Which groups have nice compactifications ? - MathOverflow

Which groups have nice compactifications ?

2010-09-22T14:58:46Z

Given a discrete group G. Is there a nice criterion to decide, whether there is a compact Hausdorff $G$- space X, that contains the discrete space $G$ as a subspace, such that the stabilizer of every point in $X$ is (virtually) cyclic ?

For example the free group admits such a compactification (As well as any hyperbolic group I think). Is it possible to decide, whether $\mathbb{Z}^2$ admits such a compactification? .

Answer by Bill Thurston for Which groups have nice compactifications ?

2010-09-22T16:26:52Z

There are many compactifications of particular groups. For your example of $\mathbb Z^2$: one construction for a compactification is to first embed it as a subgroup of $S^1 = \mathbb R / \mathbb Z$ by picking two rationally independent numbers for the images of the generators. Now compactify $\mathbb Z^2$ by making large elements connverge toward their image points in $S^1$. The stabilizer of any point is trivial.

The same method works to get a compactification associated with any action of $G$ on a compact space $X$. Just pick a point $x \in X$, and adjoin the closure of the orbit of $X$ at infinity in $G$. If the action has no fixed points in the cloure of the orbit, then stabilizers are trivial. It's easier to avoid all but cyclic stabilizers. To make actions with small stabilizers, you can take products of examples; point stabilizes in the product become intersections of stabilizers in the factors. There are many tricks, some of them useful, for making compactifications that are Hausdorff metric spaces.

There's an ultimate (but non-constructive and of large cardinality) compactification, the Stone-Cech compactification, which has trivial point stabilizers for any group,

Answer by HW for Which groups have nice compactifications ?

2010-09-22T16:42:04Z

Regarding the question of `whether the CAT(0) boundary works, if the space doesn't contain $\mathbb{R}^2$ as a subspace' (see comments above), the Flat Plane Theorem asserts that any CAT(0) group that acts on a CAT(0) space without an isometrically embedded copy of $\mathbb{R}^2$ is word-hyperbolic. So in this case you can use the usual hyperbolic boundary. See Bridson & Haefliger for details.