Juan-Pineda–Leary conjecture on finite models for the virtually cyclic classifying space

Let $G$) be a group, and let ${\underline{\underline E}}G=E_{\mathcal{VCY}}(G)$ denote a $G$-CW-model for the classifying space for the family of virtually cyclic subgroups. The model is finite if it has finitely many orbits of cells. Juan-Pineda–Leary conjecture. If $G$ admits a finite model for ${\underline{\underline E}}G$, then $G$ is virtually cyclic. The conjecture is known for hyperbolic and elementary amenable groups, and this paper establishes it for several further classes, including one-relator groups, acylindrically hyperbolic groups, $3$-manifold groups, and relevant CAT(0) groups.