Lengths In Semi-Free Groups.
We describe finitely generated groups universally equivalent (with constants from in the language) to a given torsion-free relatively hyperbolic group with free abelian parabolics. It turns out that, as in the free group case, the group embeds into the Lyndon’s completion of the group , or, equivalently, embeds into a group obtained from by finitely many extensions of centralizers. Conversely, every subgroup of containing is universally equivalent to . Since finitely generated...
Let G be an infinite, locally soluble group which is isomorphic to all its nontrivial normal subgroups. If G/G' has finite p-rank for p = 0 and for all primes p, then G is cyclic.