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The limit cone and bounds on the growth indicator function

Lasse Lennart Wolf

TL;DR

The paper studies growth rates of discrete subgroups Γ in higher-rank real semisimple groups by analyzing Quint’s growth indicator ψ_Γ and the limit cone L_Γ. It proves that, when L_Γ avoids two distinct Weyl-chamber facets, ψ_Γ is bounded above by the half-sum of positive roots ρ, which implies the temperedness of L^2(Γackslash G); this result extends to many I-Anosov subgroups with at least two roots not exchanged by the opposition, yielding Γ-independent temperedness bounds. A central framework links growth to spectral data via the joint spectrum ˜σ_Γ and the Laplace spectrum on Γackslash G/K, establishing that μ_Γ lies in ˜σ_Γ for Zariski-dense Γ and relating the bottom of σ(Δ) to a δ'-parameter. The paper further refines the maximal growth direction through the modified indicators ψ'_Γ, δ'_μ, and μ_Γ, connecting the limit cone geometry to precise asymptotics and providing both forward and reverse implications between μ_Γ and L_Γ′, with applications to critical exponents and properties under Property (T). Overall, it advances a spectral-geometry toolkit for controlling higher-rank growth and temperedness beyond classical rank-one results, with concrete bounds for Anosov-type subgroups and new insights into the structure of the joint spectrum.

Abstract

Given a real semisimple Lie group $G$ with finite center and a discrete subgroup $Γ\subset G$ whose limit cone is disjoint from two facets of the Weyl chamber we show that Quint's growth indicator function $ψ_Γ$ is bounded by the half sum of positive roots $ρ$, i.e. it has slow growth, implying that the representation $L^2(Γ\backslash G)$ is tempered. In particular, this holds for each $I$-Anosov subgroup provided that $I$ contains at least two distinct simple roots that are not interchanged by the opposition involution.

The limit cone and bounds on the growth indicator function

TL;DR

The paper studies growth rates of discrete subgroups Γ in higher-rank real semisimple groups by analyzing Quint’s growth indicator ψ_Γ and the limit cone L_Γ. It proves that, when L_Γ avoids two distinct Weyl-chamber facets, ψ_Γ is bounded above by the half-sum of positive roots ρ, which implies the temperedness of L^2(Γackslash G); this result extends to many I-Anosov subgroups with at least two roots not exchanged by the opposition, yielding Γ-independent temperedness bounds. A central framework links growth to spectral data via the joint spectrum ˜σ_Γ and the Laplace spectrum on Γackslash G/K, establishing that μ_Γ lies in ˜σ_Γ for Zariski-dense Γ and relating the bottom of σ(Δ) to a δ'-parameter. The paper further refines the maximal growth direction through the modified indicators ψ'_Γ, δ'_μ, and μ_Γ, connecting the limit cone geometry to precise asymptotics and providing both forward and reverse implications between μ_Γ and L_Γ′, with applications to critical exponents and properties under Property (T). Overall, it advances a spectral-geometry toolkit for controlling higher-rank growth and temperedness beyond classical rank-one results, with concrete bounds for Anosov-type subgroups and new insights into the structure of the joint spectrum.

Abstract

Given a real semisimple Lie group with finite center and a discrete subgroup whose limit cone is disjoint from two facets of the Weyl chamber we show that Quint's growth indicator function is bounded by the half sum of positive roots , i.e. it has slow growth, implying that the representation is tempered. In particular, this holds for each -Anosov subgroup provided that contains at least two distinct simple roots that are not interchanged by the opposition involution.

Paper Structure

This paper contains 22 sections, 28 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Further results
  3. Spectral theory
  4. Related results
  5. Critical exponents
  6. Property (T)
  7. Examples
  8. Idea of the proofs and outline of the article
  9. Preliminaries
  10. Notation
  11. The growth indicator function
  12. Spherical dual, joint spectrum, and temperedness
  13. Spherical dual
  14. Joint spectrum
  15. Temperedness
  16. ...and 7 more sections

Key Result

Theorem 1.1

If $\mathcal{L}_\Gamma\setminus \{0\}$ is disjoint from two facets $F_\alpha$ and $F_\beta$ with $\alpha \neq \beta$ and $\alpha \neq \iota\beta$, then

Figures (1)

  • Figure 1: Visualisation of Proposition \ref{['prop:boundwallavoided']} for $G=\mathrm{SO}_0(2,n)$. $\mu_\Gamma$ must be contained in $\operatorname{conv}(W (\rho-\Theta))$ (green). If $\mu_\Gamma \in \mathbb{R}_{\geq 0} \omega_{\alpha_2}$, then the maximal $\delta'_{\omega_{\alpha_2}}$ is so that $\mu_\Gamma = \delta'_{\omega_{\alpha_2}} \omega_{\alpha_2}=\rho-\Theta$. There is no $\Gamma$-independent improvement since $\operatorname{conv}(W (\rho-\Theta))$ and $\operatorname{conv}(W \delta'_{\omega_{\alpha_2}} \omega_{\alpha_2})$ (brown) coincide in this extremal case. Contrarily, if $\mu_\Gamma\in \mathbb{R}_{\geq 0}\omega_{\alpha_1}$, then $\operatorname{conv}(W \delta'_{\omega_{\alpha_1}} \omega_{\alpha_1})$ (orange) is always smaller than $\operatorname{conv}(W (\rho-\Theta))$ as $\mu_\Gamma= \delta'_{\omega_{\alpha_1}} \omega_{\alpha_1}$ is contained in the latter.

Theorems & Definitions (52)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition : OhDichotomy
  • Theorem : LWW
  • Definition 2.1: CHH88
  • Lemma 3.1
  • proof
  • ...and 42 more