Length functions and outer space (Q1803920)

Let \(F_ n\) be the free group of rank \(n\). The outer automorphism group of \(F_ n\) acts on a certain contractible space, the outer space. An \({\mathbf R}\)-graph is a graph which is a metric space where each edge is isometric to an interval of \({\mathbf R}\). Marking a graph is an identification of the fundamental group of the graph with \(F_ n\). Two graphs are equivalent if there is an isometry between them which preserves the marking. An element of the outer space determines a hyperbolic length function on \(F_ n\), where the length of \(g\in F_ n\) is the length of the shortest (unbased) loop representing the homotopy class corresponding to \(g\). In this paper several problems concerning embeddings of the space of marked graphs are investigated.