Big free groups acting on $\Lambda$-trees

Abstract

The set of homotopy classes of based paths in the Hawaiian earring has a natural $\mathbb R$-tree structure, but under that metric the action by the fundamental group is not by isometries. Following a suggestion by Cannon and Conner, this paper defines an $\mathbb R^\omega$-metric that does admit for an isometric action by the fundamental group. The space does not become an $\mathbb R^\omega$-tree but is $0$-hyperbolic and embeds in an $\mathbb R^\omega$-tree. Cannon and Conner define big free groups $\operatorname{BF}(c)$ for cardinal number $c$ which are a generalization of the fundamental group of the Hawaiian earring. They define a big Cayley graph which coincides with the set of homotopy classes of paths in the case of the Hawaiian earring. Instead of inserting real intervals to obtain the Cayley graph, we can insert $\mathbb R^c$-intervals and obtain a new $\mathbb R^c$-tree which admits an isometric action. In fact we do not need all of $\mathbb R^c$; we can insert $\mathbb Z^c$-intervals and obtain a $\mathbb Z^c$-tree. In the case of the Hawaiian earring we give a combinatorial description of the $\mathbb Z^\omega$-tree and the corresponding action.