TY - JOUR

T1 - On the growth of torsion in the cohomology of arithmetic groups

AU - Ash, A.

AU - Gunnells, P. E.

AU - McConnell, M.

AU - Yasaki, D.

N1 - Funding Information:
The authors thank the Banff International Research Station and Wesleyan University, where some work was carried out on this paper. A. A. was partially supported by NSA grant H98230-09-1-0050. P. G. was partially supported by NSF grants DMS 1101640 and 1501832. D. Y. was partially supported by NSA grant H98230-15-1-0228. This manuscript is submitted for publication with the understanding that the United States government is authorized to produce and distribute reprints.
Publisher Copyright:
© Cambridge University Press 2018.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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.

AB - 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.

KW - Galois representations

KW - cohomology of arithmetic groups

KW - torsion in cohomology

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U2 - 10.1017/S1474748018000117

DO - 10.1017/S1474748018000117

M3 - Article

AN - SCOPUS:85044221839

SN - 1474-7480

VL - 19

SP - 537

EP - 569

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

IS - 2

ER -