{
  "id": "2003.01146",
  "version": "v1",
  "published": "2020-03-02T19:14:57.000Z",
  "updated": "2020-03-02T19:14:57.000Z",
  "title": "Central extensions and bounded cohomology",
  "authors": [
    "Roberto Frigerio",
    "Alessandro Sisto"
  ],
  "comment": "31 pages",
  "categories": [
    "math.GR",
    "math.AT",
    "math.GT"
  ],
  "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 $G$ 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. We also show that Property QITB holds for every finitely presented group if and only if a conjecture by Gromov on bounded primitives of differential forms holds as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-02T19:14:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finitely generated group",
      "bounded cohomology",
      "property qitb holds",
      "euler class",
      "differential forms holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}