2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115338Consider a hyperbolic group G and a quasiconvex subgroup H of infinite index. We construct a set-theoretic section s of the quotient map (of sets) from G to G/H such that s(G/H) is a net in G; that is, any element of G is a bounded distance from s(G/H). This section arises naturally as a set of points minimizing word-length in each fixed coset gH. The left action of G on G/H induces an action on s(G/H), which we use to prove that H contains no infinite subgroups normal in G.15 pages, 1 figure; v3: Replaced another typo; v2: Replaced minor typo in abstractGeometric TopologyGroup Theory20F67 (Primary) 20F65 (Secondary)text