Let $G_n$ be the Lie group consisting of $n \times n$ upper triangular matrices of determinant $1$ with real entries. In other words, $$G_n = \{\text{$\left(\begin{matrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ 0 & a_{22} & a_{23} & \cdots & a_{2n} \\ 0 & 0 & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & a_{nn} \end{matrix}\right)$ $$ $a_{ij} \in \mathbb{R}$ and $a_{11} \cdots a_{nn} = 1$}\}.$$ Does $G_n$ contain a lattice (i.e. a discrete subgroup of finite covolume)? The obvious thing to try is to take the subgroup consisting of matrices with integer entries, but that does not work (though it would work if we were working with strictly upper triangular matrices).
1 Answer
@Edward, here is a short proof which I wrote awhile ago for my notes on group theory.
Lemma. If a 2nd countable locally compact group $G$ contains a lattice $\Gamma$ then $G$ is unimodular.
Proof. For arbitrary $g \in G$ consider the pushforward $\nu=R_g(\mu)$ of the (left) Haar measure $\mu$ on $G$; here $R_g$ is the right multiplication by $g$: $$ \nu(E)= \mu(Eg). $$ Then $\nu$ is also a left Haar measure on $G$. By the uniqueness of Haar measure, $\nu= c\mu$ for some constant $c > 0$. The lattice $\Gamma \le G$ has a fundamental domain $D\subset G$, i.e., a measurable subset of $G$ such that $$ \bigcup_{\gamma\in \Gamma} \gamma D= G, \quad \mu(\gamma D\cap D)=0, \quad \forall \gamma\in \Gamma \setminus 1. $$ (Proof of existence of such domain uses 2nd countable assumption.) In particular, $0<\mu(D)<\infty$, since $\Gamma$ is a lattice. Then $Dg$ is again a fundamental domain for $\Gamma$ and, thus, $\mu(D)=\mu(Dg)$. Hence, $\mu(D) = \mu(Dg) = c\nu(D)$. It follows that $c=1$. Thus, $\mu$ is also a right Haar measure. qed
Another proof could be found in Chapter I of Raghunathan's book "Discrete subgroups of Lie groups".

1$\begingroup$ A reference is also in DeitmarEchterhoff "Principles of Harmonic Analysis" Chapter 9 somewhere, not sure though if they refer only to uniform lattices. Nice proof +1. $\endgroup$ Dec 5, 2012 at 9:37

$\begingroup$ It's not clear why two fundamental domains have the same measure. $\endgroup$– GuntramDec 5, 2012 at 18:21

$\begingroup$ @Guntram: Because the measure of a fundamental domain is the same as the measure of the quotient $\Gamma\backslash G$. $\endgroup$– MishaDec 5, 2012 at 20:47

$\begingroup$ If $G$ is locally compact and abelian, and $D$ is a lattice in $G$, do you know if one can prove the existence of a measurable FD without the second countability assumption?.. Should I ask this as a question, or is it OK if I ask here? $\endgroup$– Nick SFeb 15, 2014 at 20:48

$\begingroup$ @NickS: I think you should ask a separate question. I do not know an off hand answer and will not be able to think about this for a week. $\endgroup$– MishaFeb 15, 2014 at 20:57
$n=2$
why this type of solvable group isn't unimodular. I'm not sure how to sort out those solvable Lie groups which are unimodular, but an old theorem of Mostow shows that any lattice must then be arithmetic. (Venkatarama gives a few of the standard references.) $\endgroup$