Dice Question Streamline Icon: https://streamlinehq.com

Bergeron–Venkatesh conjecture on precise torsion growth for Q-rank 0

Determine, for a decreasing sequence of congruence subgroups {Γ_j} of an arithmetic subgroup Γ of G(Q) with intersection {1} and for each cohomological degree q, whether the limit lim_{j→∞} (log |H^q(Γ_j; L)_{tor}|) / [Γ : Γ_j] exists, and prove that it equals 0 unless the fundamental rank δ(G) equals 1 and q = (dim(G_∞/K) + 1)/2, in which case it equals c_{G,L} · vol(Γ \ (G_∞/K)), under the assumption that the Q-rank of G is 0.

Information Square Streamline Icon: https://streamlinehq.com

Background

The paper reviews known results on the growth of torsion in cohomology for arithmetic groups and presents new results in Q-rank 1 settings. In the compact (Q-anisotropic) case, Bergeron and Venkatesh established an averaged asymptotic formula and conjectured a precise limit for each cohomological degree without assuming strong acyclicity of the coefficient system. The authors restate this conjecture to contextualize their extensions beyond the co-compact case.

The symmetric space associated to G is X = G_∞/K, and δ(G) denotes the fundamental rank rank(𝔤ℂ) – rank(𝔨ℂ). The conjecture predicts a single degree of torsion growth when δ(G)=1, with a constant c_{G,L} depending on the group and local system L and proportional to vol(Γ\X).

References

Based on equl1, Bergeron and Venkatesh made a conjecture with a precise prediction of the growth of torsion Conjecture 1.3 without any assumption on $L$. The conjecture states that for each $q$ \begin{equation}\label{conj1} \lim_{j\to\infty}\frac{\log|Hq(\Gamma_j;L)_{tor}|}{[\Gamma\colon\Gamma_j]} \end{equation} exists and equals zero unless $\delta(G)=1$ and $q=\frac{\dim(\widetilde X)+1}{2}$. In this case it equals $c_{G,L}vol(\Gamma\backslash\widetilde X)$ with $c_{G,L}>0$. All this is under the assumption that the $Q$-rank of $G$ is 0.

conj1:

lim⁡j→∞log⁡∣Hq(Γj;L)tor∣[Γ ⁣:Γj]\lim_{j\to\infty}\frac{\log|H^q(\Gamma_j;L)_{tor}|}{[\Gamma\colon\Gamma_j]}

equl1:

lim⁡j→∞∑q(−1)q+dim⁡(X~)+12log⁡∣Hq(Γj;L)tor∣[Γ ⁣:Γj]=cG,Lvol(Γ\X~).\lim_{j\to\infty}\sum_q(-1)^{q+\frac{\dim (\widetilde X)+1}{2}}\frac{\log|H^q(\Gamma_j;L)_{tor}|} {[\Gamma\colon\Gamma_j]}=c_{G,L}vol(\Gamma\backslash\widetilde X).

— 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