{
  "id": "1608.05858",
  "version": "v1",
  "published": "2016-08-20T18:31:09.000Z",
  "updated": "2016-08-20T18:31:09.000Z",
  "title": "On the growth of torsion in the cohomology of arithmetic groups",
  "authors": [
    "Avner Ash",
    "Paul E. Gunnells",
    "Mark McConnell",
    "Dan Yasaki"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let G be a semisimple Lie group with associated symmetric space D, and let Gamma subset G be a cocompact arithmetic group. Let L be a lattice inside a Z Gamma-module arising from a rational finite-dimensional complex representation of G. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup H_i (Gamma_k; L )_tors as Gamma_k ranges over a tower of congruence subgroups of Gamma. In particular they conjectured that the ratio (log |H_i (Gamma_k ; L)_tors|)/[Gamma : Gamma_k] should tend to a nonzero limit if and only if i= (dim(D)-1)/2 and G is a group of deficiency 1. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including GL_n (Z) for n=3,4,5 and GL_2 (O) for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron--Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron--Venkatesh conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-20T18:31:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F75",
      "11F67",
      "11F80",
      "11Y99"
    ],
    "keywords": [
      "cohomology",
      "rational finite-dimensional complex representation",
      "semisimple lie group",
      "cocompact arithmetic group",
      "symmetric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}