Groups acting simply transitively on hyperbolic buildings

TL;DR

This paper constructs and classifies groups acting simply transitively on hyperbolic triangular buildings with the smallest thick generalized quadrangle, revealing 23 torsion free and 168 torsion groups with specific properties.

Contribution

It provides a complete classification of groups acting on hyperbolic buildings with minimal thickness and specific local structures, including both torsion and torsion free cases.

Findings

Identified 23 torsion free groups acting on hyperbolic buildings.

Found 168 torsion groups with similar actions.

Demonstrated existence of both torsion and torsion free groups on the same building.

Abstract

We construct and classify all groups, given by triangular presentations associated to the smallest thick generalized quadrangle, that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification shows 23 non-isomorphic torsion free groups (obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the Euclidean case, we find both torsion and torsion free groups acting on the same building.