The Nielsen reduction and P-complete problems in free groups

Abstract

The complexity of some classical algorithmic problems in free groups is studied. Problems like the generalized word problem, to determine shortest coset representatives, to decide whether two subgroups are equal or isomorphic, to decide whether a set of generators is independent or to decide whether a subgroup has finite index, are known to be decidable for free groups. We show that these problems are P-complete under log-space reducibility. This is proved by encoding the computations of a deterministic polynomially time bounded TM into a subgroup of a free group and implementing the Nielsen reduction, one of the main tools for solving algorithmic problems in free groups, in polynomial time.