  We compute the joint spectrum of $D_{\infty h}$ with respect to the left
regular representation, and finds two generators of the De Rham cohomology
group of joint resolvent set which is induced by different central linear
functionals. Through action of $D_{\infty h}$ on 4-ary trees, we get a
self-similar realization of the group $C^*$ algebra of $D_{\infty h}$.
