Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha\in\mathbb C[G]$, the set of complex functions on $G$ with finite support, $\alpha^*\in\mathbb C[g]$ is defined by $$\alpha^*(g):=\overline{\alpha(g^{-1})},$$ and for $\beta\in\mathbb C[G]$, we have the convolution product defined by $$\alpha\beta(g):=\sum_{xy=g}\alpha(x)\beta(y).$$

Let $\alpha\in\mathbb C[G]$ be nontrivial, i.e., the support of $\alpha$ contains at least two elements, and put $\beta=\alpha^*\alpha$. Assume that $$ \beta(e)=1,\quad\text{and that}\quad|\beta(g)|<1\quad\text{for all $g\neq e$.}$$ Do the following limits exist for all $g\neq e$? $$\lim_{n\to\infty}\frac{|\beta^n(g)|}{|\beta^n(e)|}$$

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha\in\mathbb C[G]$, the set of complex functions on $G$ with finite support, $\alpha^*\in\mathbb C[G]$ is defined by $$\alpha^*(g):=\overline{\alpha(g^{-1})},$$ and for $\beta\in\mathbb C[G]$, we have the convolution product defined by $$\alpha\beta(g):=\sum_{xy=g}\alpha(x)\beta(y).$$

Let $\alpha\in\mathbb C[G]$ be nontrivial, i.e., the support of $\alpha$ contains at least two elements, and put $\beta=\alpha^*\alpha$. Assume that $$ \beta(e)=1,\quad\text{and that}\quad|\beta(g)|<1\quad\text{for all $g\neq e$.}$$ Do the following limits exist for all $g\neq e$? $$\lim_{n\to\infty}\frac{|\beta^n(g)|}{|\beta^n(e)|}$$

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha$ in $\mathbb C[G]$ (complex functions on $G$ with finite support), $\alpha^*$ is defined by $\alpha^*(g):=\overline{\alpha(g^{-1})}$, $g\in G$, and for $\beta\in\mathbb C[G]$, we have the (convolution) product defined by $\alpha\beta(g):=\sum_{xy=g}\alpha(x)\beta(y)$, $g\in G$.

Let $\alpha\in\mathbb C[G]$ be nontrivial (i.e. support of $\alpha$ contains at least two elements) and put $\beta=\alpha^*\alpha$. Assume that $\beta(e)=1$ and $|\beta(g)|<1$, $g\neq e$. Do the following limits exsit for all $g\neq e$? $$\lim_n\frac{|\beta^n(g)|}{|\beta^n(e)|}$$

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha\in\mathbb C[G]$, the set of complex functions on $G$ with finite support, $\alpha^*\in\mathbb C[g]$ is defined by $$\alpha^*(g):=\overline{\alpha(g^{-1})},$$ and for $\beta\in\mathbb C[G]$, we have the convolution product defined by $$\alpha\beta(g):=\sum_{xy=g}\alpha(x)\beta(y).$$

Let $\alpha\in\mathbb C[G]$ be nontrivial, i.e., the support of $\alpha$ contains at least two elements, and put $\beta=\alpha^*\alpha$. Assume that $$ \beta(e)=1,\quad\text{and that}\quad|\beta(g)|<1\quad\text{for all $g\neq e$.}$$ Do the following limits exist for all $g\neq e$? $$\lim_{n\to\infty}\frac{|\beta^n(g)|}{|\beta^n(e)|}$$