Conjugacy growth series of groups

In this thesis we study the conjugacy growth series of several group constructions in terms of

the standard and the conjugacy growth series of the building groups, with a specific generating

set. This includes (1) groups of the form G L when L admits a Cayley graph that is a tree,

(2) graph products, (3) a specific free product of Z ∗ Z with amalgamation over Z, and (4)

some HNN-extensions of graph products over isomorphic subgraph products. For all the groups

mentioned we prove that the radius of convergence of the conjugacy growth series is the same

as the radius of convergence of the standard growth series. We give an explicit formula for the

conjugacy growth series of the groups G Z, G (C 2 ∗ C 2 ), of the graph products, of a specific

free product of Z ∗ Z with amalgamation over Z, of the HNN-extension of graph products over

isomorphic subgraph products based on disjoint subgraphs, and for an HNN-extension of a group

of the form H ∗ H over itself by swapping the factor groups. We also prove at the end that for

two infinite cardinals κ 1 and κ 2 with κ 1 < κ 2 , there exists a group of cardinality κ 2 , with κ 1 for

the cardinality of its set of conjugacy classes.