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Arithmetic representation growth of virtually free groups

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2025年5月12日

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Fabian Korthauer

摘要

We adapt methods from quiver representation theory and Hall algebra techniques to the counting of representations of virtually free groups over finite fields. This gives rise to the computation of the E-polynomials of $\textbf{GL}_d(\mathbb {C})$ ${GL}_{d} (C)$ -character varieties of virtually free groups. As examples we discuss counting of representations of the groups $\mathbb {D}_\infty $ $D_{\infty}$ , $\textbf{PSL}_2(\mathbb {Z})$ ${PSL}_{2} (Z)$ , $\textbf{SL}_2(\mathbb {Z})$ ${SL}_{2} (Z)$ , $\textbf{GL}_2(\mathbb {Z})$ ${GL}_{2} (Z)$ and $\textbf{PGL}_2(\mathbb {Z})$ ${PGL}_{2} (Z)$ .