Teichmuller dynamics give us a nonhomogeneous example of an action of SL_2(R) on a space H_g preserving a finite measure. This space is related to the moduli space of genus g curves. The SL_2(R) action on H_g has a complicated behavior: McMullen identified a collection of orbits SL_2(R).x such that the SL_2(R)-stabilizer of x is infinitely generated. 

In the talk I will study these stabilizers in terms of their critical exponent (a quantity that measures a certain density of discrete subgroups) and show that the critical exponent is uniformly bounded away from 1. That is, the stabilizers are uniformly far from being a lattice.