Nonabelian representations of 2-bridge knot groups. (English) Zbl 0549.57005

The paper studies non-abelian representations \(\Phi\) (nab-reps) of 2- bridge knot groups \(\pi K\) into \(PSL(2,{\mathbb{C}})\). The main theorem establishes a correspondence between equivalence classes of nab-reps and points of a curve \(\Phi(t,u)=0\) in \({\mathbb{C}}^ 2\) for a ”representation” polynomial \(\Phi\in {\mathbb{Z}}[t,t^{-1},u],\) constructed for a given group.
There is a more general analogue theorem of W. Thurston considering links in closed 3-manifolds, the complements of which admit an excellent hyperbolic structure. The author’s approach in the special case, though, is eminently suited for explicit computation - contrary to that of Thurston. Actually a class of groups - called ”knot groups” - is studied which comprise 2-bridge knot groups and which are defined by a certain presentation.
The paper gives some detailed information on the nab-rep curve \(\Phi (t,u)=0\). There is a symmetry \(\Phi(t,u)\equiv\Phi(t^{-1},u)\) and \(\Phi\) has no nonconstant repeated factors in \({\mathbb{Z}}[t,t^{-1},u]\). The asymptotic behaviour is studied with respect to the line at infinity of \(P^ 2({\mathbb{C}})\). Furthermore the curve is shown to contain real arcs whose points correspond to representations equivalent to representations into SO(3). If a group \(\pi K\) admits faithful nap-reps, then it admits faithful representations into SO(3).
For genuine 2-bridge knot groups the following theorem is proved: Theorem 3. Let \(\Phi\) be the nap-rep polynomial for the 2-bridge knot group \(\pi\) K. There is a monic factor, say \(\Phi_ 1\) of \(\Phi\) in \(\Lambda[u]\), \(\Lambda ={\mathbb{Z}}[t,t^{-1}]\), such that the kernel of a generic representation for \(\Phi_ 1\) is the centre of \(\pi K\). (A generic representation corresponds to a zero \((t_ 0,u_ 0)\) of a monic \(\Phi_ j\) of \(\Phi\) where \(t_ 0\) or \(u_ 0\) is transcendental over \({\mathbb{Q}}.)\)
The proofs use previous results of the author on parabolic representations of knot groups, results on Schottky groups and work of Schubert and Thurston. There are several further results of a more technical quality which were not mentioned in this reference.
Reviewer: G.Burde


57M25 Knots and links in the \(3\)-sphere (MSC2010)
20G05 Representation theory for linear algebraic groups
57N10 Topology of general \(3\)-manifolds (MSC2010)
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