Quasi-isometry invariants of weakly special square complexes

Abstract

We define the intersection complex for the universal cover of a compact weakly special square complex and show that it is a quasi-isometry invariant. By using this quasi-isometry invariant, we study the quasi-isometric classification of 2-dimensional right-angled Artin groups and planar graph 2-braid groups. Our results cover two well-known cases of 2-dimensional right-angled Artin groups: (1) those whose defining graphs are trees and (2) those whose outer automorphism groups are finite. Finally, we show that there are infinitely many graph 2-braid groups which are quasi-isometric to right-angled Artin groups and infinitely many which are not.

arXiv commentWe have changed the terminology `special cacti' to `simplest cacti' since the previous definition was not enough to cover the results about graph 2-braid groups in the paper. The corrigendum will appear in Topology and its applications. This is a unified version of the previous paper and its corrigendum (everything appeared in the corrigendum has been corrected)

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Cited by in corpus (2)

• Measure equivalence classification of right-angled Artin groups: the finite $\mathrm{Out}$ classes
• The quasi-isometry invariance of the Coset Intersection Complex