The conjugacy problem in $SL(2,Z)$. For matrices $M \in GL(2,Z)$ having trace of absolute value $>2$, the slope of its expanding eigenvector has an eventually periodic continued fraction expansion (it is a quadratic irrational), and the primitive period loop is a conjugacy invariant in $SL(2,Z)$. Throw in the absolute value of the trace itself and you have a complete conjugacy invariant in $SL(2,Z)$.
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