Orderability of homology spheres obtained by Dehn filling (Q6042853)

A (closed) \(3\)-manifold is said to be left orderable if its fundamental group admits a strict total ordering which is invariant under left multiplication. For a countable group, this is equivalent to the claim that the group is a subgroup of the group \(\mathrm{Homeo}^+(\mathbb{R})\) of orientation preserving homeomorphisms of the real line. For a compact connected orientable \(3\)-manifold, this is further equivalent to the existence of a homomorphism from the fundamental group into \(\mathrm{Homeo}^+(\mathbb{R})\). In fact, representations into \(\widetilde{\mathrm{PSL}_2\mathbb{R}}\), the universal covering group of \(\mathrm{PSL}_2\mathbb{R}\), are known to be more computable and useful. To study these representations, \textit{M. Culler} and \textit{N. Dunfield} [Geom. Topol. 22, No. 3, 1405--1457 (2018; Zbl 1392.57012)] introduced the translation extension locus of a compact \(3\)-manifold with torus boundary. As the main new tool, the author introduces the holonomy extension locus for a rational homology solid torus. Using this, the left orderability of rational homology \(3\)-spheres obtained by Dehn filling on rational homology solid tori is examined. Here are two main results. (1) Let \(M\) be the exterior of a knot in a rational homology \(3\)-sphere. Suppose that \(M\) is longitudinal rigid. This means that the Dehn filling \(M(0)\) along the homological longitude has few characters. That is, each positive dimensional component of the \(\mathrm{PSL}_2\mathbb{C}\) character variety consists entirely of characters of reducible representations. If the Alexander polynomial of \(M\) has a simple positive real root \(\ne 1\), then there exists a non-empty interval \((-a,0]\) or \([0,a)\) such that for any rational \(r\) in the interval, the Dehn filling \(M(r)\) is left orderable. (2) Let \(M\) be a hyperbolic integral homology solid torus. Suppose that \(M(0)\) is a hyperbolic mapping torus of a homeomorphism of a genus two orientable surface and its holonomy representation has a trace field with a real embedding at which the associated quaternion algebra splits. Then \(M(r)\) is left orderable for any rational \(r\) in an interval \((-a,0]\) or \([0,a)\). The paper also contains several nice pictures of holonomy extension loci, and unsolved problems.