The paper studies surface subgroups of SL(4, R) acting convex cocompactly on RP3 with image in the coaffine group. The boundary of the convex core is stratified, and the one dimensional strata form a pair of bending laminations.
The key insight is that the bending data take values in a flat line bundle over the bending lamination. The holonomy of this bundle encodes delicate dynamical features of the lamination and representation.
The authors prove the following main results:
The bending data for a coaffine representation is an affine measured lamination on a convex projective surface. Given a convex projective structure and a compatible affine measured lamination, there is exactly one conjugacy class of coaffine representations with this as the boundary data.
A minimal geodesic lamination can be the bending locus if and only if there is a point where the middle eigenvalue of the holonomy along the lamination has non-positive exponential growth in both directions. Every minimal, non-orientable lamination satisfies this condition.
The bending data can be described by an affine measured lamination, which is a finite collection of positive real numbers on a train track with a flat connection. This affine lamination is measurably semi-conjugate to an affine interval exchange transformation.
The space of bending data compatible with a Hitchin representation is a sphere of dimension 6g-7, where g is the genus of the surface.
The paper develops the necessary technology to understand these coaffine representations, revealing a new phenomenon in the geometry of convex cocompact actions on 3-manifolds.
לשפה אחרת
מתוכן המקור
arxiv.org
תובנות מפתח מזוקקות מ:
by M. D. Bobb, ... ב- arxiv.org 10-03-2024
https://arxiv.org/pdf/2404.14284.pdfשאלות מעמיקות