By Claude Godbillon

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3. Theorem. Let G = (X / R) where R is cyclically reduced. (a) If R is not a proper power in the free group F(X), then G is torsion-free. 7. Theorem. Any soluble subgroup of a one-relator group G is either locally cyclic or metabelian of the form (a, b I aba-’ = bm). ,c~ ). By the inductive hypothesis the problems of membership in the subgroups (ba, bl , cj : j E Z) and : j E Z) are solvable and hence the word problem for G is solv(Wwj able. Of course more must be squeezed out to continue the induction but the normal form for HNN-extensions is sufficiently powerful to obtain the desired information.

The importance of this group in topology is unquestioned and Dehn’s solutions of the word and conjugacy problems can be regarded as the coming of age of combinatorial group theory. It was also Dehn who suggested that significant results generalising those for surface groups could be obtained for arbitrary groups given by a single defining relator. The theory begins with two classic results of Magnus [Magnus 1930, 19311. 18. Let o E Aut F and let H = Fix(a). Let X be the coset graph of H - then by the theory of coverings H ” rr(X).

Similar arguments apply except for the cases g = 0, m < 3; g = 1, m 5 1. For g = 1 = m we pass to a quotient group by introducing the relations uy, sit: and obtain the presentation (tl, u1 1 tThl, UT,t~lult~lu~l) of the dihedral group &hl of order 2h1 where it is trivial to check that no proper subword of a defining relation is a relation. For g = 0, m < 3 the groups with m 5 2 are excluded (the groups are finite cyclic groups). 12 to obtain s” # 1 if k$Omodhi. 11. We will not do this here, but will use geometric arguments instead.