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Kleinian groups with discrete length spectrum. (English) Zbl 1126.57008
The real length spectrum (without multiplicities) of a Kleinian group $$\Gamma$$ is the set of all real translation lengths of elements of $$\Gamma$$ acting on $$\mathbb{H}^3$$. Excluding $$0$$, this is the set of all lengths of closed geodesics in the quotient orbifold $$\mathbb{H}/\Gamma$$. It was shown by I. Kim [Commun. Pure Appl. Math. 59, No. 5, 617–625 (2006; Zbl 1102.53028)] that if $$\Gamma$$ is the fundamental group of a closed surface and its length spectrum $$\Lambda$$ is discrete, then there are only finitely many conjugacy classes of Kleinian groups that are isomorphic to $$\Gamma$$ and have length spectrum $$\Lambda$$.
The main result of the paper under review characterizes finitely generated torsionfree Kleinian groups with discrete length spectrum: Such a group is either (1) geometrically finite, (2) the fiber subgroup of the fundamental group of a hyperbolic $$3$$-manifold that fibers over the circle, or (3) a singular fiber subgroup of a hyperbolic $$3$$-manifold that fibers over the $$1$$-orbifold $$S^1/\langle z \sim \overline{z}\rangle$$.
An ingredient of the proof leads to a refinement of the first author’s well-known Covering Theorem; the new version states that if $$\widehat{M}\to M$$ is an orbifold covering of a hyperbolic $$3$$-orbifold by a hyperbolic $$3$$-manifold with finitely generated fundamental group, and this covering is infinite-to-one on a neighborhood of a geometrically infinite end of $$\widehat{M}^0_\varepsilon$$, then $$M$$ has finite volume and is finitely covered by a manifold $$M'=\mathbb{H}^3/G$$ such that either (1) $$M'$$ fibers over the circle and $$\widehat{M}$$ is the covering associated to the fiber subgroup of $$G$$, or (2) $$M'$$ fibers over the $$1$$-orbifold $$S^1/\langle z \sim \overline{z}\rangle$$ and $$\widehat{M}$$ is the covering of $$M'$$ assocated to a singular fiber subgroup of $$G$$. Combined with the main theorem, this implies that the real length spectrum of a finitely generated torsionfree geometrically infinite Kleinian group is discrete if and only if it is a subgroup of a cofinite volume Kleinian group.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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##### References:
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