6
$\begingroup$

A group $G$ is called self-replicating, if there exists a finite index subgroup $H$, such that $H\cong G\times\dots\times G$. Maybe the most famous example of a self-replicating group is a subgroup $K$ of index 16 in the Grigorchuk group. For this group a very explicit description is given in (Grigorchuk, Just infinite branch groups, in: New horizons in pro-$p$ groups): We have that $K$ contains a normal subgroup $K_1$ with $K/K_1\cong C_4$ and $K_1\cong K\times K$. The Frattini subgroup of $K$ is of level 2, that is, it contains the normal subgroup $K_2\cong K\times K\times K\times K$. Moreover, a set of generators and their action on the infinite binary tree is given.

Unfortunately, we have not been able to find a similarly detailed description for the other famous branch groups, in particular the Gupta-Sidki and the Fabrikowsky-Gupta groups (see: Bartholdi-Grigorchuk-Sunik, Branch groups, arXiv:math/0510294, section 1.6). More precisely, we are interested in the congruence level $\ell$ of the Frattini subgroup, and the finite group $K/K_\ell$, where $K_\ell$ is the $\ell$-th principal congruence subgroup. Computing these data right from the definitions would be a huge amount of work, so we would be most thankful for a pointer to the literature.

Thank you in advance

Jan-Christoph Schlage-Puchta

$\endgroup$
1
  • $\begingroup$ Have you try to compute this with GAP? There are two packages (AutomGrp and FR) that handle automaton groups pretty well. $\endgroup$
    – thibo
    Apr 16, 2019 at 7:55

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.