For a broad class of convergence groups, we construct free subsemigroups having critical exponents arbitrarily close to, but strictly less than, the critical exponent of the ambient convergence group. I'll explain how to apply this result to a particular class of discrete subgroups of Lie groups known as transverse groups, and also explain how our result implies a counterpart to a classical gap theorem of Kevin Corlette concerning discrete subgroups of the isometry groups of quaternionic hyperbolic space and the Cayley (octonionic) hyperbolic plane. I'll introduce these notions from scratch and try to focus on their relations to (hopefully more familiar) concepts in real hyperbolic geometry.