  We consider the topological complexity of subgroups of Artin's braid group
consisting of braids whose associated permutations lie in some specified
subgroup of the symmetric group. We give upper and lower bounds for the
topological complexity of such mixed braid groups. In particular we show that
the topological complexity of any subgroup of the n-strand braid group which
fixes any two strands is 2n-3, extending a result of Farber and Yuzvinsky in
the pure braid case. In addition, we generalise our results to the setting of
higher topological complexity.
