An extended definition of Anosov representation for relatively hyperbolic groups

Abstract

We define a new family of discrete representations of relatively hyperbolic groups which unifies many existing definitions and examples of geometrically finite behavior in higher rank. The definition includes the relative Anosov representations defined by Kapovich-Leeb and Zhu, and Zhu-Zimmer, as well as holonomy representations of various different types of "geometrically finite" convex projective manifolds. We prove that these representations are all stable under deformations whose restriction to the peripheral subgroups satisfies a dynamical condition, in particular allowing for deformations which do not preserve the conjugacy class of the peripheral subgroups.

Figures (7)

• Figure 1: Illustration for the proof of \ref{['prop:relative_geodesic_automaton_from_bb']}. The group element $\alpha_n$ nests an $\varepsilon$-neighborhood of $W_{a_{n+1}}$ inside of $W_{a_n}$ whenever $\alpha_n \cdot V_{a_{n+1}}$ intersects $V_{a_n}$.
• Figure 2: By iterating the nesting procedure backwards, we produce an infinite $\mathcal{G}$-path and a sequence of subsets intersecting in the initial point $z = z_0$.
• Figure 4: For any point $w$ in a sufficiently small neighborhood $V_z$ (pink) of $z$, the expanded neighborhood $\gamma^{-1}W_z$ (red) contains a uniform neighborhood of $\gamma^{-1} w$.
• Figure 5: For any point $w \ne q$ in a neighborhood $V_q$ (pink) of the parabolic point $q$, we find some $\gamma \in \Gamma_q$ so that $\gamma^{-1} w$ lies in $K_q$ (dark gray), a fundamental domain for the action of $\Gamma_q$ on $\partial \mathbb{H}^2 - \{q\}$. The expanded neighborhood $\gamma^{-1} W_q$ (red) contains a uniform neighborhood of $K_q$, so $\gamma^{-1}W_q$ contains a uniform neighborhood of $\gamma^{-1} w$.
• Figure 6: The group element $\gamma_z^{-1}$ is "expanding" about $V_z \subset \partial(\Gamma, \mathcal{H})$: while $W_z$ has diameter $< \delta$, $\gamma_z^{-1}W_z$ contains a $\delta$-neighborhood of $\gamma_z^{-1}V_z$. At the same time, $\gamma_z^{-1}$ enlarges an $\varepsilon$-neighborhood of $\phi^{-1}(z)$ in $M$, so that it contains a $2\varepsilon$-neighborhood of $\gamma_z^{-1}\phi^{-1}(W_z)$.

Theorems & Definitions (195)

• Definition 1.1
• Definition 1.1
• Definition 1.1
• theorem 1.2
• remark 1.3
• Definition 1.4
• Corollary 1.5
• remark 1.6
• remark 1.7
• theorem 1.8