Three amalgams with remarkable normal subgroup structures

Abstract

We construct three groups $Λ_1$, $Λ_2$, $Λ_3$, which can all be decomposed as amalgamated products $F_9 \ast_{F_{81}} F_{9}$ and have very few normal subgroups of finite or infinite index. Concretely, $Λ_1$ is a simple group, $Λ_2$ is not simple but has no non-trivial normal subgroup of infinite index, and $Λ_3$ is not simple but has no proper subgroup of finite index.