Kleinian groups with discrete length spectrum.

*(English)*Zbl 1126.57008The 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.

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.

Reviewer: Darryl McCullough (Norman)

##### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

##### Keywords:

group; Kleinian; spectrum; length; discrete; geometrically infinite; fiber; circle; covering theorem##### References:

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