AGMY Conjecture 7.2 on torsion growth for non-cocompact and general level families

Ascertain, for a family {Γ_j} of congruence subgroups of a fixed arithmetic group Γ with level(Γ_j) → ∞ and for each cohomological degree, whether the lim inf liminf_{j→∞} (log |H^q(Γ_j; L)_{tor}|) / [Γ : Γ_j] exists; and determine its value in the positive Q-rank setting, namely that it equals 0 unless the fundamental rank δ(G) equals 1 and the cohomological degree lies at the top of the cuspidal range.

Background

The authors cite extensive computational evidence from Ash–Gunnells–McConnell–Yasaki (AGMY) and present AGMY conjectures that generalize the Bergeron–Venkatesh prediction to non-cocompact cases and arbitrary growth of level. This conjecture focuses on lim infs along level-increasing families and specifies when nontrivial torsion growth should occur in the presence of positive Q-rank.

In particular, AGMY Conjecture 7.2 provides a framework for sequences not necessarily forming towers, and indicates that torsion growth is concentrated in the top cuspidal degree when δ(G)=1.

References

These are Conjectures 7.1 and 7.2 in . In particular, Conjecture 7.2 predicts that for a family ${\Gamma_j}{j\in\mathbb{N}}$ of congruence subgroups in a fixed arithmetic groups $\Gamma$ with $level(\Gamma_j)\to\infty$, \begin{equation} \liminf{j\to \infty}\frac{\log|Hq(\Gamma_j;L)_{tor}|}{[\Gamma\colon\Gamma_j]} \end{equation} exists. If $G$ has $Q$-rank $>0$, then the lim-inf equals zero unless $\delta(G)=1$ and $j$ is the top degree of the cuspidal range.

— On the growth of torsion in the cohomology of some arithmetic groups of $\mathbb{Q}$-rank one (2401.14205 - Mueller et al., 25 Jan 2024) in Introduction