Highly symmetric hypertopes

TL;DR

This paper explores incidence geometries generalising polytopes, focusing on symmetry properties like regularity and chirality, and characterizes their automorphism groups as C-groups and C+-groups.

Contribution

It introduces the concept of chirality in these geometries and proves that their automorphism groups are C-groups or C+-groups depending on regularity or chirality.

Findings

Automorphism groups of regular geometries are C-groups.

Automorphism groups of chiral geometries are C+-groups.

Chirality is a natural asymmetry property in incidence geometries.

Abstract

We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytopes theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be $C$-groups in the regular case and $C^+$-groups in the chiral case.