Degeneration of Hitchin representations along internal sequences

Abstract

Using the work of Bonahon-Dreyer and Fock-Goncharov, one can construct a real-analytic parameterization for the PSL(n,R) Hitchin component of a surface S, that is explicitly analogous to the Fenchel-Nielsen coordinates on the Teichmuller space of S. Given a Hitchin representation, we give a lower bound on the "length" of any closed curve on S in terms of the parameters describing this representation. We then show that this lower bound is good enough to produce large families of sequences in the Hitchin component along which the topological entropy converges to 0.Comment: Some typos from previous version correcte

An introduction to pressure metrics for higher Teichmüller spaces

“…However, it is clear that the intersection number is not symmetric on the Hitchin component. For example, one may use the work of Zhang [94,95] to exhibit for all K > 1 and…”

“…(2) Tengren Zhang [94,95] showed that, for all d, there exist large families of sequences of Hitchin representations with entropy converging to 0. Nie [64] had earlier constructed specific examples when d = 3.…”

“…However, very little is known about the completeness of the pressure metric in "other directions." The work of Zhang [94,95] and Loftin [56] (when d = 3) should be relevant here. It may also be interesting to study the relationship between the metric completion and Parreau's compactification [65] of the Hitchin component.…”

“…We recall that a subset A of a metric space X is said to be coarsely dense if there exists D > 0 such that every point in X lies within D of a point in A. Zhang [94,95] and Nie [64] (when d = 3) produce sequences in Hitchin components where entropy converges to 0. These sequences are candidates to produce points arbitrarily far from the Fuchsian locus.…”

Maximal representations, non-Archimedean Siegel spaces, and buildings

“…For instance compactifications of spaces of representations of Γ have been introduced and studied in [Par12,Ale08,Le12]. In the context of Hitchin representations, asymptotic properties of diverging sequences are studied in [Zha13,Zha14,CL14,Lof15,Par15,KNPS15,MSWW14].…”