Marked length rigidity for Fuchsian buildings

TL;DR

This paper proves that certain negatively curved, piecewise hyperbolic metrics on quotients of Fuchsian buildings are uniquely determined by their marked length spectrum, linking geometric and spectral properties.

Contribution

It establishes marked length spectrum rigidity for finite quotients of Fuchsian buildings under specific curvature and non-singularity conditions.

Findings

Marked length spectrum determines the volume of the complex.

Rigidity holds within classes of negatively curved, piecewise Riemannian metrics.

The spectrum uniquely encodes geometric information of the quotient complex.

Abstract

We consider finite 2-complexes X that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT(-1) metrics on X which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on X. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of X.