Quasi-isometric classification of right-angled Artin groups I: the finite out case

Abstract

Let $G$ and $G^{\prime }$ be two right-angled Artin groups (RAAG). We show they are quasi-isometric iff they are isomorphic, under the assumption that $Out(G)$ and $Out(G^{\prime })$ are finite. If only $Out(G)$ is finite, then $G^{\prime }$ is quasi-isometric $G$ iff $G^{\prime }$ is isomorphic to a finite index subgroup of $G$. In this case, we give an algorithm to determine whether $G$ and $G^{\prime }$ are quasi-isometric by looking at their defining graphs.

Modifications and expansions according the referee's report. Exposition improved

References in corpus (6)

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• Anti-trees and right-angled Artin subgroups of braid groups
• Top dimensional quasiflats in $CAT(0)$ cube complexes
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Cited by in corpus (11)

• Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen
• Quasi-isometry classification of RAAGs that split over cyclic subgroups
• Groups quasi-isometric to RAAG's
• Measure equivalence classification of transvection-free right-angled Artin groups
• Groups acting on CAT(0) cube complexes with uniform exponential growth
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
• Automorphisms of contact graphs of $\mathrm{CAT}(0)$ cube complexes
• Quasi-isometry invariants of weakly special square complexes
• Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups
• Measure equivalence classification of right-angled Artin groups: the finite $\mathrm{Out}$ classes
• The quasi-isometry invariance of the Coset Intersection Complex