Positively ratioed representations

Abstract

Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain associated cross ratios satisfy a positivity property. Examples of such representations include Hitchin representations and maximal representations. Using geodesic currents, we show that the corresponding length functions for these positively ratioed representations are well-behaved. In particular, we prove a systolic inequality that holds for all such positively ratioed representations.G.

The geometry of maximal representations of surface groups into SO0(2,n)

“…Some Anosov representations of surface groups, such as Hitchin representations into real split Lie groups or maximal representations into Hermitian Lie groups, have the additional property of forming connected components of the whole space of representations. There have been several attempts to propose a unifying characterization of these representations (see [MZ16] and [GW16]). Note that quasi-Fuchsian representations into PSL(2, C) do not form components; indeed, they can be continuously deformed into representations with non-discrete image.…”

Simple root flows for Hitchin representations

“…(3) If µ is a geodesic current, then i(µ, ω ρ ) = µ | L H ρ . Our Liouville current is closely related to the symmetric Liouville currents defined by Bonahon [1], when d = 2, and Martone-Zhang [26]. In fact, one may view their Liouville currents as symmetrizations of our Liouville current.…”

Cross Ratios on Boundaries of Symmetric Spaces and Euclidean Buildings

“…Labourie [Lab07] has given one of the cross ratios in Example 3.13 ad-hoc and used it as tool to understand Hitchin representations. Moreover, Martone and Zhang [MZ17] have constructed cross ratios on boundaries of surface groups, which in particular for SL(n, R)-Hitchin representations coincide with the pullback under the boundary map of some of the cross ratios in Example 3.13.…”

“…On the boundary ∂ ∞S of the universal cover of a closed surface S there are many other cross ratios, besides the above constructed one, that parametrize classical objects associated to the surface; such as simple closed curves, measured laminations, points of Teichmüller space [Bon88], Hitchin representations [Lab07] and positively ratioed representations [MZ17] 1 -to name a few. This prominence and importance of cross ratios in negative curvature motivates us to ask if such objects also exists for non-positively curved spaces and how much information about the geometry they carry.…”

“…The natural boundary maps Anosov representations come with can be used to pullback the cross ratios that we construct to get cross ratios on the boundary of the group. This has been done for representation into SL(n, R) in [Lab07] using an ad-hoc definition of a cross ratio and this is in the spirit of [MZ17]; and we hope that the vector valued cross ratio we analyze here allows for further applications in this area.…”