For PSL2(Z), an algorithm decides H-algebraicity of a matrix and generates the full ideal of H-equations; for n at least 4 the analogous task is impossible.
Free groups and Stallings’ foldings
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Computing $H$-equations with 2-by-2 integral matrices
For PSL2(Z), an algorithm decides H-algebraicity of a matrix and generates the full ideal of H-equations; for n at least 4 the analogous task is impossible.