Train track complex of once-punctured torus and 4-punctured sphere

Details Train track complex of once-punctured torus and 4-punctured sphere Train track complex of once-punctured torus and 4-punctured sphere

2009-01-07

arxiv.org/abs/0901.0747v1

Mathematics

General Topology

19 pages, 13 figures

Scientific paper

Consider a compact oriented surface $S$ of genus $g \geq 0$ and $m \geq 0$ punctured. The train track complex of $S$ which is defined by Hamenst\"adt is a 1-complex whose vertices are isotopy classes of complete train tracks on $S$. Hamenst\"adt shows that if $3g-3+m \geq 2$, the mapping class group acts properly discontinuously and cocompactly on the train track complex. We will prove corresponding results for the excluded case, namely when $S$ is a once-punctured torus or a 4-punctured sphere. To work this out, we redefinition of two complexes for these surfaces.

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