Rank gradient and cost of Artin groups and their relatives

Aditi Kar; Nikolay Nikolov

paper

Rank gradient and cost of Artin groups and their relatives

arXiv:1210.2873

Abstract

We prove that the rank gradient vanishes for mapping class groups of genus bigger than 1, $A u t (F_{n})$ , for all $n$ , $O u t (F_{n})$ for $n \geq 3$ , and any Artin group whose underlying graph is connected. These groups have fixed price 1. We compute the rank gradient and verify that it is equal to the first $L^{2}$ -Betti number for some classes of Coxeter groups.