Free Semigroups of Large Critical Exponent

Abstract

For a convergence group equipped with an expanding coarse-cocycle, we construct finitely generated free subsemigroups, which we call Bishop--Jones semigroups, 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 non-elementary transverse subgroup $\Gamma$ of a semisimple Lie group $G$, 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 X of G with certain additional coarse-geometric properties.