Lattices and Periodic Geodesics in Pseudoriemannian 2-step Nilpotent Lie Groups

TL;DR

This paper studies lattices in pseudoriemannian 2-step nilpotent Lie groups, exploring their geometric structure and the spectrum of periodic geodesics in associated compact nilmanifolds, with a focus on degenerate tori and flat space spectra.

Contribution

It provides a foundational analysis of lattices and the structure of geodesics in pseudoriemannian nilpotent Lie groups, including the role of degenerate tori and explicit spectrum calculations.

Findings

Lattices in pseudoriemannian 2-step nilpotent Lie groups are characterized.

The submersion structure involves degenerate tori as fibers and bases.

Complete calculation of the period spectrum for certain flat spaces.

Abstract

We give a basic treatment of lattices $\Gamma$ in these groups. Certain tori $T_F$ and $T_B$ provide the model fiber and the base for a submersion of $\Gamma\backslash N$. This submersion may not be pseudoriemannian in the usual sense, because the tori may be degenerate. We then begin the study of periodic geodesics in these compact nilmanifolds, obtaining a complete calculation of the period spectrum of certain flat spaces.