Radial rapid decay does not imply rapid decay

2020/06/25 by Adrien Boyer, Boyer, Adrien, Antoine Pinochet Lobos +3
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR) #Microtubule and mitosis dynamics

paper · pdf · doi:10.48550/arxiv.2006.14446

openalex publication_date 2020/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group Γ:= SL₂(A), where A := \mathbbF_q[X,X^-1] is the ring of Laurent polynomials with coefficients in \mathbbF_q, endowed with the length function coming from a natural action of Γ on a product of two trees, to show that is has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. The criterion also applies to irreducible lattices in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn't inherited by open subgroups.

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