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Quasi-isometry invariant of weakly special square complexes

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Sangrok Oh (KAIST)

Abstract:
Given a compact weakly special square complex Y, we develop the notion of a (reduced) intersection complex for Y which is used to study the pattern of maximal subcomplexes with product structures. It turns out that if the universal covers of two compact weakly special square complexes are quasi-isometric, then their intersection complexes are isomorphic.
We then use this fact to study quasi-isometric classification of 2-dimensional right-angled Artin groups and 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.