Is there an example of an infinite, residually nilpotent, torsion group that does not contain an infinite abelian subgroup.
I am looking for particular example or a reasoning why such a group cannot exist.
The motivation is the following: I am trying to construct a relatively one-ended relatively hyperbolic group whose peripheral subgroups are residually nilpotent but not nilpotent and which does not admit parabolic splitting.
So I begin by constructing a one-ended relatively hyperbolic group $(G, \mathbb{P})$ with whose peripheral subgroups are residually nilpotent but not nilpotent (construction is due to Cordes and Hume). Now use an accessibility theorem due to Bowditch and consider the peripheral splitting of $(G, \mathbb{P})$. A component $H$ of this splitting is again relatively hyperbolic with respect to the infinite subgroups of the form $Q = H \cap P$ where $P \in \mathbb{P}$. Moreover $(H, \mathbb{Q})$ is relatively one-ended and does not admit further parabolic splitting (due to Bowditch's theorem). I need the the members of $\mathbb{Q}$ to be residually nilpotent and not nilpotent to claim that $(H, \mathbb{Q})$ is the desired example. Hence I need each infinite $H \cap P$ to be residually nilpotent and not nilpotent.