The Computational Complexity of Torsion-freeness of Finitely Presented Groups

We determine the complexity of torsion-freeness of nitely presented groups in Kleene's arithmetical hierarchy as 0 2 -complete. This implies in particular that there is no e ective listing of all torsion-free nitely presented groups, or of all non-torsion-free nitely presented groups. 0. Introduction. One way of describing a group G is to give its presentation, i.e., to write G as G = hx i (i 2 I) j Ri (where fx i j i 2 Ig is a set of \generators" and R (the set of \relators") is a set of words in fx i ; x 1 i j i 2 Ig such that G = F=H where F is the free group generated by fx i j i 2 Ig and H is the normal subgroup of F generated by R. If we can nd a free group F of nite rank and a nite set of relators R, then we call G a nitely presented group. Groups arising in applications, such as fundamental groups in topology, often are given naturally via their presentations. Unfortunately, a nite presentation does not yield very good information about the group. Novikov [No55] and Boone [Bo54-57] showed that in some nitely presented groups, one cannot even tell whether a particular word in x 1 ; : : : ; x n and their inverses is the identity in G. (Such groups are said to have unsolvable word problem.) Further work of Baumslag, Boone, and Neumann [BBN59] revealed that many other properties of elements of G also cannot be determined from words denoting the elements. 1991 Mathematics Subject Classi cation. 03D40, 20F10.