Euclidean lengths and the Culler-Shalen norms of slopes

TL;DR

This paper explores the relationship between Euclidean lengths and Culler-Shalen norms of slopes in the context of exceptional Dehn fillings, establishing inequalities and bounds relevant to boundary slope diameters.

Contribution

It introduces new inequalities linking Euclidean lengths and Culler-Shalen norms, providing bounds on boundary slope diameters in the study of Dehn fillings.

Findings

Established inequalities between Euclidean length and Culler-Shalen norm.

Provided bounds on boundary slope diameter.

Enhanced understanding of slope properties in Dehn fillings.

Abstract

In the study of exceptional Dehn fillings, two functions on slopes, called the Euclidean length on a horotorus and the Culler-Shalen norm, play important roles. In this paper, we investigate their relationship and establish two inequalities between them. As a byproduct, some bounds on the boundary slope diameter are given.