Fibered 3-manifolds and Veech groups

TL;DR

This paper investigates the properties of Veech groups linked to fibers and foliations of hyperbolic 3-manifolds, demonstrating that, under certain conjectures, these groups have specific structural characteristics.

Contribution

It establishes new results on the nature of Veech groups for fibers and foliations in hyperbolic 3-manifolds, assuming Lehmer's Conjecture.

Findings

Veech groups for fibers generally contain no parabolic elements under Lehmer's Conjecture

Veech groups for foliations are always elementary

Results connect Veech group properties with hyperbolic 3-manifold topology

Abstract

We study Veech groups associated to the pseudo-Anosov monodromies of fibers and foliations of a fixed hyperbolic 3-manifold. Assuming Lehmer's Conjecture, we prove that the Veech groups associated to fibers generically contain no parabolic elements. For foliations, we prove that the Veech groups are always elementary.