Snippets

# A quick and dirty definition of an arithmetic subgroup?

Or, at least, of an arithmetic subgroup of $G \leq \mathrm{SL}(n,\mathbb{R})$ a semisimple Lie group:

1. The “subgroup of integer points” $G_{\mathbb{Z}} = G \cap \mathrm{SL}(n,\mathbb{Z})$ is an arithmetic group …
2. … if embedded in a reasonable way which doesn’t distort the arithmetic structure—slightly more precisely, if it is [essentially] an algebraic group over the rationals.
3. Anything isomorphic to a finite extension is arithmetic too.

For a more precise definition—Witte Morris’ book is a good place to start; he fudges over a few things too, but actually has references to fill those in …