Free semigroups of large critical exponent

For a convergence group equipped with an expanding coarse‐cocycle, we construct finitely generated free subsemigroups, which we call , of critical exponent arbitrarily close to but strictly less than the critical exponent of the ambient group. As an application, we show that for any nonelementary transverse subgroup of a semisimple Lie group , there exist finitely generated free Anosov subsemigroups in the sense of Kassel–Potrie of critical exponent arbitrarily close to but strictly less than that of the ambient transverse group. Furthermore, we show that these semigroups admit quasi‐isometric embeddings into the symmetric space of with certain additional coarse‐geometric properties.