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

Abstract

We show that, under weak assumptions, the automorphism group of a $\mathrm{CAT}(0)$ cube complex $X$ coincides with the automorphism group of Hagen's contact graph $\mathcal{C}(X)$. 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 right-angled 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.

16 pages, no figures; v2: published version, open access on IMRN

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