Number of subgroups in a Bieberbach group.

Question

Assume $\Gamma$ be a Bieberbach group which acts on $\mathbb R^n$ (i.e. a discrete subgroup of isometries of $n$-dimensional Euclidean space with a compact fundamental domain). Denote by $M(\Gamma)$ the number of maximal finite subgroups (up to conjugation) in $\Gamma$. Is it true that $M(\Gamma)\le 2^n$?

Things I can do: There is a simple geometric observation (due to Perelman) which shows that if $N(\Gamma)$ is the number of orbits of isolated fixed point of some subgroups of $\Gamma$ then $N(\Gamma)\le2^n$. Clearly, each such point corresponds to a maximal finite subgroup. Thus, $N(\Gamma)\le M(\Gamma)$, but in all examples I know I still have $M(\Gamma)\le 2^n$ (and I believe it is allways true).

The formulation is completely algebraic so maybe it has a completely algebraic solution...

Answer 1

Dima, I can not write a comment (yet) so I will start an answer to my own question.

You may assume that $\Gamma$ acts by isometries, so $A=\mathbb R^n/\Gamma$ is an Alexandrov space. For each maximal subgroup $F$ one can take its fixed point set $S_F$ in $\mathbb R^n$. $S_F$ is an (affine) subspace and image (say $E_F$) in $\mathbb R^n/\Gamma$ is a singular set (so called extremal subset of Alexandrov space). The maximality of $F$ implies that $E_F$ contains no proper extremal sets (a smaller subset is fixed by bigger group). (In fact $E_F$ is a flat manifold and its has a neighborhood which isometric for a product $E_F\times Cone$.)

So the question boils down to finding maximal number of such extremal sets in $A$. A particular case of such sets are isolated singular points. Perelman's theorem states that number of "one-point extremal sets" in an Alexandrov space with curvature $\ge 0$ is at most $2^n$. The proof repeats a proof of Erdős problem: if you have $m$ points in $\mathbb R^n$ such that all angles in all triangles $\le \pi/2$ then $m\le 2^n$. We take homothety with coefficient 1/2 for each point, then images of convex hull don't have common internal points (otherwise it would occur obtuse angle), then compairing volume of convex hull and its images gives estimate for number of points.

Answer 2

That is not an answer. I want to give an example where the argument of Erdős does not work directly.

Consider an action of group $\Gamma$ on $\mathbb R^3$ generated by the reflections $r_1, r_2$ and $r_3$ correspondingly in the lines $x=z=0$ and $x+1=z=0$ and $x-y=z-1=0$.

Each of the reflections $r_i$ generate a maxiamal $\mathbb Z_2$-subgroups, all of them are nonconjugate. These groups corespond to three singular circles, say $\Sigma_i$ in the factor $X=\mathbb R^3/\Gamma$. ($X$ is homeomorphic to $S^3$ and $\Sigma_1$, $\Sigma_2$, $\Sigma_3$ form Borromean rings, but all this is not important.)

Let us try to mimic argument of Erdős. Take subsets $X_i$ of $X$ of midpoints $m$ between $x\in X$ and a closest $x_0\in\Sigma_i$ to $x$. As in the argument of Erdős we have $\mathrm{vol}\, X_i>\tfrac{1}{2^3}\cdot\mathrm{vol}\, X$. BUT $X_1\cap X_3$ has interior points and here argument brakes into parts.

Comments

• Since fixed point sets are 1-dimensional, it would be enough to take $m\in [xx_0]$ such that $\tfrac{|mx_0|}{|xx_0|}=\tfrac1{2\sqrt[3]{2}}$. But even in this case one has interior points in $X_1\cap X_2$ (the borderline in this example seems to be $\tfrac13$).

• There is a natural bisecting hyperplane for any two affine subspaces. We may use it to cut a cylinder domain around each fixed point set of a maximal subgroup. The projection of these cylinders in $X$ gives Voronoi-like domains, but they do not cover whole space in general --- that is OK as far as we have lower bound on their volumes...