Linear Divergence of Higher-Rank Lattices

Definitions and mathematical statement

Let $G$ be a semisimple Lie group of higher rank in the sense intended by the source, and let $\Gamma\leq G$ be a lattice, meaning a discrete subgroup of finite covolume. For a finitely generated $\Gamma$, use the path metric on a Cayley graph $X$ for a finite generating set. A standard divergence convention fixes $0<\delta<1$ and a base vertex $o$ and sets \[\operatorname{Div}_{X,\delta}(r)=\sup_{d(o,x)=d(o,y)=r}\inf\{\operatorname{length}(\gamma):\gamma\text{ joins }x\text{ to }y\text{ in }X\setminus B(o,\delta r)\}.\] The infimum is $+\infty$ if no such path exists. This expresses the source's avoidance of a large ball about a third point. Linear divergence means equivalence to $r$ under the usual coarse comparison: $f\preceq g$ if $f(r)\leq Cg(Cr+C)+Cr+C$ for some $C\geq1$, and $f\simeq g$ if both comparisons hold. The source question is whether every lattice in a higher-rank semisimple Lie group has \[\operatorname{Div}_\Gamma(r)\simeq r.\] The metric is intrinsic to the lattice's word metric; a statement about paths in the ambient symmetric space alone is not the same assertion for a nonuniform lattice. The extract does not specify the divergence parameters or convention, hypotheses guaranteeing the chosen finite-generation framework, or whether higher rank means total real rank at least two or rank at least two for each noncompact simple factor. It also does not impose irreducibility or distinguish uniform from nonuniform lattices. These missing conventions are not supplied as verified additional hypotheses. The 2026-08-21 triage labels the problem partially solved, reports linear divergence for several important higher-rank lattice classes including cases listed by Drutu, Mozes, and Sapir, and calls the full assertion conjectural.

Short English statement

Must every higher-rank semisimple lattice admit detours around large balls whose lengths grow only linearly, measured in its own word metric? The intended rank and divergence conventions and the precise group hypotheses need confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 30000911, source problem catalogue entry, Linear Divergence of Higher-Rank Lattices, proposed year 2008; no individual proposer supplied.
• Buildings: Interactions with Algebra and Geometry (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/03, pp. 119--172: https://doi.org/10.4171/owr/2008/03. Supplied literature label: original OWR report.
• Supplied literature label: Drutu--Mozes--Sapir, https://arxiv.org/abs/0801.4141.