On the complexity of intersection and conjugacy problems in free groups

• Published in: Theoretical computer science Vol. 32; no. 3; pp. 279 - 295
• Main Authors: Avenhaus, J., Madlener, K.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 1984
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/0304-3975(84)90046-X

Nielsen type arguments have been used to prove some problems in free group (e.g., the generalized word problem) [2] to be P-complete. In this paper we extend this approach. Having a Nielsen reduced set of generators for subgroups H and K one can solve a lot of intersection and conjugacy problems in polynomial time in a uniform way. We study the solvability of (i) ∃ h ∈ H, k ∈ K: hx = yk in F, and (ii) ∃ w ∈ F: w ̄ l Hw = K characterize the set of solutions. This leads for (i) to an algorithm for computing a set of generators for H ∩ K (and a new proof that free groups have the Howson property). For (ii) this gives a fast solution of Moldavanskii's conjugacy problem; an algorithm for computing the normal hull of H then gives a representation of all solutions. All the algorithms run in polynomial time and the decision problems are proved to be P-complete under log-space reducibility.