Let $w$ be a word over $\{x_1, x_1^{-1}, \ldots, x_k, x_k^{-1}\}$. Then, for any group $G$, $w$ defines a function $G^k\rightarrow G$. Inserting random elements of $G$ yields a distribution $\delta$ on $G$. Since $\delta$ is constant on conjugacy classes, it can be written as a linear combination of irreducible characters of $G$, that is, there exist complex numbers $\alpha_\chi$, $\chi\in\mathrm{Irr}(G)$, such that for all $g\in G$ we have
\[
\#\{(g_1, \ldots, g_k):w(g_1,\ldots,g_k)=g\} = \sum_\chi\alpha_\chi
\chi(g). \]
If $G$ acts sufficiently nice on some set, this action can be used to give bounds for $\alpha_\chi$. In this talk I present a new method to do so. As an application I show that the random walk generated by a word over $S_n$ is $L1$-uniform after 2 steps. Other applications deal with subgroup growth of one-relator groups and random walks on linear groups. |
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