I have a question about a step in a proof from Lang's "Abelian Varieties" (page 20):
By definition an abelian variety $A$ over field $k$ is a proper smooth $k$-group scheme that is irreducible.
In the Theorem 1 where we have to show that an abelian variety is commutative the author says in the red tagged line that it suffice to show that
$$\dim T \le \dim A$$
for $A,T$ as lopological spaces.
Here $T$ is the locus if $(x, yxy^{-1}) \in A \times A$.
My question is why this suffice for the proof?
Over $\Bbb{C}$ it is a matter of complex analysis :
Let $A$ be a connected compact complex manifold which is also a group whose operations are given by holomorphic functions. Let $\phi$ be a chart from a neighborhood of $0 \in \Bbb{C}^n$ to $1 \in A$.
For every $a \in A$ and $v \in \Bbb{C}^n$ let $$\rho(a) v = \lim_{t \to 0} \frac{\phi^{-1}(a\phi(tv)a^{-1})}{t}$$
Looking at the Jacobian of $\phi$ and multiplication by $a$ shows $v \mapsto \rho(a) v$ is $\Bbb{C}$-linear. Thus $a \mapsto \rho(a)$ is an holomorphic map $A \to GL_n(\Bbb{C})$. Liouville implies $\rho$ is constant, thus $\rho(a) v = v$.
Transport the $\Bbb{R}^{2n}$ metric to the neighborhood of $1 \in A$. For any $v \in\Bbb{C}^n$ there is a unique function $\exp_v : (-r,r) \subset \Bbb{R} \to A$ such that $\exp_v(0)=1, \exp_v(t+s) = \phi(tv) \exp_v(s)+O(t^2)$. Comparing the real dimension shows any element closed to $1\in A$ is of the form $x=\prod_{j=1}^J \exp_{v_j}(1)$ for some $v_j$. And since $a \exp_{v_j}(t+u) a^{-1}= \phi(t\rho(a) v) a \exp_{v_j}(u) a^{-1}+O(t^2)$ we have $a \exp_{v_j}(t) a^{-1}= \exp_{\rho(a) v_j}(t)$ and $axa^{-1} =x$
The algebraic version for compact group varieties over algebraically closed field is similar, Liouville being replaced by that a morphism from a complete variety to an affine variety must be constant.