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There is a recently discovered connection between the spectral theory ofSchr\"o-dinger operators whose potentials exhibit aperiodic order and that ofLaplacians associated with actions of groups on regular rooted trees, asGrigorchuk's group of intermediate growth. We give an overview of correspondingresults, such as different spectral types in the isotropic and anisotropiccases, including Cantor spectrum of Lebesgue measure zero and absence ofeigenvalues. Moreover, we discuss the relevant background as well as thecombinatorial and dynamical tools that allow one to establish theafore-mentioned connection. The main such tool is the subshift associated to asubstitution over a finite alphabet that defines the group algebraically via arecursive presentation by generators and relators.

Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive substitution