Sublinear Generator Growth of Higher-Rank Lattices

Definitions and mathematical statement

Let $G$ be a higher-rank simple Lie group, as in the supplied source. Fix a Haar measure on $G$. A lattice $M\leq G$ is a discrete subgroup for which the quotient $G/M$ has finite invariant volume; write its covolume as $\operatorname{vol}(G/M)$. For a finitely generated group $M$, let \[ d(M)=\min\{\lvert S\rvert:S\subseteq M,\ \langle S\rangle=M\} \] be its minimal number of generators. The displayed question presupposes that these generator numbers are finite.

The supplied question asks whether, for every such $G$ and every sequence of lattices $(M_i)_{i\geq1}$ in $G$ with unbounded covolumes, one has \[ \lim_{i\to\infty}\frac{d(M_i)-1}{\operatorname{vol}(G/M_i)}=0. \] No torsion-free hypothesis is imposed in this question.

Unresolved formulation qualifications. Literally, unbounded covolumes means \[ \sup_{i\geq1}\operatorname{vol}(G/M_i)=\infty. \] This is weaker than $\operatorname{vol}(G/M_i)\to\infty$. The supplied record does not clarify whether the latter condition was intended; the displayed question retains the supplied wording. The record also does not specify the rank convention or the connectedness and centre conventions for a simple Lie group. Under the real-rank convention, higher rank means $\operatorname{rank}_{{\mathbb{R}}}(G)\geq2$; this interpretation requires confirmation against the source.

Short English statement

The question asks whether the minimal number of generators of higher-rank lattices, minus one, divided by covolume tends to zero along every sequence described in the source. The distinction between unbounded covolumes and covolumes tending to infinity remains unresolved. Supplied dated curation labels the problem partially solved and reports a torsion-free result; it does not impose torsion-freeness on the original question.

Sources

• Local UnsolvedMath registry, permanent ID $30003210$, title: Sublinear Generator Growth of Higher-Rank Lattices. Collection: Oberwolfach Reports Open Problems. Exact source problem code: catalogue entry. The supplied proposed year is 2016; no individual proposer is identified.
• Supplied original-source citation: Measured Group Theory (2017), Oberwolfach Reports, pp. 2347--2397, DOI 10.4171/owr/2016/41. Original conjecture link: https://doi.org/10.4171/owr/2016/41. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Fraczyk--Mellick--Wilkens, https://arxiv.org/abs/2307.01194. The supplied assessment attributes to these authors sublinear generator growth for torsion-free lattices in higher-rank semisimple real Lie groups, describing this as settling the torsion-free case while leaving the arbitrary-lattice formulation only partially resolved.
• Literature assessment date: 2026-08-21. The status partially_solved and the preceding assessment are supplied dated curation, not independently verified here. The supplied record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report, preserving its hypotheses, notation, and target. This is a supplied verification claim, not an independent source check performed for this draft.