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 ∗ 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.