Rank gradient and cost of Artin groups and their relatives

Abstract

We prove that the rank gradient vanishes for mapping class groups of genus greater than 1, Aut($F_n$) for all $n$, Out($F_n$), $n\ge 3$ 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.