On the growth of torsion in the cohomology of arithmetic groups

A. Ash, P. E. Gunnells, M. McConnell, D. Yasaki

Research output: Contribution to journalArticle

Abstract

Let be a semisimple Lie group with associated symmetric space , and let be a cocompact arithmetic group. Let be a lattice inside a -module arising from a rational finite-dimensional complex representation of. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup as ranges over a tower of congruence subgroups of. In particular, they conjectured that the ratio should tend to a nonzero limit if and only if and is a group of deficiency. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including for and 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.

Original languageEnglish (US)
Pages (from-to)537-569
Number of pages33
JournalJournal of the Institute of Mathematics of Jussieu
Volume19
Issue number2
DOIs
StatePublished - Mar 1 2020

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Galois representations
  • cohomology of arithmetic groups
  • torsion in cohomology

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