Let $s(n)$ denote the number of distinct crystallographic groups in $Isom(\mathbb{R}^n)$. Apparently the best known upper bound so far is
$$ s(n)\le e^{e^{4n^2}}, $$ given by Peter Buser in $1985$. On the other hand, it is not difficult to see (using a fixed-point lemma by Burnside-Cachy) that the number of $n$-dimensional crystallographic groups in the arithmetic class of the group of diagonal matrices has the asymptotic value $$ \frac{2^{n(n-1)}}{n!}. $$ This should give a lower bound for $s(n)$. Moreover, this class seems to be quite large in general, so that I have the feeling that the upper bound of Buser is not yet optimal.

What is known about lower and upper bounds for $s(n)$ ? Is it possible to use some other results on bounds of conjugacy classes of finite groups (since $s(n)$ is related to that by the number of conjugacy classes of finite subgroups of $GL(n,\mathbb{Z})$) ?

Update: Are there upper bounds known for the number of $n$-dimensional Bieberbach groups (torsionfree crystallographic groups), better than Buser's bound ?


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