Realizations and properties of $3$-spherical Curtis-Tits Groups and Phan groups

TL;DR

This paper proves the existence of all 3-spherical Curtis-Tits and Phan groups, explores their geometric and algebraic properties, and provides examples illustrating their complexity and diversity.

Contribution

It establishes the existence of all classified 3-spherical Curtis-Tits and Phan groups and analyzes their properties, including hyperbolicity and simple quotients.

Findings

Non-orientable Curtis-Tits groups are acylindrically hyperbolic.

Some non-orientable groups have quotients that are finite simple Lie type groups.

Orientable groups are almost simple, contrasting with non-orientable cases.

Abstract

In this note we establish the existence of all Curtis-Tits groups and Phan groups with $3$-spherical diagram as classified previously and investigate some of their geometric and group theoretic properties. Whereas it is known that orientable Curtis-Tits groups with spherical or non-spherical and non-affine diagram are almost simple, we show that non-orientable Curtis-Tits groups are acylindrically hyperbolic and therefore have infinitely many infinite-index normal subgroups. However, we also provide concrete examples of non-orientable Curtis-Tits groups whose quotients are finite simple groups of Lie type.