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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.
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundam
The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible s
In his work on the Novikov conjecture, Yu introduced Property $A$ as a readily verified criterion implying coarse embeddability. Studied subsequently as a property in its own right, Property $A$ for a discrete group is known to be equivalent to exact
We show that a triangle Artin group $text{Art}_{MNP}$ where $Mleq Nleq P$ splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either $M>2$, or $N>3$. We also prove that all even three generator Artin groups are residually finite.
We show that if a right-angled Artin group $A(Gamma)$ has a non-trivial, minimal action on a tree $T$ which is not a line, then $Gamma$ contains a separating subgraph $Lambda$ such that $A(Lambda)$ stabilizes an edge in $T$.