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 - 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 -