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

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 ${\Sigma }_{1,2}$.