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Deformation of Fuchsian representations and proper affine actions
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abstract
The main goal of this article is to generalize Mess' work and using results from Labourie--Wentworth, Potrie--Sambarino and Smilga, to show that inside Hitchin representations, infinitesimal deformations of Fuchsian representations of a cocompact surface group do not act properly along the directions corresponding to the sum of a mixed odd differential and a $2m$-differential for any $1\leq m \leq \lfloor\frac{n}{2}\rfloor$. In the process, we introduce affine versions of cross ratios and triple ratios. We introduce Margulis invariants and relate them with affine crossratios and infinitesimal Jordan projections. We obtain a general equivalent criterion for existence of proper affine actions in terms of the structure of the Margulis invariant spectra. Also, using a stability argument we show the existence of proper affine actions of non-abelian free groups whose linear part is a Hitchin representation.
Forward citations
Cited by 2 Pith papers
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Bending, entropy and proper affine actions of surface groups
Every non-Fuchsian quasifuchsian surface group in an explicit open neighborhood of the Fuchsian locus admits a proper affine action on sl(2,C) with adjoint linear part, and all entropy critical points in a larger neig...
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Affine Anosov representations
This paper is a survey of affine Anosov representations, a framework in which proper affine actions of hyperbolic groups are characterized by Margulis invariant spectra, mostly quoting the author's own results.
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