## ​Projective spectrum and finitely generated groups/Complexdynamics and the infinite dihedral group

${\mathcal B}$, its {\em projective jointspectrum} $P(A)$ is the collection of $z\in {\bf C}^n$ such that themultiparameter pencil $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible. If${\mathcal B}$ is the group $C^*$-algebra for a discrete group $G$ generated by$A_1,\ A_2,\ ...,\ A_n$ with respect to a representation $\rho$, then $P(A)$ isan invariant of (weak) equivalence for $\rho$. This series of talks presentsome recent work on the projective spectrum $P(R)$ of $R=(1,\ a,\ t)$ for theinfinite dihedral group $D_{\infty}=<a,\ t\ |\ a^2=t^2=1>$ with respectto the left regular representation. Results include a description of thespectrum, a formula for the Fuglede-Kadison determinant of the pencil$R(z)=z_0+z_1a+z_2t$, the first singular homology group of the joint resolventset $P^c(R)$, and dynamical properties of the spectrum. These results give newinsight into some earlier studies on groups of intermediate growth. Moreover,they suggest a link between projective spectrum and the Julia set of dynamicalmaps. Time permitting, I will also go over some other aspects of the projectivespectrum as related to group theory, topology, complex geometry and Liealgebras.

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