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We study the asymptotic behavior of the solution curves of the dynamics ofspacetimes of the topological type $\Sigma_{p}\times \mathbb{R}$, $p>1$, where$\Sigma_{p}$ is a closed Riemann surface of genus $p$, in the regime of $2+1$dimensional classical general relativity. The configuration space of the gaugefixed dynamics is identified with the Teichm\"uller space($\mathcal{T}\Sigma_{p}\approx \mathbb{R}^{6p-6}$) of $\Sigma_{p}$. Utilizingthe properties of the Dirichlet energy of certain harmonic maps, estimatesderived from the associated elliptic equations in conjunction with a fewstandard results of the theory of the compact Riemann surfaces, we prove thatevery non-trivial solution curve runs off the edge of the Teichm\"uller spaceat the limit of the big bang singularity and approaches the space of projectivemeasured laminations/foliations ($\mathcal{PML}$ $\mathcal{PMF}$), the Thurstonboundary of the Teichm\"uller space.

Big-bang limit of $2+1$ gravity and Thurston boundary of Teichm\"uller space