Geodesic simplices of pseudo-hyperbolic space

TL;DR

This paper provides a cohomological framework for understanding geodesic simplices in pseudo-hyperbolic spaces and establishes conditions for their finite volume, with implications for ideal geodesic polytopes.

Contribution

It introduces a cohomological interpretation of geodesic simplices in pseudo-hyperbolic space and characterizes when these simplices have finite volume.

Findings

Cohomological interpretation of geodesic simplices.

Necessary and sufficient condition for finite volume simplices.

All ideal geodesic polytopes in (2,2) pseudo-hyperbolic space have finite volume.

Abstract

We give a cohomological interpretation of the geodesic simplices of the pseudo-hyperbolic space of signature $(p,q)$ and formulate a necessary and sufficient condition for such a simplex to have finite volume. As a corollary, we obtain that every ideal geodesic polytope in the pseudo-hyperbolic space of signature $(2,2)$ has finite volume.