Mathematics Colloquia and Seminars

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Let $S_g$ be a closed oriented surface of genus $g \geq 2$ and $\mathcal{C}^1(S_g)$ be its curve complex—vertices are
homotopy classes of essential simple closed curves with two vertices sharing an edge if they have disjoint representatives. It is known that
$\mathcal{C}(S_g)$ is path connected , and the
distance, $d(\alpha , \beta)$, between two vertices $\alpha , \beta \in \mathcal{C}^1(S)$ is just the minimal count of the number of edges in an
edge-path between $\alpha$ and $\beta$. One can also consider, $ i(\alpha , \beta)$, the minimal intersection between curve representatives of
$\alpha$ and $\bet$. This talk discusses how $i(\alpha , \beta)$ will grow as $d(\alpha, \beta)$ grows. This is joint work with Dan Margalit.

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