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Equidistribution on $SU_2$$\mathrm{SU}_2$

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Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant n\}$.

Can we construct a representation $\rho\colon F_{a_1,a_2}\to \mathrm{SU}_2$, such that the image is equidistributed in the following sense: for any continuous function $f$ on $\mathrm{SU}_2$, we have \begin{equation} \lim_{n\to\infty}\frac{1}{\mathrm{Card}(A_n)}\sum_{a\in A_n}f\big(\rho(a)\big)=\int_{\mathrm{SU}_2}fdv_{\text{Haar}}, \end{equation} where $dv_{\text{Haar}}$ is the normalized Haar measure with volume $1$.

I appreciate any help or reference.

Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant n\}$.

Can we construct a representation $\rho\colon F_{a_1,a_2}\to \mathrm{SU}_2$, such that the image is equidistributed in the following sense: for any continuous function $f$ on $\mathrm{SU}_2$, we have \begin{equation} \lim_{n\to\infty}\frac{1}{\mathrm{Card}(A_n)}\sum_{a\in A_n}f\big(\rho(a)\big)=\int_{\mathrm{SU}_2}fdv_{\text{Haar}}, \end{equation} where $dv_{\text{Haar}}$ is the normalized Haar measure with volume $1$.

Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant n\}$.

Can we construct a representation $\rho\colon F_{a_1,a_2}\to \mathrm{SU}_2$, such that the image is equidistributed in the following sense: for any continuous function $f$ on $\mathrm{SU}_2$, we have \begin{equation} \lim_{n\to\infty}\frac{1}{\mathrm{Card}(A_n)}\sum_{a\in A_n}f\big(\rho(a)\big)=\int_{\mathrm{SU}_2}fdv_{\text{Haar}}, \end{equation} where $dv_{\text{Haar}}$ is the normalized Haar measure with volume $1$.

I appreciate any help or reference.

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Equidistribution on $SU_2$

Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant n\}$.

Can we construct a representation $\rho\colon F_{a_1,a_2}\to \mathrm{SU}_2$, such that the image is equidistributed in the following sense: for any continuous function $f$ on $\mathrm{SU}_2$, we have \begin{equation} \lim_{n\to\infty}\frac{1}{\mathrm{Card}(A_n)}\sum_{a\in A_n}f\big(\rho(a)\big)=\int_{\mathrm{SU}_2}fdv_{\text{Haar}}, \end{equation} where $dv_{\text{Haar}}$ is the normalized Haar measure with volume $1$.