In this paper we consider group actions on generalised treelike structures (termed ``pretrees'') defined simply in terms of betweenness relations. Using a result of Levitt, we show that if a countable group admits an archimedean action on a median pretree, then it admits an action by isometries on an $ {\Bbb R} $-tree. Thus the theory of isometric actions on $ {\Bbb R} $-trees may be extended to a more general setting where it merges naturally with the theory of right-orderable groups. This approach has application also to the study of convergence group actions on continua.
1991 Mathematics Subject Classification: 20F32
Key words and phrases: tree, archimedean, order, median, pretree, betweenness
Math. Proc. Camb. Phil. Soc. 130 (2001) 383-400.