Groups, Geometry, and Dynamics


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Volume 8, Issue 4, 2014, pp. 1195–1205
DOI: 10.4171/GGD/300

Published online: 2014-12-31

Rank gradient and cost of Artin groups and their relatives

Aditi Kar[1] and Nikolay Nikolov[2]

(1) University of Southampton, UK
(2) University of Oxford, UK

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

Keywords: Rank gradient, cost, Artin groups

Kar Aditi, Nikolov Nikolay: Rank gradient and cost of Artin groups and their relatives. Groups Geom. Dyn. 8 (2014), 1195-1205. doi: 10.4171/GGD/300