Volume 127, Issue 1 p. 35-83
RESEARCH ARTICLE

Affine deformations of quasi-divisible convex cones

Xin Nie

Xin Nie

Shing-Tung Yau Center of Southeast University, Nanjing, China

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Andrea Seppi

Corresponding Author

Andrea Seppi

Univ. Grenoble Alpes, CNRS, IF, Grenoble, France

Correspondence

Andrea Seppi, Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France.

Email: [email protected]

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First published: 06 June 2023

Abstract

We study subgroups of SL ( 3 , R ) R 3 $\mathrm{SL}(3,\mathbb {R})\ltimes \mathbb {R}^3$ obtained by adding a translation part to the holonomy of a finite-volume convex projective surface. Under a natural condition on the translations added to the peripheral parabolic elements, we show that the affine action of the group on R 3 $\mathbb {R}^3$ has convex domains of discontinuity which are regular, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all such domains arising from a fixed group and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature (CAGC). The proof is based on the analysis of CAGC surfaces developed in a previous work, along with a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. We also show that the moduli space of such groups is a vector bundle over the moduli space of finite-volume convex projective structures, with rank equal to the dimension of the Teichmüller space.