Computability of Følner sets

Abstract

ABSTRACT. We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich groups, finitely presented solvable groups with unsolvable Word Problem, have computable Følner sets. We also prove computability of Følner sets for extensions -with subrecursive distortion functions-of amenable groups with solvable Word Problem by finitely generated groups with computable Følner sets . Moreover we obtain some known and some new upper bounds for the Følner function for these particular extensions.

Følner functions and the generic Word Problem for finitely generated amenable groups

“…In combination with Theorem 4.1 and the results in [5], this proves Theorem C in the Introduction. (i) Γ is amenable;…”

“…The first equality is Theorem 4.1, the other equalities follow from Theorem 3.1, the remaining relations were already proved in [4,5].…”

“…Since in general Γ has unsolvable Word Problem we "lift" the output to F X . The following notion was introduced and studied in [4,5]:…”

“…In particular for any partition P ′ of F such that P ′ ≤ P, or equivalently, such that ν 1 , ν 2 ∈ V ∈ P ′ =⇒ π Γ (ν 1 ) = π Γ (ν 2 ), using equation (4) we have (5) π…”

“…Finally, the last question that we asked in [5]: -Does subrecursivity of the Følner function imply computability of Følner sets?-can be replaced, in view of Theorem C, by the following. Question 6.3.…”