In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. addition, prove that same works relative generated We also show every wreath product free abelian group finite rank with can be embedded into exponential function.

We investigate the algebraic Kand L-theory of the group ring RG, where G is a hyperbolic or virtually finitely generated abelian group and R is an associative ring with unit.

Definition 1.1 (Direct Products). Let {Gα}α∈I be a collection of groups indexed by an index set I. We may form the Cartesian product ∏ α∈I Gα. The elements of this Cartesian product can be denoted by tuples (aα)α∈I . We refer to the entry aα as the αth component of this tuple. We define a multiplication on this Cartesian product componentwise, i.e., (aα) ⋆ (bα) = (aα ⋆α bα) where ⋆α is the grou...

We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and series holonomic, rational in case. In addition, we language geodesics blind multicounter.

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free...

In this paper we show that there exists an uncountable family of finitely generated simple groups with the same positive theory as any non-abelian free group. particular, these have infinite w-verbal width for all proper words w.

We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the Muller-Schupp theorem. More generally, if G is ...