Equidistribution and counting of periodic flat tori

TL;DR

This paper proves exponential equidistribution of flat periodic tori in higher rank symmetric spaces and derives a prime geodesic theorem, extending results to non-cocompact lattices in special linear groups.

Contribution

It establishes exponential equidistribution of flat periodic tori in higher rank symmetric spaces and derives a higher rank prime geodesic theorem, including non-cocompact cases.

Findings

Flat periodic tori equidistribute exponentially fast.

Derived a higher rank prime geodesic theorem.

Results extend to non-cocompact lattices in SL(d, R).

Abstract

Let $G$ be a semisimple Lie group without compact factor and $\Gamma < G$ a torsion-free, cocompact, irreducible lattice. According to Selberg, periodic orbits of regular Weyl chamber flows live on maximal flat periodic tori of the space of Weyl chambers. We prove that these flat periodic tori equidistribute exponentially fast towards the quotient of the Haar measure. From the equidistribution formula, we deduce a higher rank prime geodesic theorem. These counting and equidistribution results also hold in the non cocompact, finite covolume case for $G=\mathrm{SL}(d,\mathbb{R})$ and $\Gamma<\mathrm{SL}(d,\mathbb{Z})$ a finite index subgroup.