Optimal connectivity results for spheres in the curve graph of low and medium complexity surfaces

TL;DR

This paper investigates the connectivity properties of spheres in the curve graph of certain low and medium complexity surfaces, providing new results on their connectedness and component classification.

Contribution

It establishes the connectivity of spheres of any radius in specific surfaces and classifies components of radius 2 spheres in others, answering a question posed by Wright.

Findings

Spheres of any radius are connected in the curve graph of $oldsymbol{ ext{Σ}_{2,0}}$, $oldsymbol{ ext{Σ}_{1,3}}$, and $oldsymbol{ ext{Σ}_{0,6}}$.

Union of two consecutive spheres is connected in $oldsymbol{ ext{Σ}_{0,5}}$ and $oldsymbol{ ext{Σ}_{1,2}}$.

Connected components of radius 2 spheres are classified for $oldsymbol{ ext{Σ}_{0,5}}$ and $oldsymbol{ ext{Σ}_{1,2}}$.

Abstract

Answering a question of Wright, we show that spheres of any radius are always connected in the curve graph of surfaces $\Sigma_{2,0}, \Sigma_{1,3},$ and $\Sigma_{0,6}$, and the union of two consecutive spheres is always connected for $\Sigma_{0, 5}$ and $\Sigma_{1,2}$. We also classify the connected components of spheres of radius 2 in the curve graph of $\Sigma_{0, 5}$ and $\Sigma_{1,2}$.