Higher Teichmuller theory and thermodynamic formalism

Richard Sharp (University of Warwick)

Frank Adams 1, Alan Turing Building,

It is well known that the Teichmuller space of a compact oriented surface
corresponds to a connected component of the space of representations of its
fundamental group into \(PSL(2;R) \). We will discuss representations of this
group into the higher rank Lie groups \(PSL(d;R)\) for d > 3 and show that
the connected component in the representation space corresponding to the
Teichmuller space supports a real analytic Riemannian metric analogous to
the Weil-Peterssen metric. This is done via shifts of fi nite type and thermodynamic
formalism and, in particular, ergodic theorists should not be put
off by the first two sentences in the abstract. (This is joint work with Mark
Pollicott.)

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