In this paper, we develop the theory of a very general class of treelike structures based on a simple set of betweenness axioms. Within this framework, we explore connections between more familiar treelike objects, such as $ {\Bbb R} $-trees and dendrons. Our principal motive is provide tools for studying convergence actions on continua, and in particular, to investigate how connectedness properties of such continua are reflected in algebraic properties of the groups in question. The main applications we have in mind are to boundaries of hyperbolic and relatively hyperbolic groups and to limit sets of kleinian groups. One of the main results of the present paper constructs dendritic quotients of continua admitting certain kinds of convergence actions, giving us a basis for introducing the techniques of $ {\Bbb R} $-trees into studying such actions. This is a step in showing that the boundary of a one-ended hyperbolic group is locally connected. There are further applications to constructing canonical splittings of such groups, and to limit sets of geometrically finite kleinian groups, which are explored elsewhere. We proceed in a general manner, discussing other connections with $ \Lambda $-trees, protrees, pseudotrees etc. along the way.
1991 Mathematics Subject Classification: 20F32, 20F08.
Key words and phrases: Tree, continuum, cutpoint, convergence group, dendrite, betweenness.
Memoirs Amer. Math. Soc. Volume 662 (1999).