Let $\{M_i\}$ be a sequence of 2-dimensional orientable surfaces with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov compactness theorem, one can choose a subsequence converging in the Gromov-Hausdorff sense to a compact Alexandrov space with curvature at least $-1$ and Hausdorff dimension 0,1,or 2. Let us assume that the limit space has dimension 1. Then it is either circle or segment.

Whether these both possibilities (circle and segment) can be obtained in the limit, assuming that all $M_i$ have the same genus $g\geq 2$? Can one get circle for $g=0$ and segment for $g=1$?

Remark. Clearly one can get segment for $g=0$ and circle for $g=1$.