Straight thin combings

We introduce a class of thin combings and prove that it has a nice behaviour in free constructions. Among corollaries, we deduce the relative metabolicity of Sela’s limit groups. S. Gersten defined the property of metabolicity for finitely presented groups (or even, contractible spaces with bounded geometry in dimension 2) as an attempt of an algebraic approach of word-hyperbolic groups [8] [9]. A group Γ is said metabolic if its second l∞-cohomology group, H 2 (∞)(Γ, A) is trivial, for any normed abelian group A (see [8] for definitions). This condition is stronger than hyperbolicity, but the converse is an open question for groups (for spaces, there are counterexamples exposed in [9]). In general it is difficult to prove that a given group is metabolic. Free and virtually free groups are the most obvious examples. A consequence of a delicate construction of Rips and Sela is that small cancellation C(1/8) groups are metabolic. A theorem of Gersten [9] shows that all hyperbolic surface groups are metabolic, by proving that certain amalgamated free products, over so-called amiable subgroups, preserve the metabolicity. But the problem of deciding if a given subgroup is amiable is neither easy. It is not known in general whether the maximal cyclic subgroups of a metabolic group are amiable. It is neither known whether all the cocompact Kleinian groups are metabolic. There is a combinatorial characterisation of metabolicity (for a contractible complex with bounded geometry in dimension 2) given by the existence of a thin combing. A thin combing in the 1-skeleton of a 2-complex is the assignation for each vertex v of a path from the base point to v, such that whenever two vertices are neighbors, the combinatorial area of the loop defined by their combings, and by the edge between them is bounded above by an universal constant. In [9], Gersten introduces the relative notion for metabolicity. This amounts to consider the metabolicity of a relative Rips complex (see [2] and below) of a group given with a family of subgroups. We prove here that certain relatively hyperbolic groups are relatively metabolic. Our main interest is in Sela’s limit groups, and in groups acting freely on Rtrees. Theorem 0.1 Limit groups, and groups that act freely on R-trees are metabolic relative to their maximal non cyclic abelian subgroups. The author acknowledges the support of FIM, ETH Zurich.