sectional curvature is bounded away from zero, one can relate the curvature of the space to the hyperbolicity constant of the space. However, the sectional curvature of Teichmüller sapce is not bounded away from zero [@Huang]. Therefore, this technique cannot be used.

**Acknowledgments:** I would like to thank my advisor, Jeff Brock, for suggesting this problem, support, and helpful conversations. I’d also like to thank Tarik Aougab and Peihong Jiang for helpful conversations.

Preliminaries
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Hyperbolicity
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Assume $\Gamma$ is a connected graph which we equip with the metric where each edge has length 1. We give two definitions of a graph being hyperbolic. A triangle in $\Gamma$ is $k$-*centered* if there exists a vertex $c \in \Gamma$ such that $c$ is distance $\leq k$ from each of its three sides. $\Gamma$ is $k$-*centered hyperbolic* if all geodesic triangles (triangles whose edges are geodesics) are $k$-centered. We say a triangle in $\Gamma$ is $\delta$-*thin* if each side of the triangle is contained in the $\delta$-neighborhood of the other two sides for some $\delta \in {\mathbb{R}}$. A graph is $\delta$-*thin hyperbolic* if all geodesic triangles are $\delta$-thin. Note that $\delta$-thin hyperbolic and $k$-centered hyperbolic are equivalent up to a linear factor 