We prove that a one-relator group G is Kahler if and only if either G is finite cyclic or G is isomorphic to the fundamental group of a compact orbifold Riemann surface of genus g > 0 with at most one cone point of order n: . Fundamental groups of compact Kahler manifolds, or Kahler groups for short, have attracted much attention (see Amoros, Burger, Corlette, Kotschick and Toledo [2] for a survey of results and techniques). From a very different point of view, one-relator groups have been studied for a long time in combinatorial group theory (see Lyndon and Schupp [22, Chapter 2]). (A one-relator group is the quotient of a free group with finitely many generators by one relation.) It is natural to ask which groups occur in the intersection of these two classes. In fact one-relator groups have appeared as test cases for various restrictions developed for Kahler groups. Specific examples have been ruled out by Arapura [3, Section 7J]. Restrictions have been obtained from the point of view of rational homotopy theory (see Amoros [1, Sections 3 and 4] and [2, page 39, Examples 3.26 and 3.27]). Further restrictions follow from works of Gromov [18] and Green and Lazarsfeld [16].
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