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 $g=1$.