${\mathbb{Z}}^2$-free subgroups of Thompson's $T$ are virtually free

It is an open problem to determine whether surface groups can embed in Thompson’s group $V$. We prove that any finitely generated subgroup of Thompson’s group $T$ without abelian groups of arbitrary high rank is either virtually abelian or virtually free. In particular, surface groups don’t embed in $T$, answering a question of Belk and Moore.