On character varieties, sets of discrete characters, and non-zero degree maps

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 $PS{L}_2(\mathbb{C})$-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 $PS{L}_2(\mathbb{C})$. The topology of these sets is intimately related to the algebraic structure of the $PS{L}_2(\mathbb{C})$-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 $f_n:M({\alpha }_n)\to V_n$ where $M({\alpha }_n)$ is the ${\alpha }_n$-Dehn filling of $M$ and $V_n$ is either a hyperbolic manifold or $\widetilde{S{L}_2}$ manifold. We show that quite often, up to taking a subsequence, there is a knot manifold $V$, slopes ${\beta }_j$ on $\partial V$ such that $V_j\cong V({\beta }_j)$, and a non-zero degree map $M\to V$ 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.

Cited by in corpus (1)

• Simon's conjecture for 2-bridge knots