Central extensions and bounded cohomology

Frigerio, Roberto; Sisto, Alessandro

Annales Henri Lebesgue, Tome 6 (2023), pp. 225-258

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.

Gersten a montré qu’une extension centrale d’un groupe de type fini est quasi-isométriquement triviale lorsque sa classe d’Euler est bornée. On dit qu’un groupe $G$ de type fini vérifie la propriété QITB (quasi-isométriquement trivial implique borné) si la classe d’Euler de toute extension centrale quasi-isométriquement triviale de $G$ est bornée. Nous exhibons un groupe de type fini $G$ qui ne satisfait pas la propriété QITB. Cela répond à une question de Neumann et Reeves et fournit des réponses partielles à des questions reliées de Wienhard et Blank. Nous prouvons également que la propriété QITB est satisfaite par une grande classe de groupes, contenant les groupes moyennables, les groupes d’Artin à angle droit, les groupes relativement hyperboliques avec des sous-groupes périphériques moyennables et les groupes de 3-variétés.

Enfin, nous montrons que la propriété QITB est valable pour tout groupe de présentation finie si une conjecture de Gromov sur les primitives bornées de formes différentielles est également valable.

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