An extended definition of Anosov representation for relatively hyperbolic groups

This paper introduces a unified definition of Anosov representations for relatively hyperbolic groups that encompasses various existing notions of geometrically finite behavior and proves their stability under deformations satisfying specific dynamical conditions on peripheral subgroups.

The Gist

Imagine you are trying to understand a massive, complex machine. In the world of mathematics, this machine is a group (a collection of symmetries or movements), and the "parts" of the machine are the representations (ways these movements act in a geometric space).

For a long time, mathematicians have been very good at understanding machines that are "perfectly smooth" and predictable. They call these Anosov groups. Think of them like a well-oiled, high-performance sports car: every part moves in perfect harmony, and if you tweak the engine slightly, the car still drives beautifully.

But then there are machines with "rusty" or "broken" parts. In math, these are groups with peripheral subgroups (special, often messy parts that behave differently, like a cusp or a hole). For a long time, we didn't have a good way to describe machines that were mostly smooth but had these messy, "rusty" parts. We knew how to fix the smooth parts, but the rusty ones made the whole thing seem unstable.

This paper, by Theodore Weisman, introduces a new way to describe these "mostly smooth, slightly rusty" machines. He calls them Extended Geometrically Finite (EGF) representations.

Here is the breakdown using simple analogies:

1. The Problem: The "Rusty" Parts

Imagine a map of a city.

• The Smooth City: Most of the city is a perfect grid. If you walk in a straight line, you know exactly where you'll end up. This is like the "Anosov" groups.
• The Rusty Districts: But there are a few districts (the "peripheral subgroups") that are chaotic. Maybe they are swamps, or they have weird, non-Euclidean geometry. If you try to walk through them, you might get lost or your path might twist in strange ways.

Previous math definitions said: "If your machine has a rusty district, it's not a 'good' machine unless the rust is exactly like a specific type of rust we already know." This was too strict. It meant many interesting, complex machines were ignored because their "rust" didn't fit the strict mold.

2. The Solution: The "Backwards" Map

Weisman's big idea is to change how we look at the map.

• Old Way: Try to draw a perfect line from the chaotic district to the smooth city. (This is hard because the chaotic district is messy).
• Weisman's Way: Draw a line from the smooth city to the chaotic district.

Think of it like a shadow puppet show.

• The "smooth city" is your hand (the flag manifold).
• The "chaotic district" is the wall (the Bowditch boundary).
• Instead of trying to make the wall look like your hand, you just need to make sure your hand casts a shadow that covers the wall perfectly.

This is the "Extended Geometrically Finite" (EGF) definition. It doesn't demand that the messy parts look perfect. It just demands that there is a consistent, predictable relationship (a "shadow") between the smooth part and the messy part. This allows for a much wider variety of "machines" to be considered stable and understandable.

3. The Superpower: Stability

The most exciting part of this paper is the Stability Theorem.

Imagine you have a delicate sculpture made of clay.

• The Old Rule: If you touch the sculpture, it might crumble. If you change the shape of the "rusty" part even a tiny bit, the whole thing might fall apart.
• Weisman's Rule: As long as you don't change the fundamental nature of the rusty part (you can stretch it, shrink it, or change its color, but you don't turn it into a completely different material), the whole sculpture will stay standing.

In math terms: You can deform (change) the representation of the group, even changing the specific way the "rusty" peripheral subgroups behave, and the whole system will remain an EGF representation. It's like saying, "You can repaint the rusty district or even change the traffic patterns there, but as long as the connection to the smooth city remains, the whole city plan still works."

4. The Tool: The "Relative Automaton"

How did he prove this? He invented a new tool called a Relative Quasigeodesic Automaton.

Think of this as a GPS for a chaotic city.

• In a normal city (a smooth group), you can use a simple map to find the shortest path.
• In a city with swamps and weird districts, a normal map fails.
• Weisman built a special GPS (the automaton) that knows how to navigate around the swamps. It codes the path so that even if you go through the messy parts, the GPS knows you are still on a "good" path.

This GPS allows him to prove that no matter how you tweak the "rusty" parts (within reason), the GPS still works, and the path remains stable.

5. Why Does This Matter?

This is a unifying theory.

• Before: Mathematicians had to use different rulebooks for different types of "messy" groups. Some were "Relative Anosov," some were "Convex Projective," etc.
• Now: Weisman says, "Put all those rulebooks in the trash. Use this one big book (EGF)."

It explains why certain geometric shapes (like convex projective manifolds) behave the way they do. It allows mathematicians to study transitions where a "perfect" group slowly turns into a "messy" one, or vice versa, without the math breaking down.

Summary

• The Goal: Understand groups that are mostly perfect but have some messy, "rusty" parts.
• The Trick: Instead of forcing the messy parts to be perfect, we just ensure they have a consistent "shadow" connection to the perfect parts.
• The Result: We can now change (deform) these groups in many new ways, and they will still hold together.
• The Metaphor: It's like realizing that a house doesn't need to be built entirely of perfect bricks to be stable; as long as the foundation and the roof are connected correctly, you can use all sorts of weird, recycled materials for the walls, and the house will still stand.

This paper is a "glue" that holds together many different areas of geometry, allowing mathematicians to explore a much wider universe of shapes and symmetries than ever before.

Technical Summary

Here is a detailed technical summary of the paper "An Extended Definition of Anosov Representation for Relatively Hyperbolic Groups" by Theodore Weisman.

1. Problem Statement and Motivation

Context:
In the study of discrete subgroups of Lie groups, Anosov representations (introduced by Labourie and Guichard-Wienhard) serve as the natural higher-rank generalization of convex cocompact subgroups in rank-one Lie groups. They are characterized by strong dynamical properties, including the existence of an equivariant boundary map from the Gromov boundary of a hyperbolic group to a flag manifold.

The Gap:
While Anosov representations are well-understood, they are restricted to hyperbolic groups. In rank one, the broader class of geometrically finite subgroups (which includes groups with "cusps" or parabolic subgroups) is well-studied. However, extending the concept of geometric finiteness to higher-rank Lie groups has been challenging. Previous attempts, such as relative Anosov representations (Kapovich-Leeb, Zhu, Zhu-Zimmer), impose strict dynamical constraints on the peripheral (cusp) subgroups. These constraints often exclude interesting examples, such as certain convex projective manifolds with "generalized cusps" or limits of Anosov representations where the peripheral structure changes.

Objective:
The paper aims to define a new, more flexible class of representations called Extended Geometrically Finite (EGF) representations. The goal is to unify various existing notions of geometric finiteness in higher rank, allow for deformations that change the conjugacy class of peripheral subgroups, and provide a framework for studying limits of Anosov representations.

2. Methodology

The author employs a combination of topological dynamics, geometric group theory, and Lie theory. The core methodology involves:

• Reversing the Boundary Map: Unlike standard Anosov representations (which map the group boundary to the flag manifold), EGF representations are defined via a map from a closed invariant subset of the flag manifold to the Bowditch boundary of the relatively hyperbolic group. This "backwards" map is a surjective, equivariant, antipodal map, not necessarily a homeomorphism.
• Extended Convergence Actions: The paper introduces the concept of an extended convergence group action. This generalizes the standard convergence group action by allowing the group to act on a space $M$ (the flag manifold) such that the action "extends" the convergence action on the Bowditch boundary $\partial(\Gamma, H)$ via a quotient map. This involves defining "repelling strata" ($C_z$) that control the dynamics near parabolic points.
• Relative Quasigeodesic Automata: A major technical innovation is the construction of a relative quasigeodesic automaton. Inspired by symbolic coding in hyperbolic groups (Sullivan, Kapovich-Kim-Lee) and Anosov representations (Bochi-Potrie-Sambarino), this automaton encodes the dynamics of the group action on the flag manifold. It provides a "relative coding" for points in the Bowditch boundary, handling both conical limit points and parabolic fibers simultaneously.
• Metric Contraction on Flag Manifolds: The author utilizes a metric $C_\Omega$ on proper domains of flag manifolds (based on cross-ratios and Hilbert metrics, following Zimmer) to prove that sequences of group elements along paths in the automaton are contracting and P-divergent.

3. Key Definitions

Definition 1.3 (EGF Representation):
Let $\Gamma$ be a relatively hyperbolic group with peripheral structure $H$, $G$ a semisimple Lie group, and $P \subset G$ a symmetric parabolic subgroup. A representation $\rho: \Gamma \to G$ is Extended Geometrically Finite (EGF) with respect to $P$ if:

1. There exists a closed $\rho(\Gamma)$-invariant set $\Lambda \subset G/P$.
2. There exists a continuous, $\rho$-equivariant, surjective, antipodal map $\phi: \Lambda \to \partial(\Gamma, H)$.
3. The map $\phi$ extends the convergence action of $\Gamma$. This means there exist open sets $C_z \subset G/P$ (repelling strata) for each $z \in \partial(\Gamma, H)$ satisfying specific dynamical conditions (C1 and C2) ensuring that group elements moving towards $z$ contract neighborhoods of $\phi^{-1}(z)$ while expanding away from $\phi^{-1}(z^-)$.

Key Distinction:
In standard relative Anosov representations, the boundary map is an embedding (injective). In EGF representations, the map $\phi$ is a surjection that may collapse fibers over parabolic points. This flexibility allows EGF representations to capture cases where the peripheral subgroups do not act as "nice" Anosov subgroups (e.g., they may have non-diagonalizable Jordan blocks).

4. Main Results

A. Stability Theorem (Theorem 1.4):
The central result is a relative stability theorem.

• Statement: If $\rho$ is an EGF representation, then any small deformation $\rho'$ in a peripherally stable subspace $W \subseteq \text{Hom}(\Gamma, G)$ is also EGF.
• Significance: "Peripherally stable" is a weak condition requiring only that the large-scale dynamical behavior of the peripheral subgroups is preserved. Crucially, this does not require the deformed peripheral subgroups to be topologically conjugate to the original ones. This allows for deformations that change the conjugacy class of the peripheral subgroups (e.g., changing a unipotent cusp to a diagonalizable one), which was impossible under previous definitions.

B. Characterization and Generalization (Theorem 1.10 & 4.19):

• Generalization: EGF representations strictly generalize relative Anosov representations.
• Equivalence: A representation is relative Anosov if and only if it is EGF and the boundary extension $\phi$ is injective (a homeomorphism onto its image).
• Anosov Relativization (Theorem 1.16): If an EGF representation restricts to a standard Anosov representation on every peripheral subgroup, then the entire representation is a standard (non-relative) Anosov representation.

C. New Examples and Applications:

• Convex Projective Structures: The paper proves that holonomy representations of various "geometrically finite" convex projective manifolds (including those with generalized cusps classified by Ballas-Cooper-Leitner) are EGF. These were previously not covered by relative Anosov definitions.
• Limits of Anosov Representations: The paper constructs examples of EGF representations that arise as limits of Anosov representations but are not themselves Anosov (e.g., limits of triangle reflection groups in $SL(3, \mathbb{R})$ where transversality fails).
• Deformations of Jordan Blocks: Example 9.3 demonstrates that EGF representations can be deformed to change the Jordan block structure of peripheral elements (e.g., from unipotent to diagonalizable) while maintaining the EGF property.

5. Significance and Impact

1. Unification: The EGF framework unifies disparate examples of geometric finiteness in higher rank, including convex projective structures, limits of Anosov representations, and various "relativized" Anosov definitions.
2. Flexibility in Deformation: By relaxing the requirement for peripheral subgroups to remain conjugate or preserve specific algebraic structures (like unipotency), the theory opens the door to studying a much wider range of deformation spaces and transitions between different types of geometric structures.
3. Tool Development: The introduction of relative quasigeodesic automata and the specific analysis of contracting paths in flag manifolds provide powerful new tools for analyzing discrete subgroups of higher-rank Lie groups. These tools are applicable beyond the immediate scope of EGF representations (e.g., to abstract relatively hyperbolic groups).
4. Bridging Rank One and Higher Rank: The paper successfully adapts the "geometric finiteness" intuition from rank one (where cusps are well-understood) to higher rank, providing a robust definition that accommodates the complex dynamics of semisimple Lie groups.

In summary, Weisman's paper provides a robust, flexible, and technically rigorous extension of Anosov theory to relatively hyperbolic groups, resolving limitations in previous definitions and enabling the study of a broader class of discrete subgroups and their deformations.