Residual finiteness growth in virtually abelian groups (Q6588201)

A group \(G\) is residually finite if, for every \(1 \not = g \in G\), there is a normal subgroup \(N\) of \(G\) such that \(g \not \in N\). \textit{K. Bou-Rabee} [J. Algebra 323, No. 3, 729--737 (2010; Zbl 1222.20020)] studied the asymptotic behavior of the (normal) residual finiteness growth of a finitely generated residually finite group, that is of the function \(\mathrm{RF}_{G}: \mathbb{N} \rightarrow \mathbb{N}\) defined as the minimal function such that, if \(\|g\|_{G} \leq r\), then there exists \(N \trianglelefteq{G}\) such that \(|G/N| \leq \mathrm{RF}_{G}(r)\). Here \(\|g\|_{G}\) denotes a fixed word norm on \(G\), induced by a finite generating set \(S\), so satisfying \(\|g\|_{G} \leq r\) if and only if \(g\) can be written as a product of at most \(r\) elements in \(S \cup S^{-1}\).\N\NThe main result in this paper is the following:\N\NTheorem 1.1: Let \(G\) be a finitely generated virtually abelian group, then \(\mathrm{RF}_{G}\) equals \(\log^{k}\) for some \(k \geq 0\).\N\NHere, \(k\) has an explicit form, depending on an induced integral representation. Indeed, let \(K\) be any abelian torsion-free normal subgroup of \(G\) of finite index. If the rank of \(K\) is \(m\), then the group \(H = G/K\) acts via conjugation on \(K\), leading to a representation \(\phi : H \rightarrow \mathrm{Aut}(K) \simeq \mathrm{GL}(m,\mathbb{Z})\). In the case of crystallographic groups, the representation \(\phi\) is equal to the holonomy representation, getting its name from its geometric interpretation. Over \(\mathbb{C}\), the representation \(\phi\) decomposes into irreducible subrepresentations of dimensions \(\leq m\). The authors establish the following refinement of the previous result: \N\NTheorem 1.2: Let \(G\) be a virtually abelian group with finite-index torsion-free abelian subgroup of rank \(m\). Then \(\mathrm{RF}_{G}\) equals \(\log^{k}\), where \(0 \leq k \leq m\) is the maximal dimension of the irreducible subrepresentations of \(\phi\) over \(\mathbb{C}\).