Finitely generated groups with polynomial index growth

Finitely generated groups with polynomial index growth

We prove that a finitely generated soluble residually finite group has polynomial index growth if and only if it is a minimax group. We also show that if a finitely generated group with PIG is residually finite-soluble then it is a linear group. These results apply in particular to boundedly generated groups; they imply that every infinite BG residually finite group has an infinite linear quoti...

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.

Finitely generated groups with automatic presentations

A structure is said to be computable if its domain can be represented by a set which is accepted by a Turing machine and if its relations can then be checked using Turing machines. Restricting the Turing machines in this definition to finite automata gives us a class of structures with a particularly simple computational structure; these structures are said to have automatic presentations. Give...

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...