Mahler's compactness theorem

Builds on Lattice (group)Compact set Quotient space

Statement

Let $n \ge 1$ , and let $\mathcal{L}_n$ be the set of lattices in $\R^n$ . Identify $\mathcal{L}_n$ with the quotient space $\mathrm{GL}_n(\R)/\mathrm{GL}_n(\Z)$ , by sending $g$ to the lattice $g\Z^n$ , and give it the quotient topology. Let $A \subseteq \mathcal{L}_n$ . Then $A$ has compact closure in $\mathcal{L}_n$ if and only if there are constants $C > 0$ and $\varepsilon > 0$ such that for every $\Lambda \in A$ :

the covolume of $\Lambda$ is at most $C$ ;
every nonzero vector $v \in \Lambda$ has $\norm{v} \ge \varepsilon$ .

Remarks

The proof uses reduction theory. Under the two bounds, each lattice of $A$ has a reduced basis whose vectors have length bounded above and below, with angles bounded away from $0$ . Such bases form a compact set of matrices, whose image in $\mathcal{L}_n$ contains $A$ . The converse holds because covolume and the length of a shortest nonzero vector are continuous on $\mathcal{L}_n$ .

Example: the lattices $\Lambda_t = \Z (t, 0) \oplus \Z (0, 1/t)$ in $\R^2$ have covolume $1$ . Their shortest nonzero vector has length $t$ for $0 < t \le 1$ . As $t \to 0$ condition 2 fails, and indeed $\Lambda_t$ leaves every compact subset of $\mathcal{L}_2$ .

Both conditions are needed. The lattices $t\Z^2$ with $t \to \infty$ satisfy condition 2 but not condition 1, and they also leave every compact set.

Going deeper

For a student

For unimodular lattices, condition 1 is automatic, so a set of unimodular lattices is relatively compact exactly when the lattices avoid a fixed ball around $0$ . This gives the basic criterion for divergence in the homogeneous space $\mathrm{SL}_n(\R)/\mathrm{SL}_n(\Z)$ . It is used in homogeneous dynamics, for example in the non-divergence results of Margulis and Dani for unipotent flows, and in Diophantine approximation through Dani's correspondence.

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Tag	`6QY76Q5WVS`
URI	`m2://6QY76Q5WVS`
Kind	statement, theorem
Writing	W0 stub
MSC	11H06, 22E40
Status	This copy of the site has no build data. Run `m2 status 6QY76Q5WVS`.
About	Lattice (group)
Uses	Lattice (group), Compact set, Quotient space
Same as	Wikidata Q6734194 Wikipedia Mahler's compactness theorem