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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 a1-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 actsproperly discontinuously and cocompactly on the train track complex. We willprove corresponding results for the excluded case, namely when $S$ is aonce-punctured torus or a 4-punctured sphere. To work this out, we redefinitionof two complexes for these surfaces.

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