Rank gradient and cost of Artin groups and their relatives

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