TY - THES ID - 136444811 TI - Residual Finiteness Growth for group extensions AU - Matthys, Joren AU - Deré, Jonas AU - KU Leuven. Faculteit Wetenschappen. Opleiding Master in de wiskunde (Leuven) PY - 2022 PB - Leuven KU Leuven. Faculteit Wetenschappen DB - UniCat UR - https://www.unicat.be/uniCat?func=search&query=sysid:136444811 AB - A group G is a set with an operation such as addition. It suffices some axioms. The integers Z are an example of a group with addition. Each group has a zero element. Suppose we have a group H which lies in the set/group G, we call H a subgroup. The multiples of three 3Z is for example a subgroup of Z. We can now consider a new group where H accounts for one element, namely the new zero element. The group consists of H and its translations in G. Note that this construction is not always meaningful. The new group is for example the group of three elements consisting if the zero element 3Z and the multiples of three plus one (3Z+1) or plus two (3Z+2). This construction is called a quotient group. The quotient in our example is finite, since there are only three elements. Suppose G has the following property: for each non-zero element g in G, we can find a finite quotient such that g is not part of the zero element H. If so, we say G is residually finite. In this thesis we pose the next question: If G is residually finite and g is non-zero, how small can we take a finite quotient such that g does not lie in the zero-element of that quotient? If we take the number six, then it is part of the zero element 3Z of the finite quotient above. In fact, a finite quotient with six not in its zero element will need to have at least four elements. This is also sufficient: take the multiples of four (4Z) as H, we obtain a finite quotient of four elements, namely (4Z, 4Z+1, 4Z+2, 4Z+3), and six is no multiple of four. The answer to this question is captured in the notion of residual finiteness growth, a function. We calculate this function for some classes of groups. The groups we discuss are linear groups: groups that can be seen as subgroups of the invertible matrices over some field (with multiplication). Specifically, we consider groups that can be build from the m-dimensional integers Z^m via some construction named extension. We focus primarily on semidirect product Z^m with Z and finite extensions of Z^m. For the semidirect products we give a classification for the case where m is equal to one or two. For the finite extensions, we develop some tools which we then apply to find the residual finiteness growth on some crystallographic groups. ER -