Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces

As mentioned in the comments, if the limit is a circle, then by the Yamaguchi fibration theorem, $M_i$ fibers over the circle, and hence it is a torus (or Klein bottle in the non-orientable case).

If the limit is a segment, then $M_i$ is $S^2$ for all large $i$. One (somewhat heavy handed) way to see this is to apply Corollary 0.4 of Shioya-Yamaguchi's paper. Indeed, the product of $M_i$ and a unit circle, collapses to the cylinder. Hence in Corollary 0.4 we have $g=0$ and $k=2$. Hence for large $i$ the fundamental group of $M_i\times S^1$ is a free product of $\mathbb Z$ and finitely many finite cyclic groups. Such a group cannot have a center unless all the finite cyclic groups are trivial. The circle factor of $M_i\times S^1$ is central in the fundamental group, so $\pi_1(M_i\times S^1)=\mathbb Z$. Hence $M_i$ is a sphere (for large $i$).

This argument uses orientability of $M_i$ because Shioya-Yamaguchi only deal with orientable manifolds.

Consider a shortest closed geodesic $\gamma$ on the surface of length sys, and the normal exponential map of $\gamma$. Using the lower curvature bound, we obtain an upper bound on the total area as $\text{sys}\cdot \sinh(D)$ where $D$ is the diameter. This follows just by applying Rauch bounds on Jacobi fields (this is an ingredient in the proof of Toponogov). Therefore the systole is bounded below by $ \frac{\text{area}}{\sinh D}$ and the area is bounded below by Gauss-Bonnet. Furthermore the filling radius is bounded below by the 1/6 of the systole by Gromov's inequality. The least Gromov-Hausdorff distance to a graph is bounded below by the filling radius. We therefore get a quantitative lower bound and not merely nonexistence of Yamaguchi-type collapse.

This proves that hyperbolic surfaces of curvature bounded below by $-1$ with diameter bounded above by $D$ cannot collapse so that a Gromov-Hausdorff limit is necessarily 2-dimensional.

For more details see https://arxiv.org/abs/1604.06782

I remembered another reason why closed surfaces of negative Euler characteristic cannot collapse under a lower bound on sectional curvature.

Much more is true: if a sequence of $n$-dimensional closed manifolds $M_i$ of Ricci curvature $\ge -k^2$ Gromov-Hausdorff converges to a compact space of (Hausdorff) dimension $<n$, then the simplical volume of $M_i$ is zero for large $i$.

Indeed, on the bottom of p.244 of of Gromov's Volume and bounded cohomology he shows that a lower Ricci curvature bound implies that the simplicial volume is bounded above in terms of volume and the dimension. If a sequence of manifolds converges to a space of dimension $<n$ under the lower Ricci bound, then their volume goes to zero; this is due to Colding, as mentioned e.g. on p.91 of Aspects of Ricci Curvature.

The simplicial volume of a closed surface of negative Euler charactersistic $\chi$ is $2|\chi|$, see p.217 in Gromov's paper, so it cannot collapse under a lower curvature bound. There are of course many high-dimensional manifolds of nonzero simplicial volume.