Infinite torsion subgroup in a CAT(0) group with proper cocompact action

Determine whether there exists a finitely generated group that acts properly discontinuously and cocompactly by isometries on a proper CAT(0)-space and contains an infinite torsion subgroup.

Background

The paper discusses longstanding questions on the existence and behavior of finitely generated torsion groups in geometric group theory, especially in relation to actions on CAT(0)-spaces. Classical results imply strong constraints on torsion groups acting on specific nonpositively curved spaces (e.g., trees and symmetric spaces), typically yielding fixed points. The cited open problem asks whether such constraints extend to the setting of proper, cocompact actions on proper CAT(0)-spaces, specifically whether such groups can contain infinite torsion subgroups.

The authors’ results provide partial progress by showing that finitely generated torsion groups of subexponential growth cannot act on finite-dimensional CAT(0)-spaces without a fixed point, but they do not resolve the full question under the stronger hypotheses of proper and cocompact actions on proper CAT(0)-spaces.