FINITELY GENERATED GROUPS AND FIRST-ORDER LOGIC

Finitely Generated Groups and First-order Logic

We prove that the following classes of finitely generated (f.g.) groups have Π1–complete first–order theories: all f.g. groups, the n–generated groups, and the strictly n–generated groups (n > 2). Moreover, all those theories are distinct. Similar techniques show that quasi-finitely axiomatizable (QFA) groups have a hyperarithmetical word problem, where a f.g. group is QFA if it is the only f.g...

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.

Powers in Finitely Generated Groups

In this paper we study the set Γn of nth-powers in certain finitely generated groups Γ. We show that, if Γ is soluble or linear, and Γn contains a finite index subgroup, then Γ is nilpotent-by-finite. We also show that, if Γ is linear and Γn has finite index (i.e. Γ may be covered by finitely many translations of Γn), then Γ is soluble-by-finite. The proof applies invariant measures on amenable...

Combinatorics of Finitely Generated Groups

of the Dissertation Combinatorics of Finitely Generated Groups