Unravelled Abstract Regular Polytopes

TL;DR

This paper introduces the concept of unravelled abstract regular polytopes and demonstrates their existence in certain automorphism groups of SL_3(q), providing numerous examples and detailed properties.

Contribution

It defines unravelled abstract regular polytopes and proves their existence in specific groups, expanding understanding of polytope symmetries in algebraic structures.

Findings

Existence of unravelled polytopes in SL_3(q) with transpose inverse automorphism

Construction of many small examples of such polytopes

Detailed analysis of properties of these polytopes

Abstract

This paper introduces the notion of an unravelled abstract regular polytope, and proves that $\SL_3(q) \rtimes <t>$, where $t$ is the transpose inverse automorphism of $\SL_3(q)$, possesses such polytopes for various congruences of $q$. A large number of small examples of such polytopes are given, along with extensive details of their various properties.