Marked length rigidity for one-dimensional spaces

Abstract

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 [Formula: see text] (the value [Formula: see text] being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset [Formula: see text], which is the union of all non-constant minimal loops of finite length. We show that if [Formula: see text] is a compact, non-contractible, geodesic space of topological dimension one, then [Formula: see text] deformation retracts to [Formula: see text]. Moreover, [Formula: see text] can be characterized as the minimal subset of [Formula: see text] to which [Formula: see text] deformation retracts. Let [Formula: see text] be a pair of compact, non-contractible, geodesic metric spaces of topological dimension one, and set [Formula: see text]. We prove that any isomorphism [Formula: see text] satisfying [Formula: see text], forces the existence of an isometry [Formula: see text] which induces the map [Formula: see text] on the level of fundamental groups. Thus, for compact, non-contractible, geodesic spaces of topological dimension one, the marked length spectrum completely determines the subset [Formula: see text] up to isometry.