2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229161We construct a hyperbolic group with a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.18 pages, 7 figuresGroup Theory20F67; 20E45text