2022/05/24 by Adrien Le Boudec, Nicolás Matte Bon, Boudec, Adrien Le +1
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2205.11924
openalex publication_date 2022/05/24 · openalex created_date 2022/05/27 · openalex updated_date 2026/07/28
Given a finitely generated group G, we are interested in common geometric properties of all graphs of faithful actions of G. In this article we focus on their growth. We say that a group G has a Schreier growth gap f(n) if every faithful G-set X satisfies volG, X(n)\succcurlyeq f(n), where volG, X(n) is the growth of the action of G on X. Here we study Schreier growth gaps for finitely generated solvable groups. We prove that if a metabelian group G is either finitely presented or torsion-free, then G has a Schreier growth gap n2, provided G is not virtually abelian. We also prove that if G is a metabelian group of Krull dimension k, then G has a Schreier growth gap nk. For instance the wreath product Cp \wr ℤd has a Schreier growth gap nd, and ℤ \wr ℤd has a Schreier growth gap nd+1. These lower bounds are sharp. For solvable groups of finite Prüfer rank, we establish a Schreier growth gap exp(n), provided G is not virtually nilpotent. This covers all solvable groups that are linear over ℚ. Finally for a vast class of torsion-free solvable groups, which includes solvable groups that are linear, we establish a Schreier growth gap n2.