Abstract:
Isometric actions on Λ-trees have been studied by several authors, including Morgan, Shalen, Chiswell, Bass, Kharlampovich, Miasnikov, Remeslennikov and Serbin. In particular Bass showed how isometric actions of (vertex) groups on Λ₀-trees can be combined to give an isometric action (on a Z×Λ₀-tree) of the fundamental group of an associated graph of groups, provided certain compatibility conditions are met. Notably, the hyperbolic lengths of the embedded images α_e(g), α _e(g) of elements g of edge groups must match up.
Affine actions are actions by dilations: one requires d(gx,gy) = a_g d(x,y), where a_g is an order-preserving group automorphism of Λ. In this talk we will show how certain combinations of groups can be equipped with an affine action on a Λ-tree. That is, if a graph of groups is given where the vertex groups have affine actions on Λ₀-trees, the fundamental group admits an affine action on a Λ-tree where Λ = Z×Λ₀, provided certain compatibility conditions are satisfied. Focusing on the case of free actions, we show that a large class of one-relator HNN extensions of free groups admit free affine actions on Λ-trees. Such HNN extensions cannot typically act freely by isometries because of the requirement that α_e(g) and α _e(g) have the same hyperbolic length.
Using recent work by various authors, we also show that groups that admit a free affine action on a Zⁿ-tree with no inverted line are locally quasiconvex and relatively hyperbolic with nilpotent parabolic subgroups; they therefore have solvable word, conjugacy and isomorphism problems.