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**Quasigeodesic flows and Möbius-like groups.**
*(English)*
Zbl 1279.53063

Let \(M\) denote a closed, hyperbolic 3-manifold and let \(\tilde{M} = \mathbb{H}^{3}\) be its universal cover, the real hyperbolic 3-space. Let \(\Gamma\) denote the deckgroup of the covering. A nonsingular flow \(\mathfrak{F}\) in \(M\) is said to be quasi-geodesic if each flow line \(c\) of the lifted flow \(\tilde{\mathfrak{F}}\) in \(\tilde{M}\) is a quasi-geodesic; that is, there exist positive constants \(k\), \(\epsilon\) such that \(\frac{1}{k} d(c(s),c(t)) - \epsilon \leq |s-t| \leq k d(c(s),c(t)) + \epsilon\) for all \(s,t \in \mathbb{R}\), where \(d\) denotes the hyperbolic metric on \(\mathbb{H}^{3}\). For a quasi-geodesic flow \(\mathfrak{F}\) in \(M\), let \(P\) denote the space of flow lines of \(\tilde{\mathfrak{F}}\). The space \(P\) is homeomorphic to a plane. The group \(\Gamma\) permutes the flow lines of \(\tilde{\mathfrak{F}}\) and induces an action of \(\Gamma\) on \(P\). The flow \(\mathfrak{F}\) has a closed orbit \(\Leftrightarrow\) some element of \(P\) is fixed by some element of \(\Gamma\).

Each flow line of \(\tilde{\mathfrak{F}}\) determines positive and negative endpoints in \(\partial \mathbb{H}^{3}\) and gives rise to continuous maps \(e^{+},e^{-} : P \rightarrow \partial \mathbb{H}^{3}\). For fixed \(p\in \partial \mathbb{H}^{3}\), each component of \((e^{+})^{-1}(p)\) or \((e^{-})^{-1}(p)\) is unbounded. Let \(D^{+}\), \(D^{-}\) denote the space of components of \((e^{+})^{-1}(p), (e^{-})^{-1}(p), p \in \partial \mathbb{H}^{3}\). The sets \(D^{+}\), \(D^{-}\) determine natural compactifications \(\overline{P}^{+}\), \(\overline{P}^{-}\) of \(P\) that are homeomorphic to closed discs and whose boundaries are “universal” circles \(S_{u}^{+}\), \(S_{u}^{-}\). The author shows (compactification theorem) that the action of \(\Gamma\) on P extends to an action of \(\Gamma\) on \(\overline{P}^{+}\), \(\overline{P}^{-}\) and on the boundaries \(S_{u}^{+}\), \(S_{u}^{-}\).

A group of homeomorphisms \(\Gamma\) of the circle \(S^{1}\) is said to be Möbius-like if each element of \(\Gamma\) is topologically conjugate to an element of \(\mathrm{PSL}(2,\mathbb{R})\) with the natural action. The author proves the following

Theorem. Let \(\mathfrak{F}\) be a quasi-geodesic flow on a closed hyperbolic 3-manifold. Let \(\mathfrak{F}\) have no closed orbits. Then

a) The action of \(\Gamma\) on \(S_{u}^{+}\) or \(S_{u}^{-}\) is Möbius-like and each element of \(\Gamma\) has a fixed point in \(S_{u}^{+}\) (respectively \(S_{u}^{-}\)).

b) \(\Gamma\) acting on \(S_{u}^{+}\) or \(S_{u}^{-}\) is not topologically conjugate to a subgroup of \(\mathrm{PSL}(2,\mathbb{R})\) with the natural action on \(S^{1}\).

The author conjectures that for a quasi-geodesic flow \(\mathfrak{F}\) on a closed hyperbolic 3-manifold \(M\) the action of the deckgroup \(\Gamma\) on \(S_{u}^{+}\) (respectively \(S_{u}^{-}\)) is not Möbius-like. By the result above this would imply that all quasi-geodesic flows \(\mathfrak{F}\) on closed hyperbolic 3-manifolds have a closed orbit, thereby giving an affirmative answer to one part of a conjecture of Calegari.

Each flow line of \(\tilde{\mathfrak{F}}\) determines positive and negative endpoints in \(\partial \mathbb{H}^{3}\) and gives rise to continuous maps \(e^{+},e^{-} : P \rightarrow \partial \mathbb{H}^{3}\). For fixed \(p\in \partial \mathbb{H}^{3}\), each component of \((e^{+})^{-1}(p)\) or \((e^{-})^{-1}(p)\) is unbounded. Let \(D^{+}\), \(D^{-}\) denote the space of components of \((e^{+})^{-1}(p), (e^{-})^{-1}(p), p \in \partial \mathbb{H}^{3}\). The sets \(D^{+}\), \(D^{-}\) determine natural compactifications \(\overline{P}^{+}\), \(\overline{P}^{-}\) of \(P\) that are homeomorphic to closed discs and whose boundaries are “universal” circles \(S_{u}^{+}\), \(S_{u}^{-}\). The author shows (compactification theorem) that the action of \(\Gamma\) on P extends to an action of \(\Gamma\) on \(\overline{P}^{+}\), \(\overline{P}^{-}\) and on the boundaries \(S_{u}^{+}\), \(S_{u}^{-}\).

A group of homeomorphisms \(\Gamma\) of the circle \(S^{1}\) is said to be Möbius-like if each element of \(\Gamma\) is topologically conjugate to an element of \(\mathrm{PSL}(2,\mathbb{R})\) with the natural action. The author proves the following

Theorem. Let \(\mathfrak{F}\) be a quasi-geodesic flow on a closed hyperbolic 3-manifold. Let \(\mathfrak{F}\) have no closed orbits. Then

a) The action of \(\Gamma\) on \(S_{u}^{+}\) or \(S_{u}^{-}\) is Möbius-like and each element of \(\Gamma\) has a fixed point in \(S_{u}^{+}\) (respectively \(S_{u}^{-}\)).

b) \(\Gamma\) acting on \(S_{u}^{+}\) or \(S_{u}^{-}\) is not topologically conjugate to a subgroup of \(\mathrm{PSL}(2,\mathbb{R})\) with the natural action on \(S^{1}\).

The author conjectures that for a quasi-geodesic flow \(\mathfrak{F}\) on a closed hyperbolic 3-manifold \(M\) the action of the deckgroup \(\Gamma\) on \(S_{u}^{+}\) (respectively \(S_{u}^{-}\)) is not Möbius-like. By the result above this would imply that all quasi-geodesic flows \(\mathfrak{F}\) on closed hyperbolic 3-manifolds have a closed orbit, thereby giving an affirmative answer to one part of a conjecture of Calegari.

Reviewer: Patrick Eberlein (Chapel Hill)

### MSC:

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

55Q99 | Homotopy groups |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |