On the uniform equidistribution of closed horospheres in hyperbolic manifolds

TL;DR

This paper proves that large closed horospheres in hyperbolic manifolds become uniformly distributed as their size increases, extending previous results to higher dimensions with spectral methods providing convergence rates.

Contribution

It extends equidistribution results of horospheres from dimension 2 to arbitrary dimensions using spectral techniques and provides explicit convergence estimates.

Findings

Asymptotic equidistribution of large closed horospheres in hyperbolic manifolds

Extension of earlier 2D results to higher dimensions

Spectral methods yield precise convergence rates

Abstract

We prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. This extends earlier results by Hejhal and Str\"ombergsson in dimension 2. Our proofs use spectral methods, and lead to precise estimates on the rate of convergence to equidistribution.