# Hyperbolic surfaces with sublinearly many systoles that fill

## TL;DR

This paper constructs hyperbolic surfaces with a small, sublinear number of systoles that fill, challenging previous lower bounds and providing new insights into the geometry of such surfaces.

## Contribution

It introduces a method to build hyperbolic surfaces with fewer filling systoles than previously thought possible, disproving existing lower bounds.

## Key findings

- Constructed hyperbolic surfaces with at most εg filling systoles
- Surfaces are critical points of the systole function with low index
- Disproved the conjectured lower bound of 2g-1 for filling systoles

## Abstract

For any $\varepsilon>0$, we construct a closed hyperbolic surface of genus $g=g(\varepsilon)$ with a set of at most $\varepsilon g$ systoles that fill, meaning that each component of the complement of their union is contractible. This surface is also a critical point of index at most $\varepsilon g$ for the systole function, disproving the lower bound of $2g-1$ posited by Schmutz Schaller.