2023/09/22 by Jonas Deré, Deré, Jonas, Joren Matthys +1 · 3 citations
Computer Science · Mathematics · #Cellular Automata and Applications #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2309.12831
openalex publication_date 2023/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A group G is called residually finite if for every non-trivial element g ∈ G, there exists a finite quotient Q of G such that the element g is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient Q depending on the complexity of the element g, for example by using the word norm ‖g‖G if the group G is assumed to be finitely generated. The residual finiteness growth RFG: ℕ → ℕ is then defined as the smallest function such that if ‖g‖G ≤ r, there exists a morphism φ: G → Q to a finite group Q with |Q| ≤ RFG(r) and φ(g) ≠ eQ. Although upper bounds have been established for several classes of groups, exact asymptotics for the function RFG are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals logk for some k ∈ ℕ, where the value k is given by an explicit expression. As an application, we show that for every m ≥ 1 and every 1 ≤ k ≤ m, there exists a group G containing a normal abelian subgroup of rank m and with RFG ≈ logk.