Two theorems on finitely generated groups

Finitely Generated Abelian Groups

Finitely Generated Semiautomatic Groups

The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups and in particular shows that every finitely generated group of nilpotency class 3 is semiautomatic.

On finitely generated submonoids of free groups

We prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups. We also prove that it is decidable whether or not a finitely generated submonoid of a free group is graded, and solve the homomorphism and isomorphism problems for graded submonoids of free groups. This generalizes earlier results for submonoids of free monoids. MSC: 20E05; 20...

On sequences of finitely generated discrete groups

We consider sequences of discrete subgroups Γi = ρi(Γ) of a rank 1 Lie group G, with Γ finitely generated. We show that, for algebraically convergent sequences (Γi), unless Γi’s are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of G. In the case of divergent sequences (Γi) we show that the resulti...

On manifolds with finitely generated homotopy groups

Let G be an infinite group which is finitely presented. Let X be a finite CW−complex of dimension q whose fundamental group is Z × G. We prove that for some i ≤ q the homotopy group πi(X) is not finitely generated. Let M be a manifold of dimension n whose fundamental group is Zn−2×G. Then the same conclusion holds (for some i ≤ maxn2 ] , 3 } ) unless M is an Eilenberg-McLane space. In particula...