## 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 language | English (US) |
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Pages (from-to) | 537-569 |

Number of pages | 33 |

Journal | Journal of the Institute of Mathematics of Jussieu |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2020 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

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