insight - Computational Geometry - # Coaffine Representations of Surface Groups

Bending Laminations and Affine Representations of Surface Groups in Projective 3-Space

Q: How do the typical behaviors of the bending data relate to the dynamics of affine interval exchange transformations?

The bending data in the context of coaffine representations is intricately linked to the dynamics of affine interval exchange transformations (AIETs). The bending measure, which is defined on the boundary of the convex core, can be expressed as a finite collection of positive real numbers on a train track, with the holonomy of the bending measure corresponding to the dynamics of the AIET. Specifically, the geodesic flow induces a first return map to the intersection of the tangents to the bending lamination, which can be shown to produce an AIET with an involutive symmetry. This relationship highlights how the bending data encapsulates the dynamical behavior of the system, allowing for a deeper understanding of the structure of the convex core boundary. The presence of an affine measure valued in the flat line bundle over the bending lamination ensures that the AIET retains properties of the underlying geodesic flow, thus establishing a robust connection between the bending data and the dynamics of AIETs.

Q: What are the connections between the affine laminations appearing on the boundary of the convex core and half-dilation surfaces?

Affine laminations that appear on the boundary of the convex core are closely related to the concept of half-dilation surfaces. Half-dilation surfaces are characterized by their affine structures, which can be understood through the lens of bending data. The affine measured laminations, which arise from the bending data, provide a framework for analyzing the geometric and dynamical properties of half-dilation surfaces. In particular, the bending measures associated with these laminations can be interpreted as encoding the growth rates of the affine structures along the leaves of the lamination. This connection allows for the exploration of how the bending data influences the geometric properties of half-dilation surfaces, such as their curvature and topological features. Furthermore, the study of affine laminations in this context can lead to insights into the classification of half-dilation surfaces and their associated dynamical systems.

Q: Can the results be extended to understand the convex core boundary of general deformations of Hitchin representations, not just into the coaffine group?

Yes, the results presented in the study of coaffine representations can indeed be extended to understand the convex core boundary of general deformations of Hitchin representations. The techniques and theories developed in the context of coaffine representations provide a substantial toolkit for analyzing the more general case of Hitchin representations. The bending data, which is crucial for characterizing the boundary of the convex core, can be adapted to accommodate the broader class of representations. This includes examining the dynamics of the associated flat bundles and their holonomies, which remain relevant regardless of whether the representations are specifically into the coaffine group or not. The insights gained from the bending data and the associated affine measured laminations can thus be leveraged to explore the geometric and dynamical properties of the convex core boundaries for a wider range of Hitchin representations, paving the way for further research in this area.

Core Concepts

The bending data for a 3-dimensional convex cocompact coaffine representation is an affine measured lamination on a convex projective surface.

Abstract

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.

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Stats

The bending data for a coaffine representation is an affine measured lamination, which can be described by a finite collection of positive real numbers on a train track with a flat connection.

Quotes

"The key insight of this paper 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 L(η)."
"Theorem A. The bending data for a 3-dimensional convex cocompact coaffine representation is an affine measured lamination on a convex projective surface."
"Theorem 1.1. 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."

Key Insights Distilled From

Affine laminations and coaffine representations

by M. D. Bobb, ... at arxiv.org 10-03-2024

https://arxiv.org/pdf/2404.14284.pdf

Affine laminations and coaffine representations

Deeper Inquiries

How do the typical behaviors of the bending data relate to the dynamics of affine interval exchange transformations?

The bending data in the context of coaffine representations is intricately linked to the dynamics of affine interval exchange transformations (AIETs). The bending measure, which is defined on the boundary of the convex core, can be expressed as a finite collection of positive real numbers on a train track, with the holonomy of the bending measure corresponding to the dynamics of the AIET. Specifically, the geodesic flow induces a first return map to the intersection of the tangents to the bending lamination, which can be shown to produce an AIET with an involutive symmetry. This relationship highlights how the bending data encapsulates the dynamical behavior of the system, allowing for a deeper understanding of the structure of the convex core boundary. The presence of an affine measure valued in the flat line bundle over the bending lamination ensures that the AIET retains properties of the underlying geodesic flow, thus establishing a robust connection between the bending data and the dynamics of AIETs.

What are the connections between the affine laminations appearing on the boundary of the convex core and half-dilation surfaces?

Affine laminations that appear on the boundary of the convex core are closely related to the concept of half-dilation surfaces. Half-dilation surfaces are characterized by their affine structures, which can be understood through the lens of bending data. The affine measured laminations, which arise from the bending data, provide a framework for analyzing the geometric and dynamical properties of half-dilation surfaces. In particular, the bending measures associated with these laminations can be interpreted as encoding the growth rates of the affine structures along the leaves of the lamination. This connection allows for the exploration of how the bending data influences the geometric properties of half-dilation surfaces, such as their curvature and topological features. Furthermore, the study of affine laminations in this context can lead to insights into the classification of half-dilation surfaces and their associated dynamical systems.

Can the results be extended to understand the convex core boundary of general deformations of Hitchin representations, not just into the coaffine group?

Yes, the results presented in the study of coaffine representations can indeed be extended to understand the convex core boundary of general deformations of Hitchin representations. The techniques and theories developed in the context of coaffine representations provide a substantial toolkit for analyzing the more general case of Hitchin representations. The bending data, which is crucial for characterizing the boundary of the convex core, can be adapted to accommodate the broader class of representations. This includes examining the dynamics of the associated flat bundles and their holonomies, which remain relevant regardless of whether the representations are specifically into the coaffine group or not. The insights gained from the bending data and the associated affine measured laminations can thus be leveraged to explore the geometric and dynamical properties of the convex core boundaries for a wider range of Hitchin representations, paving the way for further research in this area.