Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are defined by the Bourbaki tables.
Denote by $2U_\varphi$ the set of matrices $A^2$ where $A\in U_\varphi$. The symplectic group also has a maximal torus $T$, consisting of the diagonal matrices in $\text{Sp}_{2g}(\mathbb{Z})$.
The kernel $\Gamma(2)$ of the canoncical reduction map $\text{Sp}_{2g}(\mathbb{Z})\to\text{Sp}_{2g}(\mathbb{Z}/2\mathbb{Z})$ is the principal congruence subgroup of level $2$.
The question
I want to show that $\Gamma(2)$ is the group generated by $T$ and the $2U_\varphi$. The inclusion $\langle T,2U_\varphi:\varphi\in C_g\rangle\subset \Gamma(2)$ is trivial, by definition of the root subgroups and because the only nonzero entries of any matrix in $T$ are $1$ and $-1$. How do I prove the other inclusion?
Best Answer
If we assume that $g \geq 2$, then it is known by a Theorem of Tits ( Tits, Jacques : Systèmes générateurs de groupes de congruence. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 9, Ai, A693–A695)
that the group $\Delta$ generated by $T$ and $2U_{\phi}$ (your notation) has finite index in $Sp_{2g}(\mathbb Z)$. You can now use the fact that $Sp_{2g}(\mathbb Z)$ has the congruence subgroup property, to conclude that $\Delta$ is a congruence subgroup; it is easily seen that its closure in the congruence completion $Sp_{2g}(\widehat{\mathbb Z})$ of $Sp_{2g}(\mathbb Z)$ contains $\prod Sp_{2g}(\mathbb Z_p)$, where the product is over all odd primes. At the prime 2, the group $\Delta$ and $\Gamma (2)$ both have the same closure; we then conclude that $\Gamma (2)=\Delta$ (if two congruence subgroups have the same congruence closure, then they are the same).
When $g=1$, this is a well known result (and is actually proved in Ahlfors' book on complex analysis; the group $\Gamma (2)$ modulo centre is interpreted there as the fundamental group of ${\mathbb P}^1\setminus \{0,1,\infty \}$).