Invariant Orders on Finitely Generated Infinite Groups

Definitions and mathematical statement

Let $\Gamma$ be a finitely generated infinite group. A partial order on $\Gamma$ is a relation $\preceq$ that is reflexive, antisymmetric, and transitive: for all $x,y,z\in\Gamma$, \[ x\preceq x,\qquad (x\preceq y\ \text{and}\ y\preceq x)\Longrightarrow x=y,\qquad (x\preceq y\ \text{and}\ y\preceq z)\Longrightarrow x\preceq z. \] It is a total order if, additionally, every $x,y\in\Gamma$ satisfy $x\preceq y$ or $y\preceq x$. A partial or total order is left-invariant if \[ x\preceq y\quad\Longleftrightarrow\quad gx\preceq gy \qquad(g,x,y\in\Gamma), \] and bi-invariant if \[ x\preceq y\quad\Longleftrightarrow\quad gxh\preceq gyh \qquad(g,h,x,y\in\Gamma). \] Question as supplied. For a finitely generated infinite group $\Gamma$, does there exist an invariant order relation on $\Gamma$? Consider separately the four combinations \[ \begin{array}{c|cc} & \text{partial order} & \text{total order}\\ \hline \text{left-invariant} & \text{existence?} & \text{existence?}\\ \text{bi-invariant} & \text{existence?} & \text{existence?} \end{array} \] Unresolved qualifications. Under the definitions above, the equality relation is an admissible bi-invariant partial order. The supplied record does not specify whether partial orders must be nontrivial, for example by requiring distinct comparable elements. Such a requirement is therefore not imposed here, but its absence prevents identifying a substantive intended partial-order question. The record also does not specify whether the target is universal existence for all finitely generated infinite groups, an existence criterion for a given such group, or a question restricted to a particular class of groups suggested by the workshop context. No additional restriction is inferred from that context.

The supplied literature assessment says that the report distinguished partial and bi-invariant orders and stated that nonexistence of a bi-invariant partial order had not been proved even for $\mathrm{SL}(3,{\mathbb{Z}})$, the group of determinant-one $3\times3$ integer matrices. This is retained as a supplied source qualification, not as an independently verified assertion. In particular, its intended meaning requires clarification of the partial-order convention.

Short English statement

The question asks about left-invariant and bi-invariant orders on a finitely generated infinite group, considering both total and partial orders. The intended group quantifier and any nontriviality requirement for partial orders remain unclear. The supplied open-status assessment requires independent review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001814$, title Invariant Orders on Finitely Generated Infinite Groups, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2011 and names no individual proposer.
• Supplied report citation: Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences (2011), Oberwolfach Reports, pp. 1637--1708, DOI 10.4171/owr/2011/30. Supplied source URL: https://doi.org/10.4171/owr/2011/30. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source entry, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/30. Literature checked at: 2026-08-21. This date and the accompanying assessment are supplied dated curation, not independently verified here. That assessment labels the problem open and says the general invariant-order question remains open, citing the report's distinction between partial and bi-invariant orders and its stated lack of a nonexistence proof even for $\mathrm{SL}(3,{\mathbb{Z}})$.
• The registry labels the statement corrected_verified and says its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification claim; no source visit or independent verification is asserted in this draft.