[Paper Review] On character varieties, sets of discrete characters, and non-zero degree maps

Michel Boileau, Steven Boyer|ArXiv.org|Jan 14, 2007

Geometric and Algebraic Topology46 references4 citations

TL;DR

This paper investigates non-zero degree maps between 3-manifolds using character varieties of $\mathrm{PSL}_2(\mathbb{C})$-representations. It establishes that virtual epimorphisms between fundamental groups of small knot manifolds constrain the algebraic structure of their character varieties, leading to a priori bounds on such homomorphisms. The key result is the existence of infinite families of small, closed, orientable 3-manifolds that do not admit non-zero degree maps to any hyperbolic or geometrically structured manifold except via homeomorphisms.

ABSTRACT

In this paper we use character variety methods to study homomorphisms between the fundamental groups of 3-manifolds, in particular those induced by non-zero degree maps. A {\it knot manifold} is a compact, connected, irreducible, orientable 3-manifold whose boundary is an incompressible torus. A {\it virtual epimorphism} is a homomorphism whose image is of finite index in its range. We show that the existence of such homomorphisms places constraints on the algebraic decomposition of a knot manifold's $PSL_2(\mathbb C)$-character variety and consequently determine a priori bounds on the number of virtual epimorphisms between the fundamental groups of small knot manifolds with a fixed domain. In the second part of the paper we fix a small knot manifold $M$ and investigate various sets of characters of representations with discrete image in $PSL_2(\mathbb C)$. The topology of these sets is intimately related to the algebraic structure of the $PSL_2(\mathbb C)$-character variety of $M$ as well as dominations of manifolds by $M$ and its Dehn fillings. In particular, we apply our results to study families of non-zero degree maps $f_n: M(α_n) o V_n$ where $M(α_n)$ is the $α_n$-Dehn filling of $M$ and $V_n$ is either a hyperbolic manifold or $\widetilde{SL_2}$ manifold. We show that quite often, up to taking a subsequence, there is a knot manifold $V$, slopes $β_j$ on $\partial V$ such that $V_j \cong V(β_j)$, and a non-zero degree map $M o V$ which induces $f_j$ up to homotopy. The work of the first part of the paper is then applied to construct infinite families of small, closed, connected, orientable 3-manifolds which do not admit non-zero degree maps, other than homeomorphisms, to any hyperbolic manifold, or even manifolds with infinite fundamental groups.

Motivation & Objective

• To understand the constraints imposed by non-zero degree maps on the fundamental groups of small knot manifolds using character variety techniques.
• To analyze the topology of sets of discrete, torsion-free characters in $\mathrm{PSL}_2(\mathbb{C})$-character varieties of small knot manifolds.
• To determine when a small knot manifold dominates other 3-manifolds, particularly hyperbolic or $\widetilde{SL_2}$ manifolds, via non-zero degree maps.
• To establish minimality results for knot manifolds under domination, showing that certain manifolds do not virtually dominate any non-homeomorphic manifold in $\mathcal{H}$ or $\mathcal{M}$.
• To prove the existence of infinite families of closed, orientable 3-manifolds that do not admit non-zero degree maps to any hyperbolic or geometrically structured manifold except through homeomorphisms.

Proposed method

• Uses $\mathrm{PSL}_2(\mathbb{C})$-character varieties to analyze representations of fundamental groups of small knot manifolds.
• Applies virtual epimorphism conditions to constrain the algebraic decomposition of character varieties, especially for discrete, torsion-free representations.
• Employs bending functions $\beta_{(\rho_1,A)}$ on centralizers of image groups to detect constancy, which implies rigidity in character behavior.
• Analyzes cases where $\rho(\Gamma_0)$ is abelian but not $\mathbb{Z}/2\oplus\mathbb{Z}/2$, using trace identities to determine when bending functions are constant.
• Leverages the structure of $\mathcal{D}$ (diagonal), $\mathcal{P}$ (parabolic), $\mathcal{T}_+$ (loxodromic), and $\mathcal{N}$ (normalizer) subgroups in $\mathrm{PSL}_2(\mathbb{C})$ to classify representations.
• Uses the fact that for small knot manifolds, only finitely many characters correspond to representations with trivial or parabolic peripheral images, enabling slope-based classification of discrete characters.

Experimental results

Research questions

• RQ1 What constraints do virtual epimorphisms between fundamental groups of small knot manifolds impose on the algebraic structure of their $\mathrm{PSL}_2(\mathbb{C})$-character varieties?
• RQ2 Under what conditions is the bending function $\beta_{(\rho_1,A)}$ constant, and how does this relate to the reducibility or conjugacy of the representation $\rho$?
• RQ3 Which small, closed, orientable 3-manifolds do not admit non-zero degree maps to any hyperbolic or geometrically structured manifold except via homeomorphisms?
• RQ4 How does the topology of the set of discrete, torsion-free characters relate to the existence of non-zero degree maps between Dehn fillings and target manifolds?
• RQ5 Can infinite families of 3-manifolds be constructed that are minimal under domination in the class of hyperbolic or geometrically structured manifolds?

Key findings

• There are only finitely many characters of representations $\rho: \pi_1(M) \to \mathrm{PSL}_2(\mathbb{C})$ for which $\rho(\pi_1(\partial M))$ is trivial or parabolic, when $M$ is a small knot manifold.
• For a discrete, torsion-free character $\chi_\rho$, there exists a unique slope $\alpha$ such that $\rho(\alpha) = \pm I$, which defines the slope of $\rho$.
• The bending function $\beta_{(\rho_1,A)}$ is constant if and only if $\rho(\Gamma_1) \subset \mathcal{D}$ and $A = \pm \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$, or $\rho(\Gamma_1) \subset \mathcal{T}_+$ and $A \in \mathcal{T}_+$, implying reducibility or conjugacy into $\mathcal{N}$.
• If $\rho(\Gamma_0)$ is abelian but not $\mathbb{Z}/2\oplus\mathbb{Z}/2$, the constancy of the bending function implies strong algebraic constraints on $\rho_1$ and $A$.
• The existence of non-zero degree maps $f_n: M(\alpha_n) \to V_n$ to hyperbolic or $\widetilde{SL_2}$ manifolds implies structural restrictions on the character variety and the image of peripheral representations.
• There exist infinite families of small, closed, connected, orientable 3-manifolds that do not admit any non-zero degree map to a hyperbolic manifold or to any manifold in $\mathcal{M}$, except via homeomorphisms.