Consider a connected hyperbolic $2$-manifold $M$ with cusps. Consider a simple closed geodesic in $M$, which dissects $M$ into two components. Assume that one of the components contains exactly two cusps. Can you prove me, that this component is conformal to the hyperbolic disk with two points removed?
2026-04-01 01:11:09.1775005869
Simple closed geodesic around two hyperbolic cusps.
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This isn't true. After you cut, a component with two cusps may have any genus at all.
I shall give a specific counterexample. Let $P$ be a regular ideal hyperbolic $14$-gon, and let $S$ be the hyperbolic surface obtained by gluing together pairs of sides of $P$, as shown in the following figure:
It is easy to check that $S$ is a genus-two surface with four cusps, indicated by the red, blue, yellow, and purple dots. The dashed blue line is a closed geodesic, and cutting along this geodesic separates the surface into two components, each of which is isomorphic to a torus with two cusps and a disk removed.