Bounded Conjugation-Invariant Metrics on Higher-Rank Lattices

Definitions and mathematical statement

Let $H$ be a higher-rank simple Lie group with finite center, as specified in the supplied statement, and let $G\leq H$ be a lattice: a discrete subgroup such that $H/G$ has finite $H$-invariant volume. Let $S\subseteq G$ be a finite generating set. To make the word metric symmetric, put \[ T_S=\bigcup_{h\in G}h(S\cup S^{-1})h^{-1}. \] For $g\in G$, define \[ \ell_S(g)=\min\left\{n\in{\mathbb{Z}}_{\geq 0}:g=t_1\cdots t_n,\quad t_i\in T_S\right\}, \] where the empty product is the identity, and define \[ d_S(x,y)=\ell_S(x^{-1}y)\qquad(x,y\in G). \] This is the conjugation-invariant word metric associated with $S$; adjoining inverses makes explicit the usual symmetric word-metric convention.

The question is whether, for every such $H$, every lattice $G\leq H$, and every finite generating set $S$ of $G$, there exists $N=N(G,S)\in{\mathbb{Z}}_{\geq 0}$ such that \[ \forall x,y\in G,\qquad d_S(x,y)\leq N. \] Equivalently, must every element of $G$ be a product of at most $N$ conjugates in $G$ of elements of $S\cup S^{-1}$? No bound uniform over lattices or generating sets is requested. Boundedness is independent of the choice of finite generating set, although the bound may depend on that choice.

Unresolved source qualifications. The supplied record does not explicitly specify the rank convention or connectedness assumptions for the ambient simple Lie group. These qualifications must be checked before fixing the precise class of ambient groups. The retained original excerpt asks only whether $d_S$ is bounded, refers to an unidentified reference $[1]$, and then introduces an incomplete, separate Conjecture 3. The formulation above follows the supplied clean statement; the incomplete conjecture supplies no additional mathematical assertion here.

Short English statement

For a lattice in a higher-rank simple Lie group with finite center, must there be a finite bound on the number of conjugates of generators and their inverses needed to express any group element? The bound may depend on the lattice and generating set. The ambient rank and connectedness conventions remain to be confirmed. The supplied record labels the question open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005038$, source problem code catalogue entry, entitled Bounded Conjugation-Invariant Metrics on Higher-Rank Lattices, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2022.
• Supplied report citation: Geometric Structures in Group Theory (2023), Oberwolfach Reports, pp. 517--576, DOI 10.4171/owr/2022/11. https://doi.org/10.4171/owr/2022/11. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's 2023 label and the record's 2022 year information are retained without reconciliation.
• Supplied literature source: On bi invariant word metrics. https://arxiv.org/abs/1102.0423. The record does not identify this reference explicitly with the original excerpt's reference $[1]$.
• Supplied literature assessment, checked on 2026-08-22: the question is reported as open for arbitrary lattices in higher-rank simple Lie groups, with existing theorems said to cover important special groups only. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Both literature-source links supplied in the record are retained above.
• The registry labels the statement corrected_verified and states that the official report was checked directly and that definitions were restored in the clean statement. These are supplied provenance claims; no independent source inspection is asserted by this draft.