  It was shown by Gersten that a central extension of a finitely generated
group is quasi-isometrically trivial provided that its Euler class is bounded.
We say that a finitely generated group $G$ satisfies Property QITB
(quasi-isometrically trivial implies bounded) if the Euler class of any
quasi-isometrically trivial central extension of $G$ is bounded. We exhibit a
finitely generated group $G$ which does not satisfy Property QITB. This answers
a question by Neumann and Reeves, and provides partial answers to related
questions by Wienhard and Blank. We also prove that Property QITB holds for a
large class of groups, including amenable groups, right-angled Artin groups,
relatively hyperbolic groups with amenable peripheral subgroups, and 3-manifold
groups.
  Finally, we show that Property QITB holds for every finitely presented group
if a conjecture by Gromov on bounded primitives of differential forms holds as
well.
