VIRTUAL FINITE QUOTIENTS OF FINITELY GENERATED GROUPS J.O.Button

Quotients of Nilalgebras and Their Associated Groups

We show that every finitely generated nilalgebra having nilalgebras of matrices is a homomorphic image of nilalgebras constructed by the Golod method (Golod, 1965 and 1969). By applying some elements of module theory to these results, we construct over any field non-residually finite nilalgebras and Golod groups with non-residually finite quotients. This solves Šunkov’s problem (Kourovka Notebo...

On Property (fa) for Wreath Products

We characterize permutational wreath products with Property (FA). For instance, the standard wreath product A o B of two nontrivial countable groups A,B has Property (FA) if and only if B has Property (FA) and A is a finitely generated group with finite abelianisation. We also prove an analogous result for hereditary Property (FA). On the other hand, we prove that many wreath products with here...

Controlled embeddings into groups that have no non-trivial finite quotients

If a class of finitely generated groups G is closed under isometric amalgamations along free subgroups, then every G ∈ G can be quasi-isometrically embedded in a group Ĝ ∈ G that has no proper subgroups of finite index. Every compact, connected, non-positively curved space X admits an isometric embedding into a compact, connected, non-positively curved space X such that X has no non-trivial fin...

An Isomorphism for the Grothendieck Ring of a Hopf Algebra Order

If G is a finite abelian group, R is a principal ideal domain with field of quotients an algebraic number field K which splits G, and if A is a Hopf algebra order in KG, then the Grothendieck ring of the category of finitely generated A-modules is isomorphic to the Grothendieck ring of the category of finitely generated RG-modules. The Grothendieck group a£ of the category of finitely generated...

Ring With All Finitely Generated Modules Retractable