Sub-actions for geodesic flows on locally CAT(-1) spaces

TL;DR

This paper extends sub-action results from smooth Anosov flows to geodesic flows on locally CAT(-1) spaces, enabling volume rigidity theorems for spaces like Fuchsian buildings and surface amalgams.

Contribution

It generalizes sub-action theory to a broader class of non-smooth spaces, leading to new volume rigidity results.

Findings

Established volume rigidity for certain locally CAT(-1) spaces

Extended sub-action techniques to non-smooth settings

Applied to quotients of Fuchsian buildings and surface amalgams

Abstract

We extend a result of Lopes and Thieullen on sub-actions for smooth Anosov flows to the setting of geodesic flow on locally CAT(-1) spaces. This allows us to use arguments originally due to Croke and Dairbekov to prove a volume rigidity theorem for some interesting locally CAT(-1) spaces, including quotients of Fuchsian buildings and surface amalgams.