let $\mu$ be a measure on a finite group $g$. we define the spectral gap of $\mu$ to be the operator norm of the map that sends $\phi \in l^2(g)^{\circ}$ to $\mu * \phi$. we say that $\mu$ is symmetric if $\mu(x) = \mu(x^{-1})$. now fix $g = sl_2(\mathbb{z}/n\mathbb{z}) \times sl_2(\mathbb{z}/n\mathbb{z})$, with $n \in \mathbb{n}$ being arbitrary. suppose that $\mu$ is a measure on $g$ such that it's pushforwards to the left and right component have spectral gaps lesser than $\lambda_0 < 1$ and $\mu$ takes a minimum of $\alpha_0$ on it's support. further suppose that the support of $\mu$ generates $g$. then we show that there are constants $l, \beta > 0$ depending only on $\lambda_0,\alpha_0$ such that $\mu^{(*)l\log |g|}(\gamma) \leq \frac{1}{|g|^{\beta}}$, where $\gamma$ is the graph of any automorphism of $sl_2(\mathbb{z}/n\mathbb{z})$. we will discuss this result and its implications. this talk is based on joint work with professor alireza salehi-golsefidy.