The degeneration of convex ℝℙ²structures on surfaces

Abstract

Let M be a compact surface of negative Euler characteristic and let C(M ) be the deformation space of convex real projective structures on M . For every choice of pants decomposition for M , there is a well known parameterization of C(M ) known as the Goldman parameterization. In this paper, we study how some geometric properties of the real projective structure on M degenerate as we deform it so that the internal parameters of the Goldman parameterization leave every compact set while the boundary invariants remain bounded away from zero and infinity.

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].…”