Marked length rigidity for one dimensional spaces

In a compact geodesic metric space of topological dimension one, the minimal length of a loop in a free homotopy class is well-defined, and provides a function l : π1(X) −→ R+ ∪ ∞ (the value ∞ being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset Conv(X), which is the union of all non-constant minimal loops of finite length. We show that if X is a compact, non-contractible, geodesic space of topological dimension one, then X deformation retracts to Conv(X). Moreover, Conv(X) can be characterized as the minimal subset of X to which X deformation retracts. Let X1, X2 be a pair of compact, non-contractible, geodesic metric spaces of topological dimension one, and set Yi = Conv(Xi). We prove that any isomorphism φ : π1(X1) −→ π1(X2) satisfying l2 ◦ φ = l1, forces the existence of an isometry Φ : Y1 −→ Y2 which induces the map φ on the level of fundamental groups. Thus, for compact, noncontractible, geodesic spaces of topological dimension one, the marked length spectrum completely determines the subset Conv(X) up to isometry.