Distributions of length multiplicities for negatively curved locally symmetric Riemannian manifolds

TL;DR

This paper investigates the distribution of length multiplicities in negatively curved locally symmetric Riemannian manifolds, providing bounds and estimates for both general and arithmetic cases.

Contribution

It introduces new bounds and precise estimates for length multiplicities, extending understanding in both general and arithmetic negatively curved manifolds.

Findings

Upper bounds for length multiplicities in general cases

Precise estimates for arithmetic surfaces

Analysis of power sums of length multiplicities

Abstract

The aim of the present paper is to study the distributions of the length multiplicities for negatively curved locally symmetric Riemannian manifolds. In Theorem 2.1, we give upper bounds of the length multiplicities and the square sums of them for general (not necessarily compact) cases. Furthermore in Theorem 2.2, we obtain more precise estimates of the length multiplicities and the power sums of them for arithmetic surfaces whose fundamental groups are congruence subgroups of the modular group.