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Let $G={*}_{j=1}^{q+1}{G}_{n_j+1}$ be the product of $q+1$ finite groups each having order $n_j+1$ and let $\mu$ be the probability measure which takes the value $p_j/n_j$ on each element of $G_{n_j+1}\setminus \left\{e\right\}$. In this paper we shall describe the point spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers $n_j$. We also compute the continuous spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ in several cases. A family of irreducible representations of $G$, parametrized on the continuous spectrum of $\mu$,...

Let $\mathfrak{X}$ be a homogeneous tree in which every vertex lies on $q+1$ edges, where $q\ge 2$. Let $\mathfrak{A}=Aut\left(\mathfrak{X}\right)$ be the group of automorphisms of $\mathfrak{X}$, and let $H$ be the its subgroup $PGL\left(2,F\right)$, where $F$ is a local field whose residual field has order $q$. We consider the restriction to $H$ of a continuous irreducible unitary representation $\pi$ of $\mathfrak{A}$. When $\pi$ is spherical or special, it was well known that $\pi$ remains irreducible, but we show that when $\pi$ is cuspidal, the situation is much more complicated. We then study in detail what happens when the...

We compute explicitly the number of paths of given length joining two vertices of the Cayley graph of the free product of cyclic groups of order k.

We compute explicitly the number of paths of given length joining two vertices of the Cayley graph of the free product of cyclic groups of order k.