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

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.