n , let π 1 ⁢ ( D n ′ , p 1 ) , … , π 1 ⁢ ( D n ′ , p n ) denote the point pushing subgroups of B n , and for 1 ≤ k ≤ n set

Let Γ n ⁢ [ m ] denote Sp 2 ⁢ g ⁡ ( ℤ ) ⁢ [ m ] when n = 2 ⁢ g + 1 and ( Sp 2 ⁢ g + 2 ⁡ ( ℤ ) ⁢ [ m ] ) y → g + 1 when n = 2 ⁢ g + 2 .

Theorem 5.1 describes ρ ⁢ ( K n , k ) for n ≥ 5 . The theorem separately addresses the cases where n = 2 ⁢ g + 1 and n = 2 ⁢ g + 2 . In each case, there are two statements. The first statement is that ρ ⁢ ( K n , k ) contains Γ n ⁢ [ 4 ] and the second statement describes the quotient of ρ ⁢ ( K n , k ) by Γ n ⁢ [ 4 ] . We refer to these two statements as the containment statement and the quotient statement, respectively.

The proof of the containment statement of Theorem 5.1 is correct for k = 1 and incorrect for k ≥ 2 . What our argument for the containment statement actually shows is that each ρ ⁢ ( π 1 ⁢ ( D n ′ , p i ) ) contains Γ n ⁢ [ 4 ] and hence the argument only proves the weaker statement that

contains Γ n ⁢ [ 4 ] . Since L n , 1 = ρ ⁢ ( K n , 1 ) , the argument for the containment statement is correct for k = 1 and n ≥ 5 . For k ≥ 2 we have L n , k ⊇ ρ ⁢ ( K n , k ) , but this is not an equality in general.

It should be considered an open question as to whether the containment statement of Theorem 5.1 is correct for the case k ≥ 2 . At the end of the paper, we explain how our proof of Theorem 5.1 can be extended to the case n = 3 , in particular that ρ ⁢ ( K 3 , k ) contains Γ 3 ⁢ [ 4 ] = SL 2 ⁡ ( ℤ ) ⁢ [ 4 ] . This statement, the n = 3 version of the containment statement, is not correct. In particular, the last statement in the paper, that ρ ⁢ ( K 3 , 3 ) = Γ 3 ⁢ [ 4 ] , is not correct. In fact, ρ ⁢ ( K 3 , 3 ) has infinite index in SL 2 ⁡ ( ℤ ) . To see this, we first note that K 3 , 3 is the Brunnian subgroup of B 3 . Let Z denote the kernel of ρ : B 3 → SL 2 ⁡ ( ℤ ) . The group Z is an infinite cyclic group generated by the square of the Dehn twist about the boundary of D 3 ′ . For m ≠ 0 , no element of the coset σ 1 m ⁢ Z is Brunnian, hence no power of the matrix ρ ⁢ ( σ 1 ) lies in ρ ⁢ ( K 3 , 3 ) .

The statement and proof of the quotient statement of Theorem 5.1 are correct for k = 1 . Because of the n = 3 case, we expect that the containment statement of Theorem 5.1 is not correct for any k ≥ 2 and n ≥ 5 . If this is the case, the quotient statement does not make sense for k ≥ 2 .

As in the n = 3 case, we expect that ρ ⁢ ( K n , k ) in fact has infinite index in Γ n ⁢ [ 4 ] for n ≥ 4 and k ≥ 2 . As in the n = 3 case, the k = n version of this statement can be proven by showing that if h ∈ ker ⁡ ( ρ ) then σ 1 m ⁢ h is not Brunnian. Since ker ⁡ ( ρ ) is generated by squares of Dehn twists about curves surrounding an odd number of punctures [1], we may assume that h is such a product.

What our argument for the quotient statement of Theorem 5.1 actually shows is that the image of ρ ⁢ ( K n , k ) in Γ n ⁢ [ 2 ] / Γ n ⁢ [ 4 ] is ( ℤ / 2 ) 2 ⁢ g , ℤ / 2 , or 1, according to whether k is 1, 2, or greater. In other words, ρ ⁢ ( K n , k ) modulo ρ ⁢ ( K n , k ) ∩ Γ n ⁢ [ 4 ] is the abelian group given in the previous sentence. It is also true that L n , k / Γ n ⁢ [ 4 ] is the same abelian group. The given indices of ρ ⁢ ( K n , k ) in Γ n ⁢ [ 2 ] for k ≥ 2 are the correct indices for L n , k in Γ n ⁢ [ 2 ] .

There are two other incorrect statements in Section 3 of the published paper that we would like to point out. First, we incorrectly state that ρ ⁢ ( B n ) is the semi-direct product of a symmetric group with Γ n ⁢ [ 2 ] . In fact, ρ ⁢ ( B n ) is a non-split extension of these groups. Also, we incorrectly state that ψ : Sp 2 ⁢ g ⁡ ( ℤ / 2 ) → 𝔰 ⁢ 𝔭 2 ⁢ g ⁢ ( ℤ / 2 ) is the abelianization map for Sp 2 ⁢ g ⁡ ( ℤ / 2 ) (Sato proved that the abelianization is larger [3, Corollary 10.2]). We are grateful to David Benson and Nick Salter for these corrections.