On character varieties, sets of discrete characters, and non-zero degree maps
| dc.creator | Boileau, Michel | |
| dc.creator | Boyer, Steven | |
| dc.date | 2007-01-14 | |
| dc.date.accessioned | 2026-07-07T07:41:00Z | |
| dc.date.available | 2026-07-07T07:41:00Z | |
| dc.description | 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 $f_n: M(α_n) \to V_n$ 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 $M \to V$ 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. | |
| dc.identifier | https://arxiv.org/abs/math/0701384 | |
| dc.identifier | http://arxiv.org/abs/math/0701384 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121952 | |
| dc.subject | Geometric Topology | |
| dc.subject | 57M05, 57M50, 57R65 | |
| dc.title | On character varieties, sets of discrete characters, and non-zero degree maps | |
| dc.type | text |