The curve complex has dead ends

TL;DR

This paper reveals a previously unnoticed local pathology in the curve graph of a surface, showing the existence of dead ends and double dead-ends in geodesics, with each dead end having depth 1.

Contribution

It identifies and proves the existence of dead ends and double dead-ends in the curve complex, a novel local pathology in the graph's structure.

Findings

Existence of dead ends in the curve graph

Presence of double dead-ends

Each dead end has depth 1

Abstract

It is proved that the curve graph $C^1(\Sigma)$ of a surface $\Sigma_{g,n}$ has a local pathology that had not been identified as such: there are vertices $\alpha,\beta$ in $C^1(\Sigma)$ such that $\beta$ is a dead end of every geodesic joining $\alpha$ to $\beta$. It also has double dead-ends. Every dead end has depth 1.