# Groups with infinitely many finite conjugacy classes

I've been coming across the condition "IMFCC: having infinitely many finite conjugacy classes" often in recent times and I was wondering if there is any serious difference between having "IMFCC" and having an infinite centre. More precisely:

$\mathbf{Question:}$ Is there a finitely generated group $\Gamma$ which has IMFCC and is not quasi-isometric to a group with infinite centre?

In case the above might be too hard, here is a possible variant: recall a subgroup $M<G$ is almost-malnormal if $\forall g \in G \setminus M,$ the subgroup $gMg^{-1} \cap M$ is finite

$\mathbf{Question~(variant):}$ Let $\Gamma$ be a finitely generated group which has "IMFCC". Is there a subgroup $H < \Gamma$ so that $H$ has infinite centre and if $K$ is another subgroup such that $H < K < \Gamma$ then $K$ is not almost-malnormal?

I do not claim there is any link between quasi-isometry and almost-malnormal subgroups, it's just a geometric and an algebraic condition which seem to express the fact that groups are similar.

• IFMCC is better known as "having infinite FC-center" – YCor Mar 29 '16 at 12:17
• It is not part of your questions, but in relation to your first question, I should mention that having infinite FC-center is not a quasi-isometry invariant, see Theorem 2.13 in normalesup.org/~cornulier/commable.pdf – YCor Mar 29 '16 at 12:21
• About the questions: BH Neumann in 1937 constructed a family of groups $G$ with infinite FC-center of the form $N=\bigoplus H_n\le G$, with each $H_n$ normal in $G$ and isomorphic to some finite alternating group (with $|H_n|<|H_{n+1}|$ for all $n$), and with $G/N$ isomorphic to the group of permutations of $\mathbf{Z}$ generated by even finite permutations and translations. This could be good candidates for (1). – YCor Mar 29 '16 at 12:37
• @YCor: Thanks! I did not know the official terminology... and the references are very pertinent. – ARG Mar 29 '16 at 12:51