ON THE GROWTH OF TORSION IN THE COHOMOLOGY OF ARITHMETIC GROUPS

Let G be a semisimple Lie group with associated symmetric space D, and let 0 G be a cocompact arithmetic group. Let L be a lattice inside a Z0-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 Hi.0k IL /tors as 0k ranges over a tower of congruence subgroups of 0. In particular, they conjectured that the ratio log jHi.0k IL /torsj=T0 V 0k U should tend to a nonzero limit if and only if i D.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 GLn.Z / for n D 3; 4; 5 and GL2.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.