The Følner Sequence of the Baumslag-Solitar Groups BS(1,n)

Author : Assoc. prof. Nikolay A. Ivanov

Abstract :The Baumslag-Solitar groups BS(1,n) defined by BS(1,n) = <a,b | a-1ba = bn> are metaabelian, i.e., they are extensions of abelian groups by abelian groups. Such groups are in particular amenable. Therefore one of the equivalent conditions of amenability is met: the Følner condition: ∀ɛ>0, ∃ F ⊂ BS(1,n): ♯F < ∞ & (♯∂F)/(♯F)< ɛ where ♯F is the number of elements of a set F and ∂F is the outer boundary of F defined by ∂F = {g ɛ BS(1,n) \ F | ∃ f ɛ F & ∃ t ɛ {a,b,a-1,b-1}: tf = g} An interesting question is for every positive integer k to find the number of elements Føl(k) of the smallest set Fk that satisfies (♯∂F_k)/(♯F_k )≤ 1/k

Keywords :Følner sets in Baumslag–Solitar groups and minimal Føl(k) satisfying (|∂Fₖ|)/|Fₖ| ≤ 1/k

Conference Name :International Conference on Advances in Pure & Applied Mathematics (ICAPAM-25)

Conference Place Thessaloniki, Greece

Conference Date 31st Oct 2025