𝑅-trees, small cancellation, and convergence

Abstract

Abstract.The "metric small cancellation hypotheses" of combinatorial group theory imply, when satisfied, that a given presentation has a solvable Word Problem via Dehn's Algorithm. The present work both unifies and generalizes the small cancellation hypotheses, and analyzes them by means of group actions on trees, leading to the strengthening of some classical results.

Triangles of groups

“…(a) (Existence). Whenever h £ ff and [u, v] is a segment of A(h), we have In the present context, the main theorem (Theorem 1.7) of [3] may now be stated as:…”

“…The next item of business is to set up a suitable polarization 3 . To do this, we have to recall two more notions from [3]. Namely, for any set S of vertices of Sf, the simplex spanned by S is X(S) = {J{[u,v]:u,v£S}, and we say that S is in general position if u £ _(S -{u}), for every u £ S.…”

Triangles of Groups

“…Otherwise, we assume that ff /(h) is E-free for all heK, and we put ffx = ff and ff2 = 0 . In the present context, the main theorem (Theorem 1.7) of [3] may now be stated as:…”

“…To do this, we have to recall two more notions from [3]. Namely, for any set S of vertices of Sf, the simplex spanned by S is X(S) = {J{ [u,v]:u,v£S}, and we say that S is in general position if u £ _(S -{u}), for every u £ S.…”

“…The proof of Theorem A is based on the arboreal small-cancellation theory developed in [3]. Thus, let X denote the standard tree for G, and put %' = l\J(g-lKg)\-{l}.…”