Conjecture on infinite-order elements in fixed-point-free CAT(0) actions

Prove that every finitely generated group acting without a global fixed point on a finite-dimensional CAT(0) complex contains an element of infinite order.

Background

The conjecture, attributed to Norin–Osajda–Przytycki [NOP22], predicts a structural constraint on finitely generated groups that act by isometries on finite-dimensional CAT(0) complexes without global fixed points: such actions must exhibit elements of infinite order. It is known in some special cases, including CAT(0) cubical complexes (Sageev) and certain two-dimensional CAT(0) complexes (NOP22), but remains open in general.

The present paper proves the conjecture in the class of weakly Liouville groups and derives consequences for torsion groups of subexponential growth and certain simple groups, thereby providing a partial verification of the conjecture.