Finding equations of the fake projective plane $(C18,p=3,\{2I\})$

TL;DR

This paper explicitly derives equations for a newly identified fake projective plane, expanding the classification and understanding of these complex algebraic surfaces.

Contribution

It introduces a novel method of obtaining equations for a new fake projective plane using cyclic covers and quotients from known examples.

Findings

Explicit equations for the fake projective plane $(C18,p=3, ext{\{2I\}})$

Methodology involving cyclic covers and quotients

Extension of the classification of fake projective planes

Abstract

We find explicit equations of a new pair of fake projective planes, labeled by $(C18,p=3,\{2I\})$ in the Cartwright-Steger classification. Our method involves starting with known equations of a commensurable fake projective plane $(C18,p=3,\emptyset,d_3 D_3)$ and working through a chain of cyclic covers and quotients to get to the new one.