Normalized Mod-p Cohomology in Arithmetic Lattice Towers

Definitions and mathematical statement

Let $\Gamma\leq \operatorname{SL}_2({\mathbb{C}})$ be an arithmetic lattice. Here a lattice is a discrete subgroup for which $\operatorname{SL}_2({\mathbb{C}})/\Gamma$ has finite invariant volume; arithmeticity is understood in its usual sense for lattices in this Lie group. Let $p$ be any prime, and let ${\mathbb{F}}_p={\mathbb{Z}}/p{\mathbb{Z}}$.

Interpret a normal chain of $p$-power-index subgroups as a descending sequence \[ \Gamma\geq\Gamma_1\geq\Gamma_2\geq\cdots, \qquad \Gamma_i\trianglelefteq\Gamma, \qquad [\Gamma:\Gamma_i]=p^{a_i},\quad a_i\in{\mathbb{Z}}_{\geq0}, \] with \[ \bigcap_{i\geq1}\Gamma_i=\{1\}. \] Take $H^1(\Gamma_i,{\mathbb{F}}_p)$ to mean group cohomology with trivial action on the coefficient module ${\mathbb{F}}_p$. Thus \[ H^1(\Gamma_i,{\mathbb{F}}_p)\cong\operatorname{Hom}(\Gamma_i,({\mathbb{F}}_p,+)), \] and its dimension is taken over ${\mathbb{F}}_p$.

Question. For every arithmetic lattice $\Gamma\leq\operatorname{SL}_2({\mathbb{C}})$, every prime $p$, and every chain satisfying these conditions, is it true that \[ \lim_{i\to\infty} \frac{\dim_{{\mathbb{F}}_p}H^1(\Gamma_i,{\mathbb{F}}_p)}{[\Gamma:\Gamma_i]}=0? \] No cocompactness, torsion-freeness, or congruence-subgroup hypothesis is stated in the supplied record. The conventions that every $\Gamma_i$ is normal in $\Gamma$ and that coefficients have trivial action make standard implicit terminology explicit; their agreement with the intended source formulation requires review.

The supplied dated curation labels this question open and states that results for selected $p$-adic analytic towers give asymptotics without establishing the assertion for arbitrary chains. This is a supplied assessment, not an independently verified mathematical or literature conclusion.

Short English statement

For every arithmetic lattice in $\operatorname{SL}_2({\mathbb{C}})$ and every prime $p$, must the dimension of first cohomology over ${\mathbb{F}}_p$, divided by subgroup index, tend to zero along every descending normal chain of $p$-power index with trivial intersection? Normality is interpreted relative to the original lattice, and coefficients are interpreted as trivial. The supplied open-status assessment requires independent review.

Sources

• Local UnsolvedMath registry, permanent ID $30001588$, source problem code catalogue entry, title Normalized Mod-p Cohomology in Arithmetic Lattice Towers, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2010 and names no individual proposer.
• Supplied primary citation: Actions and Invariants of Residually Finite Groups: Asymptotic Methods (2011), Oberwolfach Reports, pp. 2335--2391, DOI 10.4171/owr/2010/40. Original OWR report: https://doi.org/10.4171/owr/2010/40. The collection describes these reports as records of mathematical workshops and their open-problem discussions. No problem-specific page or named speaker is supplied.
• Supplied literature source: Calegari--Emerton, described in the record as "mod-p cohomology growth": https://math.uchicago.edu/~emerton/pdffiles/growth.pdf.
• Supplied literature source: Lackenby, described in the record as "mod-p homology growth": https://arxiv.org/abs/math/0509036.
• Literature assessment date: 2026-08-21. The supplied dated curation labels the problem open and says that asymptotics for selected $p$-adic analytic towers do not establish the arbitrary-chain assertion. The record expressly requests independent verification of time-sensitive status and cited claims; this draft does not independently verify that assessment.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. This verification claim is supplied provenance, not a verification performed for this draft.