Counting irreducible modules for profinite groups

• Ged Corob Cook ^[1] ; Steffen Kionke ^[2] ; Matteo Vannacci ^[3]
  1. [1] University of Lincoln

     University of Lincoln

     Lincoln District, Reino Unido

     
  2. [2] University of Hagen

     University of Hagen

     Kreisfreie Stadt Hagen, Alemania

     
  3. [3] Universidad del País Vasco/Euskal Herriko Unibertsitatea

     Universidad del País Vasco/Euskal Herriko Unibertsitatea

     Leioa, España

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 4, 2023, págs. 1519-1566
• Idioma: inglés
• DOI: 10.4171/RMI/1382
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• Resumen

  • This article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-based powers, we obtain some necessary and some sufficient conditions for groups to have UBERG. As an application, we prove that the class of UBERG groups is closed under split extensions but fails to be closed under extensions in general. On the other hand, we show that the closely related probabilistic finiteness property PFP1 is closed under extensions. In addition, we prove that profinite groups of type FP1 with UBERG are always finitely generated and we characterise UBERG in the class of pronilpotent groups.

    Using infinite products of finite groups, we construct several examples with unexpected properties: (1) a UBERG group which cannot be finitely generated, (2) a group of type PFP∞ which is not UBERG and not finitely generated, and (3) a finitely generated group of type PFP∞ with superexponential subgroup growth.