We construct a geometric model for the mapping class group $\mathcal{M}\mathcal{C}\mathcal{G}$ of a non-exceptional oriented surface S of genus g with k punctures and use it to show that the action of $\mathcal{M}\mathcal{C}\mathcal{G}$ on the compact metrizable Hausdorff space of complete geodesic laminations for S is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of $\mathcal{M}\mathcal{C}\mathcal{G}$ .