Automorphisms of Contact Graphs of CAT(0) Cube Complexes

We show that, under weak assumptions, the automorphism group of a $\textrm{CAT(0)}$ 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 with 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.

Revisiting Leighton’s theorem with the Haar measure

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.