On groups whose cogrowth series is the diagonal of a rational series

2023/01/18 by Alex Bishop, Bishop, Alex
Computer Science · Mathematics · #20F65 #20K35 (Secondary) #68R15 (Primary) 05A15 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2301.07589

openalex publication_date 2023/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if a group contains ℤⁿ × F_m as a finite-index subgroup, then its cogrowth series is the diagonal of a rational function for every generating set. This answers a question of Pak and Soukup on the cogrowth of virtually abelian groups; and generalises a result by Elder, Rechnitzer, Janse van Rensburg, and Wong on the cogrowth series of the Baumslag-Solitar groups BS(N,N).

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