Rank gradient and cost of Artin groups and their relatives

TL;DR

This paper investigates the rank gradient and cost properties of various groups, including Artin groups, mapping class groups, and automorphism groups, establishing vanishing results and connections to $L^2$-Betti numbers.

Contribution

It proves the vanishing of rank gradient for several important classes of groups and relates this to their fixed price and $L^2$-Betti numbers, advancing understanding of their geometric properties.

Findings

Rank gradient vanishes for mapping class groups of genus > 1.

Rank gradient vanishes for all $Aut(F_n)$ and $Out(F_n)$ for $n eq 2$.

Artin groups with connected underlying graphs have fixed price 1.

Abstract

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.