A homotopy moduli space of G2 structures with infinite fundamental group

An irreducible G2-manifold is a Riemannian 7-manifold M with holonomy group equal to the exceptional Lie group G2. When M is closed, the Teichmüller space T(M) of G2 metrics on M divided by diffeomorphisms isotopic to the identity is a smooth, finite-dimensional manifold by a result of Joyce. Yet its topology, and that of its quotient by the smooth mapping class group, remains elusive. Using ideas of Crowley, Goette, and Hertl, we exhibit the first known example of a G2-manifold M together with infinitely many diffeomorphisms that both act freely on T(M) and preserve a connected component. The diffeomorphisms are 7-dimensional analogs of diffeomorphisms of K3 surfaces constructed recently by Farb and Looijenga.