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Hyperbolic volume of representations of fundamental groups of cusped \(3\)-manifolds. (English) Zbl 1088.57015
A complete hyperbolic metric on a compact or cusped 3-manifold \(M\) induces a discrete faithful representation \(\rho: \pi_1(M)\to\text{PSL}(2,\mathbb{C})= \text{Isom}(\mathbb{H}^3)\): namely, one identifies the universal cover \(\widetilde M\) of \(M\) with \(\mathbb{H}^3\) and \(\pi_1(M)\) with the covering translations of the cover \(\mathbb{H}^3=\widetilde M\to M\). Mostow rigidity then implies that \(\rho\) is well-defined up to conjugation within \(\text{PSL}(2,\mathbb{C})\), hence that the (finite) volume \(\text{vol}(\rho) = \text{vol}(M)\) of \(M\) is a topological invariant.
The author notes that Dunfield has explained how to calculate a volume \(\text{vol}(\rho)\) for all representations \(\rho: \pi_1(M)\to\text{PSL}(2, \mathbb{C})\), discrete and faithful or not, provided that the cusp structure is respected.
This paper is devoted to showing that the Dunfield volume \(\text{vol}(\rho)\) is well-defined – that is, \(\text{vol}(\rho)\) is independent of the choices made in Dunfield’s construction. The volume can be computed by “straightening” any ideal triangulation of \(M\).
The author also proves that, if \(M\) is cusped-hyperbolic and \(\text{vol}(\rho)\geq\text{vol}(M)\), then, in fact, \(\text{vol}(\rho)= \text{vol}(M)\) and \(\rho\) is discrete and faithful.

MSC:
57M50 General geometric structures on low-dimensional manifolds
53C24 Rigidity results
53A35 Non-Euclidean differential geometry
57N10 Topology of general \(3\)-manifolds (MSC2010)
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