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On the topology of ending lamination space. (English) Zbl 1307.57012
Let \(S:=S_{g,p}\) be a hyperbolic surface of genus \(g\) with \(p\) punctures. Consider laminations \({\mathcal L}\) of \(S\) by geodesic curves which are minimal (each leaf is dense) and filling (every simple closed geodesic intersects \({\mathcal L}\)). Such laminations are called ending laminations, presumably because they occur as ending laminations of geometrically infinite ends of finitely generated Kleinian groups.
The Hausdorff topology on the set of ending laminations \({\mathcal EL}(S)\) would be totally disconnected and have Hausdorff dimension zero. A more interesting topological structure is obtained from the coarse Hausdorff topology, where a sequence converges to \({\mathcal L}\) if a subsequence converges in the Hausdorff topology to a lamination obtained by adding finitely many leaves to \({\mathcal L}\). This topology naturally occurs when identifying \({\mathcal EL}(S)\) with the Gromov boundary of the curve complex \(C(S)\).
In the paper under review it is proved that \({\mathcal EL}(S_{g,p})\) is \((3g+p-5)\)-connected and \((3g+p-5)\)-locally connected.
When \(g=0\) it is shown that \({\mathcal EL}(S_{0,p})\) is homeomorphic to the \((p-4)\)-dimensional Nöbeling space, i.e., the space of points in \({\mathbb R}^{2p-7}\) with at most \(p-4\) rational coordinates. For \(p\leq 5\) this was known before by S. Hensel and P. Przytycki [J. Lond. Math. Soc., II. Ser. 84, No. 1, 103–119 (2011; Zbl 1246.57033)].
The author also offers some conjectures about the (co)homological properties of \({\mathcal EL}(S_{g,p})\) for \(g,p>0\) and speculates that then these spaces might be homeomorphic to the space of points in \({\mathbb R}^{6g+2p-7}\) with at most \(4g+p-5\) rational coordinates.

MSC:
57M50 General geometric structures on low-dimensional manifolds
20F65 Geometric group theory
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