Systoles of Arithmetic Hyperbolic 2- and 3-Manifolds

TL;DR

This paper investigates the systoles of arithmetic hyperbolic 2- and 3-manifolds, constructing examples with fixed systoles, bounding volumes, and identifying minimal volumes for certain systole thresholds.

Contribution

It constructs infinitely many noncommensurable arithmetic hyperbolic manifolds with identical systoles and explicit volume bounds, and determines minimal volumes for specific systole constraints.

Findings

Constructed infinitely many noncommensurable manifolds with same systole.

Provided explicit volume bounds related to systole size.

Determined least volumes for certain small systole values.

Abstract

In this paper we study the systoles of arithmetic hyperbolic 2- and 3-manifolds. Our first result is the construction of infinitely many arithmetic hyperbolic 2- and 3-manifolds which are pairwise noncommensurable, all have the same systole, and whose volumes are explicitly bounded. Our second result fixes a positive number x and gives an upper bound for the least volume of an arithmetic hyperbolic 2- or 3-manifold whose systole is greater than x. We conclude by determining, for certain small values of x, the least volume of a principal arithmetic hyperbolic 2-manifold over Q or 3-manifold over Q(i) whose systole is greater than x.