Residual finiteness growth in virtually abelian groups

A group G is called residually finite if for every non-trivial element g is an element of 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 parallel to g parallel to 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 parallel to g parallel to G <= r, there exists a morphism phi:G -> Q to a finite group Q with |Q|<= RFG(r) and phi(g)not equal eQ.<br /> 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 log(k) for some k is an element of 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 approximate to log(k).