{
  "id": "2311.00620",
  "version": "v1",
  "published": "2023-11-01T16:14:59.000Z",
  "updated": "2023-11-01T16:14:59.000Z",
  "title": "Sigma invariants for partial orders on nilpotent groups",
  "authors": [
    "Kevin Klinge"
  ],
  "comment": "21 pages, comments welcome",
  "categories": [
    "math.GR"
  ],
  "abstract": "We prove that a map onto a nilpotent group $Q$ has finitely generated kernel if and only if the preimage of the positive cone is coarsely connected as a subset of the Cayley graph for every full archimedean partial order on $Q$. In case $Q$ is abelian, we recover the classical theorem that $N$ is finitely generated if and only if $S(G,N) \\subseteq \\Sigma^1(G)$. Furthermore, we provide a way to construct all such orders on nilpotent groups. A key step is to translate the classical setting based on characters into a language of orders on $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-01T16:14:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nilpotent group",
      "sigma invariants",
      "full archimedean partial order",
      "cayley graph",
      "finitely generated kernel"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}