Abstract: An important quantity in the study of discrete groups of isometries of Riemannian manifolds, Gromov hyperbolic spaces, and other interesting geometric objects is the critical exponent. For a discrete subgroup of isometries of the quaternionic hyperbolic space or octonionic projective plane, Kevin Corlette established in 1990 that the critical exponent detects whether a discrete subgroup is a lattice or has infinite covolume. Precisely, either the critical exponent equals the volume entropy, in which case the discrete subgroup is a lattice, or the critical exponent is less than the volume entropy by some definite amount, in which case the discrete subgroup has infinite covolume. In 2003, Leuzinger extended this gap theorem for the critical exponent to any discrete subgroup of a Lie group having Kazhdan’s property (T) (for instance, a discrete subgroup of SL(n,R), where n is at least 3).

    In this talk, I will present a result which shows that no such gap phenomenon holds for discrete semigroups of Lie groups. More precisely, for any Zariski dense discrete subgroup of a Lie group, there exist free, finitely generated, Zariski dense subsemigroups whose critical exponents are arbitrarily close to that of the ambient discrete subgroup.

    As an application, we show that the critical exponent is lower semicontinuous in the Chabauty topology whenever the Chabauty limit of a sequence of Zariski dense discrete subgroups is itself a Zariski dense discrete subgroup.