Let $X$ be a negatively curved symmetric space. In other words, $X$ is one of the four examples: a hyperbolic space, a complex hyperbolic space, a quaternionic hyperbolic space or the hyperbolic Cayley plane.


Let $B\subset X$ be a compact set (wlog, a large ball). Does there always exist a subgroup $\Gamma$ of the isometry group of $X$ such that $X/\Gamma$ is a compact manifold and the images of $B$ under $\Gamma$ are disjoint (i.e. $B\cap\gamma B=\emptyset$ for every nontrivial $\gamma\in\Gamma$)?


**Remark:** It would suffice to find just one (for each $X$) residually finite discrete co-compact group of isometries. I am sure there is one but I don't know where to look. (Or maybe they all are residually finite?)


**Meta-proof:** If the answer is negative, then, for some of the model geometries (in some dimension), all compact manifolds with this local geometry are "uniformly thin" - all injectivity radii are bounded above by some universal constant. Such a wonderful fact would be widely known.



Are there arbitrarily sparse "lattices" in negatively curved symmetric spaces?
The examples are arithmetic groups, constructed in general by [Borel and Harish-Chandra](http://www.ams.org/mathscinet-getitem?mr=147566). 
See also [Dave Witte Morris' preliminary book](http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html). 
However, examples in hyperbolic and complex hyperbolic spaces probably go back further to the study of quadratic forms. 


For hyperbolic lattices, one can take a quadratic form over a quadratic number field (such as $\mathbb{Q}(\sqrt{2})$), which
is Lorentzian at one place, and definite at the other places (such as $x\_1^2+\cdots +x\_n^2-\sqrt{2} x\_{n+1}^2$), and take the group of matrices $\Gamma$ in $GL(n+1,\mathbb{Z}[\sqrt{2}])$ which preserve this quadratic form. Then [Mahler's compactness theorem](http://en.wikipedia.org/wiki/Mahler%27s_compactness_theorem) (cf. Witte Morris) implies 
that the quotient $\mathbb{H}^n/\Gamma$ is compact. Then by Selberg's Lemma and residual finiteness (as Greg points out, Malcev's Theorem), you may find a torsion-free subgroup 
of finite-index with as large injectivity radius as you like. 


For hyperbolic and complex-hyperbolic spaces, there are other examples which don't come from the arithmetic construction (in fact, most hyperbolic surfaces and 3-manifolds are not arithmetic). These are attributable to [Gromov and Piatetski-Shapiro](http://www.ams.org/mathscinet-getitem?mr=932135) in the hyperbolic case in all dimensions, and there are finitely many examples in the complex hyperbolic case (at least for $\mathbb{C}$-dim >1) going back to Deligne and Mostow (see also [Thurston's paper](http://www.msp.warwick.ac.uk/gtm/1998/01/p025.xhtml)). However, Gromov and Schoen have shown that quaternionic and Cayley-hyperbolic lattices are all arithmetic, so the Borel Harish-Chandra construction is complete in these cases. 


Addendum: Related to Protsak's comment, there is a simple example of a negatively curved homogeneous space which has no lattice action. This is Thurston's "9th geometry", which is excluded as a geometry precisely for this reason. One can take the double warped product metric $$ dr^2 + e^{2a r} dx^2 + e^{2b r} dy^2, $$ for $a,b >0$. When $a=b$, this gives hyperbolic space. But when $a\neq b$, the sectional curvatures are $-a^2, -b^2, -ab$. This has a solvable transitive group of isometries, so is homogeneous. But using the solvability, one may see that there is no cocompact action. (Remark: when $a=-b$, this gives sol geometry).


