The infinite-order element conjecture for actions on finite-dimensional CAT(0) complexes

Let $G$ be a finitely generated group acting nontrivially on a finite-dimensional $\mathrm{CAT}(0)$ complex. Infinite-order element conjecture. The group $G$ contains an element of infinite order. This conjecture would give a negative answer to the question of whether a torsion group can act nontrivially on a finite-dimensional $\mathrm{CAT}(0)$ complex; it is presented as open, and the preceding results establish it for two-dimensional CAT(0) triangle complexes under additional hypotheses.