2 Sang

Given two pseudo-Anosov homeomorphisms with distinct invariant measured foliations, some powers of their isotopy classes generate a rank two free subgroup of the mapping class group of the surface [6]. This construction gives an example of all pseudo-Anosov subgroup of the mapping class group. Whittlesey [13] gives a positive answer to the natural question of the existence of all pseudo-Anosov normal subgroups by showing that the Brunnian mapping classes on a sphere with at least five punctures are neither periodic nor reducible. In this note we show that the kernel of Burau(4)⊗Zp, the reduced Burau representation with coefficients in Zp of the 4-braid group B4, consists only of pseudo-Anosov braids. By Cooper and Long [3, 4], Burau(4) ⊗ Zp for p = 2, 3 is not faithful. It is straightforward to check that there exist non-Brunnian braids in the kernels, hence giving new examples of all pseudo-Anosov normal subgroups of B4 that are not contained in the example of Whittlesey. For the proof, assume that we are given a nontrivial 4-braid that is not pseudo-Anosov. If it is periodic, it is conjugate to a rigid rotation [2], whose Burau action is clearly non-trivial. If it is reducible, then in many ways it is similar to a 3-braid so that its Burau action is fairly predictable, for which case an automaton that records the polynomial degrees suffices to prove faithfulness. Our argument is similar to that of the ping-pong lemma. We construct an automaton whose states are disjoint subsets of Zp[t, t ] and whose arrows are braid actions that map the subsets into the subsets. For braids with more than four strands, this approach is immediately confronted by various obstacles. Since Burau(4)⊗Z2 is not faithful, the kernel of Burau(5)⊗Z2 contains reducible braids. Taking other representations or taking intersection with other subgroups to get rid of such reducible braids then makes the proof more difficult. We remark that the present result is a byproduct of working on the faithfulness question of Burau(4) [8, 9, 7, 1].