Geometry and topology of closed geodesics complements in the 3-torus

TL;DR

This paper investigates the topology of complements of closed geodesics in the 3-torus, showing their homeomorphism type is determined by the orbit of their directions under a specific group action and providing sharp volume bounds based on Farey graph distances.

Contribution

It establishes a classification of geodesic complements in the 3-torus via group actions and introduces precise volume bounds related to Farey graph metrics.

Findings

Homeomorphism type determined by $PSL_3(\

Volume bounds depend only on Farey graph distance

Abstract

We show that for at most three closed geodesics with linearly independent directions, the homeomorphism type of its complement in the 3-torus is determine by the orbit of their direction vectors subspaces under the action of $PSL_3(\mathbb{Z}).$ Moreover, we provide asymptotically sharp volume bounds for a family of closed geodesics complements. The bounds depend only on the distance in the Farey graph.