Metabolicity of Surface Groups and Amalgams

A nitely presented group G is called metabolic if H 2 (1) (G; A) = 0 for all normed abelian coeecient groups A; it is equivalent to require that a Cayley graph for G admit a thin combing. Metabolicity is stronger than hyperbolicity, and it is an open problem whether all hyperbolic groups are metabolic. We prove that if H is metabolic, then H ha;b]i F(a; b) is also metabolic, where F(a;b) is free on the indicated generators. The proof involves calculation with the Mayer-Vietoris sequence for`1-cohomology of an amalgam. A consequence is that all hyperbolic surface groups are metabolic. A metabolic group is a nitely presented group G for which H 2 (1) (G; A) = 0 for all normed abelian groups A. Since hyperbolic groups were characterized in Ge1] as nitely presented groups G for which H 2 (1) (G; ` 1) = 0, it follows that every metabolic group is hyperbolic. The converse is an open problem of considerable interest 1 with applications to the isomorphism problem for hyperbolic groups Se] and to higher dimensional isoperimetric inequalities for hyperbolic groups. 2 Although this deenition is algebraic, there is an equivalent geometric characterization of a metabolic group as a nitely presented group G possessing a thin combing Ge2]. We recall here that a combing of the Cayley graph ? of G for nite set S of generators is a collection of edge-paths fp v j v 2 ? (0) g, where p v begins as the base point v 0 and terminates at the vertex v, and the combing is called thin if there is a uniform upper bound on the combinatorial areas (with respect to a nite presentation for G with generators S) of the edge-loops p @ 0 e e p ?1 @ 1 e as e runs over the oriented edges of ?. Clearly nitely generated free groups are metabolic, and since thè 1-cohomology groups are quasi-isometry invariants, it follows that virtually free groups are metabolic. It has proved to be diicult to check that any other hyperbolic groups are metabolic. An argument is given in R-S] Appendix that C 0 (1=8)-groups are metabolic (the authors there use the term \canonical system of representatives" 1 Gromov conjectures that all hyperbolic groups are metabolic and he proposes that this follows from the existence of a geodesic ow Gr] with appropriate properties. This is an area of …