Systolic Surfaces of Arithmetic Hyperbolic 3-Manifolds

TL;DR

This paper investigates the geometry of minimal surfaces in arithmetic hyperbolic 3-manifolds, providing bounds, constructions, and growth analysis relevant to their geometric and topological properties.

Contribution

It offers new bounds on totally geodesic 2-systoles, constructs infinitely many incommensurable manifolds with controlled invariants, and analyzes minimal surface genus growth across classes.

Findings

Bounds on totally geodesic 2-systoles

Construction of incommensurable manifolds with same spectrum

Analysis of minimal surface genus growth

Abstract

In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum in which volume and 1-systole are controlled, and analyze the growth of the genera of minimal surfaces across commensurability classes. These results have applications to the study of how Heegard genus grows across commensurability classes.