A short proof of a theorem of Brodski

James Howie August 3, 2000 Abstra t A short proof, using graphs and groupoids, is given of Brodski 's Theorem that torsion-free one-relator groups are lo ally indi able 1 Introdu tion In 1980, Sergei Brodski announ ed [2℄ the result, previously onje tured by Gilbert Baumslag [1℄, that every torsion-free one-relator group is lo ally indi able, that is, every nontrivial, nitely generated subgroup has an in nite y li homomorphi image. His algebrai proof was published in full in 1984 [3℄. Around the same time, I independently obtained Brodski 's theorem, and published a slightly more general version in [7℄, with a topologi al proof: a one-relator quotient of a free produ t of lo ally indi able groups is lo ally indi able, provided the relator is neither a proper power nor onjugate to an element of one of the free fa tors. A further version of the theorem was later proved by John Hempel [5℄: the quotient of a surfa e group by a single relator that is not a proper power is lo ally indi able. This paper arose as a response to requests from olleagues notably Warren Di ks for a proof of Brodski 's theorem more a essible than those in [3, 7℄. In parti ular the topology used in [7℄ seemed to ause some diÆ ulty. Here I present a straightforward proof of the theorem, using groupoids. It is essentially my proof from [7℄, restri ted to the original ase of a torsion-free one-relator group, with as mu h of the topology as possible translated into algebra. The only remaining topology is the notion of an in nite y li over of a graph or groupoid. For more detailed ba kground material on graphs and groupoids, the best referen e is [6℄, but for ompleteness I have in luded some elementary de nitions in x2 below, and a des ription of the onstru tion of in nite y li overs in x3. A knowledgements. I rst presented a version of this proof in a seminar to mark the retirement of my PhD supervisor, Philip Higgins. I am grateful to him for his support. I am also grateful to Warren Di ks for en ouraging me to publish this material, and for pointing out Corollary 3.2.