Joint Spectrum and the Infinite Dihedral Group

For a tuple A = (A(1), A(2), ... , A(n)) of elements in a unital Banach algebra B, its projective joint spectrum P(A) is the collection of z is an element of C-n such that the multiparameter pencil A(z) = z(1)A(1) + z(2)A(2) + ... + z(n)A(n) is not invertible. If B is the group C*-algebra for a discrete group G generated by A(1), A(2), ... , 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-infinity = < a, t vertical bar a(2) = t(2) = 1 > with respect to the left regular representation lambda(D infinity), and gives an in-depth analysis on its properties. A formula for the Fuglede-Kadison determinant of the pencil R(z) = z(0) + z(1)a + z(2)t is obtained, and it is used to compute the first singular homology group of the joint resolvent set P-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-infinity 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-infinity). This self-similarity of C*(D-infinity) manifests itself in some dynamical properties of the joint spectrum.