Quint's Growth Indicator Function

• Quint's growth indicator function is a vector-valued refinement of the classical critical exponent, defining exponential growth rates in prescribed directions within symmetric spaces.
• It integrates key aspects of geometry, dynamical systems, and representation theory to relate orbit distribution, Poincaré series convergence, and the spectral geometry of locally symmetric spaces.
• Its homogeneity, concavity, and precise bounds provide actionable insights into the temperedness of discrete subgroups and the analysis of Laplace–Beltrami spectra.

Quint's growth indicator function provides a vector-valued refinement of the classical critical exponent for discrete subgroups of semisimple Lie groups, encapsulating directional growth rates in higher-rank symmetric spaces. Central to current research in representation theory and geometry, it ties together the asymptotic distribution of group orbits, convergence of Poincaré-type series, spectral geometry of locally symmetric spaces, and representation-theoretic temperedness.

1. Foundational Setting and Definition

Let $G$ denote a connected noncompact semisimple Lie group with finite center, $K < G$ a maximal compact subgroup, and $X = G/K$ the associated Riemannian symmetric space of noncompact type. The Lie algebra admits a Cartan decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$, with $\mathfrak{a} \subset \mathfrak{p}$ a maximal abelian subspace and corresponding positive Weyl chamber $\mathfrak{a}_+$. The restricted root system $\Sigma \subset \mathfrak{a}^{*}$ is selected with positive roots $\Sigma^+$, and the half-sum of positive roots is $$\rho = \frac{1}{2}\sum_{\alpha \in \Sigma^+} m_\alpha \alpha \in \mathfrak{a}^*$$.

Given a torsion-free discrete subgroup $\Gamma < G$, the Cartan projection $\mu_+ : G \to \overline{\mathfrak{a}_+}$ is defined by $g = k \exp(\mu_+(g)) k'$. The growth indicator function (denoted as $\Psi_\Gamma$ or $\psi_\Gamma$, with both conventions in the literature) is constructed to quantify the exponential growth rate of $\Gamma$-orbits in prescribed directions: $$\Psi_\Gamma(H) = \|H\| \inf_{\substack{\mathcal{C} \subset \mathfrak{a}_+\ H \in \mathcal{C} \text{ open cone}}} \left\{ s \in \mathbb{R} \mid \sum_{\substack{\gamma \in \Gamma\ \mu_+(\gamma) \in \mathcal{C}}} e^{-s \|\mu_+(\gamma)\|} < \infty \right\}, \quad \Psi_\Gamma(0) = 0.$$

2. Key Properties and Structural Results

Quint’s original work and subsequent analyses established the following core properties for the growth indicator function:

• Homogeneity: $\Psi_\Gamma(tH) = t\Psi_\Gamma(H)$ for all $t \geq 0$ and $H \in \mathfrak{a}_+$.
• Concavity: $\Psi_\Gamma$ is concave and upper semicontinuous on its support.
• Support and Asymptotic Cone: The support coincides with the asymptotic or limit cone:

$$\mathcal{L}_\Gamma = \left\{ v \in \mathfrak{a}_+ : \psi_\Gamma(v) > -\infty \right\} = \left\{ \lim t_n \mu_+(\gamma_n) : t_n \to 0,\, \gamma_n \in \Gamma \right\}.$$

On the interior, $\Psi_\Gamma$ is strictly positive; outside, $\Psi_\Gamma(v) = -\infty$.

• Upper Bounds: For all $H \in \mathfrak{a}_+$, $\Psi_\Gamma(H) \leq 2\rho(H)$ in general, and under additional geometry, tighter estimates (e.g., $\Psi_\Gamma \leq \rho$).
• Pointwise (Tent) Bounds: For simple roots $\{\alpha_1, ..., \alpha_k\}$,

$$\psi_\Gamma(v) \leq \min_{1\leq i \leq k} \delta_{\alpha_i} \alpha_i(v),$$

where $\delta_{\alpha_i}$ is the critical exponent in the $\alpha_i$-direction (tent property) (Kim et al., 2021).

• Strict Concavity for Anosov Subgroups: If $\Gamma$ is $\Delta$-Anosov, then $\psi_\Gamma$ is strictly concave and achieves equality with the tent bound only in $k$ distinguished directions, corresponding to the simple roots.

3. Modified Critical Exponents and Spectral Characterizations

The growth indicator function serves as a bridge between group-theoretical growth conditions and finer spectral properties:

• Modified Critical Exponent: By considering the polyhedral norm $d_s(H)$ (dependent on $\rho$), the modified critical exponent is

$$\tilde{\delta}_\Gamma = \inf\left\{s \in \mathbb{R}: \sum_{\gamma \in \Gamma} e^{-d_s(\mu_+(\gamma))} < \infty \right\},$$

with $$0 \leq \delta_\Gamma \leq \tilde{\delta}_\Gamma \leq 2\|\rho\|$$ and $\tilde{\delta}_\Gamma = \delta_\Gamma$ in rank one (Wolf et al., 2023).

• Spectral Formulae: The bottom of the $L^2$-spectrum $\lambda_0(\Gamma \backslash X)$ of the Laplace–Beltrami operator is expressed in terms of $\Psi_\Gamma$:

$$\boxed{ \lambda_0(\Gamma \backslash X) = \|\rho\|^2 - \max \left\{ 0,\, \sup_{H \in \mathfrak{a}_+} \frac{\Psi_\Gamma(H) - \langle \rho, H \rangle}{\|H\|} \right\}^2 }.$$

When $\Psi_\Gamma(H) \leq \langle \rho, H \rangle$ for all $H$, this simplifies to $\lambda_0 = \|\rho\|^2$.

• Equivalence with Temperedness: The following are equivalent for $\Gamma$ (Wolf et al., 2023):
  • $\tilde{\delta}_\Gamma \leq \|\rho\|$
  • $\Psi_\Gamma \leq \rho$ on $\mathfrak{a}_+$
  • $\lambda_0(\Gamma \backslash X) = \|\rho\|^2$
  • $L^2(\Gamma \backslash G)$ is tempered

This demonstrates the function’s central role in encoding both geometric and representation-theoretic data.

4. Limit Cone, Structure Theorems, and Wall-Avoidance

A crucial geometric condition is the interaction of the limit cone with the Weyl chamber walls:

• Facets and Wall Avoidance: Each simple root $\alpha$ defines a facet $F_\alpha = \ker(\alpha) \cap \mathfrak{a}_+$. The limit cone $\mathcal{L}_\Gamma$ being disjoint from $F_\alpha$ and $F_\beta$ for two distinct simple roots $\alpha \neq \beta$, $\beta \neq \iota(\alpha)$ (with $\iota$ the opposition involution), guarantees the slow growth regime:

$$\psi_\Gamma(v) \leq \rho(v)\quad\text{for all }v \in \mathfrak{a}_+$$

(Wolf, 10 Nov 2025).

• I-Anosov Subgroups: If $\Gamma$ is $I$-Anosov for $I \subset \Pi$ containing at least two simple roots in different opposition classes, the chamber wall avoidance criterion is satisfied, hence $\psi_\Gamma \leq \rho$ and $L^2(\Gamma \backslash G)$ is tempered.
• Proof Scheme: The critical supporting functional $\mu_\Gamma \in \mathfrak{a}^*$ is characterized variationally, and wall-avoidance forces $\mu_\Gamma = 0$. When this holds, $\psi_\Gamma \leq \rho$ follows immediately, confirming temperedness.

5. Special Cases and Examples

Several significant classes exhibit sharp behaviors of the growth indicator function:

• Rank-One Case: Here, $\mathfrak{a} \cong \mathbb{R}$ and $\Psi_\Gamma(H) = \delta_\Gamma \|H\|$, with the spectral dichotomy reducing to classical expressions.
• Lattices: For $\operatorname{Vol}(\Gamma \backslash X) < \infty$, $\Psi_\Gamma = 2\rho$ everywhere and the bottom spectrum $\lambda_0 = 0$ (constant functions).
• Products of Rank-One Factors: For $X = X_1 \times X_2$, $\Psi_\Gamma$ decomposes coordinate-wise, allowing finer analysis. “Tempered implies bounded” still holds without requiring the Anosov condition.
• Anosov Groups: For $\Delta$-Anosov, $\psi_\Gamma$ is strictly concave, with tangency to $k$ unique simple-root directions, and the averaged strict inequality $$\psi_\Gamma(v) < (1/k)\sum_{i=1}^k \delta_{\alpha_i} \alpha_i(v)$$ for $k \geq 2$ (Kim et al., 2021).
• Hitchin Subgroups: For $\Gamma < \operatorname{PSL}(d,\mathbb{R})$ Hitchin, the tent property specializes as $$\psi_\Gamma(v) \leq \min_{1\leq i\leq d-1} (t_i - t_{i+1})$$ for $v = \operatorname{diag}(t_1, ..., t_d)$.

6. Pointwise Bounds: The Tent Property and Tensor Structure

The "tent property" offers pointwise control over the growth indicator in terms of critical exponents in simple root directions:

• For a Zariski dense discrete $\Gamma < G$,

$$\psi_\Gamma(v) \leq \min_{1 \leq i \leq k} \left[\delta_{\alpha_i} \alpha_i(v)\right]$$

(Kim et al., 2021).

• For $\Delta$-Anosov subgroups, there are precisely $k$ equality directions, and strict inequality holds in non-radial directions when $k \geq 2$. This structure is intimately related to deep distinctions between Anosov and non-Anosov (including geometrically finite but nondiscrete) cases.
• For self-joinings of convex cocompact subgroups in products $SO(n_1,1)\times\cdots\times SO(n_k,1)$, the tent property encodes the Hausdorff dimensions of boundary limit sets.

7. Implications, Applications, and Further Remarks

The growth indicator function integrates representation theory, geometric group theory, and ergodic theory:

• Temperedness and Representation Theory: The equivalence between $\psi_\Gamma \leq \rho$ and temperedness of $L^2(\Gamma \backslash G)$ connects dynamical wall-avoidance to the absence of complementary series in the spectrum (Wolf, 10 Nov 2025, Wolf et al., 2023).
• Critical Exponents and Fractal Geometry: Sharp upper bounds on Patterson–Sullivan critical exponents and Hausdorff dimensions of limit sets in flag varieties follow from tent property estimates.
• Property (T) Groups: Universal bounds such as $\psi_\Gamma \leq 2\rho - \Theta$ (where $\Theta$ is an explicit positive functional) are obtainable in settings with property (T), refining the growth profile structure (Wolf, 10 Nov 2025).
• Limitations and Counterexamples: If $\Gamma$ is only $\{\alpha\}$-Anosov, $\psi_\Gamma \leq \rho$ can fail (nontempered examples constructed), highlighting the necessity of two-wall avoidance for slow growth in higher rank.
• Product Phenomena: In product groups, even slow growth in each factor does not guarantee joint wall avoidance or temperedness unless more stringent structural conditions are met.

These results demonstrate that the full "joint" growth profile of $\Gamma$ in the Weyl chamber, as encoded by $\psi_\Gamma$, governs both convergence phenomena for natural series and deep spectral-geometric invariants.

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The above summary directly reflects the content and results documented in (Wolf et al., 2023, Wolf, 10 Nov 2025), and (Kim et al., 2021), characterizing the central position of Quint’s growth indicator function in the modern theory of discrete subgroups and the spectral analysis of locally symmetric spaces.