We construct three groups Λ1, Λ2, Λ3, which can all be decomposed as amalgamated products F9∗F81F9 and have very few normal subgroups of finite or infinite index. Concretely, Λ1 is a simple group, Λ2 is not simple but has no non-trivial normal subgroup of infinite index, and Λ3 is not simple but has no proper subgroup of finite index.

A nitely presented group G is called metabolic if H 2 (1) (G; A) = 0 for all normed abelian coeecient groups A; it is equivalent to require that a Cayley graph for G admit a thin combing. Metabolicity is stronger than hyperbolicity, and it is an open problem whether all hyperbolic groups are metabolic. We prove that if H is metabolic, then H ha;b]i F(a; b) is also metabolic, where F(a;b) is fre...

For n ≥ 2 let ∆ be a Dynkin diagram of rank n and let I = {1, . . . , n} be the set of labels of ∆. A group G admits a weak Phan system of type ∆ over C if G is generated by subgroups Ui, i ∈ I , which are central quotients of simply connected compact semisimple Lie groups of rank one, and contains subgroups Ui,j = 〈Ui, Uj〉, i 6= j ∈ I , which are central quotients of simply connected compact s...

On the classification of distance-transitive graphs