Aspects of growth in finitely generated groups

Abstract

In this thesis we study various variants of word growth in finitely generated groups, focussing on conjugacy growth. For virtually abelian groups, we prove that the conjugacy growth series, coset growth series (for any subgroup) and relative growth series of any subgroup are rational for any choice of finite weighted generating set. We draw together work of Stoll and Babenko to produce asymptotic estimates of the conjugacy growth of class 2 nilpotent groups whose derived subgroup is infinite cyclic. These results have implications for the associated series. We also study the Baumslag-Solitar groups of the form BS(1, k), proving that they have transcendental conjugacy growth series with respect to their standard generating sets, providing explicit formulae for their conjugacy growth series, and calculating their growth rates.