Haken manifolds and representations of their fundamental groups in SL(2,\({\mathbb{C}})\). (English) Zbl 0647.57007
Culler and Shalen have shown that if the fundamental group of a compact orientable irreducible 3-manifold M has a positive-dimensional SL(2,\({\mathbb{C}})\)-character variety, then M is a Haken manifold. We show however that the converse is not true. That is there exist infinitely many Haken manifolds whose fundamental groups have a finite number of representations in SL(2,\({\mathbb{C}})\) up to equivalence. In particular, they have 0-dimensional SL(2,\({\mathbb{C}})\)-character varieties. This construction relates with the bending construction.
Reviewer: K.Motegi
MSC:
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
| 57M05 | Fundamental group, presentations, free differential calculus |
| 20C15 | Ordinary representations and characters |
| 20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |
Keywords:
Haken manifolds whose fundamental groups have a finite number of representations in SL(2,\({bbfC})\); SL(2,\({bbfC})\)-character varieties; bending constructionReferences:
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