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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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