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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.

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
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