Three amalgams with remarkable normal subgroup structures

Abstract

We construct three groups ${\Lambda }_1$, ${\Lambda }_2$, ${\Lambda }_3$, 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, ${\Lambda }_1$ is a simple group, ${\Lambda }_2$ is not simple but has no non-trivial normal subgroup of infinite index, and ${\Lambda }_3$ is not simple but has no proper subgroup of finite index.

6 pages