We prove central limit theorems for the random walks on either the mapping
class group of a closed, connected, orientable, hyperbolic surface, or on
, each
time under a finite second moment condition on the measure (either with respect to the
Teichmüller metric, or with respect to the Lipschitz metric on outer space). In the
mapping class group case, this describes the spread of the hyperbolic length of a simple
closed curve on the surface after applying a random product of mapping classes. In the
case of
,
this describes the spread of the length of primitive conjugacy classes in
under
random products of outer automorphisms. Both results are based on a general
criterion for establishing a central limit theorem for the Busemann cocycle on the
horoboundary of a metric space, applied to either the Teichmüller space of the
surface or to the Culler–Vogtmann outer space.
Keywords
mapping class groups, Out(Fn), outer automorphism groups,
random walks on groups, central limit theorem
