Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes

Abstract

We show that, under weak assumptions, the automorphism group of a CAT(0) cube complex X coincides with the automorphism group of Hagen's contact graph C(X). The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim-Koberda extension graphs, which have much larger automorphism group.We also study contact graphs associated to Davis complexes of rightangled Coxeter groups. We show that these contact graphs are less well-behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex.

Large facing tuples and a strengthened sector lemma

“…Since Aut(X) acts on the set of hyperplanes of X, preserving intersection and non-intersection of carriers, we get a homomorphism Aut(X) → Aut(CX); the relationship between these two groups is studied in [Fio20].…”

Mapping class groups are quasicubical

“…The result has been an effusion of new understanding in both settings. For mapping class groups, this has included: confirmation of Farb's quasiflat conjecture [BHS21], semihyperbolicity [DMS20,HHP20], decision problems for subgroups [Bri13,Kob12], and residual properties [DHS21,BHMS20]; and on the cubical side: versions of Ivanov's theorem [Iva97,Fio20], characterisations of Morse geodesics [ABD21,IMZ21], control on purely loxodromic subgroups [KK14,KMT17], and results on uniform exponential growth [ANS `19].…”