Branch Points and Free Actions On ℝ-Trees

Abstract

Actions of groups on ℝ-trees are natural generalizations of actions of groups on ℤ-trees (simplicial trees). The latter, known as Bass-Serre Theory, has become a standard tool in combinatorial group theory. This theory says that a group action on a ℤ-tree is equivalent to a combinatorial presentation of the group. On the other hand, actions of groups on ℝ-trees arise from groups of isometries of hyperbolic spaces, and have significant applications in the study of hyperbolic manifolds. Morgan, Shalen, Culler and others define and analyze compactification of representation space R(G) of a group G in SL(2, ℂ) and SO (n,1). For a valuation of the function field of a subvariety of R(G), they define an ℝ-tree on which G acts. This is applied notably when G = π ₁(M), M a hyperbolic manifold. The tree and action then give a codimension-1 measured lamination on M. Applications include the theory of incompressible surfaces in 3-manifolds and the description of a natural boundary for Teichmüller space. The points at infinity correspond to π_l (M)-actions on ℝ-trees.