On Dehn Functions of Amalgamations and Strongly Undistorted Subgroups

We study the Dehn functions of amalgamations, introducing the notion of strongly undistorted subgroups. Using this, we give conditions under which taking an amalgamation does not increase the Dehn function, generalizing one aspect of the combination theorem of Bestvina and Feighn. To obtain examples of strongly undistorted subgroups, we define and study the relative Dehn function of pairs of groups. As a result we obtain a new method of constructing examples of pairs of groups that are relatively hyperbolic in the sense of Farb. 0. Introduction. In [BF], M. Bestvina and M. Feighn prove a combination theorem for hyperbolic groups. Among other consequences, their theorem gives conditions under which an amalgamation P = A∗ CB of hyperbolic groups A and B is itself hyperbolic. Other results in the same vein can be found in [Gi], [KM], and [Ge2]. As is well-known, hyperbolic groups are characterized by having linear Dehn functions. Therefore the combination theorem can be viewed as giving a condition for groups with linear Dehn functions under which taking an amalgamation does not increase the Dehn function. It thus seems appropriate to try to generalize the Bestvina-Feighn result by looking for a condition on groups – which are possibly not hyperbolic – under which taking an amalgamation P = A ∗C B does not result in an increase of the Dehn function, i.e, the Dehn function δP of P = A ∗C B is bounded above by the maximum of the Dehn function δA of A and the Dehn function δB of B (though strictly speaking we have to take the subnegative closure of the maximum). To that end we introduce the notion of strongly undistorted subgroups. This definition was inspired by the approach of [KM]. The condition on subgroups is a geometric one. Recall that a subgroup H of a group G is undistorted if there are finite generating sets for H and G so that there is a constant a for which 1991 Mathematics Subject Classification. Primary 20F32.