MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

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 $SL(2,Z)$.