GEOMETRY SEMINAR: How does the critical exponent see the geometry?

Given a discrete isometry group action on the hyperbolic spaces, the critical exponent of the group measures the exponential growth rate of its orbit. This quantity not only sees the dynamics of the quotient manifold but also constrains its geometry. I will present my joint work with Wang that if the critical exponent is very small, the quotient manifold is geometrically finite. Then I will focus on how the quantity changes when we deform the Fuchsian representations in the Hitchin component of PSL(3, R), which is joint work in progress with Islam, Martone, and Pozzetti.