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 satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of is bounded. We exhibit a finitely generated group 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.
Central extensions and bounded cohomology
Frigerio, Roberto;Sisto, Alessandro
2023-01-01
Abstract
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 satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of is bounded. We exhibit a finitely generated group 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
