Joint spectrum and the infinite dihedral group

For a tuple $A=(A_1,A_2,\dots,A_n)$ of elements in a unital Banach algebra $\mathcal B$, its projective joint spectrum $P(A)$ is the collection of $z\in\mathbb C^n$ such that the multiparameter pencil $A(z)=z_1A_1+z_2A_2+\dots+z_nA_n$ is not invertible. If $\mathcal B$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_1,A_2,\dots,A_n$ with respect to a representation $\rho$, then $P(A)$ is an invariant of (weak) equivalence for $\rho $. This paper computes the joint spectrum of $R=(1,a,t)$ for the infinite dihedral group $D_\infty=\langle a,t\mid a^2=t^2=1\rangle$ with respect to the left regular representation $\lambda_{D_\infty}$, and gives an in-depth analysis on its properties. A formula for the Fuglede–Kadison determinant of the pencil $R(z)=z_0+z_1a+z_2t$ is obtained, and it is used to compute the first singular homology group of the joint resolvent set $P^\mathrm c(R)$. The joint spectrum gives new insight into some earlier studies on groups of intermediate growth, through which the corresponding joint spectrum of $(1,a,t)$ with respect to the Koopman representation $\rho$ (constructed through a self-similar action of $D_\infty$ on a binary tree) can be computed. It turns out that the joint spectra with respect to the two representations coincide. Interestingly, this fact leads to a self-similar realization of the group $C^*$-algebra $C^*(D_\infty)$. This self-similarity of $C^*(D_\infty)$ manifests itself in some dynamical properties of the joint spectrum.