On FC-central extensions of groups of intermediate growth

Abstract

It is shown that FC-central extensions retain sub-exponential volume growth. A large collection of FC-central extensions of the first Grigorchuk group is provided by the constructions in the works of Erschler [Ers06] and Kassabov-Pak [KP13]. We show that in these examples subgroup separability is preserved. We introduce two new collections of extensions of the Grigorchuk group. One collection gives first examples of intermediate growth groups with centers isomorphic to Z ∞ ; and the other provides groups with prescribed oscillating intermediate growth functions.

Extensions of hyperbolic groups have locally uniform exponential growth

“…Hence Q is quasi-isometric to ⟨φ(T )⟩ and has sub-exponential growth. And ⟨T ⟩ is quasi-isometric to a central extension of Q and so also has sub-exponential growth by [Zhe20].…”