The limit cone and bounds on the growth indicator function

2025/11/10 by Lasse L. Wolf, Wolf, Lasse Lennart
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2511.06996

openalex publication_date 2025/11/10 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

Given a real semisimple Lie group G with finite center and a discrete subgroup Γ⊂ 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 L²(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.

Citations

• On the rate of exponential decay of coefficients on homogeneous spaces
• Deformations of Anosov subgroups: Limit cones and Growth indicators
• Zariski dense non-tempered subgroups in higher rank of nearly optimal growth
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• Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups
• Tempered homogeneous spaces IV
• Tempered homogeneous spaces III
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• Eigenvalues and Entropy of a Hitchin representation
• Maximal Representations of Surface Groups: Symplectic Anosov Structures

Related

• Tent property of the growth indicator functions and applications
• Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces
• Deformations of Anosov subgroups: Limit cones and Growth indicators
• On the rate of exponential decay of coefficients on homogeneous spaces
• Subgroup growth of lattices in semisimple Lie groups