Covers of generalized quadrangles

TL;DR

This paper develops a general theory for factorizing 2-covers of finite classical generalized quadrangles, addressing an open problem and exploring related geometrical structures and isomorphism issues.

Contribution

It introduces a new framework for cover factorization in generalized quadrangles and solves an existing open problem about 2-covers.

Findings

New results on semipartial geometries from θ-covers

Solution to the isomorphism problem for covers

Development of a general theory of cover factorization

Abstract

We solve a problem posed by Cardinali and Sastry [2] about factorization of $2$-covers of finite classical generalized quadrangles. To that end, we develop a general theory of cover factorization for generalized quadrangles, and in particular we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semipartial geometries coming from $\theta$-covers, and consider related problems.