Exercise 23, § 6

Let $G$ be a real Lie group and $c = (U, \phi, E)$ a chart on the manifold $G$ centred on the identity element $e$. If $x \in U$, let $|x|_c$ denote the norm of $\phi(x)$ in the Banach space $E$.

(a) Show that, if $c' = (U', \phi', E')$ is another chart centred on $e$, there exist constants $\lambda > 0$ and $\mu > 0$ such that

for all $x \in G$ sufficiently close to $e$.

(b) Show that, for all $\rho > 0$, there exists a neighbourhood $U_\rho$ of $e$ contained in $U$ such that, for $x, y$ in $U_\rho$, $(x, y) \in U_\rho$ and

(c) Suppose that $G$ is finite-dimensional. Let $\Gamma$ be a discrete subgroup of $G$. Apply (b) with $0 < \rho < 1$ and choose $U_\rho$ to be relatively compact. The set $U_\rho \cap \Gamma$ is finite. Let

be its elements, numbered so that Show that, if $i \leq m$ and $j \leq m$, the commutator $(\gamma_i, \gamma_j)$ is equal to one of the $\gamma_k$ with $k < \inf(i, j)$. Deduce that the subgroup generated by $U_\rho \cap \Gamma$ is nilpotent of class $\leq m$.

Deduce from the above the existence of a neighbourhood $V$ of $e$ such that, for every discrete subgroup $\Gamma$ of $G$, there exists a nilpotent integral subgroup $N$ of $G$ containing $V \cap \Gamma$.

(d) Suppose that $G$ is connected and finite-dimensional and contains an increasing sequence of discrete subgroups $D_n$ whose union is dense in $G$. Then $G$ is nilpotent. (Using (c), prove the existence of a central one-parameter subgroup $H$ which meets $D_n$ for sufficiently large $n$ at a point distinct from $e$. Argue by induction on $\dim G$, distinguishing two cases according to whether $H$ is closed or relatively compact (Exercise 21 (a)).)