See these notes by McMullen.
It states a nice problem on page 207.
(McMullen’s challenge)
Find a sequence of primitive matrices \(g_n \in \mathrm{SL}_2(Z)\) such that \(\log \mathrm{Tr}(g_n) \gg D^\epsilon -> oo\) where \(D = (\mathrm{Tr}(g_n)^2 - 4)/ \gcd( a-d,b,c)^2\).
Here primitive means that the matrix is not a power of some other matrix in \(\mathrm{SL}_{2}(\mathbb{Z})\).
There is a nice paper [1] that relates this problem to counting integer points on a certain variety.
This page was updated on March 26, 2026.
Main Page