On a conjecture of Papakyriakopoulos

We disprove a conjecture of Swarup which in turn disproves a well-known conjecture of Papakyriakopoulos that a certain cover is planar. Let andwhere n is a fixed integer > 2 and r is an element of the commutator subgroup of the free group F({a l9 b v . . ., a n , b n }).Further let S n be the orientable closed surface of genus n.The fundamental group of S n is K n .Papakyriakopoulos [3] put forward the following P.l.CONJECTURE, (a) J n is torsion free and (b) the cover of S n corresponding to the kernel of the natural group homomorphism K n -• J n is planar.Papakyriakopoulos [3] showed that if P.l. is true, then so is the Poincaré Conjecture.G. A. Swarup [5] has posed the following P.2.CONJECTURE.The group J n is a nontrivial free product.G. A. Swarup [5] showed that P.l.=> P.2.=> Poincaré Conjecture.THEOREM.The conjecture P.2. is not in general true.Hence the conjecture P. 1. is not in general true.PROOF.Let G x = {a v b l \ (a l9 b t c)) 9 where c is any fixed element of the commutator subgroup F({a l9 b 1 })' of the free group F({a 1 ,b 1 }) so that (a l9 b x c) is not conjugate to (a l9 b x ) ±l in F({a l9 b x }).For example one could take c= (a v b x ).