The Nielsen Method for Groups Acting on Trees

Kleinian groups and the rank problem

“…Indeed, Grushko's theorem (see the proofs, for example, in [19,33]) ensures that every tuple generating a group G 1 * G 2 is Nielsen equivalent to a tuple (T 1 , T 2 ) where T 1 generates G 1 and T 2 generates G 2 . Note also that the statement of Theorem 9.1 holds for infinite cyclic groups (and, moreover, for finitely generated free groups).…”

Acylindrical accessibility for groups acting on ?-trees

“…Not surprisingly our methods let us recover the same bound c(G, D) = 2D(k − 1) on the complexity of acylindrical accessibility splittings as the one given in [40]. The main ingredient is a theory of Nielsen methods for groups acting on hyperbolic spaces that we systematically developed in [29,30].…”

“…The main ingredient is a theory of Nielsen methods for groups acting on hyperbolic spaces that we systematically developed in [29,30]. This theory is analogous to Weidmann's treatment of actions on simplicial trees [40], but the case of arbitrary hyperbolic spaces is technically much more complicated. Note that the proof of Theorem 1.2 completely avoids the Rips machinery for groups acting on R-trees [8,27] and Theorem 1.2 makes no traditional stability assumptions about the action.…”

Generic groups acting on regular trees

“…Since then, the method has been further developed by Zieschang, Weidmann, Kapovich and others. We will need the following theorem that can be found in Weidmann's paper [Wei02,Theorem 7], setting S i = ∅.…”