# Uniform quasiconvexity of the disc graphs in the curve graphs

## TL;DR

This paper proves that the disc graph of a surface boundary component in a 3-manifold is uniformly quasiconvex within the curve graph, providing a new proof that avoids train tracks.

## Contribution

It establishes a universal quasiconvexity constant for disc graphs in curve graphs without using train tracks, advancing understanding of their geometric properties.

## Key findings

- Disc graph is K-quasiconvex in the curve graph.
- Universal constant K exists for all such surfaces.
- Proof avoids the use of train tracks.

## Abstract

We give a proof that there exists a universal constant $K$ such that the disc graph associated to a surface $S$ forming a boundary component of a compact, orientable 3-manifold $M$ is $K$-quasiconvex in the curve graph of $S$. Our proof does not require the use of train tracks.