Laminar groups and 3-manifolds

TL;DR

This paper reviews Thurston's universal circle theory for 3-manifold groups, explores laminar groups with invariant laminations, and proves new properties of laminar groups under different conditions.

Contribution

It provides a comprehensive review of universal circle actions and introduces new results on the properties of laminar groups.

Findings

Universal circle actions always admit an invariant lamination.

Laminar groups have specific properties under various conditions.

The paper extends the understanding of laminar groups in 3-manifold topology.

Abstract

Thurston showed that the fundamental group of a close atoroidal 3-manifold admitting a co-oriented taut foliation acts faithfully on the circle by orientation-preserving homeomorphisms. This action on the circle is called a universal circle action due to its rich information. In this article, we first review Thurston's theory of universal circles and follow-up work of other authors. We note that the universal circle action of a 3-manifold group always admits an invariant lamination. A group acting on the circle with an invariant lamination is called a laminar group. In the second half of the paper, we discuss the theory of laminar groups and prove some interesting properties of laminar groups under various conditions.