From Group Operations to Geometric Structures: Amalgamations, HNN-Extensions, and Twisting in Coset Geometries

TL;DR

This paper extends classical group-theoretic constructions to coset geometries, enabling new ways to combine incidence geometries while preserving key properties, and explores applications to Shephard groups and potential foundational theories.

Contribution

It introduces a general framework for gluing coset geometries using amalgamations, HNN-extensions, and twisting, advancing the understanding of their structure and applications.

Findings

Framework for combining coset geometries preserving flag-transitivity

Analysis of Shephard groups and their complexes

Indications of a Bass-Serre theory for coset geometries

Abstract

Coset incidence geometries, introduced by Jacques Tits, provide a versatile framework for studying the interplay between group theory and geometry. In this article, we build upon that idea by extending classical group-theoretic constructions (amalgamated products, HNN-extensions, semi-direct products, and twisting) to the setting of coset geometries. This gives a general way to glue together incidence geometries in various ways. This provides a general framework for combining or gluing incidence geometries in different ways while preserving essential properties such as flag-transitivity and residual connectedness. Using these techniques, we analyze families of Shephard groups, which generalize both Coxeter and Artin-Tits groups, and their associated simplicial complexes. Our results also point to the existence of a Bass-Serre theory for coset geometries and of a fundamental geometry of a graph of coset geometries.