Length Functions For Hitchin Representations

Details Length Functions For Hitchin Representations Length Functions For Hitchin Representations

2011-06-30

arxiv.org/abs/1106.6310v1

Mathematics

Geometric Topology

17 pages, 2 figures

Scientific paper

For a closed surface $S$, the representation variety $\mathrm {Hom}(\pi_{1}(S),\PSL_{n}(\R))$ admits preferred components, called the Hit-chin components, that generalize the Teichm\"uller components of the case $n=2$. For a homomorphism $\rho : \pi_1(S) \to \PSL_n(\R)$ contained in a Hitchin component, we construct $n$ continuous maps $\ell_i^\rho : \mathcal \CH(S) \to \R$, defined on the space of H\"older geodesic currents $\CH(S)$, such that for a closed curve $\gamma$ the eigenvalues of the matrix $\rho(\gamma)$ are of the form $\pm \mathrm{exp}\, \ell_i^\rho(\gamma)$. This generalizes to higher dimensions Thurston's length function for the case $n=2$. Identities and differentiability properties for these length functions $\ell_i^\rho$, and applications to eigenvalues estimates, are also considered.

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