Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation

Abstract

We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented R \mathbb R -order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.

Non-left-orderable double branched coverings

“…This conjecture is verified for Seifert fibered spaces and non-hyperbolic geometric 3-manifolds [1]. There are many examples of 3-manifolds having non-left-orderable fundamental groups (Dabkowski, Przytycki and Togha [7], Roberts and Shareshian [15], and Roberts, Shareshian and Stein [16]): many of them are confirmed to be L-spaces (Clay and Watson [6; 5] and Peters [14]). Conversely, many known L-spaces, such as the double branched covering of alternating links, or L-spaces obtained from certain Dehn surgeries are shown to have non-left-orderable fundamental group [1; 8].…”

Problems in low dimensional contact topology

“…It was once conjectured that all hyperbolic manifolds admit taut foliations. If this were true, we could perturb them into tight contact structures, but it has recently been shown that there are hyperbolic 3-manifolds without taut foliations [6,70]. It is still possible, however, that all hyperbolic manifolds admit a tight contact structure.…”

Non-Right-Orderable 3-Manifold Groups

“…As mentioned in [BRW02], some hyperbolic 3-manifolds have right orderable fundamental groups while others do not. The first examples of hyperbolic 3-manifolds with non-right-orderable fundamental groups appeared in [RSS03,DPT05,Fen07] and an early preprint version of this paper. (A group is right orderable if and only if it is left orderable.…”

“…In both [RSS03] and [Fen07], the groups considered have presentations of the form G = t, a, b|a t = a m−1 b −1 a −1 , b t = a −1 , t p [a, b] q = 1 , where m, p, q are integers and p, q are relatively prime. In [RSS03], the case that m ≤ −3 and p q ∈ [1, ∞) is analyzed. In [Fen07], the case that m ≤ −4 and |p − 2q| = 1 is examined.…”

“…A group is called a 3-manifold group if it can be realized as the fundamental group of a 3-manifold. The groups G(φ, p, q) described above are 3-manifold groups and their study in [RSS03,Fen07] and the current paper was motivated by questions arising from the study of Reebless foliations and essential laminations in the associated 3-manifolds. In general, there is an interesting interplay between the existence of Reebless foliations or, more generally, essential laminations in a 3-manifold M and the existence of nontrivial actions of π 1 (M) on associated (not necessarily Hausdorff) 1-manifolds and trees.…”