The simple loop conjecture for 3-manifolds modeled on Sol

TL;DR

This paper proves the simple loop conjecture for 3-manifolds with Sol geometry, showing that certain immersed surfaces are either injective on fundamental groups or compressible.

Contribution

It establishes the conjecture specifically for 3-manifolds modeled on Sol, extending the understanding of surface immersions in these geometries.

Findings

Proves the simple loop conjecture for Sol 3-manifolds

Shows that immersed surfaces are either injective or compressible in this setting

Extends previous results to a new class of geometrically modeled 3-manifolds

Abstract

The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop Theorem to immersed surfaces. We prove the conjecture in the case that the target 3-manifold admits a geometric structure modeled on Sol.