The complexity of ascendant sequences in locally nilpotent groups

We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x0; x1; : : : ; xN 2 G, N 2 N, then one can always …nd a uniformly computably enumerable (i.e. uniformly 1) ascendant sequence of order type ! + 1 of subgroups in G beginning with hx0; x1; : : : ; xN iG, the subgroup generated by x0; x1; : : : ; xN in G. This complexity is surprisingly low in light of the fact that the usual de…nition of ascendant sequence involves arbitrarily large ordinals that index sequences of subgroups de…ned via a trans…nite recursion in which each step is incomputable. We produce this surprisingly low complexity seqeunce via the e¤ective algebraic commutator collection process of P. Hall, and a related purely algebraic Normal Form Theorem of M. Hall for nilpotent groups.