Marked length rigidity for one dimensional spaces

TL;DR

This paper proves a version of the marked length spectrum conjecture for compact, non-contractible, one-dimensional geodesic spaces, showing that the spectrum determines the space up to isometry.

Contribution

It introduces the subspace Conv(X) for such spaces and demonstrates that the marked length spectrum uniquely determines Conv(X) and the space itself.

Findings

X deformation retracts to Conv(X) when non-contractible

Spaces with identical marked length spectra have isometric Conv(X)

The marked length spectrum determines the space up to isometry

Abstract

We prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is non-contractible, we show that X deformation retracts to Conv(X). If two such spaces X, Y have the same marked length spectrum, we prove that Conv(X) and Conv(Y) are isometric to each other.