The Teichmüller space is distorted in the Hitchin spaces

Given a closed surface S of genus at least 2 and an integer d>2, the Teichmüller space T(S) of S embeds in the space of representations of the fundamental group of S in PSL(d,R), modulo conjugation, by postcomposing with the irreducible representation of PSL(2,R) in PSL(d,R). The connected component of T(S) is the Hitchin component H(S).

A decade ago, Bridgeman, Canary, Labourie and Sambarino constructed on H(S) several metrics, called pressure metrics, that extend the Weil--Petersson metric on T(S). Their construction uses a caracterisation of the WP metric in terms of hyperbolic lengths due to Thurston and Wolpert. We will describe the pressure metrics on certain subspaces of H(S) that are transverse to T(S), consisting of deformations through the famous bending procedure. This will allow us, in genus>2, to show that T(S) is distorted in H(S), in the sense that there is a sequence of points in T(S) whose distance to the origin diverge for the WP metric, but remains bounded for the pressure metrics.