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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.

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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