On integer points in the Hitchin moduli space

Hitchin representations form a special component in the space of SL(n,R)-representations of a compact
surface group, generalizing Teichmueller space; in this talk we will discuss two notions of integrality
for such representations and show some non-finiteness phenomena  in moduli space,for n large. We will
also show how finiteness  can be restored by the use of  appropriate height functions, one based on the
minimal area, and the other on the maximal stretch of a representation.This is joint work with F. Labourie
and A. Wienhard