On upper bounds for the first ℓ²-Betti number

2022/02/03 by Carsten Feldkamp, Feldkamp, Carsten, Steffen Kionke +1
Computer Science · Engineering · Mathematics · #20F69 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary 20F05 #Secondary 20F50 #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2202.01688

openalex publication_date 2022/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article presents a method for proving upper bounds for the first ℓ²-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first ℓ²-Betti number. Our approach extends to generalizations of ℓ²-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

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