To clarify, I am not looking for a classification but rather for well-researched examples of families of (finitely generated) groups generated by a single conjugacy class.
One example that fits this description are Coxeter groups $W$ such that the graph $G$ (see below for definition) is connected. If $W$ is a Coxeter group with presentation $$\langle s_1, \dots, s_n \;|\; (s_is_j)^{m_{ij}} \rangle,$$ the graph $G$ is the graph with vertices $s_1, \dots, s_n$ and edges between $s_i$ and $s_j$ whenever $m_{ij}$ is finite and odd. Then, all the $s_i$ are conjugate and thus, $W$ is generated by a single, somehow distinguished conjugacy class. The theory of Coxeter groups is well-established and easy to learn. Specific groups in this family include dihedral groups $D_m$ for odd $m$ (the groups obtained when $n=2$) and symmetry groups $S_n$ (by letting $m_{i(i+1)} = 3$ and all other $m_{ij} = 2$).
Another cool example are the braid groups $B_n$ (or more generally Artin groups for which the same criterion as above holds). In the answers, mapping class groups of surfaces were mentioned; they also fit my description perfectly.
It is this sort of example that I am looking for. Are there any constructions / groups / results that come to mind?
from Hot Weekly Questions - Mathematics Stack Exchange
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