Finite-sided Dirichlet domains and Anosov subgroups

Abstract

We consider Dirichlet domains for Anosov subgroups $Γ$ of semisimple Lie groups $G$ acting on the associated symmetric space $G/K$. More precisely, we consider certain Finsler metrics on $G/K$ and a sufficient condition so that every Dirichlet domain for $Γ$ is finite-sided in a strong sense. Under the same condition, the group $Γ$ admits a domain of discontinuity in a flag manifold where the Dirichlet domain extends to a compact fundamental domain. As an application we show that Dirichlet-Selberg domains for $n$-Anosov subgroups $Γ$ of $SL(2n,\mathbb{R})$ are finite-sided when all singular values of elements of $Γ$ diverge exponentially in the word length. For every $d\ge 3$, there are projective Anosov subgroups of $SL(d,\mathbb{R})$ which do not satisfy this property and have Dirichlet-Selberg domains with infinitely many sides. More generally, we give a sufficient condition for a subgroup of $SL(d,\mathbb{R})$ to admit a Dirichlet-Selberg domain whose intersection with an invariant convex set is finite-sided.

Figures (4)

• Figure 1: Illustration of the intersection of a Selberg bisector and $\mathbb{RP}^2$.
• Figure 2: An illustration of $\omega_1$ for $\mathop{\mathrm{SL}}\nolimits(4,\mathop{\mathrm{\mathbb{R}}}\nolimits)$.
• Figure 3: An illustration of the positive Weyl chamber $\mathbb{P}(\mathfrak{a}^+)=\sigma_{mod}$ for $\mathop{\mathrm{Sp}}\nolimits(6,\mathop{\mathrm{\mathbb{R}}}\nolimits)$.
• Figure 4: Illustration of the disjoint half-space property.

Theorems & Definitions (127)

• Definition 1.1: KLP18bBPS19
• Theorem 1.3: Theorem \ref{['thm:InfiniteSided']}
• Definition 1.5
• Theorem 1.6
• Definition 1.7: $\omega$-undistorted subgroups
• Remark 1.8
• Proposition 1.9: Proposition \ref{['prop:limit cone is connected Appendix']}
• Theorem 1.10: Theorem \ref{['thm:omega-URU implies properly finite-sided']}
• Theorem 1.11
• Proposition 1.12: Proposition \ref{['prop:CoarseFibration']}