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Affine actions with Hitchin linear part

Banff International Research Station for Mathematical Innovation and Discovery

Properly discontinuous actions of a surface group on \(\mathbb{R}^d\) by affine transformations were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the affine action is not properly discontinuous. The key case is that of linear part in $\mathrm{SO}(n,n-1)$, so that $\mathbb{R}^d = \mathbb{R}^{n,n-1}$ is the model for flat psuedo-Riemannian geometry of signature $(n,n-1)$. Here, the translational parts determine a deformation of the linear part into $\mathrm{SO}(n,n)$ Hitchin representations and the crucial step is to show that such representations are not Anosov in $\mathrm{SL}(2n,\mathbb{R})$ with respect to the stabilizer of an $n$-plane. Joint with Tengren Zhang.

Mathematics; Group theory and generalizations; Geometry; Algebraic structures

video/mp4

Author affiliation: University of Texas - Austin

2020-06-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74660

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/