Spaces of Kleinian Groups: Train tracks and the Gromov boundary of the complex of curves

By U. Hamenstaedt. 2004

94 citations recorded by the source corpus.

We give a combinatorial proof of an unpublished result of E. Klarreich: The Gromov boundary of the complex of curves of a non-exceptional oriented surface S of finite type can naturally be identified with the space of minimal geodesic laminations on S which fill up S, equipped with a coarse Hausdorff topology.

Subfields: Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5935166caaf65e428fe929bcd9bbb4976b823c01

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Graphics and Computer-Aided Design

1. Yair N. Minsky
2. David Gabai
3. Joseph Maher
4. Jeffrey F. Brock
5. Y. Kida
6. H. Masur

In Geometry and Topology

1. H. Masur
2. Yair N. Minsky
3. Joseph Maher
4. Jeffrey F. Brock
5. David Gabai
6. Y. Kida
7. Jason A. Behrstock
8. U. Hamenstaedt

Citations in the corpus

• Random Heegaard splittings (2008, methodology): “The Gromov boundary of the complex of curves, and hence the relative space, is the set formed by taking the foliations in PMF , the space of projective measured foliations on the surface, whose trajectories contain no closed curves, and forgetting the measures, as shown by Klarreich [10]; see also H”
• Measure equivalence rigidity of the mapping class group (2006, background): “M/-equivariant map MIN! @C constructed in [29] (see also [19]).”
• Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows (2008, methodology): “To see this, note that if h + is the segment 1248 J. BROCK, H . MASUR AND Y. MINSKY GAFA”
• The Mapping Class Group from the Viewpoint of Measure Equivalence Theory (2005, background): “Klarreich [57] studied ∂C for a non-exceptional surface, and gave the following characterization of it (see also [37]):”
• Big Mapping Class Groups: An Overview (2020, unknown): “Klarreich [76] showed that the Gromov boundary of the curve graph is Map(S)-equivariantly homeomorphic to the space of ending laminations on the surface; see also [58] for a different argument, and Pho-On’s thesis [97] for an effective proof of this using the unicorn machinery of [66].”
• Almost filling laminations and the connectivity of ending lamination space (2008, background): “This work was clarified by Ursula Hamenstadt [H1], who gave a direct combinatorial argument equating ends of quasi-geodesic rays in C(S) with points of EL(S).”