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 $1$, $\mathrm{Aut}(F_n)$ for all $n$, $\mathrm{Out}(F_n)$, $n\ge 3$ 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.